56 PD13 Abstracts

IP0 Hence the need to develop a partial regularity theory: is The SIAG/Analysis of Partial Differential Equa- it true that solutions are always smooth outside a ”small” tions Prize Lecture: Weak Solutions of the Euler singular set? The aim of this talk is first to review the Equations: Non-Uniqueness and Dissipation classical regularity theory, and then to describe some re- cent results about partial regularity. There are two aspects of weak solutions of the incompress- ible Euler equations which are strikingly different to the Alessio Figalli behaviour of classical solutions. Weak solutions are not Department of Mathematics unique in general and do not have to conserve the en- The University of Texas at Austin ergy. Although the relationship between these two aspects fi[email protected] is not clear, both seem to be in vague analogy with Gro- movs h-principle. In the talk I will explore this analogy in light of recent results concerning both the non-uniqueness, IP3 the search for selection criteria, as well as the dissipation Waves in Honeycomb Structures anomaly and the conjecture of Onsager. I will discuss the propagation of waves in honeycomb- L´aszl´oSz´ekelyhidi, Jr. structured media. The (Floquet-Bloch) dispersion rela- Universit¨at Leipzig tions of such structures have conical singularities which [email protected] occur at the intersections of spectral bands for high- symmetry quasi-momenta. These conical singularities, also Camillo De Lellis called Dirac points or diabolical points, are central to the Institut f¨ur Mathematik remarkable electronic properties of graphene and the light- Universitat Zurich, Switzerland propagation properties in honeycomb structured dielectric [email protected] media. Examples of such properties are: quasi-particles which behave as massless Dirac Fermions, tunability be- tween conducting and insulating states and topologically IP1 protected edge states. Most theoretical work on honey- Coupling Between Internal and Surface Waves in a comb structures (going back to 1947) has centered on the Two-layers Fluid tight-binding approximation, a solvable discrete limit cor- responding to infinite medium contrast. We present re- Internal waves occur within a fluid that is stratifed by tem- sults (with CL Fefferman) for the Schroedinger equation perature or salinity variation. They are commonlygener- with a honeycomb lattice potential with no assumptions atedinthe oceans. They havetheform of large-amplitude, on medium contrast showing: a) the existence of conical long-wavelength nonlinear waves that propagate over large singularities in dispersion surfaces for generic honeycomb distances. In some physically realistic situations, internal lattice potentials b) the persistence of such conical singu- waves give rise to characteristic features on the surface, a larities under perturbations which preserve time-reversal signature of their presence, in the form of narrow bands of and spatial inversion symmetries, e.g. a linear strain of rough water, sometimes referred to as a rip, which prop- the honeycomb structure c) that wave-packet initial condi- agates at the same velocity as the internal wave, followed tions, which are spectrally localized about a Dirac point after its passage, by the complete calmness of the sea, the are governed, on very long time-scales, by a system of mill pond effect. Our starting point is the two or three- homogeneous 2-dimensional Dirac equations. Finally, we dimensional Euler equations for an incompressible, irrota- discuss the question of topologically protected edge states tional fluid composed of two immiscible layers of different and recent work in this direction (with CL Fefferman and densities. We propose an asymptotic analysis in a scaling J. Lee-Thorpe). regime chosen to capture the observations described above. The analysis of the asymptotic model shows that the rip MichaelI.Weinstein region of the free surface is generated by the resonant cou- Columbia University pling between an internal soliton and the free-surface wave Dept of Applied Physics & Applied Math modes while the mill pond effect is the result of a domi- [email protected] nant reflection coeffcient for free-surface waves in a frame of reference moving with the internal soliton IP4 Catherine Sulem Granular Experiments of Discrete Nonlinear Sys- University of Toronto tems [email protected] Ordered and disordered arrangements of granular particles are governed by highly nonlinear contact interactions that IP2 confer to the particles assembly unique dynamic properties. Partial Regularity for Monge-Ampre Type Equa- We study how stress waves propagate in lattices composed tions of particles in close contact, and exploit this understand- ing to create novel materials and devices at different scales Monge-Ampre type equations arise in several problems (for example, for application in energy conversion, vibra- from analysis and geometry, and understanding their regu- tion absorption and acoustic rectification). We control the larity is an important question. In particular, this kind of constitutive behavior of these new materials selecting the equations arises in the regularity theory of optimal trans- particles geometry, their arrangement and materials prop- port maps. In the 90’s Caffarelli developed a regularity the- erties. We assemble and test experimental systems and in- ory on Rn for the classical Monge-Ampre equation, which form our experiments with discrete numerical simulations. was then extended by Ma-Trudinger-Wang and Loeper to a more general class of equations which satisfy a suitable structural condition. Unfortunately, this condition is very Chiara Daraio restrictive and it is satisfied only in very particular cases. ETH Z¨urich PD13 Abstracts 57

[email protected] boundary conditions which can be combined to a Monge- Ampre equation to solve the optimal transport problem. The wide-stencil discretization technique and fast Newton IP5 solver proposed by Oberman and Froese is extended to this Tug-of-war and Infinity Laplacian with Neumann framework and allows to compute weak viscosity solution Boundary Conditions of the optimal transport problem. Numerical solutions will be presented to illustrate strengths and weaknesses of the We study a version of the stochastic ”tug-of-war” game, method. played on graphs and smooth domains, with an empty set of terminal states. We prove that, when the run- Jean-David Benamou ning payoff function is shifted by an appropriate con- INRIA Rocquencourt, France stant, the values of the game after n steps converge. Us- [email protected] ing this we prove the existence of solutions to the in- finity Laplace equation with vanishing Neumann bound- ary condition. In earlier work with Schramm, Sheffield IP8 and Wilson (http://arxiv.org/abs/math/0605002, JAMS Modelling Collective Cell Motion in Biology 2009), we related a tug of war game to the infinity Lapla- cian equation with Dirichlet boundary conditions- I will We will consider three different examples of collective cell survey that work as well as the version for the p-Laplacian movement which require different modelling approaches: in http://arxiv.org/abs/math/0607761 - Duke 2009. (Talk movement of cells in epithelial sheets, with application to based on joint work with Tonci Antunovic, Scott Sheffield, rosette formation in the mouse epidermis and monoclonal Stephanie Somersille http://arxiv.org/abs/1109.4918 , conversion in intestinal crypts; cranial neural crest cell mi- Comm. PDE 2012) gration which requires a hybrid discrete cell-based chemo- taxis model; acid-mediated cancer cell invasion, modelled Yuval Peres via a coupled system of non-linear partial differential equa- Microsoft Research tions. We show that in many cases, all these models can [email protected] be expressed in the framework of nonlinear diffusion equa- tions. We show how these models can be used to under- stand a range of biological phenomena. IP6 Philip K. Maini Models for Neural Networks; Analysis, Simulations Centre for Mathematical Biology and Qualitative Behavior University of Oxford Neurons exchange information via discharges propagated [email protected] by membrane potential which trigger firing of the many connected neurons. How to describe large networks of such CP1 neurons? How can such a network generate a collective activity? Such questions can be tackled using nonlinear A Comparison Analysis of q-Homotopy Analy- partial-integro-differential equations which are classically sis Method (q-HAM) and Variational Iteration used to describe neuronal networks. Among them, the Method (VIM) to Fingero Imbibition Phenomena Wilson-Cowan equations are the best known and describe in Double Phase Flow through Porous Media globally brain spiking rates. Another classical model is the In this paper, we consider Variational Iteration Method integrate-and-fire equation based on Fokker-Planck equa- (VIM) and q-Homotopy Analysis Method (q-HAM) to tions. The spike times distribution, which encodes more solve the partial differential equation resulted from Fingero directly the neuronal information, can also be described Imbibition phenomena in double phase flow through porous directly thanks to structured population. We will compare media. We further compare the results obtained here with and analyze these models. A striking observation is that the solution obtained in [?] using Adomian Decomposition solutions to the I&F can blow-up in finite time, a form of Method. Numerical results are obtained, using Mathemat- synchronization that can be regularized with a refractory ica 9, to show the effectiveness of these methods on our stage. We can also show that for small or large connectiv- choice of problem especially for suitable values of hand n. ities the ’elapsed time model’ leads to desynchronization. For intermediate regimes, sustained periodic activity oc- Olaniyi S. Iyiola curs compatible with observations. A common tool is the King Fahd University of Petroleum and Minerals use of the relative entropy method. [email protected] Benoit Perthame Universit´e Pierre et Marie Curie CP1 Laboratoire Jacques-Louis Lions and Weak Solutions to Lubrication Systems Describing NRIA-Paris-Rocquencourt the Evolution of Bilayer Thin Films [email protected] We prove existence of global non-negative weak solutions for coupled one-dimensional lubrication systems that de- IP7 scribe the evolution of nanoscopic bilayer thin polymer A PDE Approach to Computing Viscosity Solu- films. We consider Navier-slip and no-slip conditions at tions of the Monge-Kantorovich Problem both liquid-liquid and liquid-solid interfaces. Additionally, we show existence of positive smooth solutions when at- After a quick overview of the optimal transport for the tractive van der Waals and repulsive Born intermolecular Euclidean distance problem and available numerical meth- interactions are taken into account. ods, I will present a new technique to deal with the state constraint that binds the transport when source and tar- Sebastian Jachalski get have compact support. It takes the form of non-linear Weierstrass Institute 58 PD13 Abstracts

[email protected] moving contact lines and contact angle hysteresis through a variational inequality. We prove the well-posedness of Georgy Kitavtsev the time semi-discrete and fully discrete (finite element) Max Planck Institute for Mathematics in the Sciences model and discuss error estimates. Simulation movies will [email protected] be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as Roman Taranets future work on optimal control of these types of flows. Institute of Applied Mathematics and Mechanics NAS Shawn Walker Ukraine Louisiana State University taranets [email protected] Department of Mathematics and CCT [email protected] CP1 Stability Analysis of Thin Film Problems with CP2 Non-Constant Base States Two-Point Riemann Problem for Inhomogeneous We address the linear stability of growing rims that appear Nonconvex Conservation Laws: Geometric Con- in thin solid and liquid films as they retract from a solid struction of Solutions substrate. Despite the different transport mechanisms, the mathematical models in both cases lead to mass conserving We consider the following conservation law: ∂tu(x, t)+ ∂xf(u(x, t)) = g(u) The following boundary conditions are free boundary problems for degenerate and non-degenerate − thin film equations of the type specified: u(0 0,t)=u0− and u(X +0,t)=uX+.In addition, we specify the initial condition u(x, 0) = ux0 for n ∈ ht + ∇·(h ∇Δh)=0, x (0,X) Method of characteristics is used to show the evolution of the initial profile and the discontinuities at n with different mobilities h . The base state is time de- the boundaries as well as appearance and in some cases pendent and does not have a simple traveling wave or self- disappearance of internal discontinuities. For illustration similar form and thus prevents a straight forward linear purposes the function f(u) will be assumed to have 3 crit- stability analysis. However, large rims evolve on a slower ical points. time scale than the perturbations. We exploit this time scale separation to derive asymptotic approximations for Dmitry A. Altshuller the base states and obtain solutions of the linear stability Dassault Systemes problem. Our results enable to track the evolution of the [email protected] dominant wavelength and yield criteria for the validity of the so-called “frozen modes analysis’ often used for these types of problems. CP2 A Nonlocal Hyperbolic Pde for Chaotic Shocks Dziwnik Marion TU Berlin We propose a new model equation that describes chaotic Institut f¨ur Mathematik shock waves. The equation is a simple modification of the [email protected] Burgers equation that includes non-locality via the pres- ence of the shock-state value of the solution in the equa- Maciek Korzec tion itself. The model predicts steady-state solutions, their TU Berlin instability through Hopf bifurcation, and a sequence of [email protected] period-doubling bifurcations leading to chaos. The dynam- ics of the solutions is qualitatively identical to those in the Andreas Muench one-dimensional reactive Euler equations. We present a OCIAM Mathematical Institute complete linear stability theory as well as nonlinear nu- University of Oxford merical simulations to characterize the observed chaotic [email protected] attractor. Aslan R. Kasimov Barbara Wagner Division of Mathematical & Computer Sciences and Weierstrass Institute, Berlin Engineering TU Berlin King Abdullah University of Science and Technology [email protected] [email protected]

CP1 Luiz Faria A New Mixed Formulation For a Sharp Interface KAUST Model of Stokes Flow and Moving Contact Lines [email protected]

Two phase fluid flows on substrates (i.e. wetting phenom- Rodolfo R. Rosales ena) are important in many industrial processes, such as Massachusetts Inst of Tech micro-fluidics and coating flows. These flows include addi- Department of Mathematics tional physical effects that occur near moving (three-phase) [email protected] contact lines. We present a new 2-D variational (saddle- point) formulation of a Stokesian fluid with surface tension (see Falk, Walker in the context of Hele-Shaw flow) that CP2 interacts with a rigid substrate. The model is derived by A Riemann Problem at a Junction of Open Canals an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for We study a Riemann problem at a junction of a star-like PD13 Abstracts 59

network of open canals. The flow in the network is given nearly glancing reflections, then, it is necessary to under- by 1D Saint-Venant equations in each canal and special stand how the reflected shock diffracts nonlinearly into the conditions at the junction. We consider the case where Mach shock as its strength approaches zero. Towards this two of the canals are identical. Firstly, we show that the end, we formulate a half-space initial boundary value prob- linearised problem has always a unique solution. Secondly, lem for the unsteady transonic small disturbance equations we show that under certain condition, there is a unique that describes nearly glancing Mach reflection. We solve solution to the nonlinear Riemann problem. this IBVP numerically using high-resolution methods, and we find in the solutions a complex reflection pattern that Mouhamadou S Goudiaby closely resembles Guderley Mach reflection. This is joint LANI, UFR SAT Universit´e Gaston Berger work with John Hunter. BP 234, Saint-Louis, Senegal [email protected] Allen Tesdall Department of Mathematics Gunilla Kreiss City University of New York, College of Staten Island Division of Scientific Computing [email protected] Uppsala University [email protected] CP3 Reconstruction of Nonlinear Water Waves by Gen- CP2 eralized SfS Method Stability of Viscous Strong and Weak Detonation Waves for Ma jda’s Model We introduce a generalized nonlinear PDE for the reflec- tivity function which occurs in optical reconstruction of We give an overview of a program based on a combina- dynamical surfaces. We use a Lambertian reflectance SfS tion of analytical and numerical Evans-function techniques procedure for the reconstruction of water waves. We com- for establishing the nonlinear stability of strong and weak pared the results of our numerical processing of data with detonation-wave solutions of Majda’s simplified, “qualita- experiments in the wave lab at ERAU, for different types tive’ model for reacting mixtures of gases. One noteworthy of surface waves: linear, nonlinear, rogue, vortex induced, aspect of the analysis is the treatment of weak viscous det- etc. We double check our SfS results with laser goniometry onation waves. Due to to their nature, the stability of these and capacitive level sensors. undercompressive waves has received scant attention in the literature. Andrei Ludu Embry-Riddle Aeronautical University Gregory Lyng Mathematics Department Department of Mathematics [email protected] University of Wyoming [email protected] Nikhila Chaudhari, Muhammad Abdul Aziz, Ilteris Demirkiran Embry-Riddle Aeronautical University CP2 Dept. Electrical, Computer, Software, & Systems Transitional Wave-Dynamics for Engineering Hyperbolic Relaxation Systems [email protected], [email protected], [email protected] We consider the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation for a linearized hyperbolic re- CP3 laxation system. As a specific example, we look at a Euler- Existence and Symmetry of Ground States to the type two-phase flow model with relaxation towards phase Boussinesq Abcd Systems equilibrium. We observe that the stability of the transi- tional waves is equivalent to the sub-characteristic condi- We consider a four-parameter family of Boussinesq systems tion and identify the existence of a critical region of wave derived by Bona-Chen-Saut. We establish the existence of numbers where the sonic waves disappear. the ground states which are solitary waves minimizing the action functional of the systems. We further show that in Susanne Solem, Peder Aursand the presence of large surface tension the ground states are Norwegian University of Science and Technology even up to translation. [email protected], [email protected] Ming Chen, Shiting Bao, Qing Liu Tore Fl˚atten University of Pittsburgh SINTEF Materials and Chemistry [email protected], [email protected], [email protected] tore.fl[email protected]

CP3 CP2 Sharp Thresholds of Global Existence and Blow-Up Glancing Weak Mach Reflection for a Class of Nonlocal Wave Equations

We study the glancing limit of weak shock reflection, in We first present the local and global existence and finite- which the wedge angle tends to zero with the Mach num- time blow-up results for the nonlocal nonlinear wave equa- ber fixed. Lighthill showed that, according to linearized tions utt − Kuxx + Mutt = g(u)xx,whereK and M theory, the reflected shock strength approaches zero at the are two pseudo-differential operators. We then establish triple point in reflections of this type. To understand the thresholds for global existence versus blow-up in the case nonlinear structure of the solution near the triple point in of power-type nonlinearities. We use the potential well 60 PD13 Abstracts

method based on the concepts of invariant sets suggested solutions of the following equation for surface water waves by Payne and Sattinger. The results cover those given of moderate amplitude in the shallow water regime: for the so-called double-dispersion equation and the tra- 2 3 ditional Boussinesq-type equations, as special cases. This ut+ux+6uux−6u ux+12u ux+uxxx−uxxt+14uuxxx+28uxuxx =0. work has been supported by the Scientific and Technolog- ical Research Council of Turkey (TUBITAK) under the Our approach is based on a method proposed by Grillakis, project TBAG-110R002. Shatah and Strauss in 1987, and relies on a reformulation of the evolution equation in Hamiltonian form. We deduce Saadet Erbay stability of solitary waves by proving the convexity of a Ozyegin University scalar function, which is based on two nonlinear functionals [email protected] that are preserved under the flow. Nilay Duruk Mutlubas Albert Erkip Sabanci University Sabanci University [email protected] [email protected] Anna Geyer Husnu A. Erbay University of Vienna Ozyegin University [email protected] [email protected]

CP3 CP3 Eulerian Computation Of Complex Short Wave Stability and Instability of Solitary Waves for a Forms Propagating Over Long Distances Class of Nonlocal Nonlinear Equations We present a new Eulerian method for computation of We investigate the existence and stability/instability prop- short waveforms over long distances using Nonlinear Soli- erties of solitary waves for the general class of non- − tary Waves that are solutions to the Hamilton-Jacobi equa- local nonlinear wave equations utt Kuxx + Mutt = tion. This method overcomes limitations of conventional ± |u|p−1u , (p>1), where K and M are two pseudo- xx discretization schemes, whose accuracy depends on grid differential operators. The so-called double-dispersion resolution, making them too costly for real world problems. equation and the Boussinesq-type equations are two well- This cost can be greatly reduced by carrying the waveforms known members of this class. The relative dominance of using thin pulses, which span over 4 grid cells and implic- the two operators M and K plays an important role in itly specifying the waveforms through initial conditions. our investigations. This work has been supported by the Scientific and Technological Research Council of Turkey Subhashini Chitta (TUBITAK) under the project TBAG-110R002. UTSI [email protected] Husnu A. Erbay Ozyegin University [email protected] John Steinhoff University of Tennessee Space Inst. Goethert Pkwy Albert Erkip [email protected] Sabanci University [email protected] CP4 Saadet Erbay On Inviscid Limits for the Stochastic Navier-Stokes Ozyegin University Equations and Related Systems. [email protected] One of the original motivations for the development of stochastic partial differential equations traces it’s origins CP3 to the study of turbulence. In particular, invariant mea- On the Generalized KdV Equation sures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical prop- In this talk we consider the generalized Korteweg-de Vries erties of turbulent flows. In this talk we discuss some recent (gKdV) equation results concerning inviscid limits in this class of measures 3 k+1 for the stochastic Navier-Stokes equations and other re- ∂ u + ∂ u + μ∂ (u )=0, t x x lated systems arising in geophysical and numerical settings. where k ≥ 4 is an integer number and μ = ±1. We will This is joint work with Peter Constantin, Vladimir Sverak review the well-posedness theory and discuss some open andVladVicol. problems. Nathan Glatt-Holtz Luiz G. Farah Virginia tech Universidade Federal de Minas Gerais (UFMG) - Brazil [email protected] [email protected] CP4 CP3 Rethinking Computation of Incompressible Fluid Orbital Stability of Solitary Waves of Moderate Flow More Than a Mathematical Trick Amplitude in Shallow Water The Navier-Stokes equation is composite, orthogonally de- We study the orbital stability of solitary traveling wave composable into a (dependent) pressure and a fundamental PD13 Abstracts 61

pressure-free governing equation for incompressible flow. Walter Rusin In differential form, these involve integrals over a Greens Oklahoma State University function for the Laplacian operator, but its not immedi- [email protected] ately obvious how to solve them numerically. However, the equivalent variational forms are amenable to efficient Igor Kukavica, Yuan Pei, Mohammed Ziane numeric computation with a simple finite element method University of Southern California using discrete divergence-free velocity bases found as the [email protected], [email protected], [email protected] curl of modified Hermite elements.

Jonas T. Holdeman CP4 retired A Unilateral Open Boundary Condition for the [email protected] Navier-Stokes Equations One of the main issues in the simulations of blood flow in CP4 arteries is a proper setting of the outflow boundary condi- Compressible Viscous Navier-Stokes Flows Grazing tion at artificial boundaries. Under some physical assump- a Non-Convex Corner tions, we propose a new approach to set the outflow bound- ary condition by using a variational inequality technique. I will talk about existence and regularity of solution for the Consequently, we solve the time-dependent Navier-Stokes compressible viscous Navier-Stokes equations on a polygon equation with a boundary-penalty term. We report some having a grazing non-convex corner. I will also talk about mathematical and numerical results on this new problem, a jump discontinuity of the density function across a curve and compare it with other methods. inside the domain and the corresponding jumps in deriva- tives of the velocity. The solution comes from a well-posed Norikazu Saito boundary value problem on a non-convex polygonal do- University of Tokyo main. By the decay formula of the jump it is seen that Graduate School of Mathematical Sciences density jumps can happen in a high speed flow occuring in [email protected] a very viscous compressible fluid. Guanyu Zhou Jae Ryong Kweon Graduate School of Mathematical Sciences, POSTECH(Pohang University of Science and Technology) The University of Tokyo Korea [email protected] [email protected] Yoshiki Sugitani The University of Tokyo CP4 Graduate School of Mathematical Sciences Entropy-Stable Schemes for the Initial Boundary [email protected] Value Euler Equations

We consider numerical schemes for the nonlinear Euler CP4 equations of gas dynamics in one space dimension. Fi- Existence of Classical Sonic-Supersonic Solutions nite difference numerical schemes, which satisfy an entropy for the Steady Euler Equations stability condition, have been derived and include far-field and wall boundary conditions. We have also been derived Given a smooth curve as a sonic line in the plane, we con- a stable numerical scheme for connecting two domains via struct a classical sonic-supersonic solution on one side of an interface. Furthermore, we present numerical compu- the curve for the steady isentropic compressible Euler sys- tations for second and fourth order accurate schemes to tem in two space dimensions. The solution is obtained by demonstrate robustness of the proposed schemes. employing an innovate coordinate system. The streamlines of the equations are analyzed to obtain a physical domain Hatice Ozcan of existence. School of Mathematics, University of Edinburgh,JCMB King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ Tianyou Zhang [email protected] The Pennsylvania State University university park Magnus Sv¨ard [email protected] University of Bergen, Department of Mathematics [email protected] Yuxi Zheng Penn State University [email protected] CP4 Primitive Equations with Continuous Initial Data CP5 Weaddressthewell-posednessoftheprimitiveequationsoftheoceanwithcontinuousinitialMultiple Steady State Solutions of Some Reaction- data. We show that the splitting of the initial data into Diffusion Systems a regular finite energy part and a small bounded part is preserved by the equations thus leading to existence and We consider a class of reaction-diffusion systems leading to uniqueness of solutions. strongly indefinite energy functionals, with nonlinearities which combine concave and convex terms. By establishing Walter Rusin two new critical point theorems for strongly indefinite even University of Southern California functionals, we obtain the existence of infinitely many large [email protected] and small energy steady state solutions. Our critical point 62 PD13 Abstracts

theorems generalize the fountain theorems of Barstch and iγ)A|A|2 + ig(x)A ,whereg(x) is a localized real valued Willem to strongly indefinite functionals. function and  is small. We are interested in finding solu- tions close to patterns of the form A = e−iγtA¯.At =0the CyrilJoelBatkam linearization about the equilibrium A¯∗ =1isnotinvert- University of Sherbrooke ible in the usual translation invariant spaces. We show that [email protected] when considered as an operator between Kondratiev spaces the linearization is a Fredholm operator. These spaces con- sist of functions with algebraic localization that increases CP5 with each derivative. We use this result to construct so- Phase Transitions with Mid-Range Interactions: a lutions close to the equilibrium via the Implicit Function Non-Local Stefan Model Theorem and derive asymptotics for wavenumbers in the far field. This is joint work with Arnd Scheel. We study a non-local version of the one-phase Stefan prob- lem which takes into account mid-range interactions, which Gabriela Jaramillo lead to new phenomena which are not present in the usual University of Minnesota local version of the one-phase Stefan model: creation of [email protected] mushy regions, existence of waiting times during which the liquid region does not move, and possibility of melting nu- cleation. If the kernel is suitably rescaled, the correspond- CP5 ing solutions converge to the solution of the local one-phase Global Attractors of the Hyperbolic Relaxation Stefan problem. of Reaction-Diffusion Equations with Dynamic Boundary Conditions Cristina Brandle U. Carlos III Madrid Under consideration is the hyperbolic relaxation of the [email protected] semilinear reaction-diffusion equation, εu + u − Δu + f(u)=0, Emmanuel Chasseigne tt t U. Franois Rabelais (Tours) ε ∈ [0, 1], with the prescribed dynamic boundary condition, [email protected] ∂nu + u + ut =0. Fernando Quir´os For all singular and nonsingular values of the perturbation U. Aut´onoma Madrid parameter, we obtain global attractors with optimal regu- [email protected] larity. After fitting both problems into a common frame- work, a proof of the upper-semicontinuity of the family of global attractors is given. The result is motivated by the CP5 seminal work of J. Hale and G. Raugel in Upper semiconti- Stability and Convergence Analysis for Nonlocal nuity of the attractor for a singularly perturbed hyperbolic Diffusion and Nonlocal Wave Equations equation, J. Differential Equations 73 (1988). We consider the time-dependent nonlocal diffusion and Joseph L. Shomberg nonlocal wave equations, formulated in the nonlocal peri- Providence College dynamics setting. Initial and boundary data are given. [email protected] For nonlocal diffusion equation, the time derivatives are approximated using Forward Euler, Backward Euler and Ciprian Gal Crank–Nicholson schemes. For nonlocal wave equation, we Florida International University get the dispersion relations and use the Newmark method cgal@fiu.edu to discretize the equation. And we use finite element method to discretize the spatial domain. For both equa- tions, the standard timestep stability conditions must be CP5 reformulated, in light of the peridynamics formulation. We On λ-Symmetry Classification and Conservations have got the the stability conditions and the convergence Laws of Differential Equations theorems. Computational implications verify our results. We study λ-symmetry classification of nonlinear differen- Qingguang Guan tial equations based on the fin equation. The λ-symmetry Department of Scientific Computing approach allows us the finding of integrating factors, the Florida State University conservation forms as first integrals, similarity solutions [email protected] and reducing degrees of differential equation, which is a new approach in the theory of Lie groups. We present Max Gunzburger also the theory for computing the integrating factors from Florida State University λ-symmetries for a second order ordinary differential equa- School for Computational Sciences tions. Then the symmetry classifications are constructed [email protected] for the different forms of heat transfer coefficient and ther- mal conductivity functions of the nonlinear fin equations. In addition, as a different approach the Jacobi last mul- CP5 tiplier method is considered to determine the new forms Localized Perturbations of the Complex Ginzburg- of λ-symmetries and corresponding conservations laws and Landau Equation: Rcovering Fredholm Poperties invariant solutions of the nonlinear fin equations and the Via Kondratiev Spaces results obtained from two different methods are compared in the study. We consider perturbations to the complex Ginzburg- Landau equation in dimension 3: At =(1+iα)ΔA+A−(1+ Teoman zer,zlemOrhan,G¨ulden G¨un PD13 Abstracts 63

Istanbul Technical University [email protected], [email protected], [email protected], [email protected] [email protected], [email protected]

CP6 CP6 Mathematical Analysis on Chevron Structures in A Longwave Model for Strongly Anisotropic Liquid Crystal Films Growth of a Crystal Step In this presentation, I will talk about the mathematical A model of morphological evolution of a single growing analysis of a model for the chevron structure arising from crystal step is presented. Via a multi-scale expansion, the cooling from the Smectic-A liquid crystal to the chiral we derived a long-wave, strongly nonlinear, and strongly Smectic-C phase in a surface-stabilized cell with certain anisotropic evolution PDE for the step profile, performed boundary conditions under a given electric field. This phe- the linear stability analysis and computed the nonlinear nomenon causes severe defects in liquid crystal display de- dynamics. Linear stability depends on whether the stiff- vices and has attracted interest from both theoretical and ness is minimum or maximum in the direction of the step practical point of view. In the static analysis, the stability growth. It also depends nontrivially on the combination of the chevron structure is established through a sequence of the anisotropy strength parameter and the atomic flux of minimization problems converging to a reduced energy from the terrace to the step. Computations show forma- functional based on Chen-Lubensky model. This study es- tion and coarsening of a hill-and-valley structure superim- tablishes the existence of minimizers and provides analysis posed onto a large-wavelength profile, which independently of the minimal energy configuration in the limiting case. coarsens. Coarsening laws for the hill-and-valley structure Lei Z. Cheng are computed for both orientations of a maximum step stiff- Indiana University Kokomo ness, the increasing anisotropy strength, and the varying [email protected] atomic flux. Mikhail V. Khenner CP6 Department of Mathematics On the Existence of Absolute Weak Minimizers of Western Kentucky University Energy Functionals Associated with the Boundary [email protected] Value Problem of Nonlinear Elasticity In this communication, I will present a new necessary and CP6 sufficient condition for the existence of absolute weak min- Finite-Volume Numerical Approximations of Con- imizers that applies to scale-invariant energy functionals servation Law with Fading Memory including those appearing in the pioneering work of 1982 by J. Ball on the pure displacement boundary value prob- We study the longitudinal motion of materials where the lem of nonlinear hyper-elastiicity. This result also brings destabilizing influence of nonlinear elastic response com- new ideas to the regularity question associated with the petes with the damping effect of the fading memory. The boundary value problem of nonlinear hyper-elasticity that resulting mathematical model is a system of conservation would replace the delicate phase plane analysis. laws involving a time convolution integral. We construct, implement, and analyze a new numerical method based on Salim M. Haidar the finite volume method. The implementation relies on GRAND VALLEY STATE UNIVERSITY numerical methods for advection equations with (discrete) [email protected] delay. Finally we apply the complete algorithm to the elas- tic memory problem to make quantitatively predictions for CP6 the behaviour of the material. Multicomponent Polymer Flooding in Two Dimen- Paramjeet Singh sional Oil Reservoir Simulation Panjab University, Chandigarh, India [email protected] We propose a high resolution finite volume scheme for a (m+1)(m+1) system of nonstrictly hy perbolic conserva- tion laws which models multicomponent polymer flooding CP6 in enhanced oil-recovery process in two dimensions. In the presence of gravity the flux functions need not be mono- Acoustic Resonance of a Vapor Bounded by Its tone and hence the exact Riemann problem is complicated OwnLiquidPhase and computationally expensive. To overcome this diffi- culty, we use the idea of discontinuous flux to reduce the The pressure and temperature perturbations due to acous- coupled system into uncoupled system of scalar conserva- tic waves in an initially quiescent vapor in equilibrium with tion laws with discontinuous coefficients. High order ac- its liquid phase induce the evaporation and condensation at curate scheme is constructed by introducing slope limiter the vapor-liquid interface. The phenomenon can be math- in space variable and a strong stability preserving Runge- ematically formulated by a set of fluid-dynamics equations Kutta scheme in the time variable. The performance of for vapor and liquid and boundary conditions at the in- the numerical scheme is presented in various situations by terface. The boundary-value problem is solved and some choosing a heavily heterogeneous hard rock type medium. physical features of solution for acoustic resonance involv- Also the significance of dissolving multiple polymers in ing the evaporation and condensation at the interface are aqueous phase is presented discussed. Sudarshan Kumar Kenettinkara, Praveen c, G.D. Takeru Yano Veerappa Gowda Department of Mechanical Engineering, TIFR Centre For Applicable Mathematics,India Osaka University 64 PD13 Abstracts

[email protected] UFOP - UNIVERSIDADE FEDERAL DE OURO PRETO [email protected] CP7 The Robin Eigenvalue Problem for the p(x)- Grey Ercole, Rodney Biezuner, Breno Giacchini Laplacian As p →∞ UFMG - UNIVERSIDADE FEDERAL DE MINAS GERAIS We study the asymptotic behavior, as p →∞,ofthe [email protected], [email protected], first eigenvalues and the corresponding eigenfunctions for [email protected] the p(x)-Laplacian with Robin boundary conditions in an open, bounded domain Ω ⊂ RN with sufficiently smooth boundary. We prove that the positive first eigenfunctions CP7 converge uniformly in Ω to a viscosity solution of a prob- Global Regularity for a Class of Quasi-Linear Non- lem involving the ∞-Laplacian with appropriate bound- local Elliptic Equations ary conditions. Joint work with F. Abdullayev (Worcester Polytechnic Institute). We consider the solvability of quasi-linear elliptic equations with either nonlocal Neumann, Robin, or Wentzell bound- Marian Bocea ary conditions, defined on bounded non-smooth domains. Loyola University Chicago We develop the corresponding regularity theory for weak [email protected] solutions, and show that such weak solutions are globally H¨older continuous. Several motivations, applications, and other properties are also addressed. CP7 Well-Posedness of a Mathematical Model for Trace Alejandro Velez-Santiago Gas Sensors Department of Mathematics University of California, Riverside Trace gas sensors have great potential in many areas such [email protected] as monitoring atmospheric pollutants, leak detection, and disease diagnosis through breath tests. We present a math- ematical model of such a sensor with a modulated laser CP8 source leading to a complex system of elliptic partial differ- Fractional Differential Equations with Impulses ential equations. Here we present a proof of well-posedness for the resulting system of equations via the Lax-Milgram In this talk we discuss fractional order impulsive differen- theorem as well as error estimates of the finite element tial equations. We establish existence and uniqueness of method solution. the solution using fixed point techniques. Moreover, we also apply our analytical result to fractional order model. Brian W. Brennan,RobertC.Kirby At the end we prove the stability and perform some numer- Baylor University ical simulation to give a pictorial view of the theoretical b [email protected], Robert [email protected] findings.

Syed Abbas CP7 Inidan Institute of technology Mandi, India A Unique Solution to a Quasi-Linear Elliptic Equa- [email protected] tion

We prove the existence of a unique classical solution u to CP8 the equation −∇ · (a(u)∇u)+v ·∇u = f with boundary Stability of Solutions of Parabolic and Hyperbolic condition ∇u · n = g, where the value of the solution u(x) Systems of Partial Differential Equations is known at a point x0 in the domain. The key to proving the uniqueness of the solution is an a priori estimate for u The lecture is devoted to the finding of criteria of solutions in a Sobolev space norm. stability of parabolic and hyperbolic systems of differen- tial equations. We consider the nonlinear parabolic sys- Diane Denny tems with Riemann-Liouville fractional derivatives. The Texas A&M University - Corpus Christi technique of stability criteria finding consists of integral [email protected] transformations of the considered equations into the sys- tems of ordinary differential equations and investigation the connection between stability of the systems of ordinary CP7 differential equations and systems of parabolic equations. Eigenvalues and Eigenfunctions of the Laplacian For the investigation of solutions stability of systems of Via Inverse Iteration with Shift ordinary differential equations with time-dependend coeffi- cients we use methods given in [1]. [1] Boykov I.V. Stability We present an iterative method, inspired by the inverse of solutions of differential equations. Penza: Penza State iteration with shift technique of finite linear algebra, de- University. 2008. 244 p. signed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition Ilya V. Boykov,VladimirRyazantsev for arbitrary bounded domains Ω ⊂ RN . Uniform con- Penza State University vergence away from nodal surfaces is also obtained. The [email protected], [email protected] method can also be used in order to produce the spectral decomposition of any given function u ∈ L2(Ω). CP8 Eder M. Martins Exact Solutions of Nonlinear Fractional Partial Dif- PD13 Abstracts 65

ferential Equation Quantum Mechanics

Oldman and Spanier first considered the fractional diferen- Research into quantum foundations has led to the develop- tial equations arising in diffusion problems. The fractional ment of a model of NRQM based upon process theory. Pro- diferential equations have been investigated by many au- cesses are modeled by means of a two player, co-operative, thors. In recent years, some effective methods for fractional combinatorial, forcing game, generating a discrete collec- calculus were appeared in open literature, such as the frac- tion of elements embedding into a space-like hyspersur- 4 tional sub-equation method and the first integral method. face in R , evolving discretely forward in time. Each el- The fractional complex transform is the simplest approach, ement is associated with a locally generated sinc wavelet, it is to convert the fractional diferential equations into ordi- the NRQM wave function being the interpolation over the nary diferential equations, making the solution procedure hypersurface. This results in a model of NRQM which is extremely simple. Recently, the fractional complex trans- local and non-contextual at the primitive level, yet non- form has been suggested to convert fractional order differ- local and contextual at the process level. The structure of ential equations with modified Riemann-Liouville deriva- the game will be described. tives into integer order diferential equations, and the re- duced equations can be solved by symbolic computation. William H. Sulis The present paper investigates for the applicability and McMaster University efficiency of the exp-function method on some fractional Departments of Psychiatry and Psychology nonlinear diferential equations. [email protected]

Adem Cevikel Yildiz Technical University CP9 [email protected] Numerical Methods and Multi-Scale Analysis for the Dirac Equation in Nonrelativistic Limit Regime Ozkan Guner We compare numerically temporal/ spatial resolution of Dumlupinar University various numerical methods for solving the Dirac equation [email protected] in nonrelativistic limit regime, involving a small param- eter 0 <≤ 1 which is inversely proportional to the Ahmet Bekir speed of light. In this regime, the solution is highly os- Eskisehir Osmangazi University cillating in time, which significantly increase the com- [email protected] putational burden. Using the conventional finite differ- ence methods, the time step τ requires a −scalability: Mutlu Akar τ = O(3). Then we propose some new methods, Time Yildiz Technical University Splitting Spectral(TPPS) and Exponential Wave Integrat- [email protected] ing(EWI). The new methods are unconditionally stable and their −scalability are improved to τ = O(2). At last we provide the multi-scale method for the Dirac equation CP8 to obtain a better numerical solution. Multiscaling Modelling in Evolutionary Dynamics Xiaowei Jia We study a class of processes that are akin to the Wright- National University of Singapore Fisher model, with transition probabilities weighted in [email protected] terms of the frequency-dependent fitness of the popula- tion types. By considering an approximate weak formula- tion of the discrete problem, we are derive a corresponding CP9 continuous weak formulation for the probability density. On the Method of Numerical Integration for Fric- Therefore, we obtain a family of PDEs for the evolution of tion Boundary Conditions the probability density, and which will be an approxima- tion of the discrete process in the joint large population, Friction boundary conditions are formulated by variational small time-steps and weak selection limit. The equations inequalities of second kind, where the nonlinearity is rep- resented by an L1 norm on the boundary. We show that in this family can be purely diffusive, purely hyperbolic 1 or of convection-diffusion type, with frequency dependent some numerical integration approximation for the L norm convection. The diffusive equations are of the degener- offers nice properties such as a discrete version of comple- ate type If the fitness functions are sufficiently regular, we menting conditions. Applications to the Poisson, elasticity, can recast the weak formulation in a stronger formulation, and fluid equations are considered. without any boundary conditions, but supplemented by a Takahito Kashiwabara number of conservation laws. Graduate School of Mathematical Sciences Max O. Souza The University of Tokyo Departamento de Matem´atica Aplicada [email protected] Universidade Federal Fluminense [email protected]ff.br CP9 Finite Element Analysis of the Stationary Power- Fabio Chalub Law Stokes Equations Driven by Friction Boundary Universidade Nova de Lisboa Conditions [email protected] We are concern with the finite element approximation for the stationary power law Stokes equations driven by slip CP8 boundary condition of “friction type”. It is shown that by A Game Theoretic Approach to Non-Relativistic applying a variant of Babuska-Brezzi’s theory for mixed 66 PD13 Abstracts

problems, convergence of the finite element approximation a simple fictitious domain method with L2-penalty for el- formulated is achieved with classical assumptions on the liptic and parabolic problems. A priori estimates and the regularity of the weak solution. Solution algorithm for the error estimates for penalization problems are carefully in- mixed variational problem is presented and analyzed in vestigated. Our methods can be applied not only to FEM details. Finally, numerical simulations that validate the but to FVM. Numerical experiments are performed to con- theoretical findings are exhibited. firm the theoretical results.

Mbehou Mohamed Guanyu Zhou Department of Mathematics and Applied Maths Graduate School of Mathematical Sciences, University of Pretoria, South Africa The University of Tokyo [email protected] [email protected]

Djoko Kamdem Jules Norikazu Saito Department of Mathematics and Applied Mathematics University of Tokyo University of Pretoria, South Africa Graduate School of Mathematical Sciences [email protected] [email protected]

CP9 CP10 Non-Linear Controller of Combined Radiative Higher Order Energy Preserving Schemes for a Conduction Systems Nonlinear Variational Wave Equation

This correspondence studies the problem of finite dimen- We consider a nonlinear variational wave equation describ- sional non-linear control for a class of systems described by ing the dynamics of the director field in a nematic liquid nonlinear hyperbolic-parabolic coupled partial differential crystal. This equation has an intrinsic dichotomy between equations (PDEs). Initially, Galerkins method is applied to what is called conservative and dissipative solutions. Stan- the PDE system to derive a nonlinear ordinary differential dard finite difference schemes will, generally speaking, give equation (ODE) system that accurately describes the dy- a dissipative solution. In this talk, we discuss how to con- namics of the dominant (slow) modes of the PDE system. struct higher order energy preserving numerical schemes After, we introduce a useful non linear controller to assure for the nonlinear variational wave equation using the dis- stabilization under convex sufficient conditions. Numerical continuous Galerkin framework. examples show high performances. Peder Aursand Ghattassi Mohamed, Ghattassi Mohamed Norwegian University of Science and Technology Universit´e de Lorraine, Institut Elie Cartan de Lorraine, [email protected] [email protected], [email protected] Siddhartha Mishra ETH Zurich CP9 Zurich, Switzerland [email protected] A Second-Order Time Discretization Scheme for a System of Nonlinear Schr¨odinger Equations Ujjwal Koley We propose a new time discretization scheme for a sys- University of W¨urzburg tem of nonlinear Schr¨odinger equations which is a model of [email protected] the interaction of a non-relativistic particles with different masses. Our scheme is composed of two (complex-valued) linear systems at each time step, and the solution is shown CP10 to converge at a second order rate. We report numerical Traveling Waves of Moderate Amplitude in Shallow example to confirm the theoretical results. Our idea can Water be applied to a large class of nonlinear PDEs. We prove existence of traveling wave solutions of an equa- Takiko Sasaki tion for surface waves of moderate amplitude arising as Graduate School of Mathematical Sciences a shallow water approximation of the Euler equations for The University of Tokyo inviscid, incompressible and homogenous fluids. Our ap- [email protected] proach is based on techniques from dynamical systems and relies on a reformulation of the equation as an autonomous Norikazu Saito Hamiltonian system. We determine bounded orbits in the University of Tokyo phase plane to establish existence of the corresponding pe- Graduate School of Mathematical Sciences riodic and solitary traveling wave solutions of elevation and [email protected] depression, as well as of solitary waves with compact sup- port. Furthermore, we provide a detailed analysis of the solitary waves’ dependence on the wave speed. CP9 Fictitious Domain Method with the L2-Penalty and Anna Geyer Application to the Finite Element and Finite Vol- University of Vienna ume Methods [email protected]

The fictitious domain approach is useful to compute nu- merical solutions of PDEs defined in complex domains and CP10 time-dependent moving domains. In this paper, we study On-Site and Off-Site Solitary Waves of the Discrete PD13 Abstracts 67

Nonlinear Schr¨odinger Equation Volterra models, and that the form of the LV model de- pends on the nature of the perturbation term. Stability We construct several types of symmetric localized stand- and bifurcation analysis for the equilibrium points of the ing waves (solitons) to the d-dimensional discrete nonlinear LV models is used to stabilize soliton-sequence propagation Schr¨odinger equation (dNLS) with cubic nonlinearity for and to achieve on-off switching. Furthermore, for some se- −2 2 2 2 d =1, 2, 3: i∂tun = h (δ u)n −|un| un,whereδ denotes tups, the LV description appears to hold even outside of the discrete Laplacian on Zd, using bifurcation methods the perturbative regime. from the continuum limit. Such waves and their relative energies are thought to play a role in the propagation of lo- Avner Peleg calized states of dNLS across the lattice, which radiate en- State University of New York at Buffalo ergy until they converge to a solitary wave at a fixed lattice apeleg@buffalo.edu site. This phenomenon is known as the ”Peierls-Nabarro Barrier” in discrete systems and was first investigated for- Debananda Chakraborty mally by M. Peyrard and M.D. Kruskal. These standing Virginia Intermonte College wave solutions and the question of their stability proper- [email protected] ties are also closely related to the variational problem of seeking critical points of the Hamiltonian subject to fixed 2 Quan M. Nguyen l norm (ground states). International University, Vietnam National University-HCMC Michael Jenkinson Vietnam Columbia University [email protected] Department of Applied Physics and Applied Mathematics [email protected] Yeojin Chung Southern Methodist University Michael I. Weinstein [email protected] Columbia University Dept of Applied Physics & Applied Math Jae-Hun Jung [email protected] SUNY at Buffalo jaehun@buffalo.edu CP10 On a Nonlinear Dispersive Wave of Third Order CP10 On Models of Short Pulse Type in Continuous Me- We discuss several features of a nonlinear modulated dis- dia persive wave of third order. We show that under certain conditions, the localized smooth solution decays in time. We consider ultra-short pulse propagation in nonlinear The rate of the decay is also obtained. metamaterials characterized by a weak Kerr-type nonlin- earity in their dielectric response. We will derive short- Jeng-Eng Lin 1 3 pulse equation (SPE) u = u + (u ) in frequency George Mason University xt 6 xx band gaps. We will discuss SPE in its characteristic [email protected] coordinate and the transformation to sine-Gordon equa- tion, and robustness of various solutions emanating from CP10 the sine-Gordon equation and their periodic generaliza- tions. By adding a regularized term in the equation, Shock-Wave Analysis of Condensate Development we will discuss the connection of (regularized) SPE with in a Model of Photon Scattering the nonlinear Schrodinger equation. Finally we will dis- cuss the wellposedness of generalized Ostrovsky equation We study long-time dynamics in a simplified Kompaneets 1 u = u + (up) with small initial data. equation. The Kompaneets equation describes evolution xt 6 xx of photon energy spectrum due to Compton scattering of Yannan Shen photons by electrons, an important energy transport mech- University of Minnesota anism in certain plasmas. For our model, we prove global [email protected] existence for initial data with any finite moment, conver- gence to equilibrium in large time, and failure to conserve photon number for large solutions, due to formation of a CP11 shock at zero energy. This is joint work with Dave Lever- Nonlocal Weighted Biological Aggregation more and Hailiang Liu. The Cucker-Smale flocking model has a well-known Robert Pego Achilles heel: the diluting effect of distant mass. A simi- Carnegie Mellon University lar weakness afflicts typical nonlocal aggregation equations [email protected] with attractive-repulsive interaction. To address the lat- ter we consider a weighted-averaging approach similar to Motsch-Tadmor’s adjustment of Cucker-Smale. We shall CP10 examine how gradient-flow structure is maintained via the Relating Collision-Induced Dynamics of Soliton Se- introduction of a new metric, one that penalizes movement quences of Coupled-Nls Equations and Dynamics in in crowded configurations (nonlocally). We will interpret Lotka-Volterra Models the metric, consider its pairing with several related energy potentials under gradient flow, and discuss its influence on We show that collision-induced dynamics of sequences of aggregation dynamics. solitons of N perturbed coupled nonlinear Schr¨odinger (NLS) equations can be described by N dimensional Lotka- Jeff Eisenbeis, Robert Pego, Dejan Slepcev 68 PD13 Abstracts

Carnegie Mellon University Qiang Du [email protected], [email protected], Penn State University [email protected] Department of Mathematics [email protected]

CP11 Front-Dynamics and Pattern Selection in Semi- CP11 Bounded Domains: Pulled Vs Pushed Fronts in Different Wave Solutions Associated with Singular Cahn-Hilliard, CGL, and FHN Lines on Phase Plane

Invasion fronts govern pattern-forming processes in many The bifurcation phenomena of a compound K(m,m) equa- experimental areas such as ion bombardment, chemical nu- tion are investigated in this paper. Three singular lines cleation, and phase separation. Often, an instability is trig- have been found in the associated topological vector field, gered spatially progressively, creating an effective bound- which may evolve in the phase trajectories of the system. ary condition ahead of an invasion front. We describe the The influence of parameters as well as the singular lines on effect of the boundary condition on the front dynamics and the properties of the equilibrium points has been explored the patterns in the wake. We find two basic scenarios cor- in details. Transition boundaries have been obtained to responding to pushed and pulled fronts. Pushed fronts lead divide the parameter space into regions associated with to oscillatory nonlinear interaction, hysteresis, and multi- different types of phase trajectories. The existence con- stability in the presence of boundaries. Pulled fronts in- ditions and related discussions for different traveling wave teract monotonically, with universal leading-order terms solutions have been presented in the end. determined by absolute spectra. We prove these results, in particular establishing a complete bifurcation diagram and Yu V. Wang leagding-order asymptotics, in the complex Ginzburg Lan- York College, The City University of New York dau equation using geometric desingularization and hete- [email protected] roclinic matching. We also present numerical studies in Cahn-Hilliard and FitzHugh-Nagumo equations. Joint w/ CP12 A Scheel. Steklov Representations of Harmonic Functions on Ryan Goh Exterior Regions University of Minnesota [email protected] This talk will summarize some results about the represen- tation of solutions of Laplace’s equation in exterior regions in R3 subject to Dirichlet, Robin or Neumann boundary CP11 data, The results generalize the theory of multipole expan- sions used for the region outside a ball. This is used to Multi-Agent Control of the Generalized Viscous describe some new formulae for the electrostatic capacity Burgers Equation of bounded sets in space. This is joint work with Qi Han.

We model the dynamic properties of traffic flows by means Giles Auchmuty of the generalized viscous Burgers equation defined over Department of Mathematics finite space. Our aim is to determine the spatio-temporal University of Houston distribution of adaptive control agents for tracking the for- [email protected] mation of traffic density patterns and for distributing the control feedback throughout the spatial domain. Feedback control is then applied to stabilize the dynamics of the CP12 Burgers equation to maximize the traffic flows or trigger Differentiability of the Function Best Sobolev Con- finite-time transition along equilibrium profiles of the gov- stant erning PDE. Let Ω be a bounded smooth domain of RN ,1≤ p ≤ N Dimitrios Papadimitriou  Np ∈  → and p := N−p . We consider the function q [1,p ] λq, UGent where [email protected]    p 1,p q λq := inf |∇u| dx : u ∈ W0 (Ω) and |u| dx =1 . CP11 Ω Ω Bi-Stable Mean-Field Model We prove that this function is absolutely continuous and also given by the expression In this poster, I will present the bi-stable mean-field model     q to illustrate the rich long-time dynamics of stochastic mod- 1 s λq = λ1 exp −p |us| log |us| dxds els. The mathematical analysis of this model, including the 1 s Ω study of mean-field limits, existence of multiple equilibria, and fluctuation theory was initiated by Dawson et al. Us- for all q ∈ [1,p], where ing the large-deviations theory Papanicolaou et al. recently     studied the dynamical phase-transitions in such systems 1,p s p us ∈ Es := u ∈ W0 (Ω) : |u| dx =1 and |∇u| dx = λs due to stochastic fluctuation. We consider various gener- Ω Ω alizations of such mean-field models and study the phase  transitions between different equilibria in various settings. for each s ∈ [1,p ). As a consequence, λq is differentiable at q ∈ [1,p) if, and only if, the functional Chao Tian  Pennsylvania State University | |q | | tian [email protected] Iq(u):= uq log uq dx Ω PD13 Abstracts 69

is constant on Eq. It follows as an application of this result Here L is a homogeneous fourth order elliptic operator and 1  that the function q → λq belongs to C ([1,p )) if Ω is χΩ denotes the characteristic function. We analyze the a ball. Moreover, since q → λq is differentiable almost regularity properties of u. everywhere we obtain the following new observation: for  almost all q ∈ (p, p ) the functional Iq is constant on the Henok Mawi set Eq. Howard University [email protected] Grey Ercole UFMG - UNIVERSIDADE FEDERAL DE MINAS GERAIS CP12 [email protected] Numerical Solution of Nonlinear Elliptic Equation Using Finite Element Method

CP12 In this paper we have discussed the behavior, difficulties Bubbling Solutions for the Chern-Simons Gauged and limitations involved in solving fully nonlinear elliptic O(3) Sigma Model partial differential equations. Then we worked on a finite element method based on strong form of the differential We construct multivortex solutions of the elliptic govern- equation which works well for fully nonlinear elliptic equa- ing equation for the self-dual Chern-Simons gauged O(3) tion without much assumptions and illustrated numerical sigma model. Our solutions show concentration phenom- result with the help of MATLAB. ena at some points of the singular points as the coupling parameter tends to zero, and they are locally asymptoti- Garima Mishra cally radial near each blow-up point. Research Scholar Research Scholar Jongmin Han [email protected] Kyung Hee University [email protected] Manoj Kumar Motilal Nehru National Institute of Technology Kwangseok Choe Allahabad, India Inha University [email protected] [email protected] CP13 Chang-Shou Lin National Taiwan University Semicontinuous Viscosity Solutions for Quasicon- [email protected] vex Hamiltonians The theory of semicontinuous viscosity solutions for con- CP12 vex hamiltonians introduced by Barron and Jensen in 1990 is extended to quasiconvex hamiltonians. Equations of the A Semismooth Newton Multigrid Method for Con- form H(x, u(x),Du(x)) = 0, with p → H(x, r, p)quasi- strained convex, are shown to have a unique lower semicontinuous Elliptic Optimal Control Problems viscosity solution and such a solution is represented as the ∞ A multigrid scheme is proposed for solving the Schur com- value function to a variational problem in L . plement linear systems arising in each Newton iteration Emmanuel Barron when the semi-smooth Newton method is applied to solve Loyola University Chicago control-constrained elliptic optimal control problems. Nu- USA merical experiments are performed to illustrate the high [email protected] efficiency of our proposed method. Computation simula- tion show that the convergence rate is quite robust as the regularization parameter approaches to zero. CP13 Jun Liu A Hamilton-Jacobi Equation for the Continuum Southern Illinois University Limit of Non-Dominated Sorting [email protected] Non-dominated sorting is a fundamental problem in multi- objective optimization, and is equivalent to several impor- Mingqing Xiao tant combinatorial problems. The sorting can be viewed Southern Illinois University at Carbondale as arranging points in Euclidean space into fronts by re- [email protected] peated removal of the set of minimal elements with respect to the componentwise partial order. We prove that in the (random) large sample size limit, the fronts converge al- CP12 most surely to the level sets of the viscosity solution of a A Free Boundary Problem for Higher Order Ellip- Hamilton-Jacobi equation. tic Operators Jeff Calder n Let Ω be a domain in R with 0 ∈ ∂Ω. Suppose in B, the Department of Mathematics n unit ball in R ,uand Ω satisfy the following equation in University of Michigan the sense of distributions: [email protected]

Lu = χΩ in B Selim Esedoglu nonumber Dαu =0for |α|≤3inB \ Ω. University of Michigan 70 PD13 Abstracts

[email protected] in hydro-dynamically relevant evolution equations. In this presentation a general solution formula to (1) is derived Alfred O. Hero and its regularity for a class of smooth initial data is stud- The University of Michigan ied. Dept of Elect Eng/Comp Science [email protected] Alejandro Sarria University of Colorado, Boulder Postdoctoral appointment to begin in Fall 2013 CP13 [email protected] General Solution to Unidimensional Hamilton- Jacobi Equation CP14 A method for finding the general solution to the par- Boundary Value Problems of the System of Pdes tial differential equations: F (ux,uy)=0;F (f(x)ux,uy)= of Steady Vibrations in the Theory of Thermoelas- 0(orF (ux,h(y)uy) = 0) is presented, founded on a Leg- ticity for Solids with Double Porosity endre like transformation and a theorem for Pfaffian dif- ferential forms. As the solution obtained depends on an The boundary value problems of the system of PDEs of arbitrary function, then it is a general solution. As an steady vibrations in the theory of thermoelasticity for extension of the method it is obtained a general solution solids with double porosity are investigated. The funda- mental solution of this system is constructed by means of to PDE: F (f(x)ux,uy)=G(x),and then applied to unidi- mensional Hamilton-Jacobi equation. elementary functions. The integral representations formu- las of classical solutions are presented. The basic properties Maria Lewtchuk Espindola of potentials are established. The uniqueness and existence Mathematics Department - Universidade Federal da theorems for boundary value problems are proved by the Paraba potential method and the theory of singular integral equa- [email protected] tions. Merab Svanadze CP13 Ilia State University [email protected] The Pullback Equation: An Overview

This talk will summarize some results I obtained with var- CP14 ious authors and led to the publication of a book: [Csato- Dacorogna-Kneuss,The Pullback Equation for Differential From Micropolar Navier-Stokes Equations to Fer- Forms, Springer, 2012]. This equation consists of finding a rofluids: Analysis and Numerics map ϕ : Rn → Rn satisfying The Micropolar Navier-Stokes Equations (MNSE), is a sys- ∗ ϕ (g)=f tem of nonlinear parabolic partial differential equations coupling linear velocity, pressure and angular velocity, i.e.: where g and f are closed differential k-forms.  This turns material particles have both translational and rotational n degrees of freedom. The MNSE is a central component of out to be a nonlinear first-order system of k partial differ- ential equations. I will emphasis the cases k =2andk = n the Rosensweig model for ferrofluids, describing the linear where we obtained global results with optimal regularity velocity, angular velocity, and magnetization inside the fer- and Dirichlet condition. rofluids, while subject to distributed magnetic forces and torques. We present the basic PDE results for the MNSE Olivier Kneuss (energy estimates and existence theorems), together with a Department of Mathematics first order semi-implicit fully-discrete scheme which decou- University of California, Berkeley ples the computation of the linear and angular velocities. [email protected] Similarly, for the Rosensweig model we present the basic PDE results, together with an fully-discrete scheme com- bining Continuous Galerkin and Discontinuous Galerkin CP14 techniques in order to guarantee discrete energy stability. Long-time Behavior of Solutions to the Generalized Finally, we demonstrate the capabilities of the Rosensweig Two-Component Hunter-Saxton System model and its numerical implementation with some numer- ical simulations in the context of ferrofluid pumping by We examine regularity of solutions to the non-local system means of external magnetic fields.   1 1 Ignacio Tomas − a 2 − k 2 − k 2 − a +2 2 uxt+uuxx ux ρ = ρ dx uxdx, (1)University of Maryland, College Park 2 2 2 0 2 0 [email protected] 2 ρt + uρx = aρux, (a, k) ∈ R . System (1) generalizes the Hunter-Saxton equation which CP14 describes the orientation of waves in massive nematic liq- Damping by Heat Conduction in the Timoshenko uid crystals. It also appears as the short-wave limit System: Fourier and Cattaneo Are the Same (a = −1,k= ±1) of the Camassa-Holm equation arising in the theory of shallow water waves, the inviscid Karman- We consider the Cauchy problem of the one-dimensional Batchelor flow (a = −k = 1), the Constantin-Lax-Majda Timoshenko system coupled with heat conduction, wherein equation (a = −k = ∞), and the generalized Proudman- the latter is described by either the Cattaneo law or the Johnson equation comprising several models from fluid dy- Fourier law. We prove that heat dissipation alone is suf- namics. Moreover, (1) serves as a tool for studying the ficient to stabilize the system in both cases, so that ad- role that one-dimensional convection and stretching play ditional mechanical damping is unnecessary. However, the PD13 Abstracts 71

decay of solutions without the mechanical damping is found University of California Los Angeles to be slower than that with mechanical damping. Fur- [email protected] thermore, in contrast to earlier results, we find that the Timoshenko-Fourier and the Timoshenko-Cattaneo sys- tems have the same decay rate. The rate depends on a MS1 certain number, which is a function of the parameters of On a Free Boundary Fluid-Elasticity Interaction the system. The talk addresses the problem of minimizing turbulence Aslan R. Kasimov inside fluid flow in the case of free boundary interaction be- Division of Mathematical & Computer Sciences and tween a viscous fluid and a moving and deforming elastic Engineering body. Our problem is motivated by the issue of reducing King Abdullah University of Science and Technology turbulence inside the blood flow in stenosed and stented [email protected] arteries. The presentation will focus on (i) existence of an optimal control acting inside the fluid domain; and (ii) the belkacem said-houari first-order necessary conditions of optimality, which involve king abdullah university of science and technoloy finding a suitable adjoint problem and using it to explic- [email protected] itly compute the gradient of the cost functional. This will provide the characterization of the optimal control, paving the way for a numerical study of the problem. MS1 Lorena Bociu Concerning the Rational Decay of Certain Fluid- Structure PDE Models North Carolina State University [email protected] In this talk, we shall demonstrate how delicate frequency domain relations and estimates, associated with coupled MS1 systems of partial differential equation models (PDE’s), may be exploited so as to establish results of uniform and Fluid-Structure Interaction with Multiple Struc- rational decay. In particular, our focus will be upon decay tural Layers properties of coupled PDE systems of different character- istics; e.g., hyperbolic versus parabolic characteristics. For We study a nonlinear, unsteady, moving boundary, fluid- such PDE systems of contrasting dynamics, the attainment structure (FSI) problem in which the structure is com- of explicit decay rates is known to be a difficult problem, posed of two layers: a thin layer which is in contact with inasmuch as there has not been an established methodol- the fluid, and a thick layer which sits on top of the thin ogy to handle hyperbolic-parabolic systems. For uncoupled structural layer. The fluid flow, which is driven by the wave equations or uncoupled heat equations, there are spe- time-dependent dynamic pressure data, is governed by the cific Carleman’s multiplier methods in the time domain, 2D Navier-Stokes equations for an incompressible, viscous wherein the exponential weights in each Carleman’s mul- fluid, defined on a 2D cylinder. The elastodynamics of the tiplier carefully take into account the particular dynamics cylinder wall is governed by the 1D linear wave equation involved, be it hyperbolic or parabolic. But for coupled modeling the thin structural layer, and by the 2D equations PDE systems which involve hyperbolic dynamics interact- of linear elasticity modeling the thick structural layer. We ing with parabolic dynamics, typically across some bound- prove existence of a weak solution to this nonlinear FSI ary interface, Carleman’s multipliers are readily applicable. problem as long as the cylinder radius is greater than zero. Given that such coupled PDE systems occur frequently in The spaces of weak solutions reveal a striking new feature: nature and in engineering applications; e.g., fluid-structure the presence of a thin fluid-structure interface with mass and structural acoustic interactions, there is a patent need regularizes solutions of the coupled problem. to devise broadly implementable techniques by which one Suncica Canic can infer uniform decay for a given PDE system. As one Department of Mathematics particular example, we shall work to conclude uniform de- University of Houston cays for structural acoustic dynamics. In these PDE mod- [email protected] els, the structural component is subjected to a structural damping ranging from viscous (weak) to strong (Kelvin- Voight). The rational decay rates we derive for this prob- Boris Muha lem explicitly reflect the extent of the damping which is in Department of Mathematics, Faculty of Science play. Since the damped elastic component of the coupled University of Zagreb dynamics is present on only a portion of the boundary, [email protected] there will necessarily be assumptions imposed upon the geometry. Martina Bukac University of Pittsburgh George Avalos Department of Mathematics University of Nebraska-Lincoln [email protected] Department of Mathematics and Statistics [email protected] MS2 The Stochastic Navier-Stokes Equation and the MS1 Statistical Theory of Turbulence Flutter of a Cantilever Beam in Low Speed Axial Air Flow:Theory and Experi Ment In 1941 Kolmogorov and Obukhov proposed that there ex- ists a statistical theory of turbulence that allows the com- Abstract not available at time of publication. putation of all the statistical quantities that can be com- puted and measured in turbulent systems. In this talk, A.V. Balakrishnan we will outline how to construct the Kolmogorov-Obukhov 72 PD13 Abstracts

refined scaling hypothesis (1962) with the She-Leveque in- [email protected], [email protected] termittency corrections from the stochastic Navier-Stokes (SNS) equation. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov ’62 MS2 scaling hypothesis with the She-Leveque intermittency cor- rections. Then we compute the invariant measure of tur- Ergodic Control for Stochastic Navier-Stokes bulence writing the stochastic Navier-Stokes equation as Equations an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the After a brief introduction to solvability and ergodic behav- invariant measure. ior of stochastic Navier-Stokes equations, the existence of optimal ergodic controls will be established for the two- dimensional stochastic Navier-Stokes equation. A con- Bjorn Birnir trolled martingale problem with relaxed controls will pro- University of California vide the framework for this study. Further properties of Department of Mathematics the optimal control will be discussed. [email protected] P. Sundar Lousiana State University MS2 [email protected] Optimal Stirring and Maximal Mixing S.S. Sritharan Passive advection of a tracer by an incompressible flow Naval Postgraduate School field typically results in mixing of the scalar field, an im- Monterey portant phenomena for applications in science and engi- [email protected] neering. Mixing may be measured via suitable norms of the tracer density, allowing for the formulation of sensible questions regarding optimal stirring strategies. In this talk MS3 we discuss current research aimed at articulating effective Spectral Stability for Transition Fronts in Multidi- estimates of maximal mixing rates for specific classes of mensional Cahn-Hilliard Systems stirring flows. We consider the spectrum associated with the linear oper- Charles R. Doering ator obtained when a Cahn-Hilliard system on Rn is lin- University of Michigan earized about a planar transition front solution. In the Mathematics, Physics and Complex Systems case of a single Cahn-Hilliard equation on Rn,it’sknown [email protected] that under general physical conditions the leading eigen- value moves into the negative real half plane at a rate |ξ|3, where ξ denotes the Fourier variable corresponding with MS2 components transverse to the wave. Moreover, in the case of a single equation, it has recently been verified that this Determining Form for the 2D Navier-Stokes Equa- spectral stability implies nonlinear stability. In the current tions Via Feedback Control. analysis, we establish a framework for verifying that the spectrum for multidimensional systems follows this same Data assimilation uses a finite dimensional trajectories in cubic law, and we use this framework to show that the a feedback control term to determine the full solution to condition holds for certain example cases. the 2D Navier-Stokes equations. The data can be a gen- eral interpolant such as finite volume elements, nodal val- Peter Howard ues, or Fourier modes. We discuss the evolution of the Department of Mathematics finite dimensional data itself, through an ordinary differ- Texas A&M University ential equation on a set of bounded trajectories. This is an [email protected] alternative to an earlier approach which works for Fourier modes. MS3 Ciprian Foias Morphological Competition in Amphiphilic Sys- Texas A & M, tems Indiana University [email protected] We present an analysis of the bifurcation and competitive evolution of co-existing morphological structures of dis- Michael S. Jolly tinct co-dimension as supported in the gradient flows of Indiana University the Functionalized Cahn-Hilliard (FCH) free energy. These Department of Mathematics systems support bilayer, pore, and micelle structures, as [email protected] well as complex network formed from the union of such structures. We analyse the pearling and meander instabil- Rostyslav Kravchenko ities of bilayer structures, and derive sharp-interface limits University of Chicago for the co-evolution of bilayer (homoclinic co-dimension 1) [email protected] and pore (radial co-dimension 2) structures, showing that possibility of co-existence. Edriss S. Titi University of California, Irvine and Keith Promislow Weizmann Institute of Science, Israel Michigan State University PD13 Abstracts 73

[email protected] lems using successive convolution

We present recent results for obtaining higher order, A- MS3 stable schemes for the wave equation. The method relies on approximating derivatives with successive applications Rigorous Computation of Connecting Orbits of a certain convolution integral. The convolution stabilizes higher order schemes, and furthermore can be applied in Connecting orbits play an organizing role in dynamical O(N) operations using fast convolution techniques. We also systems described by partial differential equations. They briefly discuss extending the methodology to first order describe localized patterns (pulses, boundary layers), act hyperbolic conservation laws. as building blocks for more complicated patterns, and de- scribe the transitions between different states. The past Matthew F. Causley decade has seen enormous advances in the development of Michigan State University computer assisted proofs in dynamics. In this talks we will [email protected] discuss recent progress in the rigorous computation of con- necting orbits, and directions for further research will be outlined. Andrew J. Christlieb Michigan State Univerity Jan Bouwe Van Den Berg Department of Mathematics VU University Amsterdam [email protected] Department of Mathematics [email protected] MS4 Jean-Philippe Lessard A New Discontinuous Galerkin Finite Element Universit´eLaval Method for Directly Solving the Hamilton-Jacobi [email protected] Equations In this talk, we will introduce a new discontinuous Galerkin method for solving time-dependent Hamilton-Jacobi equa- MS3 tions. This new method works directly on the solutions, The Stability of Periodic Patterns of Localized and makes use of a new entropy fix inspired by Harten and 2 Spots for Reaction-Diffusion Systems in R Hyman’s entropy fix for Roe scheme for the conservation laws. Compared to previous works, the method has the We determine the spectral stability threshold for a peri- advantage of simplicity in implementations, and could be odic arrangment of localized spots for some singularly per- applied to compute problems with non convex Hamiltoni- turbed two-component reaction-diffusion systems includ- ans. ing the Gierer-Meinhardt, Schnakenburg, and Gray-Scott models. In the semi-strong interaction asymptotic limit Yingda Cheng,ZixuanWang where only one of the components has an asymptotically Department of Mathematics small diffusivity, the well-known leading order stability Michigan State University threshold governing amplitude instabilities is independent [email protected], [email protected] of the lattice structure. By combining a spectral approach based on Floquet-Bloch theory together with the method of matched asymptotic expansions and appropriate Fredholm MS4 solvability condition, the next order term in the expan- Optimal energy conserving local discontinuous sion of the stability threshold is calculated in terms of the Galerkin methods for second-order wave equation regular parts of certain Green’s functions. From an opti- in heterogeneous media mization of this term, the most stable lattice arrangement of spots is identified. Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of Michael Ward applications in physics and engineering. The design of nu- Department of Mathematics merical methods for such general wave propagation prob- University of British Columbia lems is challenging because the energy conserving property [email protected] has to be incorporated in the numerical algorithms in or- der to minimize the phase or shape errors after long time David Iron integration. In this talk, we will discuss multi-dimensional Dalhousie University, Canada wave problems and consider linear second-order wave equa- [email protected] tion in heterogeneous media. We will present an LDG method, in which numerical fluxes are carefully designed John Rumsey to maintain the energy preserving property and accuracy. Dalhousie University We propose compatible high order energy conserving time [email protected] integrators and prove the optimal error estimates and the energy conserving property for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of Juncheng Wei convergence, and show that the errors of the numerical so- Department of Mathematics lution do not grow significantly in time due to the energy Chinese University of Hong Kong conserving property. [email protected] Ching-Shan Chou Department of Mathematics MS4 Ohio State University Higher order O(N) schemes for hyperbolic prob- [email protected] 74 PD13 Abstracts

Yulong Xing Forward-backward-forward Diffusion Equations for Department of Mathematics Image Restoration Univeristy of Tennessee / Oak Ridge National Lab [email protected] In this talk, we present the implementation and numer- ical experiments demonstrating new forward-backward- Chi-Wang Shu forward nonlinear diffusion equations for noise reduction Brown University and deblurring, developed in collaboration with Patrick Div of Applied Mathematics Guidotti and Yunho Kim. The new models preserve and [email protected] enhance the most desirable aspects of the closely-related Perona-Malik equation without allowing staircasing. By using a Krylov subspace spectral (KSS) method for time- MS5 stepping, the properties of the new models are preserved Multiple Solutions to An Elliptic Problem Related without sacrificing efficiency. to Vortex Pairs James V. Lambers Abstract not available at time of publication. University of Southern Mississippi Department of Mathematics Yi Li [email protected] Wright State University Xian Jiaotong University Patrick Guidotti [email protected] University of California at Irvine [email protected]

MS5 Yunho Kim Front Propagation in Isothermal Diffsuion Systems Yale University [email protected] Abstract not available at time of publication.

Yuanwei Qi MS6 University of Central Florida [email protected] Denoising an Image by Denoising its Curvature Im- age

MS5 In this work we argue that when an image is corrupted by additive noise, its curvature image is less affected by it, Global Dynamical Analysis of An Age-Structured i.e. the PSNR of the curvature image is larger. We specu- Cholera Model late that, given a denoising method, we may obtain better results by applying it to the curvature image and then re- Abstract not available at time of publication. constructing from it a clean image, rather than denoising Zhisheng Shuai the original image directly. Numerical experiments con- University of Central Florida firm this for several PDE-based and patch-based denoising [email protected] algorithms. Stacey Levine MS5 Duquesne University [email protected] A Nonlinear Problem from the Flow Between Two Counter-Rotating Disks Marcelo Bertalmio Abstract not available at time of publication. Universitat Pompeu Fabra [email protected] weiqing Xie Cal Poly Pomona [email protected] MS6 The Influence of Pde Based Image Processing in Analysis: Modern Developments MS6 A Forward-backward Regularization of the Perona- Abstract not available at time of publication. Malik Equation Surya Prasath In this talk we will survey recent developments in the anal- University of Missouri-Columbia ysis of the Perona-Malik equation as well as related models [email protected] with particular emphasis on a forward-backward regular- ization which turns out to be well-posed in the sense of weak Young measure solutions. MS7 Overturning Traveling Waves in Interfacial Fluid Patrick Guidotti Dynamics University of California at Irvine [email protected] Large-amplitude traveling waves in interfacial fluid dynam- ics can, in certain settings, be overhanging. In work with Benjamin Akers and J. Douglas Wright, we introduce a new MS6 formulation for the traveling wave problem which allows for Numerical Implementation of a New Class of such waves. This formulation gives a traveling wave ansatz PD13 Abstracts 75

for a parameterized curve, making use of a normalized ar- Department of Mathematics clength parameterization. We apply this formulation, both Florida International University analytically and computationally, to interfacial flows with caoc@fiu.edu surface tension.

David Ambrose MS8 Drexel University On the Navier-Stokes Limit of the Navier-Stokes- [email protected] Voigt Equations

We consider the three-dimensional Navier-Stokes-Voigt MS7 (NSV) equations and we analyze, from the asymptotic Low Regularity Well-posedness for the 2D behavior view point, its Navier-Stokes (NS) limit as the Maxwell-Klein-Gordon Equation relaxation parameter vanishes. We show that the NSV- attractors converge to the weak NS-attractor in the Haus- We discuss progress on the low regularity local well- dorff semidistance induced by the weak L2-metric on the posedness for the Maxwell-Klein-Gordon equation in two absorbing set of the Navier-Stokes equations. Some results dimensions. related to the strong topology of L2 are also proved.

Magda Czubak Michele Coti Zelati Binghamton University Indiana University, Bloomington [email protected] [email protected]

MS7 MS8 The Discrete Schr¨odinger Equation on Triangular Vortex Stretching and Sub-Criticality for the 3D Lattices Navier-Stokes Equations

ThediscreteSchr¨odinger equation on a cubic lattice (in The goal of this talk is to present a new approach to the any number of dimensions) is easily solved by separation problem of possible singularity formation in the 3D NSE. of variables. On a two-dimensional triangular lattice, how- The first step consists of elucidating a geometric scenario ever, separation of variables is not possible. The funda- leading to criticality for large data. This is based on a mental solution decays (in supremum norm) at the rate local, geometric measure-type regularity criterion, essen- −3/4 |t| for a generic choice of coupling coefficients between tially, local, anisotropic one-dimensional sparseness of the adjacent vertices. Slower power-law decay rates are ob- regions of intense vorticity, coupled with the mathematical served if these coefficients satisfy a certain set of algebraic evidence that the scale of sparseness needed to prevent the relations. blow-up may in fact be achieved via the mechanism of vor- tex stretching. The second step consists of attempting to Vita Borovyk, Michael Goldberg brake the criticality (in the aforementioned sense/setting). University of Cincinnati Presently, this is indeed possible for a quite general class [email protected], [email protected] of blow-ups; in particular, for any blow-up rate featuring local spatially-algebraic structure.

MS7 Zoran Grujic Well-Posedness Results for a Derivative Nonlinear University of Virginia Schr¨odinger Equation [email protected] In this talk, I will discuss new results on the existence and uniqueness of solutions to a generalized derivative nonlin- MS8 ear Schr¨odinger equation. This equation, which has nonlin- | |2σ The Navier-Stokes-Voight Model for Image In- earity i u ux has recently been given some attention for painting its solitary wave solutions and blow up dynamics. How- ≤ ever, there are interesting values of σ, including 1 <σ 2, We investigate some of the advantages of the 2D Navier- for which no existence or uniqueness results are available. Stokes-Voight (NSV) turbulence model and explore its lim- Open problems will be highlighted. its in the context of image inpainting. We begin by giving a brief review of the elegant analogy between the image in- Gideon Simpson tensity function for the image inpainting problem and the School of Mathematics stream function in 2D incompressible fluid, first introduced University of Minnesota by Bertalmio et al in 2001. We show that the NSV allows [email protected] for a more efficient numerical algorithm when automating the inpainting process. MS8 Moe Ebrahimi Global Well-Posedness of Boussinesq Equations University of California, San Diego [email protected] Boussinesq equations are mathematical models of buoy- ancy driven flows. In this talk we first introduce the Boussi- nesq equations, then, we will study the global in time exis- Michael Holst tence of classical solutions to some reduced 3D Boussinesq UC San Diego equations and also the 2D Boussinesq equations with par- Department of Mathematics tial dissipation. [email protected]

Chongsheng Cao Evelyn Lunasin 76 PD13 Abstracts

US Naval Academy the resulting elliptic equation using continuous Finite El- [email protected] ements. We analyze the consistency and compare the scheme numerically with methods such as Implicit Euler. Numerical results indicate that the AP method is indeed MS9 superior to more traditional methods. We discuss exten- Asymptotically Preserving Scheme for Some Ki- sions to higher order of consistency. netic and Fluids Equation with Multi-Scale Behav- iors Jochen Sch¨utz IGPM I will present some Asymptotically Preserving schemes in RWTH Aachen some kinetic and fluids equations, including low Mach flu- [email protected] ids, a semilinear hyperbolic relaxation system with a two- scale discontinuous relaxation rate, linear kinetic equations in the diffusion limit, kinetic-fluid modeling of disperse two- MS10 phase flows, etc. I will emphasis on numerical analysis and algorithm designs for these singular and stiff problems. Wave Fronts in a Model for Gasless Combustion with Heat Loss Jian-guo Liu Dept. of Mathematics and Physics We consider a model of gasless combustion with heat loss, Duke University where the heat loss from the system to the environment is [email protected] formulated according to Newtons law of cooling. The sys- tem of partial differential equations that describe evolution of the temperature and remaining fuel contains two small MS9 parameters, a diffusion coefficient for the fuel and a heat The Unified Gas Kinetic Scheme of K. Xu Applied loss parameter. We use geometric singular perturbation to Linear Transport in Diffusion Regimes theory to show existence of traveling waves in this system and then study their spectral and nonlinear stability. I will show that the unified gas kinetic scheme (UGKS) of K. Xu et al., originally developed for gas dynamics, can be Anna Ghazaryan applied to a linear kinetic model of radiative transfer the- Department of Mathematics ory. I will show that UGKS is asymptotic preserving (AP) Miami University in both optically thick and thin regimes. Contrary to many [email protected] AP schemes, this method is based on a standard finite vol- ume approach, it does neither use any decomposition of the solution, nor staggered grids. MS10 Luc Mieussens Coherent Structures in a Model for Mussel-Algae Institut de Mathematiques de Bordeaux Interaction [email protected] We consider a model for formation of mussel beds on soft sediments. The model consists of coupled nonlinear pdes MS9 that describe the interaction of mussel biomass on the sed- High Order Semi-Implicit Runge-Kutta Finite Dif- iment with algae in the water layer overlying the mussel ference Methods and Application to Non-Linear bed. Both the diffusion and the advection matrices in the Parabolic Relaxation Problems system are singular. We use Geometric Singular Pertur- bation Theory to capture nonlinear mechanisms of pattern We study solutions to nonlinear hyperbolic systems with and wave formation in this system. fully nonlinear relaxation terms in so called diffusive limit. Suitable semi-implicit schemes based on IMEX-Runge- Vahagn Manukian Kutta are constructed for the numerical solution of the Miami University Hamilton relaxed limit equation. Another set of schemes is devised [email protected] for relaxation systems, which are able to capture the limit diffusive relaxation with no parabolic time step restriction. A new stability analisys of the schemes is proposed and MS10 applied. Threshold Phenomena for Symmetric Decreasing Giovanni Russo Solutions of Reaction-Diffusion Equations University of Catania Department of Mathematics and Computer Science We study the long time behavior of solutions of the Cauchy [email protected] problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition Sebastiano Boscarino or monostable type. We prove a one-to-one relation be- University of Catania tween the long time behavior of the solution and the limit [email protected] value of its energy for symmetric decreasing initial data in L2 under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results MS9 between propagation and extinction for monotone families On the Efficiency of Asymptotic Preserving of initial data in the considered general setting. Schemes Cyrill B. Muratov We apply the concept of Asymptotic Preserving (AP) New Jersey Institute of Technology schemes[Jin 1999] to the linearized p-system and discretize Department of Mathematical Sciences PD13 Abstracts 77

[email protected] local well-posedness of the perturbation (weighted and un- weighted) in the Bourgain X1,b space, allowing us to recre- ate the Pego-Weinstein result via iteration. By combining MS10 this result with the I-method, we expect ultimately to ob- Traveling Waves in Coupled Reaction-Diffusion tain soliton stability for KdV with initial data too rough Models with Degenerate Source Terms to be in H 1.

We consider a general system of coupled nonlinear diffusion BrianPigott,SarahRaynor equations that do not have isolated rest states. Such sys- Wake Forest University tems arise in many different applications, e.g., propagation [email protected], [email protected] of epidemics, ion transport in cellular matter, and evapo- ration and condensation in porous media. We show that the degeneracy in the source terms implies that traveling MS11 waves have a number of surprising properties that are not Talbot Effect for the Cubic Nonlinear Schr¨odinger present for systems with nondegenerate source terms. In Equation on the Torus particular, we show that traveling wave trajectories con- necting two stable rest states can exist generically only for We study the evolution of the one dimensional periodic discrete wave speeds and that families of traveling waves NLS equation with bounded variation data. For the linear with a continuum of wave speeds cannot exist. evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. Jonathan J. Wylie For rational times the solution is a linear combination of Department of Mathematics finitely many translates of the initial data. We prove that City University of Hong Kong, Kowloon, Hong Kong a similar phenomenon occurs in the case of the cubic NLS [email protected] equation.

Nikolaos Tzirakis MS11 University of Illinois at Urbana-Champaign Dispersive Estimates for Schr¨odinger Operators in [email protected] Dimension Two

In this talk we will discuss L1 → L∞ dispersive estimates MS12 for Schrodinger operators in dimension two with real and A Geometric Approach for Mean Curvature Mo- decaying potentials in the case when there are no spectral tion in 2D assumptions at the edge of the spectrum. In addition we will describe logarithmically weighted dispersive estimates In this talk I will present a new approach for mean cur- with an integrable time decay rate at infinity. vature motion in 2D, which explicitly connects the curve shortening from the differential geometry setting with the Burak Erdogan level set formulation, in the viscosity solutions theory. University of Illinois, Urbana-Champaign This provides an exact analytical framework for its nu- [email protected] merical implementation, which has an important applica- tion in image processing. It runs on line on any image at William Green http://www.ipol.im/. Rose-Hulman Institute of Technology [email protected] Adina Ciomaga L.E. Dickson Instructor University of Chicago MS11 [email protected] Relaxation of Exterior Wave Maps to Harmonic Maps MS12 We establish relaxation of arbitrary, finite energy, one- The Thin One Phase-Problem equivariant wave maps in 3 space dimensions exterior to the unit ball to the 3-sphere with a Dirichlet condition at We describe the regularity of the free boundary for the thin r = 1, to the unique stationary harmonic map in its degree one-phase problem, in which the free boundary occurs on class. This settles a recent conjecture of Bizo´n, Chmaj, a lower dimensional subspace. This problem appears as a Maliborski who proposed this problem as a model in which model of a one-phase free boundary problem in the context to study stable soliton resolution and who observed this of the fractional Laplacian. asymptotic behavior numerically. This is joint work with C. Kenig and W. Schlag. Daniela De Silva Barnard College, Andrew Lawrie Columbia University Berkeley [email protected] [email protected]

MS12 MS11 On Congested Crowd Motion A New Approach to Soliton Stability for the KdV Equation We investigate the relationship between Hele-Shaw evolu- tion with a drift and a transport equation with a drift po- In this work, we consider the KdV equation in the expo- tential, where the density is transported with a constraint nentially weighted spaces of Pego and Weinstein. We prove on its maximum. When the drift potential is convex, the 78 PD13 Abstracts

crowd density is likely to aggregate. We show that the is joint work with J. Pruss and K. Schade. evolving patch satisfies a Hele-Shaw type equation. Matthias Hieber Inwon Kim TU Darmstadt, Germany Department of Mathematics [email protected] UCLA [email protected] MS13 Yao Yao Feedback Stabilization of Boussinesq Equations Department of Mathematics and Numerical Simulations University of Wisconsin In this talk we present theoretical and numerical results for [email protected] a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and Damon Alexander controlling energy efficient building systems. In particular, UCLA we show that it is possible to locally exponentially stabilize [email protected] the nonlinear Boussinesq equations by applying finite di- mensional Neumann/Robin type boundary controllers. In the end, a 2D problem is used to illustrate the ideas and MS12 demonstrate the computational algorithms. Numerical Methods for Geometric Elliptic Partial Differential Equations Weiwei Hu University of Southern California A few important geometric PDEs will be discussed which [email protected] can be solved using a numerical method called Wide Stencil finite difference schemes. Focusing on the Monge-Ampere MS13 equation, I show how naive schemes can work well for smooth solutions, but break down in the singular case. Well-Posedness and Small Data Global Existence There are several numerical schemes which fail to converge, for An Interface Damped Free Boundary Fluid- or converge only in the case of smooth solutions. I will Structure Model present a convergent solver which is fast, comparable to solving the Laplace equation a few times. We address the well-posedness of a fluid-structure interac- tion model describing the motion of an elastic body im- Adam Oberman mersed in an incompressible fluid. The fluid-structure sys- Simon Fraser University tem consists of the incompressible Navier-Stokes equations Canada and a damped linear wave equation coupled through trans- [email protected] mission boundary conditions on the free moving interface separating the elastic body and the fluid. We provide a priori estimates for the local-in-time existence of solutions MS13 for a class of initial data which also guarantees uniqueness. The global-in-time existence of solutions relies on a key Gevrey Regularity of Strongly Damped Wave energy inequality which in addition provides exponential Equations with Hyperbolic Dynamic Boundary decay of solutions of the system for given sufficiently small Conditions initial data.

We consider a linear system of coupled PDEs which con- Mihaela Ignatova sists of a strongly damped wave equation on a bounded University of California, Riverside domain with dynamic boundary conditions governed by an- [email protected] other wave equation (via the Laplace-Beltrami operator). We show that the system generates a strongly continuous Igor Kukavica semigroup T (t) which is analytic when strong damping is University of Southern California present on the boundary, otherwise of Gevrey class. In [email protected] both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing Irena Lasiecka ≤ | |−1 estimates of the form (d/dt)T (t) C t for the first University of Virginia ≤ | |−2 case, (d/dt)T (t) C t for the second. [email protected] Jameson Graber,IrenaM.Lasiecka University of Virginia Amjad Tuffaha The Petroleum Institute [email protected], [email protected] Abu Dhabi, UAE atuff[email protected] MS13 Analysis of the Ericksen-Leslie System for Liquid MS14 Crystals Generalized Gevrey Norms with Applications to Dissipative Equations In this talk we consider the Ericksen-Leslie system describ- ing the nematic phase of liquid crystals. We discuss local The regular solutions of the Navier-Stokes equations are and global wellposedness results for various subclasses of well-known to be analytic in both space and time vari- this system. Moreover, we present a thermodynamically ables, even when one starts with non-analytic initial data. consistent version of the so called simplified model. This The space analyticity radius has an important physical in- PD13 Abstracts 79

terpretation. It demarcates the length scale below which [email protected] the viscous effect dominates the (nonlinear) inertial effect. Foias and Temam introduced an effective approach to esti- mate space analyticity radius via the use of Gevrey norms MS14 which, since then, has become a standard tool for study- Stability of Compressible Viscous Waves in a Mov- ing analyticity. We extend this approach to a class of ing Domain dissipative equations, including critical and super-critical quasigeostrophic equations, where the dissipation opera- We establish the sharp nonlinear stability criteria for com- tor is a fractional Laplacian. This necessitates the use pressible viscous surface-internal waves for two barotropic of sub-analytic Gevrey classes and ”generalized” Gevrey fluids near the equilibrium. The stability is characterized norms and development of certain commutator estimates by the jump in the equilibrium density smaller than the in Gevrey classes to exploit the cancellation properties of critical jump value. When the lower fluid is heavier than the equation. This is achieved via the Littlewood-Paley de- the upper fluid, we will establish the zero surface tension composition and the Bony paraproduct formula. Though limit. This is joint work with Ian Tice and Yanjin Wang. not essential for the Navier-Stokes equations, such commu- tator estimates become crucial for the critical and super- Juhi Jang critical quasigeostrophic equations. Applications include University of California, Riverside, USA large time decay of higher order derivatives. [email protected]

Animikh Biswas Ian Tice University of North Carolina Carnegie Mellon University at Charlotte [email protected] [email protected] Yanjin Wang Xiamen University MS14 yanjin [email protected] Bounds on Energy and Enstrophy for the 3D Navier-Stokes-α and Leray-α Models MS15 We construct semi-integral curves which bound the pro- Nonlinearity Saturation as a Singular Perturbation jection of the global attractor of the Sabra Shell model in of the Nonlinear Schr¨odinger Equation the plane spanned by the energy and enstrophy for the 3D regime and the plane spanned by the enstrophy and palin- Saturation of the Kerr nonlinearity in the nonlinear strophy for the 2D regime. The semi-integral curves divide Schr¨odinger equation can be regarded as a singular per- the plane into regions having limited ranges for the direc- turbation which avoids the ill-posedness associated with tions of the flow. This allows us to estimate the average the self-focusing singularity. An asymptotic expansion is time it would take a solution to burst into a region of large presented which takes into account multiple scale behav- enstrophy and palinstrophy, respectively. ior in both the longitudinal and transverse directions. The mechanism for interaction of solitary wave structures and Aseel Farhat an adjacent field is determined. Indiana University Bloomington [email protected] Karl Glasner The University of Arizona Michael S. Jolly Department of Mathematics Indiana University [email protected] Department of Mathematics [email protected] Jordan Allen-Flowers Program in Applied Math Evelyn Lunasin University of Arizona United States Naval Accademy [email protected] [email protected] MS15 MS14 Spectra of Functionalized Operators Arising from On Inviscid Limits for the Stochastic Navier-Stokes Hypersurfaces Equations and Related Systems Functionalized energies, such as the Functionalized Cahn- One of the original motivations for the development of Hilliard, model phase separation in amphiphilic systems, stochastic partial differential equations traces it’s origins in which interface production is limited by competition to the study of turbulence. In particular, invariant mea- for surfactant phase, which wets the interface. This is sures provide a canonical mathematical object connecting in contrast to classical phase-separating energies, such as the basic equations of fluid dynamics to the statistical prop- the Cahn-Hilliard, in which interfacial area is energetically erties of turbulent flows. In this talk we discuss some recent penalized. In binary amphiphilic mixtures, interfaces are results concerning inviscid limits in this class of measures characterized not by single-layers, which separate domains for the stochastic Navier-Stokes equations and other re- of phase A from those of phase B via a heteroclinic connec- lated systems arising in geophysical and numerical settings. tion, but by bilayers, which divide the domain of the domi- This is joint work with Peter Constantin, Vladimir Sverak nant phase, A, via thin layers of phase B formed by homo- and Vlad Vicol. clinic connections. Evaluating the second variation of the Functionalized energy at a bilayer interface yields a func- Nathan Glatt-Holtz tionalized operator. We characterize the center-unstable Indiana University spectra of functionalized operators and obtain resolvent es- 80 PD13 Abstracts

timates to the operators associated with gradient flows of on periodic domains, and validated against classical 1D-1V the Functionalized energies. This is an essential step to a test-cases. Here we include a simple linear collision oper- rigorous reduction to a sharp-interface limit. ator (elastic scattering) and extend the model to 1D-2V. Then we study the coupled dynamics of electrons and ions Gurgen Hayrapetyan in a bounded domain. Carnegie Mellon University [email protected] Yaman Guclu Department of Mathematics Keith Promislow Michigan State University Michigan State University [email protected] [email protected] MS16 MS15 A High Order Time Splitting Method Based on Spots and Stripes in NLS-type Equations with Integral Deferred Correction for Semi-Lagrangian Nearly One-dimensional Potentials Vlasov Simulations

We consider the existence of spots and stripes for a class The dimensional splitting semi-Lagrangian methods with of NLS-type equations in the presence of nearly one- different reconstruction/interpolation strategies have been dimensional localized potentials. Under suitable assump- applied to kinetic simulations in various settings. However, tions on the potential, we construct various types of waves the temporal error is dominated by the splitting error. In which are localized in the direction of the potential and order to have numerical algorithms that are high order both have single- or multi-hump, or periodic profile in the per- in space and in time, we propose to use the integral deferred pendicular direction. The analysis relies upon a spatial correction (IDC) framework to reduce the splitting error. dynamics formulation of the existence problem, together The proposed algorithm is applied to the Vlasov-Poisson with a center manifold reduction. This reduction proce- system, the guiding center model and incompressible flows. dure allows these waves to be realized as uni- or multi-pulse homoclinic orbits, or periodic orbits in a reduced system Andrew Christlieb of ordinary differential equations. Michigan State University Todd Kapitula [email protected] Calvin College [email protected] Wei Guo University of Houston Mariana Haragus [email protected] Universite de Franche-Comte [email protected] Maureen Morton Kent State University [email protected] MS15 Stability of Line Solitons for the KP-II Equation Jingmei Qiu University of Houston I will talk on nonlinear stability of line soliton solutions [email protected] for the KP-II equation with respect to transverse pertur- bations that are exponentially localized as x →∞.The local amplitude and the phase shift of the crest of the line MS16 solitons are described by a system of 1D wave equations High Order Operator Splitting Methods Based on with diffraction terms and jumps of the local phase shift of an Integral Deferred Correction Framework the crest propagate in a finite speed toward y = ±∞. Integral deferred correction methods have been shown to Tetsu Mizumachi be an efficient way to achieve arbitrary high order accu- Faculty of Mathematics racy and possess good stability properties. In this paper, Kyushu University, Japan we construct high order operator splitting schemes using [email protected] the integral deferred correction (IDC) procedure to solve two dimensional initial-boundary-value problems. Further- more, we present analysis which establishes that the IDC MS16 method can correct for both the splitting and numerical Convected Scheme Solution of the Boltzmann- errors, lifting the order of accuracy by r with each cor- Poisson System with Spectrally Accurate Phase- rection, where r is the accuracy of the method used to Space Resolution solved the correction equation. Numerical examples in 2D on linear and nonlinear problems, with periodic and Dirich- The Boltzmann-Poisson system describes the evolution of let boundary conditions, are presented to demonstrate the multiple species of charged particles that interact through theoretical results. a self-consistent electric field and various collisional pro- cesses. Semi-Lagrangian methods like the Convected Andrew Christlieb, Yuan Liu Scheme (CS) are often used to solve the transport terms, Department of Mathematics because of their ability to take large time-steps (no CFL Michigan State University limit) and their low numerical diffusion. The CS is mass [email protected], [email protected] conservative and positivity preserving, and was recently ex- tended to arbitrarily high order of accuracy in phase-space: Zhengfu Xu the new scheme was applied to the Vlasov-Poisson system Michigan Technological University PD13 Abstracts 81

Dept. of Mathematical Sciences in one and two dimensions. [email protected] Jean-Frederic Gerbeau, Damiano Lombardi INRIA Paris-Rocquencourt MS16 [email protected], [email protected] Analysis of Optimal Superconvergence of Discon- tinuous Galerkin Method MS17 We study the superconvergence of the error for the discon- Rectification Procedure for Improved Accuracy in tinuous Galerkin method for linear conservation laws. If Reduced Basis Context we apply piecewise kth degree polynomials, the error be- tween the DG solution and the exact solution is (k+2)th or- Reduced basis techniques, as other model order reduction der superconvergent at the downwind-biased Radau points. approaches use the fact that the solution of the problem Moreover, the DG solution is (k+2)th order superconver- we are interested in lies in a low complexity space, that, gent both for the cell averages and for a particular error. e.g. can be characterized by a small Kolmogorov n-width. The idea is not based on Fourier analysis. Numerical exper- A typical application frame is the solution to some param- iments demonstrate the optimality of the superconvergence eter dependent family of PDE. In such cases, the proce- rate. dure consists in computing accurately some solutions of the ”low complexity space” and extrapolate in the parameter Yang Yang space by some approach (EIM, GEIM, Galerkin, coloca- Brown university tion..) when the solutions is required at some other values [email protected] of the parameter. In some applications, the reduced basis solution, for those parameters where an accurate solution Chi-Wang Shu has been computed is not equal to that approximation. An Brown University a posteriori ”rectification” process can then be proposed Div of Applied Mathematics and solve this deviation. It appears that the global solu- [email protected] tion procedure is the much improved by this rectification tool. We shall present a variety of cases where such a rec- tification is useful, explain the reason why it improves the MS17 accuracy and show the efficiency of this post processing Reduced-Order Models for Nonlinear Systems with action Time-Periodic Solutions Yvon Maday In this talk, we discuss the use of approximate Floquet Universit´e Pierre et Marie Curie transformations to develop reduced-order models for com- and Brown university plex flows that exhibit periodic solutions. Our strategy in- [email protected] corporates the proper orthogonal decomposition using sen- sitivity analysis and an optimal selection of subintervals for which to build snapshots. Results will include reduced- MS18 order models for Navier-Stokes and Boussinesq flows in- A Nonlocal Vector Calculus with Applications to cluding vortex shedding and driven cavity problems. Diffusion and Mechanics

Jeff Borggaard Abstract not available at time of publication. Virginia Tech Department of Mathematics Max Gunzburg [email protected] School of Computational Science Florida State University [email protected] MS17 Reduced-Order Modeling for Adjoint Equations MS18 Abstract not available at time of publication. Quadrature Methods in Peridynamics Florian De Vuyst Ecole Normale Sup´erieure Cachan Peridynamics is a nonlocal extension of classical solid me- [email protected] chanics suitable for modeling failure and fracture. Peridy- namic models require evaluation of an integral, performed numerically by quadrature over a mesh. The domain of MS17 integration in general does not align with element edges, Reduced Order Model Based on Approximated Lax meaning that one must integrate over fractions of an ele- Pairs ment. In this presentation, I survey previous approaches to quadrature in peridynamics, and present a new approach A reduced-order model algorithm, called ALP, is proposed for accurate 3D quadrature for peridynamics models. to solve nonlinear evolution partial differential equations. It is based on approximations of Lax pairs. Contrary to MichaelL.Parks other reduced-order methods, like Proper Orthogonal De- Sandia National Laboratories composition, the space where the solution is searched for [email protected] evolves according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progres- sive waves or front propagation. Numerical examples are MS18 shown for the linear advection, KdV and FKPP equations, Asymptotic Behavior for Solutions in Diffusion 82 PD13 Abstracts

Models with Peridynamic-Type Nonlocality Boussinesq Abcd System

In this talk I will present some recent results regarding the Abstract not available at time of publication. long time behavior of solutions to nonlocal wave equations with damping. These equations could appear as models in Ming Chen nonlocal diffusion, when one uses the Cattaneo-Vernotte University of Pittsburgh flux to replace the Fourier law. Several methods suitable [email protected] for this investigation will be presented, some of which are reminiscent of the local setting, and I will also point out some of the major challenges encountered in the nonlocal MS19 case. Coherent Frequency Dynamics for the Periodic Nonlinear Schrodinger Equation Petronela Radu University of Nebraska-Lincoln We consider the cubic nonlinear Schr¨odinger equation on a Department of Mathematics periodic box. By taking an appropriate large box limit of [email protected] the equation (in a spirit similar to wave turbulence theory), we derive a (new) continuum equation on R2,whichturns out to enjoy rather surprising symmetries, stability prop- MS18 erties, and even explicit solutions. This equation is then Origin and Effect of Nonlocality in a Composite used (via a rigorous approximation theorem) to better un- derstand the frequency dynamics of the original cubic NLS The purpose of this talk is to demonstrate from simple equation. mechanical arguments that nonlocality is a basic prop- erty of heterogeneous media. By deriving a model for the Zaher Hani smoothed displacement field in a simple composite, it is NYU shown that nonlocality emerges naturally. The nonlocal [email protected] model is put into the form of a peridynamic material model. Computations demonstrate improved agreement with ex- Pierre Germain perimental data on composite laminates when nonlocality New York Univeristy is included. [email protected] Stewart Silling Sandia National Laboratories Erwan Foau [email protected] INRIA [email protected]

MS19 MS19 Well-posedness of the Linear KdV Equation on a Real Line Approximation of Polyatomic FPU Lattices by KdV Equations In this talk wellposedness of a linear KdV equation Famously, the Korteweg-de Vries equation serves as a 3 model for the propagation of long waves in Fermi-Pasta- j ∂tu + aj (x, t)∂xu = f Ulam (FPU) lattices. If one allows the material coeffi- j=0 cients in the FPU lattice to vary periodically, the “clas- sical” derivation and justification of the KdV equation go is investigated. This equation is a model problem, whose awry. By borrowing ideas from homogenization theory, we understanding has applications to linear and nonlinear can derive and justify an appropriate KdV limit for this 3 KdV-type equations. The third derivative term ∂x makes problem. This work is joint with Shari Moskow, Jeremy this equation dispersive and in the non-degenerate case Gaison and Qimin Zhang. a3 ≈ 1, the wellposedness is similar to the case of a3 ≡ 1. Namely wellposedness in L2 based Sobolev spaces depends J. Douglas Wright on the balance of the regularizing dispersive effect from Department of Mathematics 3 2 a3∂x and the destabilizing backward heat term a2∂x.This Drexel University balance was investigated in [T.A., A sharp condition for [email protected] the well-posedness of the linear KdV-type equation, (2012)]. However, when the dispersive coefficient a3 degenerates the equation can be illposed without or despite the heat term, MS20 e.g. [Craig and Goodman, Linear dispersive equations of On a Stochastic Leray-Alpha Model of Euler Equa- Airy type,(1990)]. The techniques that applied in the non- tions degenerate case can be fruitfully applied in the degenerate setting. In particular I will discuss the modified energy We deal with the 3D inviscid Leray-alpha model. The well method for the wellposedness arguments and the geomet- posedness for this deterministic problem is not known; by rical optics constructions for illposedness. adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term Timur Akhunov formally preserves conservation of energy. The result holds University of Calgary for initial velocity of finite energy and the solution has finite [email protected] energy almost surely. These results are easily extended to the 2D case.

MS19 Hakima Bessaih Existence and Symmetry of Ground States to a University of Wyoming PD13 Abstracts 83

[email protected] [email protected]

MS20 MS21 An Asymptotic-Preserving Scheme for the Semi- Asymptotic Algebraic Spiral Solutions to the 2D conductor Boltzmann Equation toward the Energy- Incompressible Euler Equations Transport Limit

Self similar solutions to the 2d incompressible Euler equa- We design an asymptotic-preserving scheme for the semi- tions can exhibit spiral behavior. Classical works of conductor Boltzmann equation which leads to an energy- Kaden, Pullin, and Moore sought out asymptotic expan- transport system for electron mass and internal energy as sions through various methods and provided conjectures mean free path goes to zero. To overcome the stiffness for the exponents of the leading terms in these expansions. induced by the convection terms, we adopt an even-odd We will prove that certain classes of self-similar solutions decomposition to formulate the equation into a diffusive have these asymptotic expansions and hence streamlines relaxation system. New difficulty arises in the two-scale exhibit spiral behavior in self-similar coordinates. This is stiff collision terms, whereas the simple BGK penalization joint work with Volker Elling. does not work well to drive the solution to the correct limit. We propose a clever variant of it by introducing a threshold Michael Dabkowski, Volker Elling on the stiffer collision term such that the evolution of the University of Michigan solution resembles a Hilbert expansion at the continuous [email protected], [email protected] level. Formal asymptotic analysis and numerical results are presented to illustrate the efficiency and accuracy of the new scheme.

MS20 Jingwei Hu Well-Posedness of Euler-Like Evolution Equations The University of Texas at Austin in Nonsmooth Domains and Relations to the Verti- [email protected] cal Normal Modes Expansion of the Inviscid Prim- itive Equations Li Wang University of California, Los Angeles In previous joint work with C. Bardos and R. Temam, [email protected] we show well-posedness of the Euler equations on a two- dimensional domain with acute corners, for weak solutions MS21 with bounded initial vorticity. This problem is an approx- imation, near the zero flow, to the barotropic mode of Asymptotic Preserving Scheme Based on Hyper- the inviscid primitive equations, that is, the zeroth mode bolic Decomposition for Compressible Euler Equa- of the vertical normal modes expansion. As a further tions step towards the understanding of the coupled barotropic- We propose an asymptotic preserving, that is, stable and baroclinic system, we study the analogous approximated sufficiently accurate for all Mach number, central-upwind problem for more general background flows nearing the scheme for the compressible Euler equations of gas dynam- Brunt-Vaisala regime. The resulting system still resem- ics. It is well-known that the main difficulty associated bles the two-dimensional Euler equation, though with non- with numerical simulations of such models is amplification standard boundary conditions, calling for further advance- of the numerical viscosity for small Mach numbers, which ment of the elliptic theory tools developed in previous leads to a severe restriction on the size of time steps and work. thus substantially affects the efficiency of the method. We propose to decompose the underlying system of equations Francesco Di Plinio into two well-posed hyperbolic systems to remove the stiff- University of Rome Tor Vergata ness and thus to reduce the numerical viscosity and in- [email protected] crease the size of time steps. The proposed discretization provides a consistent approximation of both the hyper- Roger M. Temam bolic compressible regime and the elliptic incompressible Inst. for Scientific Comput. and Appl. Math. regime. We conduct a number of one- and two-dimensional Indiana University numerical experiments that demonstrate the asymptotic- [email protected] preserving ”all-speed” properties of the proposed central- upwind scheme.

MS20 Alina Chertock North Carolina State University Ill-Posedness for a Class of Equations Arising in Department of Mathematics Hydrodynamics [email protected]

We prove a linear ill-posedness result in L∞ for a transport Shi Jin equation with a non-local (singular integral) source term. Shanghai Jiao Tong University, China and the Because we only use minimal assumptions on the velocity University of Wisconsin-Madison field, we are able to prove non-linear ill-posedness results [email protected] for a large class of equations arising in hydrodynamics. Alexander Kurganov Tarek M. Elgindi Tulane University New York University Department of Mathematics Courant Institute [email protected] 84 PD13 Abstracts

Jian-Guo Liu by an attractive gravitational potential. We present sharp Duke University Liouville-type theorems on nonexistence of positive super- [email protected] solutions of such equations in exterior domains. We also discuss existence, positivity, symmetry and optimal decay properties of ground state solutions under various assump- MS21 tions on the decay of the external potential and the shape Asymptotic Preserving Imex-Type Finite Volume of the nonlinearity. In particular, we obtain a sharp decay Methods for Some Geophysical Flows estimate of the ground state solution which was discovered by E.Lieb in 1977. This is a joint work with Jean Van We present new large time step methods for some hyperboli Schaftingen (Louvain-la-Neuve, Belgium) balance laws. In particular we consider the Euler equa- tions in low Mach number limit and the shallow water Vitaly Moroz flows in the low Froude number limit. In order to take Department of Mathematics into account multiscale phenomena that typically appear Swansea University in geophysical flows nonlinear fluxes are split into a linear [email protected] part governing the acoustic or gravitational waves and the nonlinear flow advection. We propose to approximate fast linear waves implicitly in time and in space by means of MS22 a genuinely multidimensional evolution operator. On the other hand, the nonlinear advection can be approximated Dynamics of Fronts in Multistable Reaction- explicitly in time and in space by means of the method diffusion Equations of characteristics or some standard numerical flux func- tion. Time integration is realized by the implicit-explicit We study the dynamics of fronts for some reaction-diffusion (IMEX) method. We apply the IMEX Euler scheme, two equations having multi-stable reaction term. A typical ex- step Runge Kutta Cranck Nicolson scheme, as well as ample of such equations is the overdamped sine-Gordon the semi-implicit BDF scheme and prove their asymptotic equation. We will discuss generation of multiple fronts preserving property in the low Mach/Froude number lim- (with stairs-like shape), generation of metastable fronts its. Numerical experiments demonstrate stability, accuracy and slow motion of fronts. This is a joint work with Toshiko and well-balanced behaviour of these new large time step Ogiwara (Josai University) finite volume schemes with respect to small Mach/Froude numbers. This work has been done in cooperation with Ken-Ichi Nakamura S. Noelle, K.R. Arun, L. Yelash and G. Bispen. Finan- Faculty of Mathematics and Physics cial support of the German Science Foundation under LU Kanazawa University 1470/2-2 is gratefully acknowledged. [email protected]

Maria Lukacova University of Mainz MS22 Institute of Mathematics A Local Regularity Theorem for Varifold Mean [email protected] Curvature Flow

A family of k-dimensional surfaces in the n-dimensional MS21 Euclidean space or Riemannian manifold (0

Sebastian Noelle Yoshihiro Tonegawa IGPM - Institute for Geometry and Practical Hokkaido University Mathematics [email protected] RWTH - Aachen University of Technology [email protected] MS22 Jochen Schuetz A Reaction-Diffusion Approach to the Existence of Institute for Geometry and Practical Mathematics Gravity Water Waves RWTH - Aachen University of Technology [email protected] While variational techniques have been successfully applied to capillary gravity water waves in order to prove existence, there are no corresponding existence proofs working in the MS22 physical/original variables in the case of zero surface ten- Existence and Qualitative Properties of Grounds sion. In this talk we will present a singular perturbation States to the Non-Local Choquard-Type Equations approach to the existence of two-dimensional gravity water waves, which uses the original variables. The Choquard equation, also known as the Hartree equa- tion or nonlinear Schrodinger-Newton equation is a station- Georg Weiss ary nonlinear Schrodinger type equation where the nonlin- Mathematical Institute earity is coupled with a nonlocal convolution term given Heinrich Heine University PD13 Abstracts 85

[email protected] counterexample, namely a minimal blow-up solution;(2)if we further assume that a potential is repulsive, this coun- terexample does not exist, and the criteria for global well- MS23 posedness thus imply scattering. Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Non- Younghun Hong linear Schrdinger Equation University of Texas at Austin [email protected] We consider the dynamics of N bosons in one dimension. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N tends to infinity limit MS23 of the N-body dynamic and hence justify the mean-field The Gross-Pitaevskii Hierarchy on the Three- limit and prove the propagation of chaos for the focusing dimensional Torus quantum many-body system. In this talk, we will study the Gross-Pitaevskii hierarchy Xuwen Chen on the spatial domain T3. In the first part of the talk, we Brown University will prove a conditional uniqueness result for the hierarchy. [email protected] As a result of our analysis, we will obtain a sharp range of integrability exponents in the key spacetime estimate. Justin Holmer In the second part of the talk, we will add randomness Department of Mathematics, Brown University, into the problem by randomizing the collision operators on Box 1917, 151 Thayer Street, Providence, RI 02912, USA the Fourier domain. In this randomized setting, we will [email protected] study the limiting behavior of Duhamel iteration terms. The first part is based on joint work with Philip Gressman and Gigliola Staffilani. The second part is based on joint MS23 work with Gigliola Staffilani. Local Well-posedness of KdV with Potential Vedran Sohinger, Philip Gressman We prove local well-posedness of the Korteweg-de Vries University of Pennsylvania (KdV) equation on the line with potential [email protected], [email protected]

2 2 ∂tu = ∂x(−∂xu − 3u + Vu) Gigliola Staffilani 1 MIT in H (R). This is achieved by introducing a suitable [email protected] parametrix approximation to the linear propagator, and then proving local smoothing and maximal function esti- mates for this parametrix analogous to those obtained in MS24 the free case by [C.E. Kenig, G. Ponce, and L. Vega, Well- Geometric Partial Differential Equations in Ran- posedness of the initial value problem for the Korteweg-de dom Media Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, pp. 323–347.] Global well-posedness follows easily from the en- We present some results on the geometry and regularity ergy. This equation arises as a model of shallow water wave of some geometric variational problems in random media. propagation in a channel over a bottom of varying depth, As for the stationary problem, we study area minimiz- and the H 1(R) well-posedness is needed in the study of ers with random boundary condition, with focus on the soliton dynamics. Plateau Problem. For the evolutionary case, we consider ameancurvatureevolutioninrandommediaandrelated Justin Holmer problems that model population growth and phase transi- Department of Mathematics, Brown University, tions in random media. Box 1917, 151 Thayer Street, Providence, RI 02912, USA [email protected] Betul Orcan-Ekmekci G.C. Evans Instructor Mamikon Gulian [email protected] Brown University mamikon [email protected] MS24 Stochastic Homogenization for Porous Medium MS23 Type Equations Scattering for Nonlinear Schr¨odinger Equation with a Potential In a recent paper, Ambrosio, Frid, and Silva study the homogenization problem for a class of porous-medium type We study global well-posedness and scattering for a 3d cu- equations where the flux function is a stationary random bic focusing nonlinear Schr¨odinger equation process on a compact probability space. We extend their result to the general stochastic framework, removing the 2 1 iut +Δu − Vu+ |u| u =0; u(0) = u0 ∈ H , compactness assumption on the probability space. Passing from the small to the large scale, we show the convergence where V is a real-valued short-range potential with a small in probability of the weak solutions of rescaled problem to negative part. First, we find criteria for global well- the weak solution of an homogenized deterministic porous- posedness from the maximization problem related to the medium type equation. Gagliardo-Nirenberg inequality. Next, we investigate scat- tering of such global solutions. Following Kenig-Merle’s Stefania Patrizi program, we prove that (1) if there are non-scattering so- University of Rome lutions obeying the criteria, then there exists a special Rome, Italy 86 PD13 Abstracts

[email protected] [email protected]

MS24 MS25 Boundary Regularity for a Class of Degenerate Nonlinear Stability for Fluid-Structure Interaction Monge-Ampere Equations Models

We discuss boundary regularity for the Monge-Ampere The classical one-phase Stefan problem models the tem- equation when the right hand side behaves as a positive perature distribution in a homogeneous medium undergo- power of the distance function near the boundary of the ing a phase transition, such as ice melting to water. This domain. As a consequence we obtain global C2 estimates is accomplished by solving the heat equation on a time- of the eigenfunction for the Monge-Ampere equation. Sim- dependent domain whose free-boundary is transported by ilar results were obtained in 2D by J.X. Hong, G. Huang, the normal derivative of the temperature along the evolv- W. Wang. ing and a priori unknown free-boundary. We establish a global-in-time stability result for small temperatures, using Ovidiu Savin a novel hybrid methodology, which combines energy esti- Columbia University mates inspired by the Euler equations, decay estimates, [email protected] and Hopf-type inequalities. This is joint work with M. Hadzic. MS24 Steve Shkoller A New PDE Approach for Large Time Behavior of Univiversity of California, Davis Hamilton-Jacobi Equations Department of Mathematics [email protected] I will present a new PDE approach to obtain large time behavior of Hamilton- Jacobi equations. This applies to Mahir Hadzic usual Hamilton-Jacobi equations, as well as the degenerate Massachusetts Institute of Technology viscous cases, and weakly coupled systems. The degenerate [email protected] viscous case was an open problem in last 15 years.

Hung Tran MS25 University of Chicago Shape Optimization with Nonlinear PDE Con- [email protected] straints. Compressible Navier-Stokes Equations

Filippo Cagnetti Abstrat not available at time of publication. Department of Mathematics Instituto Superior Tecnico, Lisbon, Portugal Jan Sokolowski [email protected] Institut Elie Cartan Universite Henri Poincare Nancy I Diego Gomes [email protected] Instituto Superior Tecnico, Lisbon, Portugal [email protected] MS25 Stability Analysis of Fluid-structure Interaction Hiroyoshi Mitake Models Fukuoka University Fukuoka, Japan We consider an established fluid-structure interaction [email protected] model. We present recent joint results on polyno- mial/rational decay which are obtained as an interplay of functional analytic techniques, PDE estimates, and micro- MS25 local analysis, to obtain the optimal parameter of decay. Uniform Stability of a Fluid-Structure Interaction with Interface and Interior Controls in the Fluid Inside Structure Setting Roberto Triggiani University of Virginia We consider the uniform stability of finite energy solu- [email protected] tions to a nonlinear fluid-structure interaction model in a bounded domain Ω ∈ R2. In this model, Ω is occupied by two environments: fluid and solid, with fluid flown inside MS26 the solid. The interaction occurs at the static interface. Vortex Sheets in Domains with Boundaries If an interior viscous nonlinear damping is placed on the solid, the energy of the overall system decays to zero at This talk concerns the interaction of incompressible 2D a uniform rate. If additional boundary damping is placed flows with compact material boundaries under vortex sheet on the interface and a geometric condition is satisfied, the regularity. We focus on the dynamic behavior of the cir- energy decays at an exponential rate. culation of velocity around boundary components and the possible exchange between flow vorticity and boundary cir- Yongjin Lu culation in flows with vortex sheet initial data. We re- Virginia State University cast 2D Euler evolution using vorticity and the circula- [email protected] tion (around boundary components) as dynamic variables and provide a weak formulation. Our main results are the Irena M. Lasiecka equivalence between the weak velocity and weak vorticity University of Virginia formulations of the 2D Euler equations on domains with PD13 Abstracts 87

boundary and a criterion for computing the resulting force recent results on such ”wild solutions” and discuss conceiv- on boundary components for flows with little regularity. able uniqueness criteria.

Dragos Iftimie Emil Wiedemann Universite de Lyon I Dept. Math - University of British Columbia [email protected] [email protected]

Milton Lopes Filho, Helena J. Nussenzveig Lopes Universidade Federal do Rio de Janeiro MS27 [email protected], [email protected] Stationary Co-Dimension One Structures in the Functionalized Cahn-Hilliard Model Franck Sueur The functionalized Cahn-Hilliard energy models interfacial Laboratoire Jacques-Louis Lions energy in amphiphilic phase separated mixtures. Its mini- Universit´e Pierre et Marie Curie mizers encompass a rich class of morphologies with detailed [email protected] inner structure, including bilayers, pore networks, pearled pores, and micelles. We address the existence and stability MS26 of a class of single-curvature bilayer structures in d>1 space-dimensions. The existence problem involves the con- Mathematical Models of Intermittency in Fully De- struction of homoclinic solutions in a perturbed 4th-order veloped Turbulence Hamiltonian system. The linear stability involves an anal- Physical models of intermittency in fully developed turbu- ysis of meander and pearling modes. lence use many phenomenological concepts such as active Arjen Doelman volume, region, eddy, energy accumulation set, etc, used Mathematisch Instituut to de- scribe non-uniformity of the energy cascade. In this [email protected] talk we give these terms a precise mathematical meaning in the language of the Littlewood-Paley analysis. With our definitions we establish some of the deterministic counter- Keith Promislow parts of the classical scaling laws for the energy spectrum Michigan State University and second order structure function with proper intermit- [email protected] tency correction. Brian R. Wetton Roman Shvydkoy University of British Columbia University of Illinois at Chicago Department of Mathematics [email protected] [email protected]

MS26 MS27 Inviscid Limit for Invariant Measures of the 2D Existence and Stability of Solutions in the Multi- Stochastic Navier-Stokes Equations dimensional Swift–Hohenberg Equation

We discuss recent results on the behavior in the infinite The existence of stationary localized spots for the pla- Reynolds number limit of invariant measures for the 2D nar and the three-dimensional Swift–Hohenberg equation stochastic Navier-Stokes equations. We prove that the lim- is proved using geometric blow-up techniques. The spots iting measures are supported on bounded vorticity solu- have a much larger amplitude than that expected from a tions of the 2D Euler equations. Invariant measures pro- formal scaling in the far field. One advantage of the geo- vide a canonical object which can be used to link the fluids metric blow-up methods is that the anticipated amplitude equations to the heuristic statistical theories of turbulent scaling does not enter as an assumption into the analysis flow. but emerges during the construction. The stability of these solutions will also be addressed. Nathan Glatt-Holtz Virginia tech Scott McCalla [email protected] University of California Los Angeles [email protected] Vladmir Sverak University of Minnesota Bjorn Sandstede [email protected] Brown University bjorn [email protected] Vlad C. Vicol Dept. Math - Princeton University [email protected] MS27 Pattern Selection in the Wake of Fronts

MS26 Motivated by the formation of complex patterns in the Non-Uniqueness Phenomena for Weak Solutions of wake of growth processes in bacterial colonies, we’ll study the Euler Equations more generally patterns formed in the wake of moving fronts. We discuss a number of mathematical problems It has been known since V. Scheffer’s groundbreaking work that arise when one tries to predict which wavenumbers twenty years ago that the Cauchy problem for the incom- and what type of pattern is created by the growth process. pressible Euler equations admits non-unique weak solutions We therefore describe in some detail how to derive linear with highly pathological behaviour. I will survey various predictions and analyze nonlinear mechanisms that lead to 88 PD13 Abstracts

failure of these predictions. This is joint work with Matt prove asymptotics for such solutions, showing a nonlinear HolzerandRyanGoh. behavior as time goes to infinity. This is joint work with A. Ionescu. Arnd Scheel University of Minnesota Fabio Pusateri School of Mathematics Princeton University [email protected] [email protected]

MS27 MS28 Small-amplitude Grain Boundaries of Arbitrary On the Finite-time Splash and Splat Singularities Angle in the Swift-Hohenberg Equation for the 3-D Free-surface Euler Equations

We study grain boundaries in the Swift-Hohenberg equa- e prove that the 3-D free-surface incompressible Euler tion. Grain boundaries arise as stationary interfaces be- equations with regular initial geometries and velocity fields tween roll solutions of different orientations. Our analysis have solutions which can form a finite-time splash (or splat) shows that such stationary interfaces exist near onset of singularity, wherein the evolving 2-D surface, the moving instability for arbitrary angle between the roll solutions. boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of Qiliang Wu a breaking wave falls unto its trough, or in the study of drop University of Minnesota impact upon liquid surfaces. Our approach is founded upon [email protected] the Lagrangian description of the free-boundary problem, combined with a novel approximation scheme of a finite collection of local coordinate charts; as such we are able to MS28 analyze a rather general set of geometries for the evolving Global Existence for Water Wave Equations 2-D free-surface of the fluid.

Abstract not available at time of publication. Steve Shkoller Univiversity of California, Davis Pierre Germain Department of Mathematics New York Univeristy [email protected] [email protected] Daniel Coutand MS28 Heriot-Watt University A Boundary Perturbation Method for Reconstruc- [email protected] tion of Layered Media via Constrained Quadratic Optimization MS29 The scattering of linear waves by layered media constitute On the Bound of DG and Central DG Operators a fundamental model in oceanography, acoustics, electro- magnetics, and the geosciences. In this talk we exam- Discontinuous Galerkin (DG) and central DG methods are ine the problem of detecting the geometric properties of two families of high order methods, and it was observed a two-dimensional acoustic medium where the fields are that the latter allows larger time steps in stable numerical governed by the Helmholtz equation. Building upon the simulation especially when the accuracy is relatively high. success of our previous Boundary Perturbation approach In this paper, we estimate the bounds of DG and central (implemented with the Operator Expansions formalism) DG spatial operators for linear advection equation. Based we derive a new approach which augments this with a new on such estimates and Kreiss-Wu theory, time step condi- ”smoothing” mechanism. With numerous numerical exper- tions are obtained to ensure numerical stability when the iments we demonstrate the enhanced stability and accuracy methods are coupled with locally stable time discretiza- of our new approach which further suggests not only a rig- tions. In particular, with a fixed time discretization, the orous proof of convergence, but also a path to generalizing time step grows quadratically with k when the DG method the algorithm to multiple layers, three dimensions, and the is used in space (here k is the polynomial degree of the dis- full equations of linear elasticity and Maxwell’s equations. crete space), while in the case of the central DG method, the dependence of the time step on k is linear. In addition, the analysis provides new insight into the role of a param- David P. Nicholls eter in the central DG formulation. We verify our results University of Illinois at Chicago numerically, and also discuss the extension of the analysis [email protected] to some related discretiztions.

Alison Malcolm Fengyan Li MIT Rensselaer Polytechnic Institute [email protected] [email protected] Matthew Reyna MS28 Rensselaer Polytechnic Institute Global Existence for Gravity Waves in 2 Dimension Department of Mathematical Sciences [email protected] We consider the water waves system for the evolution of a perfect fluid with a free surface in 2d, under the influ- ence of gravity. For sufficiently smooth and localized initial MS29 data we prove global existence of small solutions. We also High-order Multiderivative Time Integrators for PD13 Abstracts 89

Hyperbolic Conservation Laws Univeristy of Tennessee / Oak Ridge National Lab [email protected] Time stepping methods for hyperbolic conservation laws often fall into one of two distinct categories: a method of lines formulation, and a single-step Taylor (Lax-Wendroff) MS30 formulation. The existence of so-called multiderivative Conditioning of Nonlocal Operators in Fractional time integrators for ordinary differnetial equations indicate Sobolev Spaces that this is indeed a false dichotomy. In fact, multideriva- tive methods have a long history of development for or- We study the conditioning of the weak from of the non- dinary differential equations dating back to at least the local operator associated to nonlocal diffusion and peri- 1950’s. However, only the extreme versions of these meth- dynamics in fractional Sobolev spaces. For both maximal ods have been explored as a numerical tool for solving par- and minimal eigenvalues, we provide sharp estimates which tial differential equations. In this work, we demonstrate involve explicit quantifications of the horizon, mesh size, that explicit Runge-Kutta methods as well as high-order and the regularity of the fractional Sobolev space. For Taylor (Lax-Wendroff) methods are special cases of multi- these estimates, we use an interesting combination of tools derivative methods, and in addition, we demonstrate how from functional analysis and numerical linear algebra. The these methods can be applied for solving hyperbolic conser- sharpness results are also supported with numerical exper- vation laws using the weighted essentially non-oscillatory iments. (WENO) method, as well as the discontinuous Galerkin (DG) method. Burak Aksoylu TOBB University of Economics and Technology, Dept of David C. Seal, Yaman Guclu Math Department of Mathematics Louisiana State University, Dept of Math Michigan State University [email protected] [email protected], [email protected] Zuhal Unlu Andrew J. Christlieb Louisiana State University Michigan State Univerity [email protected] Department of Mathematics [email protected] MS30 Adaptive Numerical Methods for Periydnamics MS29 A High Order Weno Scheme for Detonation Waves Abstract not available at time of publication.

Abstract not available at time of publication. Qiang Du Penn State University Wei Wang Department of Mathematics Florida International University, Miami, FL [email protected] weiwang1@flu.edu

MS30 MS29 Differentiability for Solutions of Nonlocal Models Asymptotic Preserving Discontinuous Galerkin with Weakly Singular Kernels Method for the Radiative Transfer Equation Abstract not available at time of publication. Many kinetic equations converge to macroscopic models, known as the asymptotic limit of the kinetic equations, Mikil Foss when  (the ratio of mean free path over macroscopic University of Nebraska-Lincoln size) − > 0. Asymptotic preserving (AP) methods are Lincoln, NE designed to preserve the asymptotic limits from micro- [email protected] scopic to macroscopic models in the discrete setting. In the asymptotic regimes (when <<0) , they become a con- sistent discretization of the macroscopic solvers automati- MS30 cally, while remain microscopic otherwise. In this presen- Surface Effects and Affects on Ordinary Isotropic tation, we consider the radiative transfer equation which Peridynamics Models models isotropic particle scattering in a medium. Many finite volume (FV) methods do not have the AP prop- Abstract not available at time of publication. erty. Discontinuous Galerkin (DG) (with upwind flux and at least piecewise linear polynomial) methods are known John A. Mitchell to have such property, but require much more degree of Sandia National Laboratories freedom, especially in high dimensional applications. We [email protected] will present an AP mixed DG-FV method which has com- parable computational cost and memory as FV method. Rigorous analysis will be provided to show that the pro- MS31 posed methods are consistent with the limit equation in Acoustic Propagation in a Saturated Piezo-Elastic, the asymptotic regimes. Some numerical examples are pre- Porous Medium sented to demonstrate the performance of our methods. We study the problem of derivation of an effective model Yulong Xing of acoustic wave propagation in a two-phase, non-periodic Department of Mathematics medium modeling a fine mixture of linear piezo-elastic solid 90 PD13 Abstracts

and a viscous Newtonian, ionic bearing fluid. Bone tis- [email protected] sue is an important example of a composite material that can be modeled in this fashion. We develop two-scale ho- mogenization methods for this system, and discuss also a MS31 stationary random, scale-separated microstructure. The Dynamic Brittle Fracture as a Small Horizon Limit ratio e of the macroscopic length scale and a typical size of Peridynamics of the microstructural inhomogeneity is a small parameter of the problem. Another possibly small parameter is the We consider peridynamic formulations with unstable con- Peclet number which in?uences the type of e?ective equa- stitutive laws that soften beyond a critical stretch. Here tions which are obtained. we discover new quantitative and qualitative information that is extracted from the peridynamic formulation us- Robert P. Gilbert ing scaling arguments and by passing to a distinguished Department of Mathematical Sciences small horizon limit. In this limit the dynamics correspond University of Delaware to the simultaneous evolution of elastic displacement and [email protected] fracture. The displacement fields are shown to satisfy the wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture MS31 path inside the media. The limit evolutions have bounded energy expressed in terms of the bulk and surface energies Homogenization of Rigid Suspensions with Highly of brittle fracture mechanics. Oscillatory Velocity-Dependent Surface Forces Robert P. Lipton We study particulate flows or suspensions of solid particles Department of Mathematics in a viscous incompressible fluid at the presence of highly Louisiana State University oscillatory surface forces. The flow at a small Reynolds [email protected] number is modeled by the Stokes equations coupled with the motion of absolutely rigid particles arranged in a pe- riodic array. The objective is to perform homogenization MS32 for the given suspension and obtain an equivalent descrip- On the Well-Posedness of the 3D Navier-Stokes tion of a homogeneous (effective) medium whose viscosity is Equations in the Largest Critical Space determined using the analysis on a periodicity cell. Math- ematical tools that the main technical construct is built In this talk I will discuss various well-posedness and ill- upon are based on Γ-convergence theory. posedness results for the 3D Navier-Stokes equations in the largest critical space. In particular, I will describe a recent Yuliya Gorb norm inflation result for the hyperdissipative Navier-Stokes Department of Mathematics equations that occurs in critical and supercritical spaces University of Houston even in the case where the global regularity is known. We [email protected] will see how a new non-critical scaling becomes an obstacle in proving small initial data results in critical spaces. Florin Maris KAUST Alexey Cheskidov fl[email protected] University of Illinois Chicago [email protected] Bogdan M. Vernescu Worcester Polytechnic Inst Dept of Mathematical Sciences MS32 [email protected] Initial-Boundary Layer in the Nonlinear Darcy- Brinkman System

MS31 We study the interaction of initial layer and boundary layer in the nonlinear Darcy-Brinkman system in the vanishing Nonlinear Neutral Inclusions: Assemblages of Darcy number limit. In particular, we show the existence Spheres and Confocal Coated Ellipsoids of a function of corner layer type (so called initial-boundary layer) in the solution of the nonlinear Darcy-Brinkman If a neutral inclusion is inserted in a matrix containing system. An approximate solution is constructed by the a uniform applied electric field, it does not disturb the method of asymptotic expansion. We establish the opti- field outside the inclusion. The well known Hashin coated mal convergence rates in various Sobolev norms via energy sphere is an example of a neutral coated inclusion. In this method. talk, we consider the problem of constructing neutral in- clusions from nonlinear materials. In particular, we discuss Daozhi Han assemblages of coated spheres and ellipsoids. Florida State University [email protected] Silvia Jimenez Bolanos Department of Mathematical Sciences Xiaoming Wang Worcester Polytechnic Institute FSU [email protected] [email protected]

Bogdan M. Vernescu Worcester Polytechnic Inst MS32 Dept of Mathematical Sciences Gevrey Regularity of the Critical and Supercritical PD13 Abstracts 91

Quasi-Geostrophic Equations Mark [email protected]

The dissipative quasi-geostrophic (QG) equation is an im- portant model in geophysical fluid dynamics, but also MS33 mathematically important since it can be considered as a An Element Conservative Dpg Method for 2D model of the 3D incompressible Navier-Stokes equa- Convection-Dominated Problems tions. Consequently, it has received much attention in recent years. Studies of this equation are usually sepa- Abstract not available at time of publication. rated into three cases: supercritical, critical, and subcrit- ical, where in each case the order of dissipation increases. Truman E. Ellis By now, the subcritical case is well understood; global ex- ICES istence of weak solutions was established in [Resnick, ’95], University of Texas at Austin while global existence of a unique regular solution with ini- [email protected] tial data in Lp-spaces, for instance, was established in [Wu, ’01]. Recently, global well-posedness for arbitrary smooth MS33 initial data was established by several authors indepen- dently: [Caffarelli-Vasseur, ’06], [Kiselev-Nazarov-Volberg, Dissipative Methods for Wave Equations in Second ’07], and [Constantin-Vicol, ’11]. However, there is still Order Form much that is not known in the supercritical and even, crit- ical case. Our paper explores higher-order regularity of the We consider the construction and analysis of high-order supercritical and critical QG equation. In particular, we dissipative elements for solving time-domain wave equa- establish Gevrey regularity of solutions in Lp-based Besov tions in second order form. These include natural gener- spaces. The techniques we use are based on the ideas of alizations of dissipative methods for first order hyperbolic [Miura, ’06] and an extension of those in [Biswas, preprint], systems including upwind discontinuous Galerkin methods where the main idea is to establish control on the nonlinear as well as Hermite methods.The essential ingredient in the term via a suitable commutator estimate. Since we seek Lp generalization is to consider the standard energy form for bounds, we adapt the approach of [Lemarie-Rieusset, ’99] second order wave equations as well as its space derivatives. and view the commutator as a bilinear multiplier operator. We demonstrate that the proposed methods can be stably However, to establish these bounds, we must also adopt the hybridized. techniques of Michael Lacey and Christoph Thiele used to Thomas M. Hagstrom the bound the Bilinear Hilbert Transform. As an applica- Southern Methodist University tion, we also deduce higher-order decay estimates on the Department of Mathematics solution. [email protected] Animikh Biswas Daniel Appelo University of North Carolina Department of Mathematics and Statistics at Charlotte University of New Mexico [email protected] [email protected] Vincent Martinez, Prabath Silva Indiana University, Bloomington MS33 [email protected], [email protected] Mixed Finite Elements for Wave Equations

I will survey several of the issues facing numerical approx- MS33 imation of wave equations by finite element methods – before considering stability and accuracy, we must decide High Order Finite Elements for Wave Propagation: whether to use first- or second-order forms of the equa- Like It Or Lump It? tions. We must deal with numerical dispersion, invert- ing mass matrices at each time step and energy conserva- High order finite element methods have long been used tion/dissipation properties of our numerical solutions. Af- for computational wave propagation for both first order ter setting the stage for many of the presentations of the and second order equations alike. A key issue with com- minisymposium, I will present some recent results along putational wave propagation is the phase accuracy of the these lines for mixed finite element discretization of the methods: getting the waves to propagate with the right acoustic wave and linear shallow water equations. These speed is often at least as, if not more, important than the include nearly-conservative symplectic time-stepping, con- convergence rate. The main variants are the finite element version between first- and second-order, and the effect of a method (FEM) and spectral element method (SEM) with drag term on energy balances. each technique having its band of advocates. SEM can be viewed as a higher order mass-lumped FEM. Despite the Robert C. Kirby predominance of these methods, there is comparatively lit- Baylor University tle by way of hard analysis on what each variant offers Robert [email protected] in terms of phase accuracy, and there is considerable mis- information and confusion in the literature on this topic. In this presentation, we shall attempt to shed some light MS34 on the matter, and also briefly mention new methods that Analysis of the Two-Phase Flow in Rotating Hele- improve on both FEM and SEM Shaw Cells with Coriolis Effects

Mark Ainsworth The free boundary problem of a two phase flow in a rotating Division of Applied Mathematics Hele-Shaw cell with Coriolis effects is studied. Existence Brown University and uniqueness of solutions near spheres is established, and 92 PD13 Abstracts

the asymptotic stability and instability of the trivial solu- plus given ambient vector field with low regularity. This is tion is characterized in dependence on the fluid densities. a joint work with Keisuke Takasao.

Patrick Guidotti Yoshihiro Tonegawa University of California at Irvine Hokkaido University [email protected] [email protected]

Joachim Escher MS35 Lebiniz Universitat Hannover (Germany) [email protected] Unconditional Uniqueness for the Cubic Gross- Pitaevskii Hierarchy via Quantum de Finetti - Part II Christoph Walker Leibniz University Hanover The derivation of nonlinear dispersive PDE, such as the [email protected] nonlinear Schr¨odinger (NLS) or nonlinear Hartree (NLH) equations, from many body quantum dynamics is cen- tral topic in mathematical physics, which has been ap- MS34 proached by many authors in a variety of ways. In par- Crystal Facets: From Microscale Motion to ticular, one way to derive NLS is via the Gross-Pitaevskii Singular-Diffusion Pdes (GP) hierarchy, which is an infinite system of coupled lin- ear non-homogeneous PDE. The most involved part in such Continuum theories of epitaxial growth usually break a derivation of NLS consists in establishing uniqueness of down in the vicinity of macroscopically flat surface re- solutions to the GP. That was achieved in seminal papers of gions (facets). In surface diffusion, such facets give rise Erd¨os-Schlein-Yau. Recently, with Hainzl and Seiringer we to free-boundary problems with non-trivial microstructure obtained a new, simpler proof of the unconditional unique- for fourth-order singular-diffusion PDEs. In this talk, I ness of solutions to the cubic Gross-Pitaevskii hierarchy in will present recent progress in coupling facet motion with R3. One of the main tools in our analysis is the quan- discrete (microscale) schemes for surface atomic defects tum de Finetti theorem. In the first talk, Pavlovic will (steps) under surface diffusion. present a brief review of the derivation of NLS via the GP, describing the context in which the new uniqueness result Dionisios Margetis appears. In the second talk, Chen will illustrate how quan- University of Maryland, College Park tum de Finetti’s theorem can be used to obtain uncondi- [email protected] tional uniqueness of the GP. Thomas Chen MS34 University of Texas at Austin Examples of Singular Diffusion Equations in One [email protected] and Two Dimension: Facets and More

We focus on a simple singular diffusion equation (i.e. a MS35 total variation flow in 1-d) and its counterpart regularized The 2d Schr¨odinger Equation on Irrational Tori by additive linear diffusion. We study these two equations in one and two dimensions. After quickly establishing ex- Abstract not available at time of publication. istence, uniqueness and regularity we concentrate on cre- ation, evolution and interaction of facets. We also exhibit Seckin Demirbas a number of explicit solutions. University of Illinois Urbana-Champaign Mucha Piotr [email protected] Warsaw University Mathematics Department Mathematics Department MS35 [email protected] Unconditional Uniqueness for the Cubic Gross- Pitaevskii Hierarchy via Quantum de Finetti-Part Muszkieta Monika I Wroclaw University of Technology Institute of Mathematics and Computer Science The derivation of nonlinear dispersive PDE, such as the [email protected] nonlinear Schr¨odinger (NLS) or nonlinear Hartree (NLH) equations, from many body quantum dynamics is cen- Piotr Rybka tral topic in mathematical physics, which has been ap- Warsaw University proached by many authors in a variety of ways. In par- Mathematics Department ticular, one way to derive NLS is via the Gross-Pitaevskii [email protected] (GP) hierarchy, which is an infinite system of coupled lin- ear non-homogeneous PDE. The most involved part in such a derivation of NLS consists in establishing uniqueness of MS34 solutions to the GP. That was achieved in seminal papers On Singular Perturbation Limit of the Allen-Cahn of Erd¨os-Schlein-Yau. Recently, with Chen, Hainzl and Equation Seiringer we obtained a new, simpler proof of the uncondi- tional uniqueness of solutions to the cubic Gross-Pitaevskii We present some results on the characterization of energy hierarchy in R3. One of the main tools in our analysis is concentration measure for the Allen-Cahn equation as the the quantum de Finetti theorem. In the first talk, Pavlovic thickness of interface approaches to zero. The limiting in- will present a brief review of the derivation of NLS via GP terface moves with velocity equals to its mean curvature describing the context in which the new uniqueness result PD13 Abstracts 93

appears. In the second talk, Chen will illustrate how quan- form. I will give an example of a reflector that projects a tum de Finetti’s theorem can be used to obtain uncondi- famous Dutch painting on a wall from a uniform parallel tional uniqueness of the GP. light beam.

Natasa Pavlovic Corien Prins University of Texas at Austin Philips Research Delft. [email protected] [email protected]

Jan Ten Thije Boonkkamp MS35 Department of Mathematics and Computer Science Blow Up Solutions in the Focusing NLS Eindhoven University of Technology [email protected] Abstract not available at time of publication.

Svetlana Roudenko Wilbert IJzerman The George Washington University Philips Lighting [email protected] [email protected]

Teus Tukker MS36 Philips Research Wide-Stencil and Filter Schemes for Monge Am- [email protected] pere Equations Jarno van Roosmalen Abstract not available at time of publication. Eindhoven University of Technology [email protected] Adam Oberman McGill U. [email protected] MS37 Exact Controllability of a Membrane Immersed in aPotentialFluid MS36 Optimal Transport with Proximal Splitting We consider the problem of controlling a membrane or plate that is either immersed in a (linearized) potential We here present the use of first order convex optimization fluid,or forms a portion of the boundary of a potential schemes to solve the discretized dynamic optimal transport fluid. The control acts along a portion of the boundary problem, initially proposed by Benamou and Brenier. We of the membrane. We show that for sufficiently small fluid develop a staggered grid discretization that is well adapted 2 density, and time large enough (related to the geometry to the computation of the L optimal transport geodesic of the membrane), exact controllability holds in the finite between distributions defined on a uniform spatial grid. energy space. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. Scott Hansen A specific instantiation of this method on a centered grid Iowa State University corresponds to the initial algorithm developed by Benamou Department of Mathematics and Brenier. We also show how more general cost functions [email protected] can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold. MS37 Nicolas Papadakis On Incompressible Two-Phase Fluid Flows with LJK grenoble Phase Transitions [email protected] We consider a sharp interface model for two-phase flows with surface tension undergoing phase transitions. The MS36 model is based on conservation of mass, momentum and Optimal Transportation with Infinitely Many energy, and hence is physically exact. It further employs Marginal the standard constitutive law of Newton for the stress ten- sor, Fourier’s law for heat conduction, and it is thermody- Abstract not available at time of publication. namically consistent. We establish well-posedness of the model and study the qualitative behavior of solutions. In brendan Pass particular, we characterize all equilibria and study their U. Alberta stability properties. The total entropy turns out to be an [email protected] important quantity in the stability analysis.

Gieri Simonett MS36 Vanderbilt University Optimal Transport for Illumination Optics [email protected]

The design of freeform reflectors and lenses for illumination systems has become very relevant with the rise of LEDs. MS37 The problem is closely related to optimal transport. We On the Two-phase Navier-Stokes Equations in developed a method to design convex or concave reflec- Cylindrical Domains tors based on a novel numerical method for the Monge- Amp`ere equation and a numerical Legendre-Fenchel trans- We consider the dynamics of two incompressible fluids in 94 PD13 Abstracts

a cylindrical domain, separated by a sharp interface. The [email protected] contact angle is assumed to be constant and equal to 90 degrees. We prove the well-posedness of the corresponding free boundary problem and we present results concerning MS38 the stability of equilibria (Rayleigh-Taylor instability). Fi- Planar Limits of 3D Helical Flows nally we present a result on bifurcation at multiple eigen- values. We study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular Mathias Wilke pipe, with respectively no-slip and no-penetration bound- University of Halle ary conditions, as the step approaches infinity. We show [email protected] that, as the step becomes large, the three-dimensional he- lical flow approaches a planar flow, which is governed by the so-called two-and-half Navier-Stokes and Euler equa- MS37 tions, respectively. The step or pitch is the magnitude of Min-Max Game Problem for Elastic and Visco- the translation after rotating one full turn around the sym- Elastic Fluid Structure Interactions metry axis.

We present the features of a min-max game theory prob- Anna Mazzucato lem for a linear fluid-structure interaction model in both Department of Mathematics, Penn State University elastic and visco-elastic case, with (possibly) control and [email protected] disturbance exercised at the interfce between the two me- dia. The sought-after saddle solutions are expressed in a Milton C. Lopes Filho pointwise feedback form, which involves a Riccati opera- Universidade Federal do rio de Janeiro tor; that is, an operator satisfying a suitable non-standard [email protected] Riccati differential equation. Dongjuan Niu Jing Zhang Capital Normal University, Beijing Virginia State University [email protected] [email protected] Helena J. Nussenzveig Lopes Irena M. Lasiecka, Roberto Triggiani Universidade Federal do Rio de Janeiro University of Virginia [email protected] [email protected], [email protected] Edriss S. Titi University of California, Irvine and MS38 Weizmann Institute of Science, Israel On the Behavior of Bounded Vorticity, Bounded [email protected], [email protected] Velocity Solutions to the 2D Euler Equations

I will present recent work characterizing all possible weak MS38 solutions to the 2D Euler equations in the full plane or the Remarks on the Question of Dissipation Anomaly exterior of a single obstacle having bounded velocity and for the Navier-Stokes and Euler Equations bounded vorticity. The class of all such solutions general- izes the solutions obtained originally by Phillipe Serfati in Dissipation Anomaly is one of the well-defined mathemati- 1995 for the full plane, which have sublinear growth of the cal problems in turbulence theory. It states that the mean pressure at infinity. For more general solutions a condition rate of dissipation of energy of viscous turbulent flows re- at infinity, in terms of the velocity or the pressure, holds mains bounded away from zero, as the viscosity tends to weakly, and the circulation about the obstacle can vary for zero. We will provide simple examples demonstrating this an exterior domain. These results build on those of joint phenomenon, and explaining the idea behind the available work with Ambrose, Lopes Filho, and Nussenzveig Lopes. rigorous results. These examples illustrate that the current mathematical formulation might not be adequate; and an- James P. Kelliher other formulation is proposed ruling out these examples. UC Riverside [email protected] Edriss S. Titi University of California, Irvine and MS38 Weizmann Institute of Science, Israel On Vorticity Formulation for the Navier-Stokes [email protected], [email protected] Equations in the Half Plane and Its Applications to the Inviscid Limit Problem Claude Bardos Laboratory J.L. Lions, Universit{´e} Pierre et Marie Curie We consider the Navier-Stokes equations for viscous incom- Paris, 75013, France pressible flows in the half plane under the no-slip boundary [email protected] condition. By using a vorticity formulation we prove time- local convergence of the Navier-Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in MS39 the boundary layer at the inviscid limit when the initial Title Not Available at Time of Publication vorticity is located away from the boundary. Abstract not available at time of publication. Yasunori Maekawa Dept. Math , Tohoku University Simon Arridge PD13 Abstracts 95

Department of Computer Science by describing a variational framework that I recently de- University College London veloped with J. Bronski and determine instability to long [email protected] wavelengths perturbations for a general class of Hamil- tonian systems, allowing for nonlocal dispersion. I will explain an asymptotic approach that M. Johnson and I MS39 adapted for Whitham’s equation of surface water waves, Deriving the Kubelka-Munk Equations from Ra- qualitatively reproducing the Benjamin-Feir instability of diative Transport Stokes waves. I will discuss on how to extend the results to the exact, water wave problem. Kubelka-Munk theory provides a simple model to describe for light propagation in turbid media. We derive this the- Vera Mikyoung Hur ory systematically from the theory of radiative transport University of Illinois at Urbana-Champaign theory by analyzing the system of equations resulting from [email protected] applying the double spherical harmonics method. From this systematic derivation, we establish theoretical basis of Jared Bronski the Kubelka-Munk equations which has been an outstand- University of Illinois Urbana-Champaign ing issue. Moreover, we extend this theory to study several Department of Mathematics problems of practical interest. [email protected]

Arnold D. Kim Mathew Johnson University of California, Merced University of Kansas [email protected] [email protected]

Chris Sandoval Applied Mathematics MS40 University of California, Merced Pressure Beneath a Traveling Wave with Constant [email protected] Vorticity

In this talk I will give a brief overview of two-dimensional MS39 traveling gravity waves with constant vorticity, with and Case’s Method in Three Dimensions without stagnation points. Following this, I will present a direct relationship between the surface profile of the travel- Case’s method obtains solutions to the radiative transport ing wave and the pressure at the bottom of the fluid. This equation as a linear combination of elementary solutions relationship is obtained as a direct consequence of the full or singular eigenfunctions. In this talk, we will extend equations describing the traveling wave without approxi- the method to three dimensions with arbitrary anisotropic mation. Using this relationship, we reconstruct both the scattering. We evaluate singular eigenfunctions in refer- pressure and streamlines throughout the fluid domain. In ence frames whose z-axesaretakeninthedirectionofeach particular, we can recover local minima of the pressure (be- wave vector. By doing so, the three-dimensional equation low the atmospheric value) beneath the crest in the pres- reduces to the one-dimensional equation. ence of a stagnation point.

Manabu Machida Vishal Vasan Department of Mathematics The Pennsylvania State University University of Michigan [email protected] [email protected] Katie Oliveras Seattle University MS39 [email protected] Filtered Spherical Harmonics Expansions of the Radiative Transfer Equation MS40 Recent work has shown that various filtering schemes can Mathematical Theory of Wind-generated Water greatly improve the robustness of moment-based methods Waves for solving the radiative transfer equation. In this talk I will outline several approaches used to filter spherical It is easy to see that wind blowing over a body of water can harmonics moments and apply these filters to the transport create waves. But this simple observation leads to a more solutions for problems of linear particle transport as well fundamental question: Under what conditions on the veloc- has nonlinear x-ray transport. I will also detail how filters ity profile of the wind will persistent surface water waves be can be employed to improve discrete ordinates methods generated? This has been problem has been studied inten- and talk about nonlinear extensions. sively in the applied fluid dynamics community since the first efforts of Kelvin in 1871. In this talk, we will present Ryan G McClarren a mathematical treatment of the predominant model for Texas A & M wind-wave generation, the so-called quasi-laminar model [email protected] of J. Miles. Essentially, this entails determining the (lin- ear) stability properties of the family of laminar flow solu- tions to the two-phase interface Euler equation. We give MS40 a rigorous derivation of the linearized evolution equations On the Benjamin-Feir Instability about an arbitrary steady solution, and, using this, we give a complete proof of the celebrated instability criterion of I will speak on the Benjamin-Feir (or modulational) insta- Miles. In particular, our analysis incorporates both the ef- bility of Stokes periodic waves on deep water. I will begin fects of surface tension and a vortex sheet on the air–sea 96 PD13 Abstracts

interface. We are thus able to give a unified equation con- tracted from the numerical simulations of ideal Magneto- necting the Kelvin–Helmholtz and quasi-laminar models of hydrodynamics (MHD) equations, which contain nonlinear wave generation. hyperbolic equations as well as a divergence-free condition for the magnetic field. Numerical methods without satisfy- Samuel Walsh ing such condition may lead to numerical instability. One New York University class of methods called constrained transport schemes is [email protected] widely used as a divergence-free treatment. However, why these schemes work is still not fully understood. In this Oliver Buhler, Jalal Shatah work, we take the exactly divergence-free schemes proposed Courant Institute of Mathematical Sciences by Li and Xu as a candidate of the constrained transport New York University schemes, and analyze the stability and errors when solving [email protected], [email protected] magnetic induction equations. This is the most significant part to understand the divergence-free treatment in MHD Chongchun Zeng simulations. Our result can not only explain the stability Georgia Tech mechanism of this particular scheme and the role of the ex- [email protected] actly divergence-free condition in that, but also provide the insight to understand other constrained transport schemes.

MS40 Fengyan Li Large-Amplitude Solitary Waves Generated by Rensselaer Polytechnic Institute Surface Pressure [email protected] We study solitary traveling waves in a two-dimensional in- viscid incompressible fluid bounded below by a horizontal He Yang bed and above by a free surface on which a non-constant Rensselaer Polytechnic Institute pressure is applied. By varying the surface pressure, we Department of Mathematical Sciences construct large-amplitude waves of elevation and depres- [email protected] sion. We assume that the wave and surface pressure are symmetric, and that the Froude number of the underlying flow is sufficiently large (supercritical). MS41 High Order Fast Sweeping and Homotopy Methods Miles Wheeler with Linear Computational Complexity for Solving Department of Mathematics Steady State Problems of Hyperbolic PDEs Brown University [email protected] In this talk, I present our recent studies on developing effi- cient high order numerical methods for solving steady state problems of two classes of hyperbolic PDEs: Eikonal equa- MS41 tions and hyperbolic conservation laws. The methods we On the Parametrized Maximum Principle Flux propose have linear computational complexity, namely, the Limiters for Hyperbolic Conservation Laws: Ac- computational cost is O(N)whereN is the number of grid curacy and Application points of the computational mesh. For Eikonal equations, we design a third order fast sweeping method with linear Maximum principle preserving is an important property of computational complexity. This iterative method utilizes the entropy solution, the physically relevant solution, to the Gauss-Seidel iterations and alternating sweeping strat- the scalar hyperbolic conservation laws. One would like egy to cover a family of characteristics of the Eikonal equa- to imitate the property on the numerical level. On the tions in a certain direction simultaneously in each sweeping other hand, preserving the maximum principle also pro- order. The method is based on a third order discontinuous vides a stability to the numerical schemes for solving the Galerkin (DG) finite element solver. Novel causality indi- conservation law problem. In this talk, we will discuss cators are designed to guide the information flow directions how this problem is addressed by introducing a series of of the nonlinear Eikonal equations. Numerical experiments maximum principle constraint. By decoupling those con- show that the method has third order accuracy and a lin- straints, a parametrized flux limiting technique is devel- ear computational complexity. For hyperbolic conservation oped to make sure the numerical solution preserves maxi- laws, a homotopy continuation method is applied to solve mum principle in the conservative and consistent manner polynomial systems resulting from a third order weighted while the scheme is still high order. Generalization of the essentially non-oscillatory (WENO) discretization of the technique to convection-diffusion problem will also be dis- system. Via numerical experiments for both scalar and sys- cussed. tem problems in one and two dimensions, we show that this new approach has linear computational complexity and is Zhengfu Xu free of the CFL condition constraint. Michigan Technological University Dept. of Mathematical Sciences Yongtao Zhang [email protected] University of Notre Dame [email protected]

MS41 Stability Analysis and Error Estimates of an Ex- MS41 actly Divergence-Free Method for the Magnetic In- Energy-conserving Discontinuous Galerkin Meth- duction Equations ods for the Vlasov-Amp`ere System

In this talk, we consider an exactly divergence-free scheme We propose energy-conserving numerical schemes for the to solve magnetic induction equations. This problem is ex- Vlasov-Amp`ere (VA) systems. The VA system is a model PD13 Abstracts 97

used to describe the evolution of probability density func- [email protected] tion of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, includ- ing the total energy which is comprised of the kinetic and MS42 electric energy. Unlike the total particle number conserva- Finite Element Calculations in One-Dimensional tion, the total energy conservation is challenging to achieve. Peridynamics and Multiscale Mono-models For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In our work, we develop the first Eu- The peridynamics theory of solid mechanics is a general- lerian solvers that can preserve fully discrete total energy ization of classical continuum mechanics suitable for the conservation. The main components of our solvers include simulation of material failure and damage. As a contin- explicit or implicit energy-conserving temporal discretiza- uum model, peridynamics can be discretized through many tions, an energy-conserving operator splitting for the VA different schemes. However, most of its current implemen- equation and discontinuous Galerkin finite element meth- tations use the so-called meshfree discretization, which re- ods for the spatial discretizations. We validate our schemes duces the governing equation to a discrete equation rep- resenting the dynamics of a system of virtual particles by rigorous derivations and benchmark numerical exam- interacting in space; this discretization method is rela- ples such as Landau damping, two-stream instability and tively inexpensive and suitable for simple implementations bump-on-tail instability. of bond-breaking criteria yet introduces important limita- tions on the choice of horizon, affecting the potential use Yingda Cheng of the theory in multiscale modeling. It is thus of inter- Department of Mathematics est to explore other possible discretization schemes. This Michigan State University presentation will be focused on the finite element method [email protected] for peridynamics, applied to one-dimensional models. We will present element by element calculations and compare Andrew J. Christlieb nonlocal results with their local counterparts based upon Michigan State Univerity the finite element method for classical partial differential Department of Mathematics equations. We present analytical calculations for a certain [email protected] choice of peridynamic kernel and demonstrate that the non- locality of the peridynamic model depends upon the ratio Xinghui Zhong of the horizon and the finite element mesh size. This re- Michigan State University sult suggests a multiscale approach based upon different [email protected] discretizations of a single model, which we referred to as a multiscale mono-model. Analytical results are supported by numerical experiments. MS42 Pablo Seleson The Fractional Laplacian Operator as a Special The University of Texas at Austin Case of the Nonlocal Diffusion Operator [email protected]

We analyze a nonlocal diffusion operator having as a spe- cial case the fractional Laplacian operator. We demon- MS43 strate that the solution of the nonlocal equation converges to the one of the fractional Laplacian as the nonlocal inter- Homogenization of High-Contrast Brinkman Flows actions become infinite. Through several numerical exam- ples we illustrate the theoretical results and we show that Porous media flow can have many scales and is often a by solving the nonlocal problem it is possible to obtain multi physical processes. A useful tool in the analysis of accurate approximations of the solutions of fractional dif- such problems in that of homogenization as an averaged ferential equations circumventing the problem of treating description is derived circumventing the need for compli- infinite-volume constraints. cated simulation of the fine scale features. In this talk, we recall recent developments of homogenization techniques in Marta D’Elia the application of flows high contrast media. We have par- Department of Scientific Computing ticular emphasis on the Brinkmann equation and related Florida State University, Tallahassee, FL numerical methods. [email protected] Donald Brown King Abdullah University of Science and Technology Max Gunzburger Center for Numerical Porous Media Florida State University [email protected] [email protected] Guanglian Li Texas A&M University MS42 [email protected] Homogenization of the Nonlocal Navier System of Equations Yalchin Efendiev Dept of Mathematics Abstract not available at time of publication. Texas A&M University [email protected] Tadele Mengesha Penn State University Viktoria Savatorova Department of Mathematics N6 NRNU MEPhI 98 PD13 Abstracts

[email protected] [email protected]

MS43 MS44 Error Estimates for Mesoscale Continuum Models On Incompressible Flows with Singular Velocities of Particle Systems The purpose of the talk is to establish several existence, Abstract not available at time of publcation. uniqueness and ill-posedness results for two families of ac- Alexander Panchenko tive scalar equations with velocity fields determined by the Washington State University scalars through very singular integrals. The first family is [email protected] a generalized surface quasi-geostrophic equation with the velocity field u related to the scalar θ by u = ∇⊥Λα−1θ, where 0 <α≤ 1andΛ=(−Δ)1/2 is the Zygmund MS43 operator. The borderline case α = 0 corresponds to Modeling Effective Acoustic Behavior of Porous the surface quasi-geostrophic equation and the situation Media with Microstructure is more singular for α>0. We obtain the local exis- tence and uniqueness of classical solutions, the global ex- In this talk we present some recent results in mathemat- istence of weak solutions and the local existence of patch ical modeling of porous media with multi-scale structure. type solutions. The second family is a singularly modified In particular, we focus on effective acoustic properties of version of the 2D incompressible porous media equation cancellous bone. Using two-scale convergence and other ho- with the velocity field v given by the scalar ρ as follows: −∇ −2+α − α ≤ mogenization tools we derive effective material properties v = Λ ∂x2 ρ (0, Λ ρ), for 0 <α 1. Here, the of the fluid-filled porous matrix. The effective equations case α = 0 yields Darcy’s law. In contrast, for the last sin- are Biot-like, however they account for possible effects of gular active scalar equation local well-posedness does not micro-structural anisotropy. hold for smooth solutions, but it does hold for certain weak solutions. Ana Vasilic UAE University Francisco Gancedo [email protected] University of Seville [email protected] Robert P. Gilbert Department of Mathematical Sciences University of Delaware MS44 [email protected] Big Box Limit for NLS

Alexander Panchenko I will describe the derivation of a new equation from NLS Washington State University on the torus with size L, in the weakly nonlinear regime, [email protected] as L goes to infinity. I will then discuss various properties of this equation.

MS43 Pierre Germain Effect of Slip Boundary Conditions on the Effective New York Univeristy Behavior of Suspensions and Membranes [email protected]

The first to depart from no-slip conditions for viscous flu- Zaher Hani ids on a solid surface was Navier (1827) who imposed a slip NYU condition that gives a linear dependence of the tangential [email protected] velocity on the tangential stress. Threshold slip was intro- duced for viscous flow problems by Fujita (1994) who stud- Erwan Faou ied the existence and uniqueness for the Stokes problem − ∈ | | INRIA Rennes with boundary conditions of the type (σN)τ g∂uτ , [email protected] where g ≥ 0. A similar threshold-type condition was in- troduced by Fujita to model leaky boundaries: the normal velocity is zero unless the jump in the normal stress across MS44 the membrane reaches an yield limit. We study the macro- scopic effect of threshold slip for a suspension of rigid par- Persistency of the Spatial Analyticity for Nonlinear ticles in a viscous fluid, and that of threshold leak for a Wave Equations and the Cubic Szego Equation permeable membrane. For suspensions, the Navier-Fujita type slip introduces at the macroscale a new fluid viscos- Gevrey classes were introduced by Maurice Gevrey in 1918 ity. For membranes, the effective boundary conditions are to generalize real analytic functions. Functions of Gevrey of subgradient type with an effective yield limit, in the case classes can be characterized by an exponential decay of of a densely distributed solid part, or of Navier type, in the their Fourier coefficients. This characterization has been case of dilute solid part; in the intermediate case the tan- proved useful for studying analytic solutions of various non- gential slip cancels, whereas the normal velocity and stress linear PDEs, since the work by Foias and Temam (1989) are continuous. Unlike in the case of perforated walls, no on the Navier-Stokes equations. We use this technique to stress concentrations are present. investigate the persistency of spatial analyticity for nonlin- ear wave equations (joint work with Edriss S. Titi), and the Bogdan M. Vernescu cubic Szego equation (joint work with Patrick Gerard and Worcester Polytechnic Inst Edriss S. Titi). An advantage of this method is that it pro- Dept of Mathematical Sciences vides a lower bound for the radius of the spatial analyticity PD13 Abstracts 99

of the solutions. [email protected]

Yanqiu Guo The Weizmann Institute of Science MS45 [email protected] Discontinuous Petrov Galerkin Methods for Wave Propagation MS44 Discontinuous Petrov Galerkin (DPG) methods use stan- Sensitivity Analysis with Respect to Principal Pa- dard trial spaces and nonstandard test spaces. The test rameters of Long Time Dynamics in Hyperbolic spaces are automatically computed to obtain good stabil- Systems with Critical Exponents ity constants. The resulting method can also be interpreted We consider a third order in time equation which arises as as a least-squares finite element method that minimize a a model of wave propagation in viscous thermally relax- residual in a nonstandard norm. In this talk, we report ing fluids. The nonlinear PDE model under consideration our results obtained by applying the DPG methodology to displays two important characteristics: (i) it is quasilinear time-harmonic acoustic wave propagation. The main ad- and (i) it is degenerate in the principal part. The main goal vantage of the method is that it yields Hermitian positive of this talk is to present results on existence-nonexistence definite systems which are easier to solve. Other least- and decay rates for the solutions as a function of physi- squares methods also have the same advantage, but we cal parameters in the equation. This is joint work with show how DPG methods improve on standard least-squares Barbara Kaltenbacher, University of Graz. methods.

Irena M. Lasiecka Jay Gopalakrishnan University of Virginia Portland state university [email protected] [email protected]

MS45 MS45 Finite Element Exterior Calculus Methods for Geo- DGSEM-ALE Approximation of Reflection and physical Fluid Dynamics Transmission at Moving Interfaces

We show how mixed finite element methods that fall into We derive a spectrally accurate moving mesh method the finite element exterior calculus framework can be ap- for mixed material interface problems for Maxwell’s and plied to the design of numerical weather prediction models. the classical wave equation. We use a discontinuous This investigation is being driven by the need to extend Galerkin spectral element approximation with an arbitrary the properties of the C-grid finite difference staggering to Lagrangian-Eulerian mapping and derive the exact upwind arbitrary triangular/quadrilateral meshings of the sphere numerical fluxes that model the physics of wave reflection which are required for scalable massively parallel weather and transmission at jumps in material properties. Spec- forecasting. We shall discuss the following: (a) Poisson tral accuracy is obtained by placing moving material in- structure, Casimirs, (b) geometric forms of stabilisation of terfaces at element boundaries and solving the appropri- the discretisation, (c) conservation laws, (d) treatment on ate Riemann problem. We present examples showing the curved elements, (e) extension to fully three dimensional performance of the method for plane wave reflection and models. transmission at dielectric and acoustic interfaces.

Colin Cotter Andrew Winters, David A. Kopriva Imperial College London Department of Mathematics Department of Aeronautics The Florida State University [email protected] [email protected], [email protected]

MS45 MS46 Numerical Approximation of Boussinesq-Green- Naghdi Models for Near-Shore Wave Physics Stability Analysis of Flock Rings for 2nd Order Models in Swarming Boussinesq-Green-Naghdi equations model the physics of waves in the near shore. They are approximations to the In this work we consider second order models in swarm- full Navier-Stokes equations where one assumes a vertical ing in which individuals interact pairwise with a power-law profile for the wave velocity. The resulting model is a highly repulsive-attractive potentials. We study the stability for nonlinear system of PDES with higher order derivatives, al- flock ring solutions and we show how the stability of these beit in one less space dimension than the full model. The solutions is related to the stability of a first order model. In model is well-suited for discontinuous Galerkin approxi- unstable situations it is also possible to observe formation mations. We describe a general approach and discuss the of clusters and fat rings. accuracy and stability of the numerical approximation, as well as validation of the model against experimental data. Daniel Balague Departament de Matem`atiques Clint Dawson Universitat Autnoma de Barcelona Institute for Computational Engineering and Sciences [email protected] University of Texas at Austin [email protected] Giacomo Albi Dipartimento di Matematica e Informatica Nishant Panda Universit`a di Ferrara The University of Texas at Austin [email protected] 100 PD13 Abstracts

Jose Carrillo chances? This question can be reformulated in terms of a Department of Mathematics random walk of brownian particle; the answer is given in Imperial College London terms of a certain optimization problem. Its solution is [email protected] again related to vortex crystals. Joint work with: Yuxin Chen, Daniel Zhirov and Michael Ward James von Brecht University of California, Los Angeles Theodore Kolokolnikov [email protected] Dalhousie University [email protected]

MS46 MS47 Two Short Stories on Existence and Stability of N- Vortex Equilibria in Fluids Analysis of Point Defects in a Ferroelectric Liq- uid Crystal Using a Generalized Ginzburg-Landau Abstract: Vortex relative equilibria are configurations of Model vortices that maintain their basic shapes for all time, while rotating or translating rigidly in space. It is common to Defects in Smectic C* liquid crystal film cause distinc- observe these patterns in nature and experiments, where tive spiral patterns in the texture of the film. This par- the underlying dynamics are governed, for example, by the ticular ferroelectric thin film is modeled by a generalized two-dimensional Euler or Navier-Stokes equations. One Ginzburg-Landau energy functional, can often take advantage of the structure of vortices to  derive equations of motion that are much simpler than the 1 2 2 1 −| |2 2 k1(div u) + k2(curl u) + 2 (1 u ) dx. original PDE. In this work, we look at two point vortex 2 Ω 2 models to study the existence and stability of a number of relative equilibria. We complement this analysis with Given fixed boudary data of degree d>0, we study the comparisons of our results to crystals observed in various limiting configuration of a sequence of minimizers {u} as experiments and numerical simulations.  → 0. We show the limit function contains d degree one singularities and near each, the function asymptotically has Anna M. Barry either a bend or splay pattern, depending on the relative Institute for Mathematics and its Applications values of k1 and k2. We also show that this functional has a University of Minnesota renormalized energy that is minimized by the singularities [email protected] of the limit function.

Sean A. Colbert-Kelly MS46 National Institute of Standards and Technology Bifurcation Dynamics in a Nonlocal Hyperbolic [email protected] Model for Self-organised Biological Aggregations Daniel Phillips The investigation of complex spatial and spatio-temporal Department of Mathematics, Purdue University patterns exhibited by self-organised biological aggregations [email protected] has become a very active research area. Here, we focus on a class of nonlocal hyperbolic models for self-organised ag- gregations, and investigate the bifurcation dynamics of a Geoffrey McFadden variety of spatial and spatio-temporal patterns observed National Institute of Standards and Technology numerically near codimension-1 and codimension-2 bifur- geoff[email protected] cation points.

Raluca Eftimie MS47 University of Alberta, Edmonton, Canada Electric-Field-Induced Instabilities in Liquid- [email protected] Crystal Films

Simple models of liquid crystals characterize orientational MS46 properties in terms of a director field, which can be influ- Vortex Crystals, Animal Skin Patterns and Ice enced by an applied electric field. The local electric field is Fishing influenced by the director field; so equilibria must be com- puted in a coupled way. Equilibria are stationary points of I will discuss three very different topics which turn out to a free energy functional,whichcanfailtobecoercivebe- have a related mathematical formulation. The first topic is cause of the nature of the director/electric-field coupling. classical vortex dynamics in the plane. We look for relative We discuss characterizations of local stability and anoma- equilibria (lattice crystals) in the limit of large number of lous behavior in such systems. vortices. Many results for the steady state and its local stability can be obtained taking the mean-field limit. Next, Eugene C. Gartland we consider hot-spot solutions to certain reaction-diffusion Kent State University PDE system. Similar PDE’s are often used to model spots [email protected] on the animal skins. We derive a reduced system of ODE’s which describe the motion and locations of these spots. In certain regimes, we show that these ODE’s are related MS47 to the relative equlibria of the vortex model. Finally, we Regularity Properties for Nematic Liquid Crystals describe the problem of mean first passage time. Suppose you live in Canada and you want to catch a fish in a lake We investigate regularity properties for and bounds on lo- covered by ice. Where do you drill a hole to maximize your cal minimizers to an energy for nematic liquid crystals with PD13 Abstracts 101

a Maier-Saupe potential. Nicola Garofalo Universita’ di Padova Patricia Bauman [email protected] Purdue University [email protected] Arshak Petrosyan Purdue University Daniel Phillips Department of Mathematics Department of Mathematics, Purdue University [email protected] [email protected] Tung To MS48 Purdue University [email protected] Free-boundary Problems in Surface Evolution by Curvature Flows MS48 We will discuss the qualitative properties of solutions to fully non-linear parabolic equations that arise from the evo- Regularity of Solutions and of the Free Boundary of lution of hyper-surfaces by functions of their principal cur- the Obstacle Problem for the Fractional Laplacian vatures. Such examples include evolutions by powers of the with Drift Gaussian curvature or evolution by the harmonic mean of the principal curvatures. Since such equations become de- We study the elliptic obstacle problem defined by the frac- generate or singular the classical regularity results fail. We tional Laplacian with drift, will discuss the optimal regularity of solutions and related n free-boundary problems. min{Lu(x),u(x) − ϕ(x)} =0, ∀x ∈ R ,

Panagiota Daskalopoulos where the operator L is defined by Columbia University [email protected] Lu(x)=(−Δ)su(x)+b(x) ·∇u(x)+c(x)u(x), ∀x ∈ Rn.

MS48 In joint work with Arshak Petrosyan, we develop a new monotonicity formula and we establish the optimal regu- A Classical Perron Method for Existence of Smooth 1 larity of solution, C ,s(Rn), and Lipschitz regularity of the Solutions to Boundary Value and Obstacle Prob- free boundary in neighborhoods of regular points, in the lems for Degenerate-elliptic Operators via Holo- case 1/2

based on the theory for poroelastic materials, will be dis- [email protected], [email protected] cussed and its numerical solutions will be presented for various clinically-relevant scenarios. MS50 Giovanna Guidoboni Regularity of Solutions to the Axisymmetric Euler Indiana University-Purdue University at Indianapolis Equations Department of Mathematical Sciences [email protected] We investigate properties of solutions to the axisymmetric Euler equations when initial vorticity belongs to a space of Besov type. In particular, we study regularity of solutions MS49 when a certain Besov norm of initial vorticity diverges in Intrinsic Decay Rates for the Energy of Second Or- a controlled way. der Nonlinear Evolutions with Viscoelasticity Elaine Cozzi Nonlinear second order evolutions subject to viscoelastic Dept. Math - Oregon State University damping are considered. It is well known that the stability [email protected] characteristics of finite energy solutions are dependent on the decay properties of the relaxation kernel. The aim of the talk is to present sharp decay rates of solutions corre- MS50 sponding to relaxation kernels without quantified asymp- H¨older Continuous Euler Flows totic behavior. The obtained decay rates are described by solutions to appropriately constructed ordinary differential Motivated by the theory of hydrodynamic turbulence, Lars equations. Applications to nonlinear waves and von Kar- Onsager conjectured in 1949 that solutions to the incom- man systems of dynamic elasticity with memory will be pressible Euler equations with H¨older regularity below 1/3 given. may fail to conserve energy. I will discuss Onsager’s conjec- ture and recent progress towards this conjecture, including Irena M. Lasiecka the construction of solutions to Euler which are (1/5 − )- University of Virginia H¨older continuous and fail to conserve energy. [email protected] Phil Isett Dept. Math - Princeton University MS49 [email protected] Model Development and Uncertainty Quantifica- tion for Systems with Nonlinear and Hysteretic Ac- MS50 tuators Nonlinear Stability of Vortex Pairs Smart material actuators and sensors provide unique con- trol capabilities for a range of applications involving fluid- In this talk, we discuss nonlinear orbital stability for structure and flow-structure interactions. These include steadily translating vortex pairs, a family of nonlinear PZT-based macrofiber composites which are being consid- waves that are exact solutions of the incompressible, two- ered for flow control and shape memory alloys which are be- dimensional Euler equations. We use an adaptation of ing tested for use as catheters employed for laser treatments Kelvin’s variational principle, maximizing kinetic energy of atrial fibrillation. In this presentation, we will discuss penalised by a multiple of momentum among mirror- the development of a modeling framework for these systems symmetric isovortical rearrangements. This formulation that facilitates subsequent design, uncertainty quantifica- has the advantage that the functional to be maximized tion, and real-time control implementation for transducers and the constraint set are both invariant under the flow of operating in highly nonlinear and hysteretic regimes. the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence Ralph C. Smith yields a wide class of examples to which our result applies. North Carolina State Univ Dept of Mathematics, CRSC Geoffrey Burton [email protected] Univ. of Bath [email protected]

MS49 Milton Lopes Filho, Helena J. Nussenzveig Lopes New Advances and Open Problems in Nonlinear Universidade Federal do Rio de Janeiro Flow-Plate Interactions [email protected], [email protected]

We address nonlinear PDE models arising in the study of flow-plate interactions and the associated phenomenon MS51 (e.g. flutter). We will review the recent well-posedness and Algebraic Vortex Spirals long-time behavior results for a clamped, non-rotational plate immersed in a supersonic flow. Additionally, we ad- Vortex spirals are ubiquitous in fluid flow, for example as dress novel mathematical challenges associated to the well- turbulent eddies or as trailing vortices at aircraft wings. posedness theory for other physical configurations of re- However, there are few proofs of existence for any of the cent experimental and numerical interest: specifically, the common fluid models. We consider solutions of the in- Kutta Joukowsky flow condition and free plate boundary compressible Euler equations that have vorticity stratify- conditions. ing into algebraic spirals. The solutions are selfsimilar: velocity v(t, x)=tm−1v(t−mx), for similarity exponent 1 ∞ Justin Webster,IrenaM.Lasiecka 2

tractable boundary value problems. The key to the exis- Institut f¨ur Angewandte Analysis und Numerische tence proof is an coordinate change which is implicit, de- Simulation pending on the a priori unknown solution. We will also [email protected] discuss the importance of the program for showing non- uniqueness in the initial-value problem for the 2d incom- pressible Euler solutions. MS51 The Onset of Cavitation for the Equations of Poly- Volker W. Elling convex Elasticity University of Michigan [email protected] Cavitation refers to the opening of holes during the dy- namic deformation of an elastic material and is a phe- nomenon that lies at the boundary of continuum model- MS51 ing. We will describe a solution constructed by Pericak- Linear and Nonlinear Reflection Patterns in Gas Spector and Spector describing a cavity opening from a Dynamics homogeneously deformed state. This poses challenges to the existence theory of multi-d conservation laws, as the The analysis of self-similar, non-potential flow for the equa- constructed cavitating solution turns out to decrease the tions of gas dynamics involves the interaction of the non- mechanical energy. We will analyze this example from the linear family of acoustic waves with the linear, degenerate perspective of a theory that accounts for the singular layers family or families of shear (and possibly entropy) waves. of the cavitating solution. We call the resulting notion of We examine a simple pattern, regular reflection of a shock solution as singular limiting induced from continuum so- at wedge near the reflection point. Behind a transonic lution. It turns out that this perspective can account for shock, the self-similar equations change type, with the the cost of energy associated with producing the cavity, acoustic waves now corresponding to a second-order elliptic and that the resulting energy of the cavitating solution is equation coupled with a hyperbolic component correspond- larger than that of a homogeneously deformed state. ing to the shear waves (in isentropic flow). The elliptic solution operator is compact, while the transport system Athanasios Tzavaras gives rise to a contraction. We show how to couple these University of Crete to produce a local solution. [email protected]

Barbara Keyfitz The Ohio State University MS52 bkeyfi[email protected] Asymptotic Approximation of the Dirichlet to Neu- mann Map of High Contrast Conductive Media Katarina Jegdic University of Houston-Downtown I will present an asymptotic study of the Dirichlet to Neu- [email protected] mann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result Suncica Canic is an explicit characterization of the map in the asymptotic Department of Mathematics limit of the distance between the particles tending to zero. University of Houston Liliana Borcea [email protected] Rice University Dept Computational & Appl Math Hao Ying [email protected] The Ohio State University [email protected] MS52 Anomalous Diffusion of a Tracer Particle in Fast MS51 Cellular Flow Hyperbolic Techniques for Multidimensional Inter- face Dynamics It is well known that a diffusive tracer particle in the pres- ence of an array of strong opposing vortices (aka cellular We are interested in interfacial dynamics in continuum me- flow) behaves like an effective Brownian motion on long chanics which are governed by hyperbolic evolution equa- time scales. On intermediate time scales, however, a ro- tions in the bulk and appropriate transfer conditions across bust anomalous diffusive behaviour has been numerically the interface. Instances of bulk systems are provided observed. This talk is a first step towards understanding by compressible Euler equations for liquid-vapour flow, this anomalous behaviour. We√ will show that the variance nonlinear elasticity for solid-solid transitions, or Buckley- of the particle behaves like O( t) on ”intermediate” time Leverett type formulations for in porous media flow. Trans- scales; in contrast, the long time behaviour of the variance fer conditions have not only to gurantee conservation prop- is like O(t). erties but account for e.g. curvature, phase transition or connect different types of bulk physics. To track the inter- Gautam Iyer faces’ local dynamics IVPs in normal directions are consid- Carnegie Mellon University ered. This leads to generalized Riemann problems which [email protected] are hardly to solve exactly. We will introduce approxi- mations which rely on regularization or hyperbolic-elliptic relaxation. The convergence of the approximations and the MS52 (non)validity of entropy conditions are discussed. Asymptotic Expansion for Wave Propagation in a Media with a Small Hole Christian Rohde University of Stuttgart In this talk, I will discuss the asymptotic expansion of the 104 PD13 Abstracts

acoustic field under the effect of a small hole in the quasi- [email protected] static regime. As a consequence, one can prove that the gradient of the field is ”bounded”. This is joint work with Reitich and Lin. MS53 Engineering Anisotropy to Amplify a Long Wave- Hoai Minh Nguyen length Field Without a Limit University of Minnesota School of Mathematics We present a simple design of circular or spherical shells ca- [email protected] pable of amplifying a long wavelength or static field. This design makes use of only two isotropic materials and is optimal restricted to the prescribed geometric and materi- MS52 als constraint. Further, it is shown that the amplification Title Not Available at Time of Publication factor of the structure can be made arbitrarily large as the ratio of the radius of the inner sphere to that of the Abstract not available at time of publication. outer sphere decreases to zero. It is anticipated that the presented design will be useful for high gain antenna for Lenya Ryzhik telecommunication, magnets generating strong, local uni- Stanford University form fields, and thermoelectric devices harvesting thermal [email protected] energy.

MS53 Liping Liu Department of Math, Mechanical Engineering Tight-Binding Approximations and Edge States in Rutgers University Honeycomb Optical Lattices [email protected] Over the last several years, a great deal of interest has emerged over honeycomb optical lattices, which are mod- MS53 eled by a Gross-Pitaevskii (GP) equation with a periodic, two-dimensional potential. Using a tight-binding approxi- Pathological Scattering in the Resonant Slow-Light mation in the semi-classical limit, a two-dimensional non- Regime linear discrete system has been derived as an approxima- Layered media consisting of alternating magnetic and tion to the GP equation. We present results that establish anisotropic components were shown by Figotin and Viteb- the validity of this approximation on asymptotically long skiy to admit an electromagnetic mode of zero energy flux time scales. By introducing an edge into the discrete sys- at a frequency in the interior of a propagation band; oper- tem, we then present results on the propagation of modes ation near this frequency is called the “slow-light” regime. localized along the edge, or edge modes. Ignoring nonlin- We consider scattering of waves by a planar defect em- earity, one can find a plethora of linear edge modes, but bedded in a slow-light ambient medium. The scattering a central question is what is the impact of nonlinearity on problem is pathological because three eigenmodes coalesce these problems. In the case of weak nonlinearity, we have into the one zero-flux mode as branches of a Puiseux series developed a rigorously justified approximation describing in a perturbation parameter ?, and the completeness and the impact of nonlinearity on the linear modes. The most independence of the rightward and leftward modes breaks important result from this is that the nonlinearity does down. It is even more pathological when the defective layer not cause delocalization away from the edge. In the case admits a guided resonant mode exactly at the slow-light of strong nonlinearity, we present numerical results which parameters (?=0). The unusual scattering phenomena fit show the nonlinearity does not cause delocalization, and into a perturbative linear-algebraic setting; the players are thus edge modes should be a stable feature for optical lat- 4 a fixed indefinite flux form in C , an analytic flux-unitary tices with edges. transfer matrix T(?), and an analytic flux-selfadjoint ma- Chris Curtis trix K(?), where K(0) has a non-trivial Jordan block of size Department of Applied Mathematics 3. Joint Work with Aaron Welters, MIT. University of Colorado at Boulder [email protected] Stephen P. Shipman Department of Mathematics Louisiana State University MS53 [email protected] Elastic Cloaking for Civil Structures

The talk will present some mathematical framework for MS54 elastic cloaking and will be followed by some numerical On Forced Turbulence in Physical Scales of 3D NSE illustration and an experiment on seismic metamaterial. We apply the physical scales framework to prove that Kol- Such concepts have been developed in conjunction with the 3 earlier works by Milton, Briane Willis (cloaking preserv- mogorov Dissipation Law ε ∼ U /L implies energy cascade ing the symmetry of the elasticity tensor), Norris (acoustic in the 3D periodic NSE, provided the force is large enough. cloaking with pentamode materials) and Brun, Guenneau, We also obtain restrictive scaling properties of the solutions Movchan (cloaking with an elasticity tensor without the in this case, and compare our results to the ones obtained minor symmetries). The large-scale seismic test held on a in the classical Fourier framework. soil metamaterial using vibrocompaction probes is the first application of cloaking to civil engineering. Joint work with Radu Dascaliuc S. Enoch, A. Diatta, S. Brˆul´e and E. Javelaud. Oregon State University [email protected] Sebastien Guenneau Institut Fresnel, UMR CNRS 6133 Zoran Gruji´c PD13 Abstracts 105

University of Virginia in this study. In particular we show how suitable boundary [email protected] conditions make the initial and boundary problem mildly dissipative and well-posed.

MS54 Roger M. Temam The Well-posedness Linear Hyperbolic Partial Dif- Inst. for Scientific Comput. and Appl. Math. ferential Equations in a Rectangle Indiana University [email protected] In this article, we consider the linear hyperbolic Initial Boundary Value Problems (IBVP) in a rectangle in both Aimin Huang the constant and variable coe cients cases. We use semi- Indiana University group method instead of Fourier analysis to achieve the [email protected] well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two simple modes, which we call hyperbolic and elliptic modes in the system. MS55 The hyperbolic system in consideration is either symmetric Nonstandard Dispersive Estimates and Long Time or Friedrichs-symmetrizable. We also give some applica- Existence for the 2d Water Wave Problem tions of our results.

Aimin Huang In this talk, I will presents recent results on the relationship Indiana University Bloomington between the decay of a solution to the linearized water Department of Mathematics wave problem and its initial data and implications for the [email protected] long time existence of gravity waves. Certain new decay bounds for a class of 1D dispersive equations (including the linearized water wave) display a surprising growth factor, Roger M. Temam which we show is sharp. A further exploration leads to a Inst. for Scientific Comput. and Appl. Math. result relating singularities of the initial data at the origin Indiana University in Fourier frequency to the regularity of solutions to these [email protected] dispersive equations.

Jennifer Beichman MS54 University of Michigan An Appropriate Notion of Attractor for Semi- [email protected] Dissipative Equations

The large-time behavior of a dissipative system can often Sijue Wu be understood by studying its global attractor, which can University of Michigan at Ann Arbor contain deep information about its underlying structure. [email protected] We will show that the notion of attractor can be extended to systems with only partial dissipation, and apply the con- cept to fluid dynamics. We will see that this generalized MS55 attractor not only has a rich structure, but also encodes a Finite Time Blow-Up Versus Global Existence of wealth of turbulent phenomena in a single object. Solutions for a 1D Fluid Model

Animikh Biswas We discuss the finite time blow-up versus global well- University of North Carolina posedness for a 1D fluid model. This model may be de- at Charlotte rived by reducing the Navier-Stokes equations, the prim- [email protected] itive equations, Burgers equations and many other fluid models. This is a joint work with K. Nakanishi and E. S. Ciprian Foias Titi. Texas A & M, Indiana University Slim Ibrahim [email protected] University of Victoria [email protected] Adam Larios Department of Mathematics MS55 Texas A&M University [email protected] Scattering for Small Solutions of NLS with Sub- critical Nonlinearity

MS54 We prove global existence and scattering for small scale Existence and Uniqueness of Solutions for the In- invariant data for a model close to the standard cubic NLS. viscid Shallow Water Equations The proof is based on the method of normal forms. This is joint work with Vladimir Georgiev, University of Pisa. Motivated by the equations of the large scale oceans and at- mosphere (primitive equations), we discuss the issue of ex- Atanas Stefanov istence and uniqueness of solutions for the linearized shal- University of Kansas low water equations in space dimension two in a rectangle. [email protected] We also study the nonlinear shallow water equations in some subcritical and supercritical situations. The choice of the suitable boundary conditions and the fact that the MS55 domain (rectangle) is not smooth, are two essential issues Justifying the Modulation Approximation of the 106 PD13 Abstracts

Full Water Wave Problem Domain

In this joint work with Sijue Wu (U. Mich.), we consider In this work we study nonlinear parabolic equations of the solutions to the inviscid infinite depth water wave problem p-Laplace type degenerating on a part of the domain. The neglecting surface tension which are to leading order wave degeneration is controlled by a small parameter. For the packets with small amplitude and slow spatial decay that fixed value of this parameter, the results we discuss fall are balanced. It has been known formally since Zakharov within the scope of E. DiBenedetto’s results. We obtain in the 60s that such solutions have modulations that evolve estimates uniform with respect to ”epsilon” , thus extend- according to a focusing cubic NLS equation on very slow ing the results, obtained earlier by Yu.A. Alkhutov and time scales. Justifying this rigorously is a real problem, V.A. Liskevich in the linear case. since standard existence results for the water wave prob- lem do not yield solutions which exist for long enough to Mikhail D. Surnachev see the NLS dynamics. Nonetheless, given initial data close Insitute of applied mathematics, to a wave packet in a suitable L2 Sobolev space, we show Russion Academy of Science, that there exists a unique solution to the water wave prob- [email protected] lem which is close to the formal approximation for appro- priately long times. This is done by applying the energy Mikhail D. Surnachev method to formulations of the evolution equations for the Keldysh Institute of Applied Mathematics water wave problem in 2D and 3D developed by Sijue Wu Russion Academy of Science with quadratic nonlinearities that are either absent or in [email protected] some sense mild.

Nathan Totz MS56 Department of Mathematics, Duke University Some Recent Progress on the Study of Behavior of [email protected] Solutions of Degenerate Viscous Hamilton-Jacobi Equations

MS56 I will give a summary on a new approach to study large time behavior of solutions of degenerate viscous Hamilton- On a Chemotaxis Model with Saturated Chemo- Jacobi equations. Then I will describe its applications to tactic Flux the usual cases, the obstacle problems, and the fully non- linear cases. We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) sys- Hung Tran tem. Our modification is based on a fundamental physical University of Chicago property of the chemotactic flux function – its bounded- [email protected] ness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradi- MS57 ent tends to infinity the cell velocity saturates. Unlike the Stationary States and Asymptotic Behaviour of original PKS system, the solutions of the modified model Aggregation Models with Nonlinear Local Repul- do not blow up. After obtaining local and global existence sion results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; We consider a continuum aggregation model with nonlin- we also study the stability of bifurcating steady states. Fi- ear local repulsion given by a degenerate power-law diffu- nally, we numerically verify these analytical results. Joint sion with general exponent. The steady states and their work with A. Kurganov, X. Wang and Y. Wu properties in one dimension are studied both analytically and numerically, suggesting that the quadratic diffusion is Alina Chertock a critical case. The focus is on finite-size, monotone and North Carolina State University compactly supported equilibria. We also investigate nu- Department of Mathematics merically the long time asymptotics of the model by simu- [email protected] lations of the evolution equation. Issues such as metasta- bility and local/ global stability are studied in connection to the gradient flow formulation of the model. MS56 G-Convergence of Elliptic Operators with Nonstan- Martin Burger dard Growth University of Muenster Muenster, Germany We consider a class of monotone elliptic operators with [email protected] nonstandard growth condition. We present a preliminary result on G-compactness under certain additional technical Razvan Fetecau assumptions. Simon Fraser University [email protected] Alexander Pankov Department of Mathematics Yanghong Huang Morgan State University, Baltimore Imperial College London [email protected] [email protected]

MS56 MS57 Parabolic Equations Degenerating on a Part of the Modeling Selective Local Interactions with Mem- PD13 Abstracts 107

ory [email protected]

In this talk we present our recent results on stochastic par- ticle methods for describing the local interactions between MS58 cyanobacteria. In these models, particles selectively choose Higher-Order Gradient Theories for Nematic Liq- one of their neighbors as the preferred direction of motion. uid Crystals Memory is incorporated into the model by allowing persis- tence. One- and two-dimensional models are studied. We explore the connection between mathematical theories describing defects in nematic liquid crystals and crystalline Doron Levy solids. The parallels between these theories allow us to University of Maryland propose an alternative framework for modeling nematic de- [email protected] fects. Amit Acharya MS57 Civil and Environmental Engineering Carnegie Mellon University Minimization of an Energy Defined via an [email protected] Attractive-Repulsive Interaction Potential Related to Nonlocal Aggregation Models Dmitry Golovaty The University of Akron In this talk I will consider the constrained minimiza- Department of Mathematics tion of an energy defined via an attractive-repulsive in- [email protected] teraction potential. This energy is related to aggrega- tion models given by the equation ρt + ∇·(ρ(−∇K ∗ ρ)) where ρ is the density of the aggregation and K is the MS58 interaction potential. Looking at the energy E(ρ)= K(x − y)ρ(x)ρ(y)dxdy for interaction potentials of the Stability of Radially Symmetric Solution in form K(|x|)=|x|q/q −|x|p/p in the parameter regime Landau-De Gennes Theory of Liquid Crystals −N

MS57 Lei Zhang Cheng Title Not Available at Time of Publication School of Sciences Indiana University Kokomo Abstract not available at time of publication. [email protected]

James von Brecht MS59 University of California, Los Angeles [email protected] An Interface Capturing Method for Simulating Dy- namic Gel-Fluid Interaction

MS58 Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid sol- Mathematical Models for a Soap Opera vent (water). In some applications, free boundaries sep- arate regions of gel and regions of pure solvent, resulting In this talk, I will put forth a new energy for the descrip- in a degenerate system. To overcome this difficulty, we tion of large distortions of lipid bilayers, which constitute develop a simple interface-capturing method that consists the base component of cell membranes. This energy ac- of a regularization procedure to solve the two-fluid model counts for the soap-like liquid crystalline nature of lipid bi- equations describing gels composed of two viscous fluids, layers and the coupling between the deformations of lipid in the situation in which the volume fraction of one fluid molecules and layers. The analogies between soap-like liq- vanishes in part of the domain. uid crystals, with an infinite number of layers, and lipid bilayers, with only two layers, will be further discussed. Jian Du Department of Mathematics Raffaella De Vita Florida Institute of Technology Engineering Science and Mechanics jdu@fit.edu Virginia Tech [email protected] Robert D. Guy Mathematics Department Iain W. Stewart University of California Davis University of Strathclyde [email protected] 108 PD13 Abstracts

Aaron L. Fogelson tion of an incompressible, generalized Newtonian fluid with University of Utah a linearly elastic Koiter shell whose motion is restricted [email protected] to transverse displacements. The moving middle surface of the shell constitutes the mathematical boundary of the Grady B. Wright three-dimensional fluid domain. I show that weak solu- Department of Mathematics tions exist as long as one can exclude selfintersections of Boise State University, Boise ID the shell. [email protected] Daniel Lengeler University of Regenesburg James P. Keener [email protected] University of Utah [email protected] MS60 MS59 A Constructive Existence Proof for a Moving Boundary Fluid-Multi-Layered Structure Interac- Toward a Theory of Mutual Attractions and Re- tion Problem pulsions of Floating Objects We consider a nonlinear, unsteady, moving-boundary prob- The horizontal force experienced by two infinite parallel lem of parabolic-hyperbolic-hyperbolic type modeling the plates of possibly differing solid materials, situated verti- interaction between a Newtonian fluid and an elastic struc- cally and partially immersed in an infinite fluid bath under ture composed of two layers: a thin layer modeled by the gravity, is characterized in the context of classical surface 1D wave equation, and a thick layer modeled by the 2D tension theory. Varying kinds of behavior are encountered, equations of linear elasticity. The thin layer is in contact depending on contact angles and on plate separation. The with the fluid, thereby serving as a fluid-structure inter- particular case, in which the two contact angles on the face with mass. We will present a constructive proof of an plate sides directly facing each other are supplementary, existence of a weak solution. leads to a striking limiting instability. Boris Muha Robert Finn Department of Mathematics, Faculty of Science Stanford University University of Zagreb fi[email protected] [email protected]

MS59 Suncica Canic Department of Mathematics Coupling Between Fluid Equations and Fabric Sur- University of Houston face Model [email protected] Abstract not available at time of publication. MS60 Xiaolin Li Department of Applied Math and Stat The Motion of the Rigid Body with Collisions in a SUNY at Stony Brook Bounded Domain. Global Solvability Result [email protected] We consider the problem of motion of a rigid body in an in- compressible viscous fluid, filling a bounded domain. This MS59 problem was studied by many autors. They considered classical non-slip boundary conditions, which gave them Compressible Navier-Stokes System with Temper- very PARADOXICAL result of no collisions of the body ature Dependent Viscosity and Heat Conductivity with the boundary of the domain. In this work we study The full system of compressible Navier -Stokes equations when Navier slip conditions are prescribed on the boundary is the model for the compressible fluids with heat transfer. of the body (instead of non-slip conditions).We prove for Particularly, Chapman-Enskog theory suggests the viscos- this model the GLOBAL existence of weak solution, which ity and heat conductivity of a fluid are powers of the tem- permits COLLISIONS with the boundary of the domain.It perature. Feireisl’s type weak solution is constructed for is joint work with N. Chemetov. this model. Sarka Necasova Mathematical Institute of Academy of Sciences weizhe zhang Czech Republic Georgia Institute of Technology [email protected] [email protected]

Ronghua Pan MS60 Georgia Tech. Global Existence and Nonlinear Stability for Fluid- [email protected] structure Problems

I will discuss both viscous an inviscid fluid-structure inter- MS60 action problems which couple either the Navier-Stokes or Weak Solutions for an Incompressible, Generalized Euler equations to fully nonlinear thin elastic shell struc- Newtonian Fluid Interacting with a Linearly Elas- tures. tic Koiter Shell Steve Shkoller In this talk I will present an existence result for the interac- Univiversity of California, Davis PD13 Abstracts 109

Department of Mathematics Spaces in Theory of PDE [email protected] We are interested in the existence of solutions to strongly nonlinear partial differential equations, which come from MS61 dynamics of non-Newtonian fluids of a nonstandard rhe- Global Existence of Solutions to Fluid Structure ology and abstract theory of parabolic equations. The Model with Moving Interface and Boundary Dissi- nonlinear highest order term is monotone and coerciv- pation ity/growth conditions are given by a general convex func- tion. We study both types of nonlinearity: sub- and super- Fluid structure interaction model comprising of a N-S linear. Such a formulation requires a general framework, equation coupled to a 3-D system of dynamic elasticity therefore we work with non-reflexive and non-separable Or- is considered. The coupling occurs on an interface be- licz and Musielak-Orlicz spaces. tween the two media and the model accounts for a free boundary of the solid which is moving within the fluid. Aneta Wroblewska-Kaminska The moving interface is subjected to a frictional damp- Institute of Mathematics PAS ing. It is proved that solutions corresponding to sufficiently [email protected] small initial data are unique and they exist globally. This result is established with minimal regularity assumptions imposed on the data and without any interior dissipation MS62 imposed on the structure. The only source of added dis- Coupled Systems of Nonlinear Dispersive Wave sipation is the interface (boundary) damping. This is, to Equations our best knowledge, the first global existence result for fluid structure interaction in the 3 dimensional space. This is Systems of wave equations which are coupled through non- joint work with Michaela Ignatova (Stanford University), linear and dispersive effects arise in a number of applica- Igor Kukavica (University of Southern California) and Am- tions. Unlike single equations, systems throw up far more jad Tuffaha (The Petrolium Institute, Abu-Dhabi, UAE) diverse dynamics. This lecture will discuss a few of these in the context of coupled Korteweg-de Vries systems. Irena M. Lasiecka University of Virginia Jerry Bona [email protected] University of Illinois at Chicago [email protected]

MS61 Steady Compressible Navier-Stokes-Fourier Sys- MS62 tem with Temperature Dependent Viscosities The Regularity Criteria for 3D Magneto- hydrodynamic System We consider the system of partial differential equations de- scribing the steady flow of a compressible heat conducting This work studies the regularity problem for the 3D Newtonian fluid in a bounded three-dimensional domain. Magneto-Hydrodynamic (MHD) system. We apply an ap- We assume the pressure law of the form p(, ϑ) ∼ γ + ϑ proach of splitting the dissipation wavenumber combined with γ>1 and the viscosities ∼ (1 + ϑ)α, α ∈ [0, 1]. We with an estimate of the energy flux to obtain a new reg- show the existence of a weak or variational entropy solu- ularity criterion. This splitting approach was originally tion for the above model in dependence on the parameters introduced by Cheskidov and Shvydkoy. The regularity γ, α and the form of the heat flux. criterion presented here is weaker than the existing criteria for the 3D MHD system. Milan Pokorny Charles University in Prague Alexey Cheskidov [email protected]ff.cuni.cz University of Illinois Chicago [email protected]

MS61 Mimi Dai,MimiDai Generalized Stokes System and Implicit Constitu- University of Colorado Boulder tive Relations Department of Applied Mathematics [email protected], [email protected] We are interested in flows of incompressible fluids in which the deviatoric part of the Cauchy stress and the symmetric part of the velocity gradient are related through an implicit MS62 equation. Although we restrict ourselves to responses char- The Atmospheric Equations of Water Vapor with acterized by a maximal monotone graph, the structure is Saturation rich enough to include power-law type fluids, stress power- law fluids, Bingham and Herschel-Bulkley fluids, etc. We In this lecture we will recall the atmospheric equations of study Stokes-like problems wherein the inertial effects are water vapor with saturation, appearing as a nonlinear mul- neglected. tivalued system of partial differential equations. We will address the issues of the definition of the solutions, and Agnieszka Swierczewska-Gwiazda the questions of existence, uniqueness and regularity of so- University of Warsaw lutions. [email protected] Roger M. Temam Inst. for Scientific Comput. and Appl. Math. MS61 Indiana University Non-Newtonian Fluids - Applications of Orlicz [email protected] 110 PD13 Abstracts

Michele Coti Zelati University of Pittsburgh Indiana University, Bloomington [email protected] [email protected] Dehua Wang University of Pittsburgh MS62 Department of Mathematics Averaging and Spectral Properties of the 2D [email protected] Advection-diffusion Equation in the Semi-classical Limit for Vanishing Diffusivity MS63 Abstract not available at time of publication. Shock-free Transonic Solutions to the Steady Euler Jesenko Vukadinovic System Department of Mathematics College of Staten Island, CUNY We focus on the Ringleb exact solutions and build a family [email protected] of solutions nearby for the steady Euler system in two space dimensions. For large amplitude flows, we implant a shock where the Ringleb-like solution would form a cusp. MS63 Relative Entropy in Hyperbolic Relaxation of Bal- Yuxi Zheng ance Laws The Pennsylvania State University [email protected] We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the Tianyou Zhang analysis lies on the construction of suitable relaxation sys- The Pennsylvania State University tems and the derivation of a relative entropy identity. We university park provide a direct proof of convergence in the smooth regime [email protected] for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges MS64 and requires delicate treatment. Our analysis is in the spirit and continuity of the previous work of A. Tzavaras Long Time Influence of Small Perturbations (Comm. Math. Sci. 2005) for systems of hyperbolic con- servation laws. I will consider deterministic and stochastic perturbations of dynamical systems and stochastic processes with mul- Alexey Miroshnikov tiple invariant measures. Under certain conditions, the University of Massachusetts Amherst perturbed system in this case has a fast and a slow com- [email protected] ponents. The slow component, which actually determines the long time behavior is a motion in the cone of invariant Konstantina Trivisa measures. The limiting slow motion can be described using University of Maryland the large deviation theory or by a modified averaging prin- Department of Mathematics ciple. I will demonstrate this approach in the case of the [email protected] Landau-Lifshitz equation and for diffusion and wave fronts in narrow random channels.

MS63 Mark I. Freidlin On Multiphase Flow Models: Global Existence of University of Maryland Weak Solutions [email protected]

This work is part of a research program whose objective is the analysis of nonlinear systems via the construction MS64 of suitable approximations and the establishment of ap- Asymptotic Behaviors for the Random Schrdinger propriate compactness of the approximate solutions. Well- Equation with Long-Range Correlations posedness results are presented for a class of such nonlinear systems. Wave propagation in random media with long-range cor- relations was recently stimulated by data collections show- Konstantina Trivisa ing that propagation media can exhibit long-range correla- University of Maryland tion effects. Under the forward scattering approximation, Department of Mathematics the wave equation can be reduced to a random Schrdinger [email protected] equation. In this talk, we will study the Schrdinger equa- tion with a slowly decorrelating random potential, and show that the energy density of the wave function exhibits MS63 some anomalous diffusion phenomena. Global Weak Solutions to the Inhomogeneous Navier-Stokes-Vlasov Equations Christophe Gomez Aix-Marseille University In this talk, we are going to talk about the global weak [email protected] solutions to the coupled system by the incompressible in- homogeneous Navier-Stokes equations and Vlasov equation in the three dimensional space. MS64 Cheng Yu Linear Relaxation to Planar Travelling Waves in PD13 Abstracts 111

Inertial Confinement Fusion this leads to topical effects such as cloaking/ invisibility, flat lensing, negative refraction and to inducing directional The models of ICF usually couple hydrodynamical features behaviour of the waves within a structure. A general the- in plasmas with reaction-diffusion equations for the tem- ory will be described and applications to continuum, dis- perature. One of the main challenges is the non-linear crete and frame lattice structures will be outlined. The heat diffusion of the porous media type, i-e ∇·(λ∇T ) results and methodology are confirmed versus various illus- −1 with λ = λ(T )=T m for some conductivity exponent trative exact/ numerical calculations showing that theory m>1. In this talk we will consider an approximate non- captures, for instance, all angle negative refraction, ultra linear parabolic equation for which there exist planar wave refraction and localised defect modes. solutions having a physical meaning. Stability of these waves with respect to wrinkled perturbations is crucial Richard Craster for the ICF process to be effective. We show here that Department of Mathematics a diffusion-induced mechanism forces relevant perturba- Imperial College London tions to become planar exponentially fast in the long time [email protected] regime, and also derive an asymptotic dispersion relation relating this rate of relaxation to some small temperature MS65 ratio ε = Tmin/Tmax → 0. In this limit the problem de- generates into a free boundary one, and the proof involves Quantum Graph Spectral Analysis of Graphyne a rescaled singular principal eigenvalue problem. Structures

L`eonard Monsaingeon We will discuss spectral features (band-gap structure, Carnegie Mellon University Dirac points, etc.) of various graphene and graphyne struc- [email protected] tures. This is a joint work with N. T. Do and O. Post. Peter Kuchment MS64 Department of Mathematics, Texas A&M University USA Classical Limit for a System of Random Non-Linear [email protected] Schr¨odinger Equations

This work is concerned with the semi-classical analysis MS65 of mixed state solutions to a Schr¨odinger-Poisson equa- Geometric Optimization of Laplace-Beltrami tion perturbed by a random potential with weak ampli- Eigenvalues tude and fast oscillations in time and space. We show that the Wigner transform of the density matrix con- Let (M,g) be a compact Riemannian surface and denote by verges weakly and in probability to solutions to a Vlasov- lambda (M,g)thek-th eigenvalue of the Laplace-Beltrami Poisson-Boltzmann equation with a linear collision kernel. k operator, Δg. In this talk, we consider the dependence of A consequence of this result is that a smooth non-linearity the mapping (M,g) → lambdak(M,g). In particular, we such as the Poisson potential (repulsive or attractive) does propose a computational method for solving the eigenvalue not change the statistical stability property of the Wigner optimization problem of maximizing lambdak(M,g)overa transform observed in linear problems. We obtain in addi- conformal class [g]offixedvolume,vol(M,g) = 1. We also tion that the local density and current are self-averaging, consider the more general problem of letting M vary over which is of importance for some imaging problems in ran- all orientable compact surfaces of fixed genus. This is joint dom media. The proof brings together the martingale work with Chiu-Yen Kao and Rongjie Lai. method for stochastic equations with compactness tech- niques for non-linear PDE in a semi-classical regime. It Braxton Osting partly relies on the derivation of an energy estimate that is University of California, Los Angeles straightforward in a deterministic setting but requires the [email protected] use of a martingale formulation and well-chosen perturbed test functions in the random context. MS65 Olivier Pinaud Exponential Asymptotics for Line Solitons in Two- Colorado State University Dimensional Periodic Potentials [email protected] As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal con- MS65 tinuous waves, an analytical theory is presented for line High Frequency Homogenization: Connecting the solitons, whose envelope varies only along one direction, Microstructure to the Macroscale in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on It is highly desirable to be able to create continuum equa- a certain recurrence relation for going beyond all orders tions that embed a known microstructure through effective of the usual multiscale perturbation expansion, a key step or averaged quantities such as wavespeeds or shear moduli. of the exponential asymptotics procedure previously used The methodology for achieving this at low frequencies and for solitons in one-dimensional problems. Instead, we pro- for waves long relative to a microstructure is well-known pose a more direct treatment which not only overcomes and such static or quasi-static theories are well developed. the recurrence-relation limitation, but also simplifies the However, at high frequencies the multiple scattering by the exponential asymptotics process. Using this modified tech- elements of the microstructure, which is now of a similar nique, we show that line solitons with any rational line scale to the wavelength, requires a dynamic homogeniza- slopes bifurcate out from every Bloch-band edge; and for tion theory. Many interesting features of, say, periodic each rational slope, two line-soliton families exist. Fur- media: band gaps, localization etc occur at these higher thermore, line solitons can bifurcate from interior points of frequencies. The materials exhibit effective anisotropy and Bloch bands as well, but such line solitons exist only for 112 PD13 Abstracts

a couple of special line angles due to resonance with the Benard Convection Bloch bands. In addition, we show that a countable set of multiline-soliton bound states can be constructed an- We discuss the scaling of the rate of heat transport in the alytically. The analytical predictions are compared with vertical direction, quantified as the Nusselt number, within numerical results for both symmetric and asymmetric po- the Rayleigh-Benard model for convection. How the Nus- tentials, and good agreement is obtained. This is joint work selt number scales in terms of the parameters of the system, with Sean Nixon and T.R. Akylas. Rayleigh number and Prandtl number, is an outstanding open problem in classical physics. Both the case of finite Jianke Yang Prandtl number, and the case of infinite Prandtl number Department of Mathematics and Statistics will be discussed. Classical no-slip boundary condition as University of Vermont well as geophysically relevant free-slip boundary conditions [email protected] will be investigated. The singular limit of infinite Prandtl number limit will also be discussed both in terms of finite- time trajectory convergence and in terms of convergence of MS66 long time statistical properties. On the Temperature Variance Cascade in Surface Quasi-geostrophic Turbulence Xiaoming Wang Using a multi-scale, dynamical averaging methodology, we FSU rigorously establish the existence of a temperature variance [email protected] cascade across a range of physical scales in surface quasi- geostrophic turbulence. MS67 Zachary Bradshaw, Zoran Grujic University of Virginia Two Dimensional Water Waves In Holomorphic [email protected], [email protected] Coordinates This article is concerned with the in nite bottom water MS66 wave equation in two space dimensions. We consider this On Dynamics of Fluid Flows in Porous Media problem expressed in position-velocity potential holomor- phic coordinates. Viewing this problem as a quasilinear We study the degenerate parabolic equation with time- dispersive equation, we establish two results: (i) local well- dependent flux boundary condition for generalized Forch- posedness in Sobolev spaces, and (ii) almost global solu- heimer (non-Darcy) flows of slightly compressible fluids in tions for small localized data. Neither of these results are porous media. The solution is estimated, particularly for new; they have been recently obtained by Alazard-Burq- large time, in L∞-norm, W 1,r-norm for r ≥ 1, and W 2,2−δ- Zuily, respectively by Wu using different coordinates and norm for δ>0. The L∞-estimates of the solution’s time methods. Instead our goal is improve the understanding of derivative are also obtained. We establish the continu- this problem by providing a single setting for both results, ous dependence of the solution on initial and boundary as well as new, somewhat simpler proofs. data. The De Giorgi and Ladyzhenskaya-Uraltseva itera- tion techniques are combined with uniform Gronwall-type Mihaela Ifrim estimates, specific monotonicity properties and suitable McMaster University parabolic Sobolev embeddings. This is joint work with [email protected] Thinh Kieu and Tuoc Phan. Daniel Tataru Luan Hoang University of California, Berkeley Texas Tech University [email protected] Department of Mathematics and Statistics [email protected] John Hunter University of California, Davis MS66 [email protected] On the Existence of Pullback Attractors for the 3D Navier-Stokes Equations MS67 The existence of attractors for the 2D Navier-Stokes equa- On the Spectral Stability of Kinks in Some PT- tions has been thoroughly researched. In the nonau- symmetric Nonlinear Equations tonomous case, when the uniform attractor exists, its sec- tions have attraction properties which are in a pullback sense, when the initial conditions go to −∞. I will present We consider the introduction of PT-symmetric terms in an abstract framework for studying the existence of pull- the context of classical Klein-Gordon field theories and ex- back attractors for the nonautonomous 3D Navier-Stokes plore their implications on the spectral stability of coher- − equations, where uniqueness of solutions is yet unresolved. ent structures. These models take the form utt uxx + εγ(x)ut + f(u)=0,whereγ(x) is an odd function. In par- ticular, for the corresponding static kink solutions of the sine-Gordon equation and the Klein-Gordon equation with Landon Kavlie 3 University of Illinois at Chicago nonlinearity f(u)=2(u − u) we prove spectral stability [email protected] for all small enough ε.

Milena Stanislavova MS66 University of Kansas, Lawrence Rate of Vertical Heat Transport in Rayleigh- Department of Mathematics PD13 Abstracts 113

[email protected] Texas Tech [email protected]

MS67 Yalchin Efendiev Global Exact Controllability of Semilinear Plate Dept of Mathematics Equations Texas A&M University [email protected] I will discuss some exact controllability results for semi- linear hyperbolic PDE’s. In early 1990’s it was demon- strated by Lasiecka and Triggiani that the global exact MS68 interior and boundary controllability holds for many wave Two-phase Generalized Forchheimer Flows and plate models with nonlinear source terms, provided the source feedback maps are globally Lipschitz. In the We study two-phase generalized Forchheimer flows of in- special case of localized interior controls these theorems compressible fluids in porous media. A family of non- have been extended by Li and Zhang to sources that are constant symmetric steady states are obtained. We prove at most ”logarithmically superlinear” at infinity. I will that they are linearly asymptotically stable in case of present some nascent developments concerning global ex- bounded domains. For unbounded domains, we obtain the act controllability for plate equations with more general stability results and the behavior of the solutions of the lin- polynomially bounded sources. earized system at space-infinity. The lemmas of growth of Landis-type are established with explicit barrier functions Matthias Eller using the particular structure of the steady states. This is Georgetwon University joint work with Luan Hoang and Akif Ibragimov. Department of Mathematics [email protected] Thinh Kieu Texas Tech University Daniel Toundykov [email protected] University of Nebraska-Lincoln [email protected] Luan Hoang Texas Tech University MS68 Department of Mathematics and Statistics [email protected] Upscaling of Forchheimer Flows

In this work we propose upscaling method for nonlin- Akif Ibragimov ear Forchheimer flow in highly heterogeneous porous me- Texas Tech dia. The generalized Forchheimer law is considered for in- [email protected] compressible and slightly-compressible single-phase flows. We use recently developed analytical results by E.Aulisa, L.Bloshaskaya, L.Hoang, and A.Ibragimov [J. Math. Phys. MS68 50, 103102 (2009)] and formulate the resulting system in Complementary Media in Metamaterials terms of a degenerate nonlinear flow equation for the pres- sure with the nonlinearity depending on the pressure gradi- In this talk, I will discuss the frame work of complementary ent. The coarse scale parameters for the steady state prob- media in metamaterials. Resonant and localized resonant lem are determined so that the volumetric average of ve- phenomena will be discussed. Applications in cloaking will locity of the flow in the domain on fine scale and on coarse be given. scale are close. A flow-based coarsening approach is used, Hoai-Minh Nguyen where the equivalent permeability tensor is first evaluated University of Minnesota following streamline methods for linear cases, and modified [email protected] in order to take into account the nonlinear effects. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities MS68 in the domain of computation. Direct numerical computa- tions for average velocity and productivity index justify the On Passage to the Limit in Nonlinear Elliptic and usage of the coarse scale parameters obtained for the spe- Parabolic Equations cial steady state case in the fully transient problem. For Passing to the limit in nonlinear terms one encounters the nonlinear case analytical upscaling formulas in stratified fundamental problem of ”weak convergence of fluxes to domain are obtained. Numerical results were compared to a flux”. The term ”flux” refers to the vector A(x, ∇u ) these analytical formulas and proved to be highly accu- ε which stands in the elliptic equation under the divergence rate. This is the joint work with E.Aulisa, A.Ibragimov sign. Usually one has the weak convergence u uin and Y.Efendiev. ε some Sobolev space and the weak convergence of the fluxes ∇ Lidia Bloshanskaya A(x, uε) zin some Lebesgue space. This leads to the problem of the equality z = A(x, ∇u). Let us state one re- SUNY at New Paltz, Department of Mathematics 1,α [email protected] sult in this direction. Assume that 1. uε uin W (Ω), α>1; 2. The symbol A is monotone:

Eugenio Aulisa (A(x, ξ) − A(x, η)) · (ξ − η) ≥ 0; Department of Mathematics and Statistics. Texas Tech University 3. divA(x, ∇u ) = 0 in the sense of distributions, [email protected] ε

  β A ≡ A(x, ∇u ) zin Lβ (Ω),β= ; Akif Ibragimov ε ε β − 1 114 PD13 Abstracts

4. 1 <α≤ β<α∗, with Heterogeneities and Boundaries  α(N−1) − −1− for α

Abstract not available at time of publication. MS70 David T. Uminsky The Existence of Global Solutions to the Ericksen- University of San Francisco Leslie System in R2 Department of Mathematics [email protected] In this talk, I will discuss the regularity and existence of global weak solution of the general Ericksen-Leslie system in dimension two under a set of condition of Leslie coef- MS69 ficients, which assure that the total energy is decreasing Nonlocal Interaction Equations in Environments under the flow. This is based on joint work with Jinrui PD13 Abstracts 115

Huang and Fanghua Lin. [email protected]

Changyou Wang department of mathematics MS71 university of kentucky Monotonicity Formulas in the Signorini Problem [email protected] We will discuss some new monotonicity formulas which play a crucial role in the study of the Signorini problem. MS70 Nicola Garofalo Models for Active Liquid Crystals and Their Ap- Universita’ di Padova plications to Complex Biological Systems [email protected]

Abstract not available at time of publication. MS71 Qi Wang On a Thermodynamically Consistent Stefan Prob- University of South Carolina lem with Variable Surface [email protected] A thermodynamically consistent two-phase Stefan prob- lem with temperature-dependent surface tension and with MS71 or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined The Wiener Test for the Removability of the Loga- state manifolds. If a solution does not exhibit singularities, rithmic Singularity for the Elliptic PDEs with Mea- it is proved that it exists globally in time and converges to- surable Coefficients, and Its Measure-Theoretical, wards an equilibrium of the problem. In addition, stability Topological, and Probabilistic Consequences and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are un- This talk introduces a notion of log-regularity (or log- stable if surface heat capacity is small; however, if kinetic irregularity) of the boundary point ζ (possibly ζ = ∞)of undercooling is absent, they are stable if surface heat ca- arbitrary open subset Ω of the Greenian deleted neigbor- pacity is sufficiently large. (Joint work with J. Pr¨uss and hood of ζ in R2 concerning second order uniformly elliptic M. Wilke). equations with bounded and measurable coefficients, ac- cording as whether the log-harmonic measure of ζ is null Gieri Simonett (or positive). A necessary and sufficient condition for the Vanderbilt University removability of the logarithmic singularity, that is to say [email protected] for existence of a unique solution to the Dirichlet problem in Ω in a class of solutions with possibly logarithmic growth rate at ζ, is established in terms of the Wiener test for the MS72 log-regularity of ζ. From the topological point of view, the Modeling Arterial Walls As a Multi-Layered Struc- Wiener test at ζ presents minimal thinness criteria of sets ture, and Their Interaction with Pulsatile Blood near ζ in minimal fine topology. Precisely, the open set Ω Flow is a deleted neigbourhood of ζ in minimal fine topology if and only if ζ is log-irregular. From the probabilistic point We study the interaction between an incompressible, vis- of view Wiener test presents asymptotic law for the log- cous fluid and a multilayered structure, which consists of Brownian motion near ζ conditioned on the logarithmic a thin and a thick elastic layer. The thin layer is modeled kernel with pole at ζ. using the linearly elastic Koiter shell model, while the thick layer is modeled using the 2D equations of linear elastic- Ugur G. Abdulla ity. The system is coupled via the kinematic and dynamic Department of Mathematical Sciences coupling conditions. We present a loosely-coupled numer- Florida Institute of Technology ical algorithm, and stability results. The results will be abdulla@fit.edu supported with numerical examples. Martina Bukac University of Pittsburgh MS71 Department of Mathematics The Elliptic Obstacle Problem When the Coeffi- [email protected] cients Are Only in Vmo Boris Muha We study the elliptic obstacle problem in both divergence Department of Mathematics, Faculty of Science and nondivergence form. In both cases, although we can University of Zagreb show a measure theoretic version of Caffarelli’s Theorem [email protected] from his famous 1977 Acta paper, we can also give exam- ples of nonuniqueness in blowup limits. Suncica Canic Department of Mathematics Ivan Blank, Zheng Hao University of Houston Kansas State University [email protected] [email protected], [email protected]

Kubrom Teka MS72 SUNY Oswego Validation of An Open Source Framework for the 116 PD13 Abstracts

Simulation of Blood Flow in Deformable Vessels general framework capable to accommodate strongly and loosely coupled schemes. We apply it to design FSI algo- Wediscussinthevalidationofanopensourceframe- rithms for multilayered poroelastic arteries. We will com- work for the solution of problems arising in hemodynamics. bine the analysis of the equations with direct simulations The framework is assessed through a numerical benchmark to understand how poroelasticity affects blood flow in ar- dealing with the propagation of a pressure wave in a fluid- teries. filled elastic tube. The computed pressure wave speed and frequency of oscillations, and the axial velocity of the fluid Martina Bukac on the tube axis are close to the values predicted by the University of Pittsburgh analytical solution associated with the benchmark. Department of Mathematics [email protected] Tiziano Passerini Department of Math & CS Ivan Yotov Emory University Univeristy of Pittsburgh [email protected] Department of Mathematics [email protected] Annalisa Quaini Department of Mathematics, University of Houston Rana Zakerzadeh [email protected] Department of Mechanical Engineering and Materials Science Umberto E. Villa University of Pittsburgh Dept. of Mathematics and Computer Science [email protected] Emory University [email protected] Paolo Zunino University of Pittsburgh Alessandro Veneziani Department of Mathematics MathCS, Emory University, Atlanta, GA [email protected] [email protected]

Suncica Canic MS73 Department of Mathematics Singular Limits of Compressible Fluids University of Houston [email protected] We discuss several singular limits arising in the scale anal- ysis of the Navier-Stokes(-Fourier) system describing the MS72 motion of a general compressible, viscous, (and heat con- ducting) fluid. We consider rapidly rotating stratified flu- Reduced Models for Solving Inverse Fluid- ids in the incompressible and/or inviscid limit. The effect Structure Interaction Problems of simultaneous scalings is considered. To this end, we use the concept of relative entropy, together with careful anal- We are interested in solving the data assimilation prob- ysis of certain oscillatory integrals arising in the study of lem: given the displacement of a pipe where an incom- Poincar´ewaves. pressible fluid flows, identify the wall compliance (Young modulus E). We resort to a variational approach. E is re- garded as control variable in the minimization of the mis- Eduard Feireisl match between measured and computed displacement. In Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1 solving this inverse fluid-structure interaction problem, we Czech Republic face formidable computational costs. We resort to reduced [email protected] models based on a Proper Orthogonal Decomposition ap- proach. MS73 Luca Bertagna Compressible Navier-Stokes System with Inflow Dept. Math & CS Condition Emory University [email protected] I will talk about the issue of stability of Poiseuille type flows in regime of compressible Navier-Stokes equations Alessandro Veneziani in a three dimensional finite pipe-like domain. We prove MathCS, Emory University, Atlanta, GA the existence of stationary solutions with inhomogeneous [email protected] Navier slip boundary conditions admitting nontrivial in- flow condition in the vicinity of constructed generic flows. Our techniques are based on an application of a modifi- MS72 cation of the Lagrangian coordinates. Thanks to such ap- Enforcing Interface Conditions for Fsi Problems proach we are able to overcome difficulties coming from hy- Using Nitsche’s Method. Derivation of Explicit perbolicity of the continuity equation, constructing a max- Coupling Strategies for Multilayered Poroelastic imal regularity estimate for a linearized system and apply- Arteries. ing the Banach fixed point theorem. The talk is based on the joint result with Tomasz Piasecki. Nitsche’s method was developed to weakly enforce Dirichlet boundary conditions for PDEs approximated with FEM. Piotr Mucha Recently, it has been applied to the transmission conditions Warsaw University of fluid-structure interaction (FSI) problems. It provides a Mathematics Department PD13 Abstracts 117

[email protected] Olivier Saut Institut de Math´ematiques de Bordeaux, CNRS [email protected] MS73 Well-Posedness of the Primitive Equations of the Thierry Colin Ocean with Continuous Initial Data Institut de Math´ematiques de Bordeaux Institut Polytechnique de Bordeaux We address the well-posedness of the primitive equations of [email protected] the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the equations MS74 thus leading to existence and uniqueness of solutions. On the Euler-Poincare Equation with Non-Zero Dispersion Walter Rusin University of Southern California The Euler-Poincare equation was introduced by Holm, [email protected] Marsden and Ratiu to study ideal fluids with nonlinear dis- persion. It can be viewed as a natural multi-dimensional Walter Rusin generalization of the popular Camassa-Holm equations. I Oklahoma State University will discuss some recent results on the sharp characteri- [email protected] zation of a special family of solutions based on some new inverse-Soblev type inequalities. In particular this settles Igor Kukavica, Yuan Pei, Mohammed Ziane an open problem raised by Chae and Liu. University of Southern California Dong Li [email protected], [email protected], [email protected] Department of Mathematics University of British Columbia [email protected] MS73 Existence Analysis of the Navier-Stokes System for Xinwei Yu, Zhichun Zhai Multicomponent Mixtures University of Alberta [email protected], [email protected] We will present a model of motion of compressible, heat- conducting mixture of arbitrary large number of species. We consider a special form of density-dependent viscosity MS74 coefficients and a singular behaviour of the cold compo- nent of the internal pressure near vacuum. Under these Laminar Boundary Layers in Rayleigh-Benard hypotheses we prove global-in-time existence of weak so- Convection lutions. This result is based on several joint papers with We study Rayleigh-Benard convection in the high- P.B. Mucha and M. Pokorn´y. Rayleigh-number and infinite-Prandtl-number regime. While the dynamics in the bulk are characterized by a Ewelina Zatorska chaotic convective heat flow, the boundary layers at the Institute of Applied Mathematics and Mechanics horizontal container plates are essentially conducting and Warsaw University, Poland thus the fluid is motionless. Consequently, the average [email protected] temperature exhibits a linear profile in the boundary lay- ers. In this talk, I present new rigorous local bounds on MS74 temperature field and fluid velocity in the boundary layers, and show that the temperature profile is indeed essentially Pde Model of Gbm: Insights Into The Shortcom- linear. All results are uniform in the system parameters ings of Anti-Angiogeneic Therapy (e.g. the Rayleigh number) up to logarithmic correction terms. The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncov- Christian Seis ered unusual tumor responses. Here, we derive a new University of Toronto mathematical model that takes into account the ability of [email protected] proliferative cells to become invasive under hypoxic condi- tions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necro- MS75 sis. The model is validated by replicating and justifying Weak Solutions for Pressureless Gas Dynamics and the clinical observation of rebound growth when AA ther- Other Hyperbolic Systems Through Energy Mini- apy is discontinued in some patients. The model is inter- mization rogated to derive fundamental insights in cancer biology and on the clinical and biological effects of AA drugs. In- We introduce a new and very simple method to obtain vasive cells promote tumor growth, which in the long-run weak solutions to some hyperbolic problems, including gas exceeds the effects of angiogenesis alone. Furthermore, AA dynamics in dimension 1 (any gamma between 1 and 3) and drugs increase the fraction of invasive cells in the tumor, pressureless gas dynamics in any dimension. The method which explain progression by FLAIR and the rebound tu- itself simply consists in minimizing the kinetic energy over mor growth when AA is discontinued. the considered time interval with appropriate constraints.

Hassan M. Fathallah-Shaykh University of Alabama at Birmingham Pierre-Emmanuel Jabin [email protected] University of Maryland 118 PD13 Abstracts

[email protected] lar perturbation theory (Fenichel’s theorem and Exchange Lemma), our proof relies on matched asymptotics tech- Jean-Francois Jabir niques and Fredholm properties of differential operators University of Santiago, Chile on suitable Banach spaces (Spectral Flow and Nonlocal [email protected] Exchange Lemma). Gregory Faye Alexis F. Vasseur University of Minnesota, USA University of Texas at Austin [email protected] [email protected]

Amina Amassad Arnd Scheel University of Nice University of Minnesota [email protected] [email protected]

MS75 MS77 Title Not Available at Time of Publication Deterministic Solvers to Describe DG-MOSFET Devices Abstract not available at time of publication. Statistical models are used to describe electron transport Alexis F. Vasseur in semiconductors at a mesoscopic level. The basic model University of Texas at Austin is given by the Boltzmann transport equation for semicon- [email protected] ductors in the semi-classical approximation coupled with Poissons equation, since the electric ?eld is self-consistent due to the electrostatics produced by the electrons and the MS75 dopants in the semiconductor. However, for short devices, Incompressible Limits for Magnetohydrodynamics this mathematical description has to be modified, since the electrons behave as particles along the long dimension, Incompressible limits for magnetohydrodynamics will be and as waves along the short dimension. In these cases, discussed. In particular, the zero Mach limit and the hy- the devices can be modelled by the Boltzmann-Schrdinger- drodynamic limit will be presented. Poisson system. In this talk we show deterministic solvers to describe long and short Double Gate Metal Oxide Semi- Dehua Wang conductor Field E?ect Transistors (DG-MOSFET) devices. University of Pittsburgh Department of Mathematics [email protected] Maria Caceres Universidad de Granada, Spain [email protected] MS75 Convex Entropy, Hopf Bifurcation, and Viscous and Inviscid Shock Stability MS77 We discuss relations between one-dimensional inviscid and Charge Transport in Bulk-Heterojunction Organic viscous stability/bifurcation of shock waves in continuum- Solar Cells: Adaptivity, High-Throughput Com- mechanical systems and existence of a convex entropy. In puting and Optimal Morphologies particular, we show that the equations of gas dynamics admit equations of state satisfying all of the usual assump- Abstract not available at time of publication. tions of an ideal gas, along with thermodynamic stability- Baskar Ganapathysubramanian i.e., existence of a convex entropy- yet for which there oc- Iowa State University cur unstable inviscid shock waves. For general 3x3 systems [email protected] (but not up to now gas dynamics), we give numerical evi- dence showing that viscous shocks can exhibit Hopf bifur- cation to pulsating shock solutions. Our analysis of inviscid MS77 stability in part builds on the analysis of R. Smith charac- terizing uniqueness of gas dynamical Riemann solutions in Multiscale Modeling and Computation of Nano terms of the equation of state of the gas, giving an analo- Optical Responses gous criterion for stability of individual shocks. We introduce a new framework for the multiphysical Kevin Zumbrun modeling and multiscale computation of nano-optical re- Indiana University sponses. The semi-classical theory treats the evolution of USA the electromagnetic field and the motion of the charged [email protected] particles self-consistently by coupling Maxwell equations with Quantum Mechanics. To overcome the numer- ical challenge of solving high dimensional many body MS76 Schrodinger equations involved, we adopt the Time Depen- Coherent Structures of a Nonlocal FitzHugh- dent Current Density Functional Theory (TD-CDFT). In Nagumo Equation the regime of linear responses, this leads to a linear system of equations determining the electromagnetic eld as well We prove the existence of coherent structures for a class of as the current and electron densities simultaneously. A FitzHugh-Nagumo equations with nonlocal diffusion. Un- self-consistent multiscale method is proposed to deal with like the dynamical systems approach via geometric singu- the well separated space scales. Numerical examples are PD13 Abstracts 119

presented to illustrate the resonant condition. geneous Medium

Di Liu I’ll describe the asymptotic behavior of solutions to a Michigan State University reaction-diffusion equation with KPP-type nonlinearity. Department of Mathematics One can interpret the solution in terms of a branching [email protected] Brownian motion. It is well-known that solutions to the Cauchy problem may behave asymptotically like a traveling wave moving with constant speed. On the other hand, for MS77 certain initial data, M. Bramson proved that the solution First-Principles Prediction of Charge and Energy to the Cauchy problem may lag behind the traveling wave Transport in Disordered Semiconductors for Pho- by an amount that grows logarithmically in time. Using tovoltaic Applications PDE arguments, we have extended this statement about the logarithmic delay to the case of periodically varying Abstract not available at time of publication. reaction rate. We also consider situations where the de- lay is larger than logarithmic. This new PDE approach Gang Lu involves the study of the linearized equation with Dirichlet California State University Northtridge condition on a moving boundary. [email protected] Francois Hamel Universit´e d’Aix-Marseille [email protected] MS78 Homogenization for Problems with Oscillatory James Nolen Boundary Conditions Duke University Mathematics Department In this talk I will discuss two new homogenization results, [email protected] one concerning a growing free boundary in a randomly perforated domain (with Inwon Kim) and another deal- ing with a fully nonlinear Neumann problem with periodic Jean-Michel Roquejoffre data (with Russell Schwab). In both instances, the main Universit´e Paul Sabatier result is that the oscillations of the boundary data or of jean-michel.roquejoff[email protected] the random perforations resolve themselves in such a way that the problems converge in the high frequency limit to Lenya Ryzhik a deterministic and homogeneous problem. Although both Stanford University problems are of a very different nature, their broad anal- [email protected] ysis centers on two key themes: the derivation of uniform pointwise estimates and the identification of an effective equation via auxiliary correctors. In the case of the fully MS78 nonlinear Neumann problem, the effective equation at the Sharp Asymptotic Growth Laws of Turbulent boundary can be identified thanks to a new representation Flame Speeds in Cellular Flows by Inviscid formula for nonlinear operators enjoying a global compar- Hamilton-Jacobi Models ison principle. To determine the turbulent flame speed is a fundamental Nestor Guillen problem in turbulent combustion. A specific project is to Department of Mathematics UCLA understand its growth law with respect to the flow inten- [email protected] sity. In this talk, we will present the sharp growth law of turbulent flame speed in two dimensional cellular flows predicted by a model proposed by Majda and Souganidis MS78 in 90’s. This answers an open problem posed in a paper by Embid, Majda and Souganidis. Anomalous Spreading in a System of Coupled Fisher-Kpp Equations Yifeng Yu University of California This talk will concern anomalous spreading in a system of Irvine reaction-diffusion equations. When a single parameter is [email protected] set to zero the system consists of a pair of uncoupled Fisher- KPP equations. Anomalous spreading occurs when one Jack Xin component of the system spreads at a faster speed in the Department of Mathematics coupled system than it does in isolation, while the speed of UCI the second component remains unchanged. We will discuss [email protected] the role of the pointwise Green’s function in anomalous spreading and discuss mechanisms leading to anomalous spreading in both the linear and nonlinear regime. MS79 Landis-type Lemma of Growth in the Layered Matt Holzer Cylinders for Solution of the Parabolic Degenerate School of Mathematics Equation and Applications University of Minnesota [email protected] We will consider a parabolic equation of second order in un- bounded domain with Dirichlet type boundary conditions. Corresponding operator can degenerate at infinity with re- MS78 spect to x and t. Lemma of growth will be proved for class The Logarithmic Delay of Kpp Fronts in a Hetero- of the solution of the equation in layered cylinder. Using 120 PD13 Abstracts

this Lemma we will prove parabolic analog Phragmen´ - differential equations. The regularization and the numeri- Lindelof¨ type principle in x and t directions. cal solution of this system is considered. It is shown that in- corporation of a curvature- dependent surface tension into Akif Ibragimov fracture modeling eliminates the integrable crack-tip stress Texas Tech and strain singularities of order 1/2 present in the classi- [email protected] cal linear fracture mechanics solutions. Directions for the future research in this area will be presented. Luan Hoang Texas Tech University Anna Zemlyanova Department of Mathematics and Statistics Department of Mathematics [email protected] Texas A&M University [email protected] Thinh Kieu Texas Tech University [email protected] MS80 Global Regularity for Some Oldroyd-B Type Mod- els of Viscoelasticity MS79 An Ersatz Existence Theorem for Fully Nonlinear We investigate some critical models for visco-elastic flows Parabolic Equations without Convexity Assump- of Oldroyd-B type in dimension two. We use a transfor- tions mation which exploits the Oldroyd-B structure to prove an L∞ bound on the vorticity which allows us to prove We show that for any uniformly parabolic fully nonlinear global regularity for our systems. This is a joint work with second-order equation with bounded measurable “coeffi- Frederic Rousset cients’ and bounded “free’ term in the whole space or in any cylindrical smooth domain with smooth boundary data Tarek Elghindi one can find an approximating equation which has a contin- Courant Institute uous solution with the first and the second spatial deriva- USA tives under control: bounded in the case of the whole space [email protected] and locally bounded in case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the sec- MS80 ond spatial derivatives of the unknown function. This is different from a previous work of Hongjie Dong and the Title Not Available at Time of Publication author where the modification was done for large values of the unknown function and its spatial derivatives. Abstract not available at time of publication.

Nicolai Krylov Taoufik Hmidi University of Minnesota University of Rennes 1 [email protected] [email protected]

MS79 MS80 Ultrasound-modulated Optical Tomography: A The Numerical Approach of the Singularly Per- Nonlinear Model turbed Problems

Let Ω be an optical body with several obstacles and let In this talk, I will present the numerical solutions of sin- a∗ denote the unknown absorption coefficient of Ω. Such gularly perturbed problems in a circular domain and pro- obstacles can be imaged using the knowledge of a∗.In vide as well approximation schemes, error estimates and this talk, by using near-infrared light to illuminate Ω and numerical simulations. To resolve the oscillations of classi- spherical acoustic waves to purturb Ω, we can establish a cal numerical solutions due to the stiffness of our problem, nonlinear equation for a∗. This equation can be solved by we construct, via boundary layer analysis, the so-called applying the optimal control method. boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space en- Loc Nguyen riched with the boundary layer elements, we obtain an ac- Ecole Normale Superrieure, Paris curate numerical scheme in a quasi-uniform mesh. The talk [email protected] includes a joint work with C.-Y. Jung and R. Temam.

MS79 YoungJoon Hong Indiana University A New Curvature-dependent Surface Tension [email protected] Model in Fracture Mechanics

A new approach to modeling of a brittle fracture based on an extension of a continuum mechanics to the nanoscale MS80 will be considered. The classical Neumann boundary Title Not Available at Time of Publication condition on the fracture surface is augmented with a curvature-dependent surface tension. The model is stud- Abstract not available at time of publication. ied on an example of a plain-strain mixed mode fracture. Using complex variable methods, the mechanical problem Slim Ibrahim is reduced to a system of coupled Cauchy singular integro- University of Victoria PD13 Abstracts 121

[email protected] Florida Institute of Technology jgoldfar@my.fit.edu

MS81 Magic Toplitz Matrix for Mixed Boundary Laplace MS81 Equation and Its Application to Breast Cancer De- Recent Progress on Local Behavior of the Logarith- tection mic Diffusion Equation

Laplace equation naturally arises in modeling of electrical This talk describes some recent progress on the local behav- properties of the media and can be used to recover conduc- ior of the equation ut =Δlnu including a Harnack-type tivity and permittivity inside the body through measure- inequality, L1 form Harnack inequality and local special ment of currents and voltages on the periphery of the body. analyticity. Connection with the porous medium equation m In electrical impedance tomography (EIT), currents are in- ut =Δ(u /m) and stability of all the estimates as m tends jected through the electrodes located on the boundary of to 0 is presented. the body with no current between electrodes which leads to a mixed boundary value problem. The inverse problem on Naian Liao the reconstruction of conductivity and permittivity at each Vanderbilt University point inside the body is multidimensional and ill-posed. [email protected] We suggest that for cancer detection/screening one needs to prove that the breast tissue is non-homogeneous. This Emmanuele DiBenedetto leads to the problem of derivation of the exact solution of Department of Mathematics the Laplace equation with mixed-type boundary conditions Vanderbilt University, Nashville (Robin-type on the electrodes and Neumann-type between [email protected] electrodes). Under simplifying assumption, this solution is derived for the disk that gives rise to Magic Toeplitz Ugo P. Gianazza (circulant) matrix which can be viewed as a Dirichlet-to- Department of Mathematics Neumann map, or in the context of physics as a general- University of Pavia ization of the Ohms law on the disk. It is shown that the [email protected] elements of the Magic Toeplitz matrix can be represented through new special function that is expressed via Fourier integral or series. We apply Magic Toeplitz matrix to de- MS81 tect bad electrode contact on the breasta typical problem Title Not Available at Time of Publication of EIT in a clinical setting. Abstract not available at time of publication. Eugene Demidenko Darthmouth College Jean-Paul Zolesio [email protected] CNRS and INRIA Sophia Antipolis, France [email protected] MS81 On the Optimal Control of Free Boundary Bound- ary Problems for the Second Order Parabolic PDEs MS82 Kinetic Formulation and Regularizing Effects for We develop a new variational formulation of the inverse Degenerate Parabolic Equations Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the In this talk, we present how to obtain regularity of solutions temperature and free boundary. We employ optimal con- to degenerate parabolic equations. The main idea is to trol framework, where boundary heat flux and free bound- identify the renormalization property of solutions from the ary are components of the control vector, and optimality structure of the nonlinearity. We then apply the velocity criteria consists of the minimization of the sum of L2-norm averaging to their underlying kinetic formulations, which declinations from the available measurement of the temper- leads to the regularizing effect of solutions. Our results ature flux on the fixed boundary and available information extend previous regularity statements and provide a new on the phase transition temperature on the free bound- regularity result of renormalized solutions. ary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, Hantaek Bae and is available through measurement with possible error. University of California, Davis It also allows for the development of iterative numerical [email protected] methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved Eitan Tadmor in a fixed region instead of full free boundary problem. University of Maryland We prove well-posedness in Sobolev spaces framework and USA convergence of discrete optimal control problems to the [email protected] original problem both with respect to cost functional and control. MS82 Ugur G. Abdulla Critical Regularity Estimates for Transport Prob- Department of Mathematical Sciences lems Florida Institute of Technology abdulla@fit.edu A new method to obtain regularity for solutions to trans- port equations is introduced. This is applied to problems in Jonathan Goldfarb biology (such as some chemotaxis models) but also to com- 122 PD13 Abstracts

pressible fluid dynamics where it significantly extends the at Charlotte existing existence theory (non monotone pressure terms, [email protected] temperature dependent viscosity... In addition of provid- ing compactness for the macroscopic density, the method also yields explicit, critical regularity estimates. MS83 Nonlinear Diffusion Equation Model of Bacterial Pierre-Emmanuel Jabin Dynamics University of Maryland [email protected] Abstract not available at time of publication.

Mark S. Alber MS82 University of Notre Dame Regularizing Effects on 1D Nonlinear Scalar Con- Department of Computational and Applied Mathematics servation Laws in Fractional Bv Spaces and Stat [email protected] In 1D, fractional BV spaces are introduced to obtain smoothing effects for conservation laws. Theses spaces have trace properties like BV . This is not the case for Sobolev MS83 spaces with similar regularity. For nonlinear degenerate Fully Anisotropic Diffusion Equations and Appli- convex fluxes the optimal smoothing effect is proven. Some cations to Glioma Growth and Wolf Movement hints will be given on the non convex case. The important notion of nonlinear flux will be discussed. Furthermore, Abstract not available at time of publication. the fractional BV regularity implies the maximal regular- ity conjectured by Lions-Perthame-Tadmor in 1994. Thomas J. Hillen University of Alberta Stephane Junca Department of Mathematical and Statistical Sciences Universite de Nice [email protected] France [email protected] MS83 Christian Bourdarias Scale Invariance in Mathematical Models of Bio- Universit´edeSavoie logical Pattern Formation [email protected] Abstract not available at time of publication.

Pierre Castelli Hans G. Othmer Univerist´edeNice,LJAD University of Minnesota [email protected] Department of Mathematics [email protected] Marguerite Gisclon Universit´edeSavoie [email protected] MS83 Local Kinetics of Morphogen Gradients

MS82 Abstract not available at time of publication. Dissipation Vs. Quadratic Nonlinearity: from En- ergy Bound to High-Order Regularizing Effect Stanislav Shvartsman Department of Chemical Engineering We consider a general class of evolutionary PDEs involv- Princeton University ing dissipation (of possibly fractional order), which com- [email protected] petes with quadratic nonlinearities on the regularity of the overall equation. This includes the prototype mod- els of Burgers’ equation, Navier-Stokes equations, the MS84 surface quasi-geostrophic equations and the Keller-Segel Regularity and Uniqueness for a Class of Weak So- model for chemotaxis. Here we establish a Petrowsky lutions to the Hydrodynamic Flow of Nematic Liq- type parabolic estimate of such equations which entail uid Crystals a precise time decay of higher-order Sobolev norms for this class of equations. To this end, we introduce as We establish an -regularity criterion for any weak solution a main new tool, an “infinite order energy functional’, (u, d) to the nematic liquid crystal flow such that (u, ∇d) ∈ p q n n nθ/2 ≥ ≥ − ∗ 2 Lt Lx for some p 2andq n satisfying the condition + E(t):= n ant ( Δ) u( ,t) L , which captures the q ∗ 2 regularizing effect of all higher order derivatives of u( ,t), p = 1. As consequences, we prove the interior smoothness by proving — for a carefully, problem-dependent choice of and uniqueness of any such a solution when p>2and weights {an},thatE(t) is non-increasing in time. q>n.

Eitan Tadmor Tao Huang, Changyou Wang University of Maryland University of Kentucky USA [email protected], [email protected] [email protected]

Animikh Biswas MS84 University of North Carolina Existence and Decay of Solutions of the 2dqg Equa- PD13 Abstracts 123

tion in the Presence of An Obstacle [email protected]

We study the 2D dissipative quasi-geostrophic equation in MS85 the presence of an obstacle, where the dissipative term is given by a fractional power of the Laplacian. Our main Well-Posedness for Variational Wave Equations tools are the generalized Fourier transform due to Ikebe We discuss the singularity formation, global existence and and Ramm, and the localized version of the fractional uniqueness for energy conservative solutions for variational Laplacian due to Caffarelli and Silvestre as improved by wave systems from modelling nematic liquid crystals and Stinga and Torrea. We give applications to the problem of other models. Gradient blowup happens in general for so- existence of weak solutions and the decay of these solutions 2 lutions in these systems, which causes the major difficulty. in the L -norm. Especially, a recent progress got with Alberto Bressan on the uniqueness of energy conservative solution for a varia- Leonardo F. Kosloff tional wave equation University of Southern California holmoff[email protected] utt − c(u) c(u) ux =0, with 0

MS84 Geng Chen Mathematics Department, Penn State University Dynamical Aspects of the Cubic Instability in University Partk, State College, PA a Landau-De Gennes Energy for Nematic Liquid [email protected] Crystals

MS85 We consider a four-elastic-constant Landau-de Gennes en- ergy characterizing nematic liquid crystal configurations. Boussinesq System with Nonlinear Heat Diffusion It is known that certain physical considerations require on a Bounded Domain the presence of a cubic term, which nevertheless makes On a bounded 2D domain with smooth boundary, we prove the energy unbounded from below. We study the dynam- the global existence of a unique smooth solution to the in- ical effects produced by the presence of this cubic term 2 viscid heat conductive Boussinesq Equations with nonlin- by considering the L gradient flow generated by this en- ear diffusion, along with homogeneous Dirichlet boundary ergy. We work mostly in dimension two and our objective condition for temperature and slip boundary condition for is to provide an understanding of the relations between the velocity. Such a solution is also shown to be the vanish- physicality of the initial data and the global well-posedness ing viscosity limit for the corresponding viscous and heat of the system. conductive Boussinesq equation with the same boundary condition for temperature, and Navier boundary condition Xiang Xu,GautamIyer for the velocity. This talk is based on the joint work with Carnegie Mellon University Huapeng Li, Hongjun Yuan, and Weizhe Zhang. [email protected], [email protected] Ronghua Pan Arghir Zarnescu Georgia Tech. University of Sussex, [email protected] [email protected] MS85 Title Not Available at Time of Publication MS84 Abstract not available at time of publication. Regularizing Effect of the Forward Energy Cascade in the Inviscid Dyadic Model Mikhail Perepelitsa Univ. of Houston, USA We study the inviscid dyadic model of the Euler equations [email protected] and prove some regularizing properties of the nonlinear term that occur due to forward energy cascade. This leads MS85 to several regularity results the inviscid case. We conjec- ture that every blow up has Onsager’s scaling and discuss Conservation Laws with no Classical Riemann So- recent progress toward achieving this goal. lutions: Existence of Singular Shocks The basic tool in the construction of solutions to the Karen Zaya Cauchy problem for conservation laws with smooth ini- University of Illinois at Chicago tial data is the Riemann problem. We review the results [email protected] obtained for the solutions to the Riemann problem and present a system of two equations derived from isentropic Alexey Cheskidov gas dynamics with no classical solution. We then use the University of Illinois Chicago blowing-up approach to geometric singular perturbation 124 PD13 Abstracts

problems to show that the system exhibits unbounded so- the the Hall Magneto-Hydrodynamic equations. We first lutions (singular shocks) with Dafermos profiles. provide a derivation of this system from a two-fluids Euler- Maxwell system for electrons and ions, through a set of Charis Tsikkou scaling limits. We also propose a kinetic formulation for West Virginia University the Hall-MHD equations which contains as fluid closure [email protected] different variants of the Hall-MHD model. Then, we will present some well-poseness results for the incompressible Barbara Lee Keyfitz resistive Hall-MHD model, including global weak solutions, The Ohio State University local existence of smooth solutions for large data, global Department of Mathematics smooth solutions for small datas well, as a Liouville the- bkeyfi[email protected] orem for the stationary solutions. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions MS86 in the form of axisymmetric purely swirling magnetic fields Competitive Geometric Evolution of Complex and propose some regularization of the Hall equation. Structures in Fuel Cell Membranes Jian-Guo Liu Surfactant molecules are amphiphilic: composed of a hy- Duke University drophilic and a hydrophobic moiety. When mixed with [email protected] a wettable material the amphiphilic component forms molecular-width bilayer interfaces and pore structures which interpenetrate the bulk regions of the second phase. MS87 These structures are manifest as pore networks in blends of Eventual Self-Similarity of Solutions for Diffusion- functionalized polymer and solvent, and lipid-filled cellular Absorption Equation with a Singular Source membranes in biological settings. The competition for the We consider a class of diffusion-absorption equations with surfactant phase generates interfacial dynamics fundamen- n tally different from those obtained in mixtures of mutu- a singular source in R . We establish existence of a weak ally hydrophobic phases. Using the Functionalized Cahn- self-similar solutions for these equations which arise in a Hilliard free energy as a model for amphiphilic mixtures, limit of an infinitely strong singular source. We prove we employ a multiscale analysis to derive the curvature uniqueness of such solution in a suitable weighted en- driven flow of smooth closed bilayers and closed-loop pore ergy spaces. Moreover, we show that the obtained self- structures, which we extend to a sharp-interface reduction similar solutions are the long-time limits of the solutions for the competitive evolution of collections of bilayer inter- of the initial value problem for considered class of diffusion- faces and closed-loop pores. In particular, for a mixture absorption problems with zero initial data and a singular of spherical bilayers and circular closed pores we explicitly source of arbitrary constant strength. This is a joint work identify two regimes: one in which spherical bilayers ex- with Cyrill Muratov. tinguish and the circular pores arrive at a common radius, and a complimentary regime in which spherical bilayers of Peter Gordon differing radii stably coexist with common-radius, closed- Department of Mathematics loop, circular pores. The University of Akron [email protected] Shibin Dai Michigan State University Cyrill B. Muratov [email protected] New Jersey Institute of Technology Department of Mathematical Sciences [email protected] MS86 Master Equation Model for Charge Transport in Disordered Semiconductors MS87 Front Propagation in Sharp and Diffuse Interface Charge transport plays a key role in the performance of Models of Stratified Media bulk heterojunction solar cells. This type of solar cells contains a mixture of donor and acceptor molecular species We study front propagation problems for forced mean cur- with a certain degree of phase segregation. I will present vature flows and their phase field variants that take place a Pauli master equation model for the charge transport in in stratified media, i.e., heterogeneous media whose char- these systems and address how the charge mobility is af- acteristics do not vary in one direction. We consider phase fected by the energy offset between the transporting levels change fronts in infinite cylinders whose axis coincides with of the two molecular species, the partial volume occupied the symmetry axis of the medium. Using the recently de- by each species and their typical domain size. Our results veloped variational approaches, we provide a convergence also pertains partially crystalline samples made of a single result relating asymptotic in time front propagation in the material. diffuse interface case to that in the sharp interface case for suitably balanced nonlinearities of Allen-Cahn type. Jos´eFreire The result is established by using Γ-convergence type argu- Universidade Federal do Parana, Brazil ments to obtain a correspondence between the minimizers [email protected] of an exponentially weighted Ginzburg-Landau-type func- tional and the minimizers of an exponentially weighted area-type functional. These minimizers yield the fastest MS86 moving traveling waves in the respective models and de- Hall-Mhd and Related Plasma Problem termine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts In this talk, I will present some derivations and analysis of are the exponentially stable global attractors for this kind PD13 Abstracts 125

of initial data and give sufficient conditions under which Lame System with Infinite Memories in Bounded complete phase change occurs via the formation of the con- Domain sidered fronts. In this work, we consider the Lame system in 3-dimension Annalisa Cesaroni bounded domain with infinite memories. We prove, un- Dipartimento di Matematica der some appropriate assumptions, that this system is well Universita di Padova posed and still stable, and we get a general and precise [email protected] estimate on the convergence of solutions to zero at infinity in term of the growth of the infinite memories. Cyrill B. Muratov New Jersey Institute of Technology Ahmed Bchatnia Department of Mathematical Sciences University of Dammam [email protected] [email protected]

Matteo Novaga MS88 University of Padova (Italy) [email protected] Vanishing Viscosity Limit of Some Symmetric Flows

MS87 Abstract not available at time of publication. Quasi-Static Evolution and Congested Crowd Mo- tion Gung-Min Gie Indiana University In this talk we investigate a transport equation with a drift USA potential, where a constraint on the L∞ norm is imposed [email protected] on the density. This model, in a simplified setting, de- scribes the congested crowd motion with a density con- straint. When the drift potential is convex, the crowd MS88 density is likely to aggregate, and thus if the initial den- Singular Perturbation Analysis for Convection- sity starts as a patch (i.e. if it is a characteristic function diffusion Equations with Corners of some set), then the density evolves like a patch. We show that patch evolves according to a quasi-static evolu- We study the asymptotic behavior at small diffusivity of tion equation, which is a free boundary problem and has the solutions to a convection-diffusion equation in a rect- some connection with the Hele-Shaw equation. To show angular domain ?. The diffusive equation is supplemented this result we make use of both viscosity solutions theory with a Dirichlet boundary condition, which is smooth along as well as the gradient flow structure of the problem. the edges and continuous at the corners. To resolve the dis- crepancy between the exact solutions and the correspond- Damon Alexander ing limit solution on the boundaries, we propose asymp- UCLA totic expansions of at any arbitrary, but fixed, order. In [email protected] order to manage some singular effects near the four corners of ?, the so-called elliptic and ordinary corner correctors are Inwon Kim added in the asymptotic expansions as well as the parabolic Department of Mathematics and classical boundary layer functions. Then, performing UCLA the energy estimates on the difference of the exact solutions [email protected] and the proposed expansions, the validity of our asymp- totic expansions is established in suitable Sobolev spaces. Yao Yao Department of Mathematics Chang-Yeol Jung University of Wisconsin Ulsan National Institute of Science and Technology [email protected] South Korea [email protected] MS87 Discrete Motion by Mean Curvature MS88 We will discuss some results on the motion of interfaces How Do Flows in Conduit and Porous Media In- in a discrete environment. The model takes into account teract? nearest and next nearest neighbor spin interactions lead- ing to interfaces between patterns. Intricate stick-slip phe- Multiphase flow phenomena are ubiquitous. Common ex- nomena also arises through different spatial and temporal amples include coupled atmosphere and ocean system (air discretization. This is joint work with Andrea Braides and and water), oil reservoir (water, oil and gas), cloud and fog Marco Cicalese. (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental Aaron Yip science applications. In some applications such as flows in Purdue University unconfined karst aquifers, karst oil reservoir, proton mem- Department of Mathematics brane exchange fuel cell, multiphase flows in conduits and [email protected] in porous media must be considered together. How free flows in conduit/channel interact with flow in porous me- dia is a challenge. In this talk we present a family of phase MS88 field models that couples two phase flow in conduit with Well-Posedness and Asymptotic Stability for the two phase flow in porous media. These models together 126 PD13 Abstracts

with the associated interface boundary conditions are de- rived utilizing Onsager’s extremum principle. The models derived enjoy physically important energy law. Numerical scheme that preserves the energy law will be presented as well.

Xiaoming Wang FSU [email protected]