36 PD07 Abstracts

CP1 CP1 Energy Functional of the Dbvp and the First Eigen- The Boundary Value Problems with Strong Singu- value of the Laplacian larity of Solution.

n n Let B1 be a ball of radius r1 in S (H ) with r1 <π We consider the boundary value problems with strong sin- n for S . Let B0 be a smaller ball of radius r0 such that gularity. Depending on the coefficients and right-hands B0 ⊂ B1. Let u be a solution of the problem −Δu =1in sides of the equation and of the boundary conditions, the Ω:=B1 \ B0 vanishing on the boundary. It is shown that solution can be infinite at some points (the case of strong the energy functional E(Ω) is minimal if and only if the singularity, u ∈ H1(Ω)). For these problems we offer to de- balls are concentric. It is also shown that first Dirichlet fine the solution as Rν -generalized one. Such a new concept eigenvalue of the Laplacian on Ω is maximal if and only if of solution led to distinction of two classes of the boundary the balls are concentric. value problems: the problems with coordinated and non- coordinated degeneration of initial date; and also made Anisa Chorwadwala it possible to study the existence and uniqueness of solu- The Institute of Mathematical Sciences tions as well as coercivity and differential properties in the [email protected] weighted Sobolev spaces. Victor Rukavishnikov CP1 Russian Academy of Scinces Singularities, Not Existing Solutions, Mac and [email protected] PDE

The non existing solution of Dirichlet problem for simple CP1 bounded domain is analyzed. If the domain is a ring for Uniqueness and Asymptotic Stability for the Ra- Laplace equation then the solution exist for finite inner and dial Solutions of Semilinear Elliptic Equations outer radius but it can have singularity in the solution for small inner radius. The solution does not exist for zero in- Less is known of the uniqueness for the radial solutions u = n ner radius. To obtain the physically acceptable solution the u(r) of the problem Δu = f(u+)=0inR (n>2),u(ρ)= method of additional conditions (MAC) is applied. Then 0,u(0) = 0, besides the cases where limr→∞u(r)=0;and the bounded solution can be obtained. for the cases based only on the evolution of the functions d f(t) f(t) and dt t . This paper proves uniqueness for the prob- Igor Neygebauer lem Da + f(u+)=0(r>0),u(ρ)=0,u(0) = 0 based Associate Professor 1 on the assumption that f ∈ C ([0, ∞)) and that ρ satisfies National University of Rwanda a boundedness condition. Furthermore, we prove asymp- [email protected] totic stability for Da + φ(r)f(u+) = 0 based only on the evolution of u(r) and u − φ(r)f(u). CP1 Sikiru Adigun Sanni A Theory of Infinite-Dimensional Evans Function University of Uyo, Uyo, Nigeria and the Associated Bundle. [email protected] The purpose of this talk is to introduce a topological framework to treat elliptic eigenvalue problems in a channel or CP2 bounded domain in higher dimension. The main objects Ultra-Fast Soliton Propagation in Nonlinear Opti- are generalizations of the Evans function and the bundle cal Media construction which are originally used in the stability analysis of traveling wave in R. Motivated by the recent interest in optical materials with embedded nanostructures (so-called metamaterials), Shunsaku Nii we study stability of ultra-fast pulses governed by the Kyushu University Maxwell-Duffing model. A solitary wave is obtained as a [email protected] simulated annealing solution for the two-point boundary- value problem of the 6 by 6 system of ODEs for the trav- Jian Deng eling wave. Stability of solitary waves is demonstrated Central University of Finance and Economics through a series of DNS as well as through a numerical [email protected] linearized stability analysis.

Yevgeniy Frenkel, Victor Roytburd CP1 Department of Mathematical Sciences Singular Limit Problem for Some Elliptic Systems RPI [email protected], [email protected] For the sharp interface problem arising in the singular limit of some elliptic systems, we prove the existence and the nondegeneracy of solutions whose interface is a distorted CP2 circle in a two-dimensional bounded domain without any Evaluation of the Finite Impulse in Plastic-Elastic assumption on the symmetry of the domain. Wave Propagation

Yoshihito Oshita We consider a wave picture in semi-inﬁnite bar in case of Department of Mathematics plastic-elastic loading and elastic unloading. The stress in Okayama University the endpoint of the bar represents the ﬁnite impulse. The [email protected] boundary between loading and unloading zones is called the unloading wave. In case under consideration the initial PD07 Abstracts 37

part of the unloading wave is the strong wave, but a jump Goethert Pkwy of strains decreases. We obtain the condition for impulse [email protected] amplitude, such that this jump equals zero within finite time. After this jump has equaled zero the unloading wave becomes the weak wave. We obtain the form of this wave CP2 analytically. Numerical Simulation on Optimal Quantum Sys- tem Control Alexander A. Kosmodemyanskiy MIIT (Moscow Institute of Railways) Amazing quantum world is the most frontier realm. Among Comp. Math Department this, quantum system control, with its supprisingly devel- [email protected] oping and rapidly growing, is extensively demanded area. It attracts numerous attention of worldwide researchers and scientists. Fortunately, laboratory simulation provides CP2 the essential step to realize quantum control in advanced A Variational Approach to Inverse Recovery of Co- laser technology. With this motivation, the novel point of efficient Functions in the Acoustic Wave Equation this work focus on exploring substantial stage in the direction of real quantum control via computational experiment. We present a full waveform, fully nonlinear minimization approach for the reflection seismology problem of recover- Quan-Fang Wang ing the subsurface density ρ, sound speed κ, and absorption Mechanics and Automation Engineering γ from direct measurements of pressure over a part of the The Chinese University of Hong Kong water surface. Our approach consists of transforming the [email protected] underlying hyperbolic partial differential equation 1 1 CP3 (ptt + γ(x)pt) −∇. ∇p = F (x, t) ρ(x)κ2(x) ρ(x) The Stability and Stabilization of Solutions of the Nonlinear Differential Equations And Their Appli- to the self-adjoint elliptic equation cations

2 −λ −∇.(A(x)∇u)+(λ B(x)+λC(x))u = −e (λR(x)+S(Thex)) presentation consists of the three parts. In the first part we give a review of the works of the author devoted ˆ + F (x, λ) to Liaponov stability of the motion. Given new criterion’s of stability of nonlinear differential and difference equa- via the finite Laplace transform and minimizing a functions in Banach spaces, of stability of solution of nonlinear tional that is believed to possess a unique stationary point. systems of differential equations with lateness, nonlinear The minimization procedure is based on a conjugate gra- systems of differential equations with a small parameter at- dient descent algorithm using Sobolev gradients. tached to derivative, nonlinear systems of partial differen- Ian W. Knowles tial equations. Stability of solution of nonlinear systems of University of Alabama at Birmingham differential equations with discontinuous right-hand sides Department of Mathematics is investigated also. We investigate the Turing stability [email protected] of solution of partial differential equations. Criterion’s of Liapunov stability are applied to regular case and to all possible critical cases simultaneously. In the second part Mandar S. Kulkarni of the presentation we give some new criterion’s of stabi- Department of Mathematics lization of solutions of differential equations. In the third University of Alabama at Birmingham part of the presentation we give applications of these crite- [email protected] rions to some tasks of physics, ecology and economics. In particular, we offer new criterion of stability of Hotelling- CP2 Scellam models in ecology and economics, dissipative and Schrodinger systems. New criterion of stability of Kol- Discrete Nonlinear Waves As Solutions of Linear mogorov model in ecology are given too. Wave Equation Ilya V. Boykov A new method is described to compute thin pulse solu- Professor tions to the linear wave equation, discretized on a uniform [email protected] lattice. Unlike conventional Eulerian numerical solutions, these waves remain narrow, several grid cells (h) in width, and do not spread due to discretization error even over CP3 indefinite distances. These “pulses” represent a traveling The Extremal Solutions of Nonlinear Singular wave front as a codimension 1 surface in the limit h-¿0. The Boundary Value pulse is essentially a nonlinear solitary wave that “lives” on the lattice. It accurately represents the total amplitude of We consider the nonlinear boundary value problems for an arriving pulse at each lattice point in one or three di- partial differential equations, that are not satisfy the co- mensions, as well as the arrival time. Multiple arrival times ercivity condition and for which there are not the global can also be computed. solvability results. By using the method of the insertion of spurious control function the given problem is regularized. Subhashini Chitta For each value of control functions the obtained problem UTSI has at least one global solution. We form the discrepancy [email protected] functional, then we prove some theorems for existence of optimal control. The obtained solutions are regularizators John Steinhoff UTSI 38 PD07 Abstracts

for original problem. ear Viscoelasticity Victor Ivanenko In this paper it will be studied the boundary integral Institute for Applied and System Analysis NAS of formulation of the initial-boundary value problem of a Ukraine Volterra type integro-diﬀerential equation [email protected] t ∂u(x, t) (t − s)α−1 − μΔu(x, t) − Δu(x, s)ds =0, Pavlo Kasyanov ∂t 0 Γ(α) Kyiv National Taras Shevchenko University, Ukraine in Ω u = f, [email protected] on ∂Ω u(x, 0) = 0. Valeriy Melnik Institute for Applied and System Analysis NAS of We present the boundary integral formulation of the prob- Ukraine, K lem by using the fundamental solution of the integro- [email protected] diﬀerential operator in question. The mapping properties of the boundary integral operators in anisotropic Sobolev spaces will be given. Furthermore we show the validity of CP3 the G˚arding inequality. These properties are needed in the The Multivalued Penalty Method for Evolution analysis of the numerical approximation methods. Finally Variational Inequalities with 0-Pseudomonotone we will show that the Galerkin and collocation methods Multivalued Maps are stable and convergent.

We consider the evolution variational inequalities with 0- Keijo M. Ruotsalainen pseudomonotone maps. The main properties of the given Professor, University of Oulu maps have been investigated. By using the finite differ- [email protected].fi ences method, the strong solvability for the class of evolution variational inequalities with 0-pseudomonotone map CP3 has been proved. Due to the penalty method for multi- valued maps, the existence of weak solutions for evolution Reduction of Dynamical Systems Under Stochastic variation inequalities on closed convex sets has been shown. Parametric Forcing The class of multivalued penalty operators has been con- A novel model reduction strategy for stochastic ODE/PDE structed. We have been considered one model example to systems with parametric forcing is discussed and illustrated illustrate the given theory. with a SEIR system. In particular: 1) We show how Victor Ivanenko Markov multiplicative parametric noise can be accurately Institute for Applied and System Analysis NAS of modeled using additive noise provided non static effects are Ukraine taken into consideration in estimating its variance, in con- [email protected] trast with the standard approach (Freidlin and Wentzell, Skorokhod). 2) We present results on how the nature of the stochastic/harmonic forcing affects the dynamics in various Pavlo Kasyanov parameter regimes. Kyiv National Taras Shevchenko University, Ukraine [email protected] Bruno D. Welfert Arizona State University Luisa Toscano Department of Mathematics & Statistics Second University of Naples, Aversa, Italy [email protected] [email protected] Reynaldo Castro, Juan Lopez Arizona State University CP3 [email protected], [email protected] Krylov Subspace Spectral Methods for Hyperbolic Systems CP4 It has been demonstrated that for time-dependent variable- Blade-Vortex Interaction Analysis Using Potential coefficient scalar equations, Krylov subspace spectral Flow Theory methods are capable of simultaneously achieving high- order accuracy in time, along with remarkable stability, Abstract not available at time of publication. considering that they are explicit methods. In this talk, we explore the generalization of these methods to hyper- Marcel Ilie bolic systems of PDE. We also examine the use of homoge- Carleton University nization via unitary similarity transformations to enhance Department of Mechanical and Aerospace Engineering accuracy and stability even further. [email protected] James V. Lambers Stanford University CP4 Department of Energy Resources Engineering Exact Solution of the 3D Linear Eulers Equations [email protected] As a Benchmark Test

We obtain an exact solution of the three-dimensional linear CP3 Eulers equations containing a combination of entropy, vor- Boundary Integral Approach for a Problem in Lin- ticity, and acoustic waves in a mean ﬂow. The solution is a good benchmark test for numerical schemes developed for PD07 Abstracts 39

Computational Aeroacoustics problems. We examine the porous media consisting of geological features, such as accuracy of a sliding mesh interface method for problems faults and fractures, that are challenging from a grid- involving moving geometry. The sliding interface method ding standpoint. We demonstrate an approach for ob- uses an optimized interpolation scheme with coefficients taining an unstructured grid, and accompanying finite vol- obtained by minimizing interpolation error in wave num- ume scheme, using spatially-varying multi-point flux sten- ber space. cils computed from local fine-scale solutions in such a way as to simultaneously achieve accuracy and robustness. Philip P. Lepoudre Florida State University Elizabeth J. Spiteri [email protected] Chevron Energy Technology Company [email protected]

CP4 James V. Lambers Stable Interface Conditions for Discontinuous Stanford University Galerkin Approximations of the Navier-Stokes Department of Energy Resources Engineering Equations [email protected] The lecture presents a methodology for constructing well-posed boundary and interface conditions for one- CP5 dimensional Discontinuous Galerkin approximations of A Positivity Preserving Central-Upwind Scheme advection-diffusion equations. While Riemann solvers have for Chemotaxis and Haptotaxis Models traditionally been used at element interfaces to compute the interface flux for the case of pure advection, most dis- The paper is concerned with development of a new finite- cretizations of the Navier-Stokes equations use an average volume method for a class of chemotaxis models and for a of the viscous fluxes on the two sides of an interface. Re- closely related haptotaxis model. In its simplest form, the quiring the norm of the solution to not increase in time, a chemotaxis model is described by a system of nonlinear set of boundary/interface conditions that can be thought PDEs: a convection-diffusion equation for the cell density of as ”viscous” Riemann solvers can be devised, which are coupled with a reaction-diffusion equation for the chemoat- compatible with the pure advection limit. tractant concentration. The first step in the derivation of the new method is made by adding an equation for the Arunasalam Rahunanthan chemoattractant concentration gradient to the original sys- Department of Mathematics, University of Wyoming tem. We then show that the convective part of the result- Laramie, Wyoming 82071-3036 ing system is typically of a mixed hyperbolic-elliptic type [email protected] and therefore straightforward numerical methods for the studied system may be unstable. The proposed method Dan Stanescu is based on the application of the second-order central- Department of Mathematics, University of Wyoming, upwind scheme, originally developed for hyperbolic sys- Laramie, Wyoming 82071-3036 tems of conservation laws in [A. Kurganov, S. Noelle, [email protected] and G. Petrova, SIAM J. Sci. Comput., 21 (2001), pp. 707–740], to the extended system of PDEs. We show that the proposed central-upwind scheme is positivity pre- CP4 serving, which is a very important stability property of The PDE2D Collocation Finite Element Method the method. The scheme is applied to a number of two- dimensional problems including the most commonly used PDE2D (www.pde2d.com, sold through Visual Numerics’ Keller-Segel chemotaxis model and its modern extensions e-commerce site) is a finite element program which solves as well as to a haptotaxis system modeling tumor inva- very general PDE systems in 1D intervals, arbitrary 2D re- sion into surrounding healthy tissue. Our numerical results gions and a wide range of simple 3D regions. Both Galerkin demonstrate high accuracy, stability, and robustness of the and collocation options are available for 1D and 2D prob- proposed scheme. lems, only collocation for 3D problems. The collocation algorithms have some novel features, particularly relating Alina Chertock to how non-rectangular regions are handled. Some advan- North Carolina State University tages and disadvantages of these collocation algorithms are Department of Mathematics discussed, relative to the more standard Galerkin meth- [email protected] ods. A new GUI now allows extremely easy access to the PDE2D collocation methods, in fact, the new GUI is im- Alexander Kurganov possible to beat in terms of ease-of-use because it requires Tulane University the user to supply the absolute minimum of information Department of Mathematics necessary to describe the PDEs and boundary conditions. [email protected] Granville Sewell Mathematics Dept. CP5 University of Texas El Paso [email protected] Traveling Waves in Reaction-Diffusion Equations with Density-Dependent Diffusion

CP4 We consider a coupled system of reaction-diffusion equations with density-dependent diffusion, which arises as a Accurate and Robust Transmissibility Upscaling on model of spreading of a bacterial population in a surface Unstructured Grids environment. Traveling wave solutions of the system and In this talk, we consider coarse-scale modeling of flow in their asymptotical behaviors are investigated. We further study the traveling wave speed dependence on various pa- 40 PD07 Abstracts

rameters. Numerical examples are presented as an illus- For the ODE system, we prove permanence as well as uni- tration of the above results. The results are in agreement form boundedness and show that the effect of the car- with experimental data. casses leads to chaotic dynamics for biologically reasonable choices of parameters by numerical simulations. Then we Haiyan Wang consider the spatial diffusion problem as the PDE system Arizona State University and propose that the diffusion speed depends only on the [email protected] diffusion coefficient of ’invader species’.

Zhaosheng Feng Sungrim S. Lee University of Texas-Pan American Graduate school of environmental sciences [email protected] Okayama university [email protected]

CP5 Global Stability for Lotka-Volterra Systems CP5 Anguilliform Swimming Through Activation in An We study Cauchy problem for the n-species Lotka-Volterra Elastic Rod tree systems of reaction-diffusion equations. We obtain a set of sufficient conditions for the globally asymptotic sta- I will discuss a model for anguilliform swimming that in- bility of the solutions to Cauchy problem . The criteria corporates a simple model for muscle forces involving the in this paper are in explicit forms of the parameters, and release of calcium ions from the sarcoplasmic reticulum, thus, are easily verifiable. Moreover, this criteria is appli- and coupled to an activated elastic rod. Most fish swim by cable to competition model, cooperation model, as well as rhythmically passing a wave of muscle activation from head to predator-prey model. to tail, which creates a corresponding wave of curvature. I will provide an explanation for the fact that the wave of Xinhua Ji activation travels faster than the wave of curvature. Institute of Mathematics Academia Sinica Tyler McMillen [email protected] California State University, Fullerton Department of Mathematics [email protected] CP5 A Competition Model for Two Invasive Species and Philip Holmes a Native Species Princeton University MAE Dept. The dynamics of three competing species is examined in [email protected] space-time. The model is a system of Lotka-Volterra type partial differential equations, with one equation lacking a Thelma Williams diffusion term. The system is analyzed in terms of traveling St. Georges Hospital Medical School waves, including conditions on the propagation speed of [email protected] the wave front. Comments include material applicable to ecological invasions by multiple species. CP6 Daniel L. Kern University of Nevada, Las Vegas Nondegenerate Isentropic and Anisentropic Dept of Mathematical Sciences One-Dimensional Wave-Wave Regular Interaction [email protected] Structures: Details of a Parallel We use two concurrent approaches [“algebraic” (Burnat; CP5 essential for isentropic multidimensional extensions) and “differential” (Martin)] to characterize nondegenerate one- Dependence of the Speed of Traveling Wave on dimensional wave-wave regular interaction structures. The Invader Species and Dynamics in Two-Prey, One- anisentropic “differential” characterization results from a Predator Commensal Effect Model significant [Monge−Ampère] extension and is proven to be We propose the relation between invader species and the strictly “non-algebraic” (the isentropic “differential” and diffusion speed of traveling wave in three species prey- “algebraic” characterizations are proven to coincide). A pedator model. It is important problem biologically what parallel between two contexts [isentropic, anisentropic], fo- affects the spreding speed (diffusion speed) by invasion of cused on significant contrasts between the “differential” outsider species. We apply the problem to two-prey, one- types of wave-wave regular interaction structures respec- predator model involving the effect of carcasses. The model tively associated, is finally considered. has a commensalism interaction that resident prey species Marina Ileana Dinu eat the remains of carcasses of invader species after preda- Polytechnical University of Bucharest, Dept. of Math. III tor species have consumed it. Under some biological as- ROMANIA sumptions, we propose the model as ODE system and also [email protected] consider the spatial diffusion effect as PDE system. The PDE system is given as the following: Liviu Florin Dinu 2 ∂ ∂ h1 Institute of Mathematics of the Romanian Academy h1 = d1 2 h1 + e1 1 − h1 − α1h1p + ωh2ph1, ∂t ∂x k1 [email protected], [email protected] 2 ∂ ∂ h2 2 2 2 2 2 − 2 − 2 2 ∂t h = d ∂x h + e 1 k2 h α h p, ∂ ∂2 − ∂t p = d3 ∂x2 p γp + β1h1p + β2h2p. PD07 Abstracts 41

CP6 system converge to zero as t →∞. Coupled Mode Theory of Resonant Scattering of Water Waves by a Two Dimensional Array of Ver- Ihsan A. Topaloglu tical Cylinders Indiana University [email protected] We study the Bragg resonance of surface water waves by a two-dimensional array of vertical cylinders covering a large Alp Eden area of the sea. Using the resonance criterion known in Bogazici University the solid- state and crystallography, we employ asymptotic Feza Gursey Institute techniques to derive two-dimensional coupled-mode equa- [email protected] tions for the envelopes of scattered waves resonated by a plane incident wave. Explicit analytical solutions are obtained for a long strip of cylinder array. Examples of both CP7 two-wave and three-wave resonances are discussed in de- Numerical Analysis for the Nonlocal Allen-Cahn tail. Roles of the bandgaps are examined. The analysis Equation and results may also apply to sound or optical waves in periodic media. We propose a stable, convergent and linear ﬁnite diﬀer- ence scheme to solve numerically the nonlocal Allen-Cahn Chiang C. Mei equation which can model a variety of physical and biolog- Dept of Civil and Environmental Engineering ical phenomena involving media with properties varying in MIT space. Also we prove that the scheme is uniquely solvable [email protected] and the numerical solution will approach the true solution in the L∞ norm. Yile Li Shell International Exploration & Production Sarah M. Brown, Jianlong Han Houtson TX Southern Utah University [email protected] brown [email protected], [email protected]

CP6 CP7 Wave-Breaking Dissipation in Wave/Current In- Title Not Available at Time of Publication teractions Abstract not available at time of publication. Wave breaking dissipates energy in oceanic waves and it is Ming Chen agreed that it affects currents as well. The question to be University of Minnesota answered are what aspects of the dynamics of waves, cur- [email protected] rents and their interactions are affected and how. Using multi-scale ideas and a stochastic parametrization of the Lagrangian fluid path it is possible to model the determin- CP7 istic interactions and include in these dissipative effects. Multiscale Phase Field Modeling of Martensitic The resulting model is then capable of answering some of Microstructure Evolution and Nucleation the phenomenological questions, and further, it can be used to make specific and testable predictions. The talk will fo- A three-dimensional Landau theory for multivariant stress- cus on the modeling strategy and on preliminary results induced martensitic phase transformations was recently de- derived from the analysis of the model. veloped (V. I. Levitas, D. L. Preston and D. W. Lee, Phys. Rev. B, 134201, 2003) This new theory describes correctly Juan M. Restrepo the stress and temperature effect on PTs. The time de- University of Arizona pendent Ginzburg-Landau equation is used to simulate mi- [email protected] crostructures evolution. The barrierless martensitic nucleation at various dislocation configurations (single disloca- CP6 tion, dislocation dipole and low angle grain boundary) is also considered. Standing Waves for a Generalized Davey-Stewartson System: Re- Dong-Wook Lee visited Mechanical Engineering Texas Tech University The existence of standing waves for a generalized Davey– [email protected] Stewartson (GDS) system was shown in [A. Eden and S. Erbay, Standing waves for a generalized Davey–Stewartson system, J. Phys. A: Math. Gen. 39 13435–13444 (2006)] CP7 using an unconstrainted minimization problem. Here, we The Numerical Computation of the Critical Bound- consider the same problem but relax the condition on the b ary Displacement for Radial Cavitation parameters to χ + b<0orχ + m1 < 0. Our approach is to use a constrained minimization problem and utilize Lions’ We study radial solutions of the equations of isotropic elas- concentration compactness lemma. When both methods ticity in two dimensions (for a disc) and three dimensions apply we show that they give the same minimizer and we (for a sphere). This is equivalent to a nonlinear (quasi- obtain a sharp bound for a Gagliardo–Nirenberg type in- linear) boundary value problem depending parametrically equality. This leads to a global existence result for small- on the boundary displacement. We describe a numerical mass solutions. Moreover, we show that when p>2, the scheme for computing the critical boundary displacement p L -norms of solutions to the Cauchy problem for a GDS for cavitation. We give examples for specific materials and compare our numerical computations with some previous 42 PD07 Abstracts

analytical results. We give a characterization in the phase ﬂow is governed by the Navier-Stokes equations. The so- plane of the nonconstant solutions of the corresponding lution is found by the successive approximations. Every Euler–Lagrange equations for radial solutions in terms of approximation is the solution to the linearized problem in the curves of constant normal component of Cauchy stress the domain with known boundary. This solution is con- for general classes of stored energy functions. structed by the ﬁnite element method. Pablo Negron Ilya S. Mogilevskiy University of Puerto Rico - Humacao Tver university [email protected] Mathematical Department [email protected] Jeyabal Sivaloganathan Department of Mathematical Sciences CP8 University of Bath, Bath 2 [email protected] An H -Compatible Finite Element Solver for the Navier-Stokes Equations

CP7 Liu, Liu, and Pego have developed an unconditionally stable algorithm for finite element discretization of the Navier- Analysis of Microbial Depolymerization of Xenobi- 2 otic Polymers Based on Mathematical Models and Stokes equations in the setting of the H Sobolev spaces. Experimental Data The algorithm enforces conservation of mass without a need for the inf-sup condition that is common in H1 formulations. In domains with reentrant corners the solution A microbial depolymerization process falls into one of two 2 categories: exogenous type and endogenous type. Assum- of the Navier-Stokes equations need not be in H , there- ing that a degradation rate is a product of a molecular fore the algorithm needs to be modified. We have modified factor and a time factor, a time dependent depolymeriza- the algorithm by recasting it in weighted Sobolev spaces tion model is transformed into a time independent model. where appropriate weights at reentrant corners compensate Given the weight distribution before and after degrada- for the solution’s singularities. The numerical results obtained this way show good agreement with those obtained tion, the molecular factor is determined with techniques 1 developed in the previous studies. These techniques and by the usual H methods. techniques to determine the time factor will be discussed. Ana Maria Soane Masaji Watanabe University Of Maryland, Baltimore County Graduate School of Environmental Science, Okayama Department of Mathematics and Statistics University [email protected] [email protected] CP8 Fusako Kawai Research Institute for Bioresources On a 1D Model of the 3D Euler/Navier-Stokes Okayama University Equations [email protected] In 1985 a simple 1D equation ωt = H(ω)ω is proposed by P. Constantin, P. Lax and A. Majda as a model of the vortex CP8 streteching phenomenon in 3D incompressible Euler equations. This model equation blows up in finite time. Later S. Mathematical Model of Filtration in Membrane Schochet discovered a counter-intuitive phenomenon: the We study a general mathematical model of permeate flux in ”viscous” model ωt = H(ω)ω + ωxx blows up faster for two variables. First, we state the fundamental differential certain initial values. In this talk I will present new under- equation for the inner flow and the permeate flux, which are standing of the interaction between stretching and dissipa- obtained from the continuity equation. The flux satisfies a tion. With this understanding, the phenomenon discovered first order linear partial differential equation and the inner by Schochet is no longer counter-intuitive. flow it obtained from a second order non-linear differential Xinwei Yu equation.We solve all the equations.For the linear one, the UCLA method of characteristics is used. [email protected] Ezio Marchi IMASL-UNSL CP9 [email protected] Multiscale Particle Method for Hyperbolic PDEs

Carlos Tarazaga A meshless particle method in a multiscale framework is Universidad Nacional de SanLuis,San Luis,5700, investigated. The aim is to obtain numerical solutions for Argentina PDEs by avoiding the mesh generation and by employing a [email protected] set of particles arbitrarily placed in problem domain. The elimination of a mesh combined withthe properties of di- lation and translation of scaling and wavelets functions is CP8 particularly suitable for problems with large deformations Finite Element Method for the Plane Stationary and high gradients. Numerical examples are presented to Free Boundary Flow illustrate the flexibility of the proposed approach. We consider the stationary flow in a bounded domain 2 Elisa Francomano Ω ⊂ R with boundary Γ = Γ1 ∪ Γ2.Γ1 is a fixed part Dipartimento di Ingegneria Informatica of the boundary, Γ2 is an unknown free boundary. The Università degli Studi di Palermo - Italia PD07 Abstracts 43

[email protected] degree equal one at inﬁnity.

Adele Tortorici Leonid Berlyand Universita degli Studi di Palermo - Italia Penn State University [email protected] Eberly College of Science [email protected] Elena Toscano Universita degli Studi di Palermo Dmitry Golovaty [email protected] The University of Akron Department of Theoretical and Applied Mathematics [email protected] CP9 Non-Oscillatory Central Schemes for a Traffic Flow Yaniv Almog Model with Arrhenius Look-Ahead Dynamics Louisana State University [email protected] We develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics. This Itai Shafrir model, takes into account interactions of every vehicle with Technion - I.I.T other vehicles ahead (“look-ahead” rule) and can be writ- [email protected] ten as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the first-order staggered Lax-Friedrichs scheme and the second- MS1 order Nessyahu-Tadmor scheme, which belong to a class of On a Minimization Problem with a Mass Con- Godunov-type projection-evolution methods, but does not straint Involving a Potential Vanishing on Two require any (approximate) Riemann problem solver, which Curves is unavailable for conservation laws with global fluxes. Our numerical experiments demonstrate high resolution, stabil- We study a singular perturbation type minimization prob- ity, and robustness of the proposed method, which is used lem with a mass constraint on a domain involving a poten- to numerically investigate both dispersive and smoothing tial vanishing on two curves in the plane. We analyse the effects of the global flux. limiting behavior of both the minimizers and their energies when the small parameter epsilon goes to zero and demon- Alexander Kurganov strate the effect of the domain geometry on this behavior. Tulane University Department of Mathematics Itai Shafrir [email protected] Department of Mathematics, Technion- I.I.T [email protected]

MS1 Nelly Andre Vortex Solutions of a ”Semi-Stiﬀ” Boundary Value Department of Mathematics Problem for the Ginzburg-Landau Equation University of Tours, France [email protected] We study solutions of 2D Ginzburg-Landau equation − 1 | |2 − Δu + ε2 u( u 1) = 0 subject to the so-called ”semi- stiﬀ” boundary conditions: the Dirichlet condition for the MS1 | | modulus, u = 1, and the homogeneous Neumann condi- Superconducting Wires Subjected to Electric Cur- tion for the phase. The principal result of this work shows rents that there are stable solutions of this problem with vortices that are located near the boundary and have bounded en- We consider a one-dimensional time-dependent Ginzburg- ergy as ε → 0. The proof is based on the introduction of a Landau model for the current in a superconducting wire notion of the approximate bulk degree which is stable with driven by an applied current. We discuss the co-existence respect to the weak H1 convergence unlike the standard of normal and supercurrent in the wire and the bifurcation degree boundary conditions. This is a joint work with V. from the normal state of both steady-state supercurrents Rybalko. and periodic solutions possessing vortices. Leonid Berlyand Kevin Zumbrun Penn State University Indiana University Eberly College of Science USA [email protected] [email protected]

Volodymir Rybalko Peter Sternberg Institute for Low Temperature Physics Indiana University Kharkov, Ukraine Department of Mathematics [email protected] [email protected]

MS1 Jacob Koby Rubinstein Indiana University and The Technion, Israel Global Minimizers for a P-Ginzburg-Landau En- Department of Mathematics ergy Functional [email protected] For a p-Ginzburg-Landau energy on the plane we prove the existence of global minimizers in the class of functions with 44 PD07 Abstracts

MS2 is a viscoelastic liquid with first and second normal stresses Modeling the Inhomogeneous Response in Steady and shear-thinning. We implement a volume-of-fluid algo- and Transient Flows of Wormlike Micellar Solu- rithm with a parabolic re-construction of the interface for tions the calculation of the surface tension force (VOF-PROST). Numerical results are checked against small deformation Surfactants are amphiphilic molecules that self-assemble theory for a viscoelastic matrix liquid, an Oldroyd-B ex- in solution. Under certain conditions they self-assemble tensional flow simulation, and experimental data on vis- into long worm-like micelles. Wormlike micellar solutions coelastic drop/liquid systems from S. Guido and P. Molde- have important uses in the petroleum industry and the naers (private communication). Reasons for the difference health-care products industry. In solution these long flexi- between Newtonian and viscoelastic drop shapes are ex- ble wormlike structures entangle thus exhibiting viscoelas- plored with numerical simulations. tic properties like polymers; but, in contrast to polymers, they also break and reform continuously thus presenting an Yuriko Renardy additional relaxation mechanism. Experimental observa- Virginia Tech tions show that these solutions exhibit distinctive (inhomo- Department of Mathematics geneous) behaviors under different deformation conditions; [email protected] for example, in steady shear flow these surfactant solutions develop shear bands (or more accurately shear rate bands). Concurrent with this banding is the development of a shear MS2 stress plateau in the flow curve over a broad range of shear Singularities and Transport in Viscoelastic Flows rates. In this talk, several constitutive models (VCM, PEC, PEC+M) are investigated numerically and analytically in In the past several years it has come to be appreciated that a cylindrical Taylor Couette geometry. These models are in low Reynolds number flow the nonlinearities provided described through coupled systems of nonlinear partial dif- by non-Newtonian stresses of a complex fluid can provide ferential equaitons. The VCM model is a two- species net- a richness of dynamical behaviors more commonly asso- work model that incorporates a discrete version of Cates’ ciated with high Reynolds number Newtonian flow. For breaking and reforming dynamics. The PEC and PEC+M example, experiments by V. Steinberg and collaborators models are, respectively, a one mode and a two mode ap- have shown that dilute polymer suspensions being sheared proximation of the full VCM model (without the effects of in simple flow geometries can exhibit highly time depen- stress-concentration coupling). The VCM model has been dent dynamics and show efficient mixing. The correspond- studied in viscometric (homogeneous) flow in Vasquez et. ing experiments using Newtonian fluids do not, and in- al. JNNFM.07. The rheological predictions of these con- deed cannot, show such nontrivial dynamics. To better stitutive model are investigated analytically and computa- understand these phenomena we study numerically the 2D tionally in steady and transient shear and in ”step strain” Oldroyd-B Viscoelastic model at low Reynolds number. A and the results are compared with experiment. The flows background force is used to create a periodic cell with four- described are inhomogeneous and do exhibit shear band- roll mill vertical structure around a hyperbolic fixed point. ing. The effects of curvature, of diffusive terms in the stress We consider both steady and time-periodic forcing. For low equations, and of inertia are discussed. This work was sup- Weissenberg number (Wi) the elastic stresses are bounded ported by the NSF-DMS #0405931 and .slaved. to the forcing, with mixing confined to small sets near the hyperbolic point. At larger Wi an analog to L. Pamela Cook the coil-stretch transition occurs yielding large stresses and Department of Mathematical Sciences stress gradients concentrated on sets of small measure, per- University of Delaware haps indicating the development of singularities. The flow [email protected] then becomes very sensitive to perturbations in the forcing and there is a transition to global mixing in the fluid. Lin Zhou Becca Thomases ENSUREDMAIL, INC. Courant Institute Wilmington, DE New York University [email protected] [email protected] Paula Vasquez Dept of Mathematical Sciences MS2 University of Delaware Mathematical Modeling and Computational Issues [email protected] The properties of viscoelastic fluids are governed by the Gareth McKinley flow-induced evolution of molecular configurations. In ad- Dept of Mechanical Engineering dition to the usual balance equations this requires consti- Massachusetts Institute of Technology tutive equations for the configuration and stress tensor. [email protected] Modeling and numerical simulation of viscoelastic fluids is not an easy task and improvements are needed in both areas. Recently, various new constitutive equations have MS2 been derived from microscopic theory. We discuss some Numerical Simulations of Drop Deformation for traditional and recently proposed constitutive equations. Viscoelastic Liquid-Liquid Systems The focus will be on mathematical modeling and computational issues. We investigate the motion of the interface separating a drop and its surrounding liquid (matrix) in a large channel Peter Wapperom driven by wall-induced shear. The governing equations are Department of Mathematics,Virginia Tech the momentum equation, incompressibility, and constitu- [email protected] tive equations for the Giesekus model. The Giesekus liquid PD07 Abstracts 45

MS3 others. Semi-Strong Pulse Dynamics Tasso J. Kaper Pulse interactions play a central role in a variety of pat- Boston University tern formation phenomena. They can be distinguished into Department of Mathematics three types: weak interactions, in which the pulses are as- [email protected] sumed to be sufficiently far apart, the fully strong interactions, and the intermediate concept of semi-strong interac- Arjen Doelman tions, that has been introduced in the context of singularly CWI Amsterdam, the Netherlands perturbed systems, in which only some components of the [email protected] pulse interact weakly. In this talk, an overview will be given of the dynamics of semi-strong pulse interactions, as well Peter van Heijster as the methods by which they can be studied analytically. CWI [email protected] Tasso J. Kaper Boston University Department of Mathematics MS3 [email protected] A Numerical Method for Coupled Surface and Grain Boundary Motion Arjen Doelman CWI Amsterdam, the Netherlands We study the coupled surface and grain boundary motion [email protected] in a bi-crystal in the context of the “quarter loop” geometry. Two types of normal curve velocities are involved in Keith Promislow this model: motion by mean curvature and motion by sur- Michigan State University face diffusion. Three curves meet at a single point where [email protected] junction conditions are given. A numerical formulation that describes the coupled normal motion of the curves and preserves arc length parametrization up to scaling is pro- MS3 posed. The formulation is shown to be well-posed in a sim- Asymptotic Behavior Near Planar Transition ple, linear setting. Equations and junction conditions are Fronts for the Cahn-Hilliard Equation approximated by finite difference methods. Numerical convergence to exact traveling wave solutions is shown. Other I will discuss recent results on the stability of stationary formulations of the problem and their numerical properties solutions for the Cahn-Hilliard equation in Rd, d ≥ 1. are discussed. For the case d = 1, there are precisely three types of non-constant bounded stationary solutions, periodic solu- Brian R. Wetton, Zhenguo Pan tions, pulse-type (reversal) solutions, and monotonic tran- University of British Columbia sition fronts. These solutions can be categorized as follows: Department of Mathematics the periodic and reversal solutions are both spectrally un- [email protected], [email protected] stable, while the transition fronts are nonlinearly (phase- asymptotically) stable. The cases d ≥ 2 are more complicated, and I will discuss what is known about stationary MS4 solutions in these cases. Particular emphasis will be placed Compactness for the Landau State in Thin Films on planar transition front (or “kink”) solutions. Micromagnetics Peter Howard We study a model for the magnetization in thin ferromag- netic films. It comes as a variational problem, depend- Texas A&M 2 [email protected] ing on two parameters η and ε, for S -valued maps m (the magnetization). We are interested in the behavior of minimizers as η, ε → 0. The minimizers are expected MS3 to concentrate on lines at a scale η with an energy of or- Pulse Dynamics in a Three-Component Model der O(1/η| log η|) and around a vortex at a scale ε with an energy of order O(| log ε|). We prove compactness of mag- We analyze the 3-component RD system developed by netizations in the regime where a vortex energy is more Schenk, Or-Guil, Bode, and Purwins. The system consists important than a line energy. of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard in- Radu Ignat hibitor. It is a prototype 3-component system that gen- Universite Paris 11 (Orsay) erates rich pulse dynamics and interactions. We establish [email protected] the existence and stability of stationary 1-pulse and 2-pulse solutions, as well as of travelling 1-pulse solutions. Also, Felix Otto we analyze various bifurcations, including the SN bifurca- University of Bonn tions in which they are created, as well as the bifurcation Germany from a stationary to a travelling pulse. For 2-pulse solu- [email protected] tions, we show that the third component is essential, since the reduced bistable 2-component system does not support them. Finally, as we illustrate with numerical simulations, MS4 these solutions form the backbone of the rich pulse dynam- 2-D Stability of the Néel Wall ics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among We are interested in thin–film samples in micromagnetism, where the magnetization m is a 2–d unit–length vector 46 PD07 Abstracts

field. More precisely we are interested in transition layers the 1-d Allen Cahn equation. The central ingredient is a which connect two opposite magnetizations, so called Néel nonlinear energy-energy-dissipation relationship. walls. We prove stability of the 1–d transition layer under Maria G. Westdickenberg 2–d perturbations. This amounts to the investigation of Georgia Institute of Technology the following singularly perturbed energy functional: [email protected],edu 2 1 −1/2 2 E2d(m)= |∇m| dx + |∇ ∇·m| dx. Felix Otto 2 University of Bonn Germany The topological structure of this two–dimensional problem [email protected] allows us to use a duality argument to infer the optimal lower bound. The lower bound relies on an –perturbation MS5 of the following logarithmically failing interpolation inequality Numerical Stencils for Large Scale Wave Propaga- tion Problems 1/2 2 |∇ φ| dx ≤ C sup |φ| |∇φ| dx. We have developed numerical schemes for the acoustic wave equation that minimize the dispersion error. To control complexity in large 3D computations we are forced to use only a few grid points per wavelength. The resulting dis- Hans Knuepfer persion is a larger source of error than order. In geophysics, University of Bonn the wave equations have discontinuous velocity and den- [email protected] sity functions, for instance, at the water-solid interface and also rock-salt interface. When the numerical stencil crosses a discontinuity, we need to consider modifications to the MS4 weights in the stencil, so that we get the correct reflection Z-Invariant Micromagnetic Configurations in coefficients at the discontinuities. We will review analytic Cylindrical Domains and numerical results. In this presentation, we have proposed a rather complete Phillip Bording study of the micromagnetic model when the configurations Memorial Univeristy of Newfoundland are assumed to be vertically invariant, the (fixed) domain Department of Earth Sciences is cylindrical and when the exchange constant vanishes to [email protected] 0. The result is given as a Γ− convergence theorem and when the external magnetic field has a small enough mag- Andreas Atle nitude, it fully explains the Van den Berg construction. Dept of Earth Sciences Moreover, the hypothesis of a small exterior field enables Memorial University of Newfoundland us to catch the (form of) the next order term in energy. [email protected] This term is achieved with a kind of Bloch walls instead of Néel walls for thin films. From a physical point of view, it is known that typical walls in thick films are the so-called MS5 “assymetric Bloch walls” which consist in a Bloch wall in- Maximization of the Quality Factor of an Optical side the thickness of the sample closed by Néel caps on Resonator the top and bottom parts of wall. Thus, our results may seem physically meaningless. On the other hand, the nu- We consider resonance phenomena for the scalar wave merical computations given in show that in thick films, the equation in an inhomogeneous medium. Resonance is a magnetization is vertically invariant in the core part of the solution to the wave equation which is spatially localized wall and the top and bottom Néel caps are small in thick- while its time dependence is harmonic except for decay ness. Hence, our model could well explain the core part due to radiation. The decay rate, which is inversely proof the (asymetric) Bloch wall, the Néel parts being caught portional to the qualify factor Q, depends on the material energetically by the fact that our configurations are not properties of the medium. In this work, the problem of de- tangent to the (horizontal) boundaries on the top and bot- signing a resonator which has high Q (low loss) is consid- tom parts. Indeed, if most of the configuration is vertically ered. High Q resonators are desirable in a variety of appli- invariant, our ansatzes become relevant. cations, including photonic band gap devices. We demonstrate a numerical optimization approach to find high Q Stephane Labbe structures. Universite Paris-Sud Orsay [email protected],fr Chiu-Yen Kao Dept. of Mathematics Ohio State University MS4 [email protected] Slow Motion of Gradient Flows Fadil Santosa Sometimes physical systems exhibit “dynamic metastabil- School of Mathematics ity,” in the sense that states get drawn toward so-called University of Minnesota metastable states and are trapped near them for a long [email protected] time. A familiar example is the one-dimensional Allen Cahn equation. In this work, we give sufficient conditions for a gradient flow system to exhibit metastability. We MS5 then apply the abstract result to give a new analysis of A Fast Phase Space Method for Computing Creep- PD07 Abstracts 47

ing Waves Recent results in the analysis of solutions to the benchmark problem in blood flow will be presented and recent develop- Creeping waves can give an important contribution to the ments in the numerical algorithm design will be mentioned. solution of medium to high frequency scattering problems. Applications involving certain cardiovascular interventions They are generated at the shadow lines of the illuminated will be shown. Colalborators: Dr. Z. Krajcer and Dr. scatterer by grazing incident waves and propagate along D. Rosenstrauch (Texas Heart Institute), Dr. C. Hartley geodesics on the scatterer surface, continuously shedding (Baylor College of Medicine), Prof. R. Glowinski, Prof. diffracted waves in their tangential direction. In this talk T.W. Pan, Prof. G. Guidoboni (University of Houston), we show how the wave propagation problem can be formu- Prof. A. Mikelic (University of Lyon 1, FR), Prof. J. Tam- lated as a partial differential equation (PDE) in a three- baca (University of Zagreb, CRO) dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains informa- Suncica Canic tion about all possible creeping waves. This information Department of Mathematics includes the phase and amplitude of the field, which are University of Houston extracted by a fast post-processing. Computationally the [email protected] cost of solving the PDE is less than tracing all rays individ- ually by solving a system of ordinary differential equations. We consider an application to mono-static radar cross sec- MS6 tion problems where creeping waves from all illumination Proof of the Existence of a Solution to a Fluid- angles must be computed. The numerical results of the fast Structure Interaction Problem in Blood Flow phase space method and a comparison with the results of ray tracing are presented. We prove the existence of a solution to a free boundary problem modeling blood flow in viscoelastic arter- Olof Runborg ies.The model equations are obtained using asymptotic KTH, Stockholm, Sweden reduction from the incompressible, viscous Navier-Stokes [email protected] equations,coupled with the linearly viscoelastic membrane equations. The reduced, effective equations are defined on Mohammad Motamed the cylindrical geometry. The resulting problem is nonlin- Department of Computer Science and Communication ear with a free boundary. The existence of a weak solution Royal Institute of Technology (KTH) is obtained by using the fixed point arguments. [email protected] Taebeom Kim Department of Mathematics MS5 University of Houston Efficient Numerical Methods Based on Wiener [email protected] Chaos Expansion for Stochastic Maxwell Equations

We propose a new stochastic model for general spatially MS6 incoherent sources with applications in photonic crystal. Title Not Available at Time of Publication The model naturally incorporates the incoherent property and leads to stochastic Maxwell equations. Fast numeri- Abstract not available at time of publication. cal methods based on Wiener Chaos Expansions (WCE), Alfio Quarteroni which convert the random equations into coupled system of Ecole Pol. Fed. de Lausanne deterministic equations, are developed. The new methods Alfio.Quarteroni@epfl.ch can achieve 2 order of magnitude faster computation time over the standard method in photonic crystal spectrometer design. MS6 Haomin Zhou A Fictitious Domain Method for Simulating Vis- Georgia Tech cous Flow in a Constricted Elastic Channel Subject School of Mathematics to a Uniform External Pressure [email protected] Cardiovascular illness is most commonly caused by a constriction, called a stenosis. A nonlinear mathematical MS6 model with a free moving boundary is used to model elastic wall with constriction subjected to a uniform external pres- Fluid-Structure Interaction in Blood Flow: An sure and a prescribed pressure drop. A fictitious domain Overview method combined with operator splitting techniques and The focus of this minisymposium and of this talk will be finite element method has been developed to simulate vis- on the analysis and computation of fluid-structure inter- cous incompressible flow in the above channel. Numerical action in blood flow. Understanding solutions to moving- results will be presented in this talk. boundary problems describing fluid-structure interaction Tong Wang between blood flow and arterial walls is important in under- Department of Mathematics standing the mechanisms leading to various complications University of Houston in cardiovascular function. Although fascinating progress [email protected] has been made in some areas of modeling and simulation of the human cardiovascular system many of the basic difficulties remain open and will continue to present major MS7 challenges in the years to come. The speaker will give an Free Surface Problems in Irrotational 2D and 3D overview of the main problems and difficulties associated with the study of fluid-structure interaction in blood flow. 48 PD07 Abstracts

Fluids known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of We consider the irrotational water wave and the vortex those infinite dimensional manifolds. The surface tension sheet. We describe a formulation of the problems using produces stronger conservative forces than the instabili- special parameterizations and convenient variables. In 2D, ties and thus regularizes the surface evolution. A scale of the parameterization is by arclength and in 3D the pa- functionals as ”energies” are defined and they bound high rameterization is isothermal. The formulation leads to a Sobolev norms of the velocity field as well as the mean quasilinear hyperbolic system of evolution equations for the curvature of the fluid boundary. Thus we establish the vortex sheet (with surface tension) or for the water wave regularity of the solutions for a short time depending on (either with or without surface tension). Having written the initial data. In the one fluid case, the energy esti- the evolution equations in this way, we are then able to mates is uniform as the surface tension tends to zero if make energy estimates. In the water wave case, these es- the Rayleigh-Taylor sign condition is satisfied. This justi- timates are uniform in surface tension, allowing us to take fies the zero surface tension problem as the limit of small the limit as surface tension tends to zero. This gives a new surface tension problem. proof of well-posedness of the irrotational water wave in 3D. Most of this is joint work with Nader Masmoudi. If Chongchun Zeng possible, there will be some discussion of singularity for- Georgia Tech mation for some of these problems. [email protected]

David Ambrose Jalal Shatah Clemson University New York University [email protected] [email protected]

MS7 MS8 Well-Posedness of the Free-Boundary Compress- AeroElastic Flutter Boundary Expansion with ible Euler Equations in Vacuum Boundary Control

We prove well-posedness for the free-boundary compress- No aeroelastic system is stable; and cannot be be stabilized ible Euler equations in vacuum. In this setting, the density with controls (Linear or Non-Linear) on the structure. The vanishes on the boundary, and leads to a degenerate non- possibility of increasing the flutter speed with self-straining linear wave equation for the Jacobian determinant of the actuators is discussed. Lagrangian coordinate. This is joint work with D. Coutand and H. Lindblad. A. V. Balakrishnan University of California, Los Angeles Steve Shkoller Flight System Research Center Univiversity of California, Davis [email protected] Department of Mathematics [email protected] MS8 MS7 Boundary Control and Stability in Dynamic Fluid Structure Interaction Models Effect of Vorticity on Steady Water Waves Fluid structure interaction comprising of Navier-Stokes We compute 2D finite-depth steady periodic water waves and dynamic system of elasticity is considered . The cou- with general vorticity and large amplitude. In particular, pling holds on interface between fluid and solid and is pre- the effect of a shear surface layer, which could be produced scribed via matching conditions imposed on Cauchy Polya by the action of wind, is investigated. We find that the stress tensors and velocities. Problems to be discussed are: maximum amplitude increases as a function of vorticity. Stability and boundary uniform stabilization of the non- This occurs along the connected set of water waves up to a linear model. Optimal Bolza boundary control problem possible stagnation point in the interior or at the bottom. defined for linearization of the model Optimal rates of sin- Walter Strauss,JoyKo gularity (blow up) of controls and of transfer function will Brown University be given. [email protected], [email protected] Irena Lasiecka University of Virginia, Charlottesville MS7 Department of Mathematics [email protected] Free Boundary Problems of the Euler Equation and Hydrodynamical Instabilities MS8 We consider the evolution of free surfaces of incompressible and invicid fluids. Neglecting the gravity, we are in- Nonlocal Models of Transport in Multiscale Porous terested in the cases of 1.) the motion of a droplet in the Media vacuum with or without surface tension and 2.) the mo- We discuss ”usual” model for advection-diffusion- tion of the interface between two fluids with surface ten- dispersion in porous media and its validity in the presence sion. The evolution of these fluid boundaries and the ve- of multiple scales especially when they are not well sepa- locity fields is determined by the free boundary problem rated. Problem is motivated by recent experimental results of the Euler’s equation. Each of these problems can be by Zinn, Haggerty et al. We present a new model based on considered in a Lagrangian formulation on an infinite di- variational duality principles extending the classical dou- mensional Riemannian manifold of volume preserving dif- ble porosity models. Model preserves mass and has discrete feomorphisms. In the absence of surface tension, the well- PD07 Abstracts 49

and continuous variants. Some computational realizations - band drift diﬀusion equations with an inter - band col- emphasizing transition between scales and nonlocal eﬀects lision operator which dissipates the quantum mechanical similar to non-Fickian dispersion will be given. equivalent of a thermodynamic entropy.

Malgorzata Peszynska, Ralph Showalter Christian Ringhofer Department of Mathematics Arizona State University Oregon State University [email protected] [email protected], [email protected] MS10 Electrical Impedance Tomography with Resistor MS8 Networks Possio Integral Equation in Aeroelasticity (Asymp- totical Version) We present an inversion method that handles the se- vere ill-posedness of Electric Impedance Tomography by The Possio integral equation in Aeroelasticity relates the proper sparse parametrization of the unknown conductiv- pressure distribution over a slender wing in subsonic com- ity. Specifically, we consider finite volume grids, which pressible air flow to the normal velocity (downwash). De- are resistor networks in disguise, where the resistors are rived by C.Possio in 1938, it is crucially important in sta- averages of the conductivity over grid cells. By interpret- bility analysis. In spite of enormous amount of numerical ing the resistor network that matches the measurements as papers with strong results, the fundamental problem of such discretization we are able to efficiently estimate the solvability of the equation has not been solved. We report conductivity. Our method can also incorporate a priori preliminary results towards solvability. information about the solution, if available. Marianna Shubov Liliana Borcea University of New Hampshire Rice University Department of Mathematics and Statistics Dept Computational & Appl Math [email protected] [email protected]

Vladimir Druskin MS9 Schlumberger-Doll Research Quantum Fokker-Planck Models: Global Solutions, [email protected]ﬁeld.slb.com Steady States, and Large-Time Behavior

Dissipative open quantum systems like the quantum- Fernando Guevara Vasquez Fokker-Planck (QFP) model are important for quantum Stanford University Brownian motion and numerical simulations of nano- [email protected] semiconductor devices. The evolution equation can be written in the Wigner phase-space framework or equaiva- MS10 lently for density matrix operators. We discuss global-in- time wellposedness of the nonlinear QFP-Poisson model Solving the Electrical Impedance Tomography using density matrices, steady states of the linear QFP Problem with Optimization Techniques equation in a given confinement potential in the Wigner Electrical Impedance Tomography is an inverse problem framework, and establish large-time convergence (with an of determining the conductivity function inside a conduc- exponential rate). tive domain from the simultaneous measurements of volt- Anton Arnold ages and currents on the boundary of the domain. The Vienna University of Technology (Austria) problem is solved in three steps by first obtaining an op- [email protected] timal finite difference grid for discretization of the PDE, then solving a discrete EIT problem for a resistor network, and finally finding a parameterized continuous conductiv- MS9 ity. All three subproblems can be formulated as uncon- Title Not Available at Time of Publication strained optimization problems, so that we can apply a modified Gauss-Newton method. Certain regularization Abstract not available at time of publication. techniques are considered including adaptive SVD trunca- tion of the Jacobian and a penalty function approach. We Maria Gualdani test our method on a well-studied case of a conductive disk The University of Texas at Austin in 2D for which a direct method of finding optimal grids [email protected] and resistors in a network is available. Then we generalize our approach to a quasi-3D setting with domain of interest being an infinite cylinder and the conductivity which does MS9 not depend on Z coordinate. Semiclassical Transport Models in Thin Slabs Alexander Mamonov Quantum transport in simulation domains with a small Rice University aspect ratio is often approximated by semiclassical limit [email protected] equations, so called sub - band models, which assume that the transport picture is classical only in one direction. This talk is concerned with sub - band models in the presence of MS10 strong forces in the quantum mechanical direction. After On Combining Model Reduction and Gauss- giving an overview of the general approach to sub - band modeling, we derive and discuss a system of coupled sub 50 PD07 Abstracts

Newton Algorithms for Inverse PDE Problems MS11 Solitary Waves and Their Linear Stability in We suggest an approach to speed up the Gauss-Newton so- Weakly Coupled KdV Equations lution of inverse PDE problems by minimizing the number of forward problem calls. The acceleration is based on ef- Abstract not available at time of publication. fective incorporation of the information from the previous iterations via a reduced order model (ROM). It is designed Douglas Wright with the help of Galerkin and pseudo-Galerkin methods School of Mathematics for self-adjoint and complex symmetric problems respec- University of Minnesota tively. The constructed ROM generates effective multivari- [email protected] ate rational interpolation matching the forward olutions and the Jacobians from the previous iterations. Numerical examples for the inverse conductivity problem for the 3D MS11 Maxwell system show significant accelerations. New Problems in Stability of Viscous Conservation Laws Vladimir Druskin Schlumberger-Doll Research Recent Lyapunov-type theorems established by Mascia- [email protected]field.slb.com Zumbrun, Gues-Metivier-Williams-Zumbrun, Howard- Raoofi-Zumbrun, Lyng-Raoofi-Texiier-Zumbrun, Texier- Mikhail Y. Zaslavsky Zumbrun, and others have reduced the study of nonlinear Schlumberger-Doll Research, stability, asymptotic behavior, and bifurcation of viscous Boston traveling waves in gas dynamics combustion and MHD to mzaslavsky@ridgefield.oilfield.slb.com determination of point spectra of the corresponding linearized operator about the wave: that is, the study of the associated eigenvalue ODE. We discuss the mathematical MS11 issues arising in this new landscape and propose general Galloping Instability of Viscous Shock Waves strategies for their analysis, by a balanced approach combining numerical computation, asymptotic ODE/singular Abstract not available at time of publication. perturbation theory, and classical energy estimates. Benjamin Texier Kevin Zumbrun Institut de Mathématiques de Jussieu Indiana University Université Paris 7/Denis Diderot USA [email protected] [email protected]

MS11 MS12 Long Tails in the Long Time Asymptotics of Quasi- Title Not Available at Time of Publication Linear Hyperbolic-Parabolic Systems of Conserva- tion Laws Abstract not available at time of publication.

We consider the long-time behaviour of solutions of sys- Andrew J. Christlieb tems of conservation laws. Liu and Zeng have given a Michigan State Univerity detailed exposition of the leading order asymptotics close Department of Mathematics to a constant background state. We extend their analy- [email protected] sis by examining higher order terms in the asymptotics in the framework of the so-called two dimensional p-system, though our methods and results should also apply to more MS12 general systems. We give a constructive procedure for ob- Well-Posedness of Spatially In- taining these terms, and we show that their structure is homogeneous Becker-Doering Kinetic System and determined by the interplay of the parabolic and hyper- Vanishing Diffusion Limit bolic parts of the problem. In particular, we prove that the corresponding solutions develop long tails that precede We prove the existence, uniqueness, and stability for the the characteristics. Becker-Doering infinite dimensional dynamic system of merging and splitting particles in the spatially inhomoge- Guillaume Van Baalen neous case with spatial transport and diffusion. Our ap- Boston University proach and new estimates based on a modified version of Dept. of Mathematics and Statistics maximum principle allow the consideration of broad classes [email protected] of unbounded kinetic coefficients. The results hold both for parabolic, or hyperbolic, or mixed parabolic-hyperbolic Gene Wayne system. Moreover, we prove that the solution with van- Boston University ishing diffusion tends to the solution of purely convection [email protected] problem. Pavel B. Dubovski Nikola Popovic Stevens Institute of Technology Boston University [email protected] Center for BioDynamics and Department of Mathematics [email protected] MS12 A Multi-Species Kinetic Model for Microelectron- ics Manufacturing Processes in the Transition and PD07 Abstracts 51

Knudsen Regimes and mathematical questions. We will address some of these questions in the talk. We present a model for the gas flow inside chemical reactors used in the microelectronics industry to produce computer David Gérard-Varet chips. The ranges of pressures on the one and of the length ENS Paris scales of interest on the other hand necessitate the use of a [email protected] kinetic model for this problem. We will present the kinetic transport and reaction model developed for this problem and show results with a focus on multi-scale effects in time. MS13 Spectral Stability of Traveling Water Waves Matthias K. Gobbert University of Maryland, Baltimore County The motion of the free surface of an ideal fluid under the [email protected] effects of gravity and capillarity arises in a number of problems of practical interest, consequently, the reliable and accurate numerical simulation of these ”water waves” is MS12 of central importance. Recently, in collaboration with F. Title Not Available at Time of Publication Reitich, the author has developed a new, efficient, stable and high-order Boundary Perturbation scheme for the nu- Abstract not available at time of publication. merical simulation of traveling solutions of the water wave equations. In this talk we will discuss the extension of this Robert Strain Boundary Perturbation technique to address the equally Harvard University important topic of dynamic stability of these traveling wave [email protected] forms. More specifically, we describe a new numerical algorithm to compute the spectrum of the linearized water MS13 wave problem as a function of the amplitude of the traveling wave. In addition, we will present numerical results for Shallow Water Approximation: A Mathematical two and three dimensional waves subject to quite general Justification. perturbations. In this talk, we will focus on a Shallow Water approxi- Dave Nicholls mation coming from the Navier-Stokes equations with free University of Illinois, Chicago, surface. We will propose a mathematical justification of USA such simplification based on a careful study of the approxi- [email protected] mation process and based on the use of adequate estimates. This is a joint work with P. Noble (Institut Camille Jordan, Lyon 1, France). MS14 Didier Bresch Regularity of the Solutions to Higher Order Elliptic UniversitédeSavoie Equations on Non-Smooth Domains [email protected] We explore the connections between the properties of the solution to the Dirichlet problem for a higher order elliptic MS13 equation and the geometry of the domain. In the case of the biharmonic equation on an arbitrary 3-dimensional do- KdV Cnoidal Waves are Linearly Stable main we prove that the gradient of the solution is bounded. Going back to considerations of Benjamin (1974), there The result is sharp in the sense that the solution is not has been significant interest in the question of stability for necessarily continuously differentiable. In this connection, the stationary periodic solutions of the Korteweg-deVries we show that certain geometrical properties of the domain equation, the so-called cnoidal waves. In this paper, we yield necessary and sufficient conditions for the continuity exploit the squared-eigenfunction connection between the of the gradient. When the domain is convex, we show that linear stability problem and the Lax pair for the Korteweg- the second derivative of the solution is bounded in all di- deVries equation to completely determine the spectrum mensions. Moreover, we derive new results regarding the of the linear stability problem for eigenfunctions that are well-posedness of the corresponding boundary value prob- bounded on the real line. We find that this spectrum is lem on Lipschitz domains. confined to the imaginary axis, leading to the conclusion Vladimir Maz’ya of spectral stability. An additional completeness argument The Ohio State University allows for a statement of linear stability. Linköping University, University of Liverpool Bernard Deconinck [email protected] University of Washington [email protected] Svitlana Mayboroda Mathematics, Ohio State University Brown University MS13 [email protected] Wall Laws at a Boundary with Non Periodic Irreg- ularity MS14 The concern of this talk is the effect of a rough wall on Regularity for Elasticity and Laplacean and FEM an incompressible fluid. Mathematical studies of this effect have been carried out under the strong assumptions of We prove a well posedness result for anisotripic elasticity periodic irregularities. More realistic configurations (like in weighted Sobolev spaces. Then we discuss the relevance a random distribution of irregularities) raise both physical of this result for improving the convergence of the Finite 52 PD07 Abstracts

Element Method. thin gaps between neighboring particles in the suspension. Victor Nistor, Mazzucato Yuliya Gorb Pennsylvania State University Texas A&M University [email protected] [email protected]

Alexei Novikov MS14 Penn State University Quadrature for Meshless Methods and Generalized Mathematics Finite Element Methods [email protected] The creation of eective quadrature schemes for Meshless Methods and Generalized Finite Element Methods is an Leonid Berlyand important problem. We discuss such quadrature schemes, Penn State University and prove an energy-norm error estimate. The major hy- Eberly College of Science pothesis is that the quadrature stiness matrix has zero [email protected] row sums, a hypothesis that can be easily achieved by a simple correction of the diagonal elements. This talk is MS15 based on joint work with Ivo Babuska, Uday Banerjee and Qiaoluan(Helen) Li. Asymptotic and Numerical Approximation of the Resonances of Thin Photonic Structures John Osborn Department of Mathematics We study the computation of the resonances of thin dielec- University Of Maryland, College Park tric structures by using both asymptotics and PML. Of [email protected] great interest in the study of photonic band gap materials is when these structures are periodic with a defect. We derive a limiting resonance problem as a the thickness goes to MS14 zero, and for the case of a simple resonance find a first order Essential Boundary Conditions for Elliptic Equa- correction. The limiting problem and correction have one tions in the Generalized Finite Element Method dimension less, which can make the approach very efficient. Convergence estimates are proved for the asymptotics. We One of the major problems in the implementation of the then proceed to compute these resonances by two very dif- Generalized Finite Element Method is the enforcement of ferent methods: the asymptotic method by solving a dense, essential (Dirichlet type) boundary conditions. In this talk but small, nonlinear eigenvalue problem; and PML, which we address this topic in the case of boundary value prob- yields a large, but linear and sparse generalized eigenvalue lems involving second-order elliptic operators. In particu- problem. Both methods reproduce a photonic band gap lar, we are interested in boundary data with low regularity type mode found previously by finite difference time do- (possibly a distribution). main methods. Nicolae Tarfulea Jay Gopalakrishnan Mathematics, Purdue University Calumet University of Florida [email protected] [email protected]fl.edu

Shari Moskow MS15 University of Florida Transport in Porous Media. Microgemetric Eﬀects Department of Mathematics in Capillary Networks [email protected]ﬂ.edu

We model porous media as capillary networks and develop Fadil Santosa methods to study the eﬀect of the network geometry (i.e. School of Mathematics “microgeometry”) on the “macroscopic” transport proper- University of Minnesota ties of the media. In particular, we study solute transport, [email protected] energy transport and ﬁnes migration and clogging.

Guillermo Goldsztein MS15 Georgia Tech Atlanta, GA Asymptotic Analysis of Boundary Layer Correctors [email protected] and Applications. Abstract not available at time of publication. MS15 Daniel Onofrei Singularity of the Overall Viscous Dissipation Rate Mathematics Department of Highly Concentrated Suspensions Rutgers University [email protected] We present a two-dimensional mathematical model of a highly concentrated suspension of rigid particles in an incompressible Newtonian ﬂuid. The overall viscous dissipa- MS16 tion rate W of such a suspension exhibits a singular behav- Mode Competition in Extended Domains ior. We obtain all singular terms in the asymptotics of W as an interparticle distance tends to zero. Our analysis al- In many extended systems, governed by PDEs, the tran- lows for a complete qualitative description of microﬂows in sition is from a spatially uniform one-dimensional state to a two-dimensional periodic pattern. Classically, these PD07 Abstracts 53

systems have been simulated in periodic domains of the [email protected] wavelength of the pattern. Nevertheless, this periodicity can be broken by secondary bifurcations or in laboratory experiments due to imperfections. Recent computations MS17 accounting for spatial non-homogenities and the transition Sharp Well-Posedness and Long-Time Bounds for to spatio-temporal complexity observed in hydrodynamic the BBM Equation systems will be discussed. We discuss recent results on well posed-ness and long time Marc Avila bounds for a certain nonlinear dispersive wave equation. Universitad Politecnica de Catalunya Applied Physics Jerry Bona [email protected] University of Illinois at Chicago [email protected]

MS16 The Role of Spatio-Temporal Symmetries in the MS17 Transition Towards Turbulence On Partial Regularity for the Navier-Stokes Equa- tions Transitions from 2D to 3D are fundamental steps towards turbulence. A classic problem is periodically shedding bluff A classical result of Caffarelli, Kohn, and Nirenberg states body wakes; the transition breaks spanwise O(2) symme- that the one dimensional Hausdorff measure of singularities try. Considering only the O(2) symmetry breaking is insuf- of a suitable weak solution of the Navier-Stokes system is 5/2+q ficient as there are additional spatio-temporal symmetries zero if the force belongs to the space L where q> with important dynamic consequences; their joint actions 0 is arbitrary. We present a shorter proof of the partial must be considered. The transitions to spatio-temporal regularity result which requires the force to belong only to complexity in a variety of problems related by their sym- L5/3+q where q>0 is arbitrary. metry group are described. Igor Kukavica Juan Lopez USC Arizona State University [email protected] Department of Mathematics and Statistics [email protected] MS17 N-Vortex Dynamics on a Rotating Sphere Coupled MS16 to a Background Field Organizing Centers and Their Connections We describe a model for the evolution of N-point vortices The competition between different instability mechanisms embedded in a background flow field that initially corre- leads to very complicated dynamics, including homoclinic sponds to solid-body rotation on the sphere. The full ‘em- and heteroclinic phenomena. The dynamics are organized bedded’ dynamical system is a discretization of the Eu- by codimension-two bifurcations, where modes due to the ler equations for incompressible flow on a rotating spheri- different instability mechanisms bifurcate simultaneously. cal shell, hence a barotropic model of the one-layer atmo- The dynamics are explored using a three-dimensional sphere. We show how the coupling of the N-point vortices Navier-Stokes solver, which is also implemented in a num- (in ring formation) to the background Rossby waves breaks ber of invariant subspaces in order to follow some unstable the integrability of the problem when the center of vortic- solution branches and obtain a fairly complete bifurcation ity vector associated with the ring is misaligned with the diagram of the mode competitions. axis of rotation of the background field. Joint work with T. Sakajo. Francisco Marques Univ. Polit Paul K. Newton ‘ecnica de Catalunya U Southern California Applied Physics [email protected] [email protected]

MS17 MS16 Time Dependent Darcy-Stokes Flow with Beavers- Travelling Waves in Modulated Rotating Convec- Joseph Interface Boundary Condition tion We show that the time dependent coupled Darcy-Stokes Recent experiments in rotating convection have shown that system is well-posed with the classical Beavers-Joseph the chaotic Kuppers-Lortz dynamics can be suppressed by interface boundary condition instead of the modified small amplitude modulations of the rotation rate. Axisym- Beavers-Joseph-Saffman-Jones interface boundary condi- metric target patterns develop into axisymmetric travelling tion. Convergence of fully discretized finite element wave states as the modulation amplitude and Rayleigh schemes will be presented. Generalization to nonlinear number are increased. Using fully 3D time-dependent models as well as applications to karst aquifer will be dis- Navier-Stokes-Boussinesq equations with physical bound- cussed. ary conditions, the experimental results are reproduced numerically gaining physical insight into the responsible Xiaoming Wang mechanism, associated with a SNIC bifurcation. Florida State University e-mail: TBA Antonio M. Rubio Arizona State University 54 PD07 Abstracts

Yanzhao Cao the Oseen-Frank theory, in which the mean orientation of Department of Mathematics the rod-like molecules is modelled through a unit vector Florida A and M U. field n. This theory has the apparent drawback that it does [email protected] not respect the head-to-tail symmetry in which n should be equivalent to -n, that is, instead of n taking values in Fei Hua the unit sphere S2, it should take values in the sphere with Department of Mathematics opposite points identified, i.e. in the real projective plane Florida State University RP2. The de Gennes theory respects this symmetry by [email protected] working with the tensor Q=s(nn- 1/3 Id). In the case of a non-zero constant scalar order parameter s the de Gennes Max Gunzburger theory is equivalent to that of Oseen-Frank when the di- Florida State University rector field is orientable. We report on a general study School for Computational Sciences of when the director fields can be oriented, described in [email protected] terms of the topology of the domain filled by the liquid crystals, the boundary data and the rate of blow-up of possible singularities. We also analyze the circumstances MS18 in which the non-orientable configurations are energetically Well-Posed Boundary Value Problems for Smectic favoured over the orientable ones. This is joint work with Liquid Crystals John Ball.

We prove existence of minimizers to variational models John Ball for smectic liquid crystals with various types of prescribed University of Oxford, United Kingdom boundary values of interest for applications. [email protected] Patricia Bauman Arghir Zarnescu Purdue University University of Oxford [email protected] [email protected]

Daniel Phillips Department of Mathematics, Purdue University MS19 [email protected] Title Not Available at Time of Publication Abstract not available at time of publication. MS18 Sujeet Bhat Energies of Ferroelectric Liquid Crystal Elastomers IMA and Filaments [email protected] Models of elastomer liquid crystals are discussed and analyzed, with emphasis on those presenting ferroelectric prop- MS19 erties. We discuss well-possedness of boundary value problems and present an application to filament geometry. Electromagnetic Transmission Spectra in Periodic Hole Arrays Maria-Carme Calderer Deparment of Mathematics Several recent experimental studies have investigated the University of Minnesota, USA transmission resonances which occur at certain frequen- [email protected] cies when a thin perforated conducting plate is illuminated by an electromagnetic wave. There is hope that this phenomenon will prove useful in constructing filters and other MS18 devices for manipulating radiation in the terahertz range. Modeling of Ferroelectricity in the Chiral Smectic We will describe an analytical and a numerical approach Liquid Crystals and Bent Core Molecules for studying this problem. Numerical results will be presented, along with some clues as to how the transmission In this presentation, we study the structure of ferroelectric spectrum depends on the pattern and shape of the holes. liquid crystals with interactions of polarizations. Such a material is very interesting with respect to practical ap- David C. Dobson plications, such as a fast switching between the active and University of Utah inactive state. We introduce a model for ferroelectric liquid [email protected] crystals including nonlocal energy term and discuss equilibrium configurations of the energy. We will focus on periodic MS19 configurations which appear in the physics literature. Title Not Available at Time of Publication Jinhae Park Purdue University Abstract not available at time of publication. [email protected] Alexei Novikov Penn State University MS18 Mathematics [email protected] Orientable and Non-Orientable Director Fields For Liquid Crystals MS19 Uniaxial nematic liquid crystals are often modelled using Modeling Brittle Fracture Through an Extension PD07 Abstracts 55

of Continuum Mechanics to the Nanoscale MS20 Stability of Large-Amplitude Viscous Shocks Described in this talk is a new approach to modeling fracture in brittle materials based upon an extension of con- By a combination of Evans function methods, spectral en- tinuum mechanics to the nanoscale. The approach accepts ergy estimates, and boundary layer analysis, we are able bulk continuum constitutive properties away from material to prove stability of (1-D) large-amplitude viscous shocks interfaces but corrects material behavior for effects due to in isentropic Navier-Stokes (or the p-system with real vis- long range intermolecular forces in the vicinity of material cosity). We then describe our latest results in generalizing interfaces which in the context of fracture are the fracture this work to other systems, including the full (ideal gas) surfaces. The correction is in the form of a nonlinear mu- Navier-Stokes equations. Finally, we give an overview of tual force in the bulk differential momentum balance equa- the strategies used and how they may be applied more tions and a nonlinear boundary condition coming from the generally. jump momentum balance across fracture surfaces. Both correction terms derive from atom-atom interaction poten- Jeffrey Humpherys tials. It is proved that this new modeling paradigm avoids Brigham Young University the crack edge singularities found in classical elastic frac- jeff[email protected] ture theories. Kevin Zumbrun Jay R. Walton Indiana University Department of Mathematics USA Texas A&M [email protected] [email protected]

K.-B. Fu MS20 Department of Mathematics On Darcy’s Law for Compressible Porous Medium Texas A&M University Flows [email protected] The motion of compressible ﬂow through porous medium J. Slattery can be modeled by compressible Euler equations with fric- Department of Aerospace Engineering tional damping. Physically, Darcy’s law is valid for large Texas A&M University time and the density obeys the porous medium equation. [email protected] In 1-d, the mathematical veriﬁcation on this conjecture was given by Hsiao & Liu in 1992 for small smooth solutions. T. Sendova We will give a proof for any physical weak solutions. Department of Mathematics Ronghua Pan Texas A&M University Georgia Tech. [email protected] [email protected]

MS20 MS20 Nonlinear Stability of Time-Periodic Viscous Shock Stability of Undercompressive Viscous Shock Pro- Waves ﬁles of Hyperbolic-Parabolic Systems

Time-period viscous Lax shocks are shown to be nonlin- We show for viscous shock profiles of arbitrary amplitude early stable. The result is obtained by formulating the and type, for hyperbolic-parabolic systems of conserva- linear evolution using a contour integral and using the com- tion laws, that necessary spectral (Evans function) condi- ponents of the Green’s function associated to the neutral tions for linearized stability are also sufficient for linearized directions to define time-dependent shifts of the phase and and nonlinear phase-asymptotic stability, yielding detailed spatial translate. Using the integral formulation of solu- pointwise estimates and sharp rates of convergence in Lp, tions, we show these shifts converge to asymptotic limits 1 ≤ p ≤∞. and the solution converges to the resulting space and time translate of the periodic shock. Kevin Zumbrun Indiana University Kevin Zumbrun USA Indiana University [email protected] USA [email protected] Mohammadreza Raoofi University of Minnesota Bjorn Sandstede raoofi@umn.edu University of Surrey [email protected] MS21 Margaret Beck Diffusive Expansion in Kinetic Theory University of Surrey Department of Mathematics In this talk, the diffusive expansion to the Vlasov-Maxwell- [email protected] Boltzmann system is discussed. Such an expansion yields a set of dissipative new macroscopic PDEs and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a uni- 56 PD07 Abstracts

fied nonlinear energy method and it guarantees the global lems in time validity of such an expansion up to any order. We develop an analytical tool which is adept for detecting Juhi Jang shapes of oscillatory functions, is useful in decomposing Institute for Advanced Study homogenization problems into limit-problems for kinetic [email protected] equations, and provides an efficient framework for the validation of multi-scale asymptotic expansions. We apply it first to a hyperbolic homogenization problem and trans- MS21 form it to a hyperbolic limit problem for a kinetic equation, Mathematical Problems in Reactive and Inert Ki- and establish conditions determining an effective equation netic Models for Dense Gases and counterexamples for the case that such conditions fail. Second, when the kinetic decomposition is applied to the In the kinetic theory of simple reacting spheres (SRS), the problem of enhanced diffusion, it leads to a diffusive limit molecules behave as if they were single mass points with problem for a kinetic equation that in turn yields the ef- two (or more) internal states of excitation. Collisions alter fective equation of enhanced diffusion. the internal states when the kinetic energy associated with the reactive motion exceeds the activation energy. Both Athanasios Tzavaras reactive and non-reactive collision events are considered to University of Maryland be hard–spheres like. Thus, one considers a four compo- [email protected] nent mixture A, B, A∗, B∗, and the chemical reaction of the type Pierre-Emmanuel Jabin ∗ ∗ University of Nice A + B A + B . [email protected] Here, A∗ and B∗ are distinct species from A and B, and we use the indices 1, 2, 3, and 4 for the particles A, B, A∗, ∗ MS22 and B respectively. I assume that mass transfer in reac- Homogenization of Polyconvex Functionals tive collisions satisfies the condition m1 + m2 = m3 + m4, where mi denotes the mass of the i-th particle, i =1,...,4. Using two-scale Young measures, I will describe the zero Reactions take place when the reactive particles are sepa- 1 level set of an effective energy density related to a peri- rated by a distance σ12 = 2 (d1 + d2), where di denotes the odic microstructure. I will use this analysis to show that diameter of the i-th particle. polyconvex energies are not closed with respect to periodic homogenization. The counter-example is based on The SRS, being a natural extension of the hard–sphere col- a rank-one laminated structure obtained by mixing two lisional model, reduces itself to the revised Enskog theory polyconvex functions with p-growth, where p ≥ 2 can be (RET) when the chemical reactions are turned off. Thus, it fixed arbitrarily. I will also discuss a counter-example to can be considered as a good candidate for the RET analog the continuity of the determinant with respect to two-scale of the reacting hard-spheres. convergence.

I will show that the kinetic theory of simple reacting Marco Barchiesi spheres has an H–function. Its functional form depends Carnegie Mellon University on a specific type of the closure relation used in the first [email protected] hierarchy equation. In addition to its physical importance, the H–function is utilized to obtain existence and stability theorems for the corresponding kinetic equations. MS22 Variational Principles in L∞ and Aronsson Equa- Jacek Polewczak tions California State University, Northridge [email protected] Connections between highly nonlinear degenerate PDEs arising as Aronsson equations associated to variational principles for supremal functionals under general PDE con- MS21 straints will be discussed. Treatment of Multiscale Hybrid Determinis- tic/Stochastic Coupled Systems Marian Bocea North Dakota State University A class of model prototype hybrid systems comprised of a [email protected] microscopic Arrhenius surface process describing adsorp- tion/desorption and/or surface diffusion of particles coupled to an ordinary or partial differential equation display- MS22 ing bifurcations triggered by the microscopic process is pre- Quasistatic Evolution in Plasticity with Softening sented and analyzed. These models arise from diverse applications ranging from surface processes and catalysis to In this talk a quasistatic evolution problem in plasticity atmospheric and oceanic sciences. with softening is considered in the framework of small strain associative elastoplasticity. The presence of a non- Alexandros Sopasakis convex term due to the softening phenomenon requires a University of North Carolina, Charlotte nontrivial extension of the variational framework for rate- Mathematics independent problems to the case of a nonconvex energy [email protected] functional. In this case the use of global minimizers in the corresponding incremental problems appears to be not jus- tified from the mechanical point of view. Thus, a different MS21 selection criterion for the solutions of the quasistatic evo- A Kinetic Formulation for Homogenization Prob- lution problem is proposed, which is based on a viscous PD07 Abstracts 57

approximation. This leads to a generalized formulation in Department of Mathematics and Statistics terms of Young measures. This is a joint work with G. Dal McMaster University Maso, A. De Simone, and M. Morini. [email protected] Maria Giovanna Mora SISSA, Trieste, Italy MS23 [email protected] Solitary Waves of the Kadomtsev-Petviashvili Equation with Eﬀects of Rotation

MS22 The rotation-modified Kadomtsev-Petviashvili (RMKP) Epitaxial Growth of Strained Crystalline Films: a equation was formally derived by Roger Grimshaw to Variational Approach describe small-amplitude, long internal waves with weak transverse effects in a rotating channel of shallow water, We consider strained epitaxial films grown on a relatively where the effects of rotation balance with weakly nonlin- thick substrate in the context of plane linear elasticity. ear and dispersive effects. A brief account is given of its Adopting a variational approach, we look at equilibrium formulation and of connections to other model equations in configurations; i.e., at local minimizers of the total energy oceanography and geophysics, for instance, the usual KP functional, which is assumed to be the sum of the energy equation and the Ostrovsky equation. Comments are made of the free surface of the film and the strain energy. In on the local and the global well-posedness of the Cauchy this talk we will focus on some qualitative properties of problem for the RMKP equation. Presented with proofs such configurations that can be rigorously inferred from are the existence of localized, traveling solutions, namely, the model. solitary waves, of the RMKP equation and their symmetry properties, stability. Emphasis is given to the effects of Massimiliano Morini rotation. SISSA Trieste,Italy Robin Ming Chen [email protected] University of Minnesota [email protected] MS22 Yue Liu Nusselt Number Scaling in Thermal Convection University of Texas at Arlington Rayleigh-Benard convection is a paradigm of nonlinear dy- [email protected] namics. One key feature is the enhancement of heat transport as characterized by the Nusselt number, a nondi- Vera Mikyoung Hur mensional parameter whose scaling with respect to the Massachusetts Institute of Technology Rayleigh number has been widely studied. The background [email protected] method is a variational method for producing an upper bound on the scaling relationship. We use the background method together with a novel background temperature pro- MS23 file and a new weighted inequality for the bi-Laplacian op- A Shallow Water Approximation for Water Waves erator to produce the best analytical result to date. This is joint work with Felix Otto In this talk we are concerned with the initial value problem for water waves in arbitrary space dimensions. We will give Maria G. Westdickenberg a mathematically rigorous justification of the shallow water Georgia Institute of Technology approximation for water waves in Sobolev spaces. One [email protected] of the main part of the analysis is to give some uniform estimates for the Dirichlet-to-Neumann map for a scaled Laplace’s equation by using a suitable diffeomorphism. MS23 Nonlinear Water Waves in Random Bathymetry Tatsuo IGUCHI Keio University This talk discusses the behavior of surface water waves in JAPAN fluid domains whose boundaries are not precisely known, [email protected] and therefore treated as random. It is and important topic to coastal engineering as well as to the theory of tsunami propagation. We consider the long-wave asymptotic limit MS23 of nonlinear surface waves over bathymetry given by a sta- Water Waves Over a Random Topography tionary ergodic process. It is a problem of homogenization theory. However it turns out that in the appropriate scal- We discuss the problem of nonlinear wave motion of the ing regime, and in the case in which the bottom variations free surface of a body of fluid over a variable bottom. The are decorrolated on length scales that are long with respect object is to describe the character of wave propagation in a to the surface waves, there remain random, realization de- long wave asymptotic regime. We assume that the bottom pendent effects which are as important as the dispersion of the fluid region can be described by a stationary random and nonlinearity of the problem. The talk will describe process whose variations take place on short length scales. the methods of Hamiltonian averaging, the probability the- It is a problem in homogenization theory. Our principal ory that goes into the analysis of these random processes result is the derivation of effective equations and a consis- and their canonical limits, and the results for surface water tency analysis. This talk constitutes one part in a series waves propagating over a random bottom. This talk should of two talks on this topics, the other talk being given by be considered joint with the presentation of C. Sulem. Walter Craig. Walter Craig Catherine Sulem 58 PD07 Abstracts

University of Toronto studied in a bounded domain with large data. The ex- [email protected] istence and large-time behavior of global weak solution are established. MS24 Dehua Wang Time-Analyticity and Backwards Uniqueness for University of Pittsburgh Weak Solutions of the Navier-Stokes Equations of Department of Mathematics Compressible Flow [email protected]

We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid- MS25 particle trajectories, can be extended as analytic functions Output Tracking Control of a PDE Model of a Fac- of complex time. One important corollary is backwards tory Production Line uniqueness: if two such solutions agree at a given time, then they must agree at all previous times as well as at Semiconductor factory production is essentially a trans- subsequent times. Additionally, analyticity yields sharp port problem that can be modelled by hyperbolic PDEs. estimates for the time derivatives of arbitrary order of so- The associated transport velocity can be modelled as a lutions along particle trajectories. Our analysis depends state equation leading to a non-local nonlinear equation. on a careful study of solutions of a suitably complexified A major practical problem is the determination of the inversion of the Navier-Stokes system written in Lagrangean flux that generates a desired outflux over a given period of coordinates. time. Using a Lagrangian approach we present several different algorithms to solve this problem for different types Eugene Tsyganov, David Hoff of state equations. Indiana University [email protected], hoff@indiana.edu Dieter Armbruster Arizona State University Dept of Mathematics MS24 [email protected] Boundary Behavior of Compressible, Viscous Fluid Flows Michael LaMarca Arizona State University In this talk we will discuss some results on the existence [email protected] of the weak solutions to the Navier-Stokes equations for the motion of compressible, viscous fluid flows in the half- space with the no-slip boundary condition. The emphasis MS25 is made on the boundary behavior of weak solutions with Optimization Techniques For Network Models II the discontinuous density field. We consider optimization problems for production net- Mikhail Perepelitsa works. This talk continues the presentation Optimization Vanderbilt University Techniques For Network Models I. We further investigate [email protected] the optimality system. We present presolve techniques inspired by the continuous dynamics to reduce the computational effort. We show that large–scale instances of the MS24 MIP can be solved only if the new presolve techniques are From the Dynamics of Gaseous Stars to the Incom- used. Furthermore. we show that under certain conditions pressible Euler Equations on the objective functional, the MIP is in fact equivalent to a linear programming problem. We present numerical A model for the dynamics of gaseous stars is introduced results for large-scale production facilities. formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and d Simone Goettlich inviscid limit of the NSP system in the torus T is in- Kaiserslautern University of Technology, FRG vestigated. The convergence of the Navier-Stokes-Poisson [email protected] system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data. MS25 Optimization Techniques For Network Models I Konstantina Trivisa University of Maryland We consider optimization problems arising in the contin- [email protected] uous modeling of production networks. We introduce an optimization problem based on recently established contin- Donatella Donatelli uous model for supply chains. We present an adjoint calcu- University of L’Aquila -Italy lus for solving the optimality system and discuss suitable [email protected] discretizations of this system. For particular discretizations the resulting problem is in fact a MIP. We present extensions to this model including a discretized priority MS24 model as well as numerical results for sample supply chain Global Solution to the Multi-Dimensional networks. Equations of Compressible Magnetohydrodynamic Flows Michael Herty Kaiserslautern University of Technology, FRG An initial-boundary value problem for the three- [email protected] dimensional compressible magnetohydrodynamic flow is PD07 Abstracts 59

MS25 [email protected] Modelling and Control of Production Networks Based on Road Network Traffic Management Methods MS26 On a Nonlocal Model of Biological Aggregation This paper is concerned with a fluid approach towards modelling of production networks inspired by the macro- I will discuss some features of a continuum model of biolog- scopic modelling of freeway networks. The manufacturing ical aggregation introduced by Topaz, Bertozzi and Lewis. process is modelled as a system whose dynamics are de- Individuals, described by the model, are attracted to each scribed by a PDE and its state is influenced by controlled other at a distance, but avoid overcrowding via a local inputs and uncontrolled disturbances. The modelling PDE dispersal mechanism. In the continuum model for the pop- expresses the principle of the manufactured items’ con- ulation density the attraction is described via a nonlocal servation over the range of product completion and can operator, while the repulsion is modeled by a differential be used for both simulation and control design purposes. operator. We show that the density profile develops in- Production systems can be controlled with the use of con- terfaces between a near-constant-density aggregate state trol measures similar to those applied in freeway networks. and the empty space. The interfaces evolve under surface- These control measures will be reviewed and their inter- tension-like “forces”. More precisely, using the gradient pretation within the manufacturing framework will be dis- flow structure and asymptotic analysis, we show that the cussed. A number of different scenarios will be examined sharp interface limit for the interfacial motion is the Hele- involving serial and parallel manufacturing lines. Shaw flow. On long time scales the interfacial motion leads to coarsening of length scales present in the system. The Apostolos Kotsialos rate of coarsening can be investigated using the Kohn-Otto Durham University, UK framework.We will describe a geometric viewpoint, which [email protected] unites the coarsening results in a variety of interfacial models. The talk is based on joint work with Andrea Bertozzi. MS26 Dejan Slepcev Gradient flow of the Aviles Giga functional Carnegie Mellon University [email protected] Aviles and Giga introduced the functional 2 ε 2 1 2 MS26 (ΔΘ) + 1 −|gradΘ| dx 2 4ε Title Not Available at Time of Publication in the context of liquid crystals. It has since appeared in I will discuss some asymptotic limits of the Ginzburg- various guises for models of thin-film blistering, micromag- Landau and Chern-Simons-Higgs energies via Gamma con- netics, and pattern formation. The dynamical analog (the vergence methods. L2 gradient flow) is also of interest, because of the implications for microstructure formation. This talk considers Daniel Spirn the singular, sharp interface limit ε → 0 which gives rise University of Minnesota to time-evolving grain boundary networks. Formal com- [email protected] putations lead to a complex free boundary problem coupling motion of network edges and junctions. Two junction types, characterized by their geometry, are identified. MS26 The gradient flow of the (conjectured) Gamma-limit is also The IVP for Euler/Euler-Poisson Type Systems considered, and is show to be equivalent. Consequences of the resulting reduced dynamics, including dynamic scaling, The ”sticky particles” model at the discrete level is em- will be discussed. ployed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler Karl Glasner and pressureless Euler-Poisson systems. We consider the The University of Arizona case of finite, nonnegative initial Borel measures with fi- Department of Mathematics nite second-order moments, along with continuous initial [email protected] velocities of at most quadratic growth and finite energy. Adrian Tudorascu MS26 Georgia Institute of Technology Singular Interfacial Energy, Faceting, and Crystal [email protected] Microstructure in Epitaxial Relaxation

The relaxation of crystal surfaces is driven by the motion MS27 of atomic line defects (“steps”). In the continuum limit, On a Diffuse Interface Model for a Binary Mixture the evolution equation can be viewed as a steepest-descent of Compressible Viscous Fluids system on the basis of a singular surface energy. Corner singularities of the energy density correspond to macro- We consider a model describing the time evolution of a bi- scopically flat regions (“facets”). In this talk I will discuss nary mixture of compressible viscous fluid consisting of the continuum solutions consistent with the motion of steps. compressible Navier-Stokes system coupled with a Cahn- In particular, I will show how the crystal microstructure, Hilliard equation for the order parameter. We show that in the form of individual steps on top of the physical facet, the problem posesses a global-in-time weak solution for any affects the slope profile macroscopically. finite energy initial data. Related questions are also discussed. Dionisios Margetis University of Maryland, College Park Eduard Feireisl 60 PD07 Abstracts

Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1 MS28 Czech Republic Threshold-Based Quasi-Static Brittle Damage Evo- [email protected] lution

Helmut Abels We will first consider a variational model for damage pro- Max Planck Institute posed by Francfort and Marigo, and then give an existence Leipzig result (for a quasi-static version) by Francfort and Gar- [email protected] roni. A case for a strain-threshold damage model will be made, and precise definitions for both an unrelaxed (i.e., the damage region is a set at each time) and relaxed (i.e., MS27 the damage ”set” is given by a density function, together Contact Discontinuity for Jin-Xin Relaxation Sys- with an effective tensor that stores the microstructure of tem the damage) quasi-static evolution will be given. It turns out that solutions of the unrelaxed variational problem are We construct the contact waves for Jin-Xin relaxation sys- solutions to the unrelaxed threshold problem, but there is tem in 1-D. Such waves are relaxation correspondence of an interesting issue in showing the same for the relaxed contact discontinuity for hyperbolic conservation laws at problems. This issue is related to a question regarding G- equilibrium. We further prove that they are stable under closure and the relative scalings of damage microstructure generic small intial perturbations. at different times. A new variational formulation, arguably more physical than the existing one, will be given, and we Ronghua Pan will show that relaxed solutions of this variational prob- Georgia Tech. lem are solutions of the relaxed threshold problem. This is [email protected] based on joint work with A. Garroni. Christopher Larsen MS27 Worcester Polytechnic Institute Stability of Perfect-Fluid Shock Waves in Special Math Dept and General Relativity [email protected] For general relativity, the persistence problem of shock fronts in perfect fluids is also a continuation problem for MS28 a pseudo-Riemannian metric of reduced regularity. We Nonuniqueness in a Free Boundary Problem From solve the problem by considerations on a Cauchy problem Combustion which combines a well-known formulation of the Einstein- Euler equations as a first-order symmetric hyperbolic sys- We will present a rigorous justification for a nonuniqueness tem and Rankine-Hugoniot type jump conditions for the example in the parabolic free boundary problem with fixed fluid variables with an extra (non-)jump condition for the gradient condition, which appears in combustion theory to first derivatives of the metric. As in non-relativistic set- describe the propagation of premixed equidiffusional flames tings, the shock front must satisfy a Kreiss-Lopatinski con- in the limit of high activation energy. dition in order for the persistence result to apply. Aaron Yip, Arshak Petrosyan Mohammadreza Raoofi Purdue University University of Minnesota, School of Mathematics Department of Mathematics raoofi@umn.edu [email protected], [email protected]

Heinrich Freistuehler Universit MS28 ”at Leipzig Infinity Laplace Equation And Related Issues [email protected] The speaker plans to introduce the basic theory of Infin- ity Laplace Equation and related topics. In particular, he MS27 plans to cover some recent progress in the theory. Title Not Available at Time of Publication Peiyong Wang Abstract not available at time of publication. Department of Mathematics Wayne State University Zhouping Xin [email protected] Chinese University of Hong Kong [email protected] MS28 Global Minimizers of Chern-Simons-Higgs Func- MS28 tional Below Critical Magnetic Field On a Free Boundary Problem Arising from Mate- rial Science For CSH functional of superconductors in an external field, we prove the global minimizer is vorticesless when the ex- In this talk, we will study a free boundary problem arising ternal field is below critical field. It is a joint work with from material science. The regularity of both the phase Daniel Spirn from Minnesota function and the free boundary will be discussed. Xiaodong Yan Huiqiang Jiang Michigan State University Department of Mathematics, University of Pittsburgh Department Of Mathematics [email protected] [email protected] PD07 Abstracts 61

MS29 UCLA Mathematics An Analytic Model in Finding Geodesic Path Be- [email protected] tween Shapes Stanley J. Osher Curve matching plays a crucial role for many problems in University of California computer vision and medical image analysis. In this paper, Department of Mathematics we built a novel model based on Hellinger distance for curve [email protected] matching. In this model, we can construct a geodesic path between two shapes analytically. Besides, this model with slight modifications is capable of finding optimal correspo- MS29 nendents among a group of curves simultaneously. Our Redistancing by Flow of Time Dependent Eikonal experiments show the effectiveness of this model. Equation

Pengwen Chen, Yunmei Chen Construction of signed distance to a given inteface is a University of Florida topic of special interest to level set methods. There are [email protected]fl.edu, [email protected]fl.edu currently, however, few algorithms that can efficiently produce highly accurate solutions. We introduce an algorithm for constructing an approximate signed distance function MS29 through manipulation of values calculated from flow of time Upper Bounds on the Coarsening Rate of Discrete, dependent eikonal equations. We provide operation counts Ill-Posed, Nonlinear Diffusion Equations and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy. We prove a weak upper bound on the coarsening rate of the Comparison with the standard level set reinitialization al- discrete-in-space version of an ill-posed, nonlinear diffusion gorithm shows ours is superior in terms of predictability equation. The continuum version of the equation violates and local construction,critical, for example, in local level parabolicity and lacks a complete well-posedness theory. In set methods. We further apply the same ideas to extension particular, numerical simulations indicate sensitive depen- of values off interfaces. Together, our proposed approaches dence on initial data. Nevertheless, models based on its can be used to advance the level set method for fast and discrete-in-space version, which we study, are widely used accurate computations of the latest scientific problems. in a number of applications, including population dynamics (chemotactic movement of bacteria), granular flow (forma- L.T. Cheng tion of shear bands), and computer vision (image denoising University of California, San Diego and segmentation). Our bounds have implications for all [email protected] three applications. They are obtained by following a recent technique of Kohn and Otto. Based on joint works with Richard Tsai John Greer and Dejan Slepcev. Department of Mathematics Selim Esedoglu Univesity of Texas Department of Mathematics [email protected] University of Michigan [email protected] MS30 A Long-Short Wave Equation in Fermi-Pasta-Ulam MS29 Lattice A Second Order Variational Model for Staircase It is well known that the Boussinesq equation, further, the Reduction KdV equation can be derived from the Fermi-Pasta-Ulam The one-dimensional version of the higher order total (FPU) lattice as a long wave continuum approximation. variation-based model for image restoration proposed by Taking into account both the acoustic and optical modes, Chan, Marquina, and Mulet is analyzed. A suitable func- we derive a coupled partial differential equations in attempt tional framework is proposed in which the minimization to study the long-short wave interaction in FPU lattice. problem is well posed, and it is proved analytically that Some numerical and analytical results of this long-short the higher order regularizing term prevents the occurrence wave equation will also be presented. of the staircasing effect. Baofeng Feng Irene Fonseca U. of Texas-Pan American Carnegie Mellon University [email protected] [email protected] MS30 MS29 Dynamics of the Localized Waves in Rayleigh- Nonlocal Operators for Image Processing Benard Convection With Heterogeneous Concen- tration Field We propose the use of nonlocal operators to define new types of flows and functionals for image processing. A main We study the Rayleigh-Benard convection in binary fluid advantage over standard algorithms is the ability to simul- mixture by an amplitude equation derived by Riecke. This taneously the texture and cartoon parts of images. Ap- equation has a variable, corresponding to the concentra- plications to denoising, segmentation and inpainting are tion field, which works as a heterogeneous driving force for given. This work builds on much research done earlier in fluid motion. This equation gives interesting dynamical a host of related fields. phenomena among localized waves(pulses), i.e., formation of bound states by both co- and counter-propagating pulses Guy Gilboa and various input-output relation by pulse collision. These 62 PD07 Abstracts

phenomena are analyzed in terms of the network topol- estimates for the coupling procedure will be provided, and ogy constructed by stable and unstable manifolds in phase numerical examples will be presented. space, forcusing on the role of the concentration field. Gang Bao Yasumasa Nishiura, Makoto Iima Michigan State University Research Institute for Electronic Science [email protected] Hokkaido University [email protected], Peijun Li [email protected] Department of Mathematics University of Michigan [email protected] MS30 Pinning and Depinning for Particle-Like Solutions in Dissipative Systems MS31 Superposition of Multi-Valued Solutions in High Particle-like dissipative patterns arise in many fields such Frequency Wave Dynamics as chemical reaction, gas-discharge system, liquid crystal, binary convection, and morphogenesis. We discuss about The weakly coupled WKB system captures high frequency the dynamics of moving particles in heterogeneous media, wave dynamics in many applications. For such a system especially we are interested in the case in which traveling a level set method framework has been recently developed pulses or spots encounter heterogeneities of bump type. to compute multi-valued solutions to the Hamilton-Jacobi A variety of dynamics are produced such as penetration, equation and evaluate position density accordingly. In this annihilation, rebound, splitting, and relaxing to an ordered paper we propose two approaches for computing multi- pattern, i.e., pinning. It turns out that global bifurcation valued quantities related to density, momentum as well as such as heteroclinic one and a basin switching controlled the energy. Within this level set framework we show that by the unstable manifolds of saddless called scattors play physical observables evaluated in [?, ?] are simply the su- a crucial role for the transition from pinning to depinning. perposition of their multi-valued correspondents. A series We also discuss a wave-generator created near the jump of of numerical tests is performed to compute multi-valued heterogeneity. quantities and validate the established superposition properties. Yasumasa Nishiura RIES, Hokkaido University Hailiang Liu Japan Iowa State University [email protected] USA [email protected] Takashi Teramoto Chitose Institute of Science and Technology [email protected] MS31 A Field Space-Based Level Set Method for Com- puting Multi-Valued Solutionsto 1D Euler-Poisson MS30 Equations Behavior of an Amoeba Crossing an Environmental Barrier We present a field space based level set method for computing multi-valued solutions to one-dimensional Euler- In this talk, we will focus on the effect of inhomogeneous Poisson equations. The system of these equations has environments on tip migration behavior in the Physarum many applications, and in particular arises in semiclassical plasmodium of true slime mold. The plasmodium migrat- approximations of the Schrödinger-Poisson equation. The ing in a narrow lane shows penetration, rebound, and split- proposed approach involves an implicit Eulerian formulating dynamics when it encounters the presence of a chem- tion in an augmented space — called field space, which ical repellent, quinine. In order to understand the dynam- incorporates both velocity and electric fields into the con- ics, a phenomenological model based on reaction-diffusion figuration. Both velocity and electric fields are captured systems is proposed. We discuss how the origin of the three through common zeros of two level set functions, which different outputs could be reduced to the hidden instabili- are governed by a field transport equation. Simultane- ties of internal dynamics of the tip. ously we obtain a weighted density f by solving again the field transport equation but with initial density as starting Kei-Ichi Ueda data. The averaged density is then resolved by the inte- Kyoto University, Japan gration of the obtained f against the Dirac delta-function ueda [email protected] of two level set functions in the field space. Moreover, we prove that such obtained averaged density is simply a linear superposition of all multi-valued densities; and the MS31 averaged field quantities are weighted superposition of cor- Coupling of Finite Element and Boundary Integral responding multi-valued ones. Computational results are Methods for a Scattering Problem in Near-Field presented and compared with some exact solutions which Optics demonstrate the effectiveness of the proposed method.

This talk is concerned with the solution of the Helmholtz Zhongming Wang equation in the modeling of a scattering problem for one of Iowa State University important experimental modes of near-ﬁeld optics, photon [email protected] scanning tunneling microscopy. The well-posedness for the continuous and discrete problems of coupling ﬁnite element and boundary integral methods will be addressed, error PD07 Abstracts 63

MS31 Roles of the bandgaps are examined. Computation of the Radiative Transfer Equation with Interface Reflection and Transmission Chiang C. Mei Dept of Civil and Environmental Engineering Abstract not available at time of publication MIT [email protected] Xu Yang Department of Mathematics Yile Li University of Wisconsin-Madison Shell International Exploration and Production Inc [email protected] Houston TX yile.li@gmailcom MS32 Variational Principles for Supremal Functionals MS32 and Applications to Polycrystal Plasticity G-Convergence and Homogenization of Fluid- Structure Interaction and Two-Fluid Flow Γ-convergence results for a general class of power-law functionals acting on divergence free fields are presented. Some We study two homogenization problems for flows with mov- consequences are described in the settings of antiplane ing interfaces: a small deformation model of fluid-structure shear and plane stress plasticity. interaction and a Stokes flow of two immiscible fluids without surface tension. Since the interfaces are moving, we Marian Bocea work in the framework of G-convergence that requires no North Dakota State University a priori geometric assumptions. To pass to the limit we [email protected] propose a new construction of divergence-free oscillating test functions. The construction involves several auxiliary MS32 funcitons that are found by solving auxiliary problems. The effective equations for fluid-structure interaction are Inverse Homogenization - Recovery of the Struc- of viscoelastic type with long memory terms not present in ture of Composite Materials the microscale problems. In the case of two fluid flow, the The talk deals with inverse homogenization problem which effective equations are of the same type as microscale prob- is a problem of deriving information about the microgeome- lems, but the effective viscosity tensor depends of time in try of composite material from its effective properties. The general. Since constitutive equations for viscosity should approach is based on reconstruction of the spectral mea- not depend on time explicitly, it would be desirable to ob- sure in the analytic Stieltjes representation of the effective tain sufficient conditions for existence of an equation of tensor of two-component composite. This representation state that would prescribe the effective viscosity as a time- relates the n-point correlation functions of the microstruc- independent function of the effective density and effective ture to the moments of the spectral measure, which con- pressure. We present some results in this direction (joint tains all information about the microgeometry. The prob- work with Elena Cherkaev). lem of identification of the spectral function from effective Alexander Panchenko measurements in an interval of frequency has a unique so- Washington State University lution, however the problem is ill-posed. The talk discusses [email protected] several stabilization techniques as well as Pade approximation used to reconstruct the spectral function. Results of reconstruction of microstructural parameters are shown for MS33 visco-elastic composites and for composites of two materi- High Confinement Limit and Time Averaging for als with different complex permittivity. The reconstructed the Gross-Pitaevski Equation spectral function can be used to compute other effective properties of the same composite; this gives solution to Abstract not available at time of publication. the problem of coupling of different effective properties of a two-component composite material with random mi- Naoufel Ben Abdallah crostructure. University of Toulouse [email protected] Elena Cherkaev University of Utah Department of Mathematics MS33 [email protected] Quantum Charge Transport in Random Media

In this talk, we discuss some recent results addressing quan- MS32 tum charge transport in a weak random potential. We Coupled-Mode Theory of Wave Resonance in a Pe- focus on the derivation of Boltzmann equations, obtained riodic Medium from a macroscopic, hydrodynamic limit, both in the one- particle, and in the fermionic manybody framework. In We study the Bragg resonance of surface water waves by a connection with the latter, we report on a recent joint work two-dimensional array of vertical cylinders covering a large with I. Sasaki. area of the sea. Employing some concepts of solid state and crystallography, we combine the techniques homogeniza- Thomas Chen tion and inner-outer expansion to derive two-dimensional Princeton University equations for the envelopes of scattered waves resonated by [email protected] a plane incident wave. Solutions for two- and three-wave scattering are obtained for a long strip of cylinder array. 64 PD07 Abstracts

MS33 Department of Mathematics 5-Moment-Corrections to a Drift-Collision Balance University of Houston Model of Quantum Transport [email protected]

We present a fluid–dynamical model which describes Tsorng-Whay Pan quantum–driven electron transport in the so-called “drift– Department of Mathematics collision balance” regime. Here, external field and inter- University of Houstofn action with a phonon bath are the dominant phenom- [email protected] ena and have comparable strength. More precisely, in the rescaled Wigner–BGK equation they appear at the Roland Glowinski same order in the parameter . The BGK term contains O ∈ University of Houston the (Wigner) (¯ )–correction to the thermal equilibrium [email protected] function, parametrized by the electron position density and the bath temperature, which models the state to which the system shall be driven by the sole interaction with the Giovanna Guidoboni crystal ions. Via a Chapman–Enskog procedure we find University of Houston O()–correction to the solution of the scaled Wigner-BGK Department of Mathematics equation. Then, by applying the moment method, we find [email protected] O()–corrections to the equations for position density, velocity flux and energy density. In these corrective terms ap- Sergey Lapin pear five moments of the O(∞)–in– solution and they are Department of Mathematics ∈ of diffusive type in the fluid unknowns. High-field O(¯ )– Washington State University corrections are present that are peculiar of quantum-driven [email protected] transport. Giovanni Frosali MS34 Dept. of Applied Mathematics Adhesion and Slough-off of Chondrocytes Under University of Florence Controlled Flow Conditions giovanni.frosali@unifi.it Cell adhesion under conditions of fluid flow plays an important role in many physiological and biotechnological pro- Giovanni Borgioli cesses. Recently auricular chondrocytes were studied for Dept. of Electronics lining a stent, which is a potential treatment of coronary University of Florence artery disease. To improve the biocompatibility, it is cru- giovanni.borgioli@unifi.it cial for cells to remain firmly on surface initially. In this presentation, we mathematically model and simulate adhe- Chiara Manzini sion and slough-off of chondrocytes in shear flow. We first Dept. of Applied Mathematics ”G.Sansone” discuss the Adhesive Dynamics for bonding between cell Universita’ di Firenze, Italy and surface; then introduce a Lagrange multiplier based chiara.manzini@unifi.it fictitious domain method to solve the fluid-cell system. Nu- merical results for the cells in shear flow and the case with deterministic adhesion will be presented. MS33 On the Global Existence of Smooth Solution to the Jian Hao 2-D FENE Model University of Houston Department of Mathematics In two space dimension, we prove the global existence [email protected] of smooth solutions to a coupled microscopic-macroscopic FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteract- MS34 ing polymer chains. Critical Thresholds in a Quasilinear Hyperbolic Model of Blood Flow Ping Zhang Beijing Academy of Science We study critical threshold phenomena in a quasilinear hy- [email protected] perbolic model of blood flow through compliant axisymmetric vessels with viscous effects. We identify critical thresholds for global existence of smooth solutions and for MS34 the finite time singularities. We prove the existence of a A Novel Algorithm for Fluid-Structrure Interac- smooth solution under some conditions on the slope of the tion in Blood Flow pressure data. We obtain that for certain physiologically relevant data one has shock formation. The fluid-structure interaction in blood flow involves two media, blood and arterial walls, whose densities are of the Tong Li same order of magnitude. This leads to a highly nonlin- Department of Mathematics ear interfacial coupling. To date, only strongly coupled University of Iowa algorithms (implicit/monolithic) seem applicable to blood [email protected] flow simulations. In this talk, we present a novel class of loosely coupled (partitioned) algorithms which combine Suncica Canic the stability properties of implicit schemes with the low Department of Mathematics computational costs of explicit schemes. University of Houston Suncica Canic [email protected] PD07 Abstracts 65

MS34 bifurcation and the Hopf bifurcation are considered. Simulating Blood Flow and Vessel Dynamics in Patient-Specific Arterial Models Tian Ma Sichuan University In recent years, image-based computational methods have / been described for modeling blood flow in the cardiovascular system and test hypotheses related to the localization Shouhong Wang of vascular disease. We will present results related to mod- Department of Mathematics eling blood flow and vessel dynamics in the systemic and Indiana University pulmonary circulations of individual subjects. We then de- [email protected] scribe a new approach for cardiovascular treatment planning in which the physician utilizes computational tools to Jerry Bona construct and evaluate a combined anatomic/physiologic University of Illinois at Chicago model to predict the outcome of alternate treatment plans [email protected] for an individual patient. In order to ensure that these predictive tools are clinically relevant, they must faith- Chun-Hsiung Hsia fully represent the anatomy and physiology of individual U Illinois Chicago patients. Recent progress in constructing subject spe- [email protected] cific anatomic models from medical imaging data, acquir- ing physiologic data under resting and exercise conditions, planning treatments, and modeling blood flow and vessel MS35 deformation will be described. Verification of FSI methods Stochastic Parametrization of Wave Breaking and with analytical solutions will be discussed. Results from Wave/Current Interactions in vitro and animal studies utilizing magnetic resonance imaging techniques to validate computational blood flow If wave breaking modifies the Lagrangian fluid orbital simulations will be presented. paths by inducing an uncertainty in the path itself and this uncertainty on wave motion time scales is observable Charley Taylor as additive noise, it is shown that within the context of Stanford University a wave/current interaction model for basin and shelf scale Mechanical Engineering motions it persists on long time scales. A model of con- [email protected] servative dynamics, developed in collaboration with Lane and McWilliams, provides the general framework for the MS35 dynamics of the wave-current interactions. In addition to the deterministic part, the vortex force, which couples the Maximum Enstrophy Production in the 3D Navier- total flow vorticity to the residual flow due to the waves, Stokes will have a part which is associated with the dissipative It is still not known whether solutions to the 3D Navier- mechanism. At the same time the wave field will expe- Stokes equations for incompressible flows in a finite peri- rience dissipation, and tracer advection is affected by the odic box can become singular in finite time. It is known appearance of a dissipative term in the Stokes drift veloc- that solutions remain smooth as long as their enstrophy ity. Consistency leads to other dynamic consequences: the (mean-square vorticity) is finite. The generation rate of en- boundary conditions are modified to take into account the strophy is given by a functional that can be bounded using diffussive process and proper mass/momentum balances at elementary functional estimates. Those estimates establish the surface of the ocean. I will present an overview of short-time regularity but do not rule out finite-time singu- the model, how wave dissipation effects are included as a larities in the solutions. In this work we formulate and stochastic parametrization, and show preliminary results solve the variational problem for the maximal growth rate on a model problem. of enstrophy and display flows that generate enstrophy at Juan M. Restrepo the greatest possible rate. Implications for questions of reg- University of Arizona ularity or singularity in solutions of the 3D Navier-Stokes [email protected] equations are discussed.

Charles R. Doering MS35 University of Michigan Dept of Mathematics On the Motion and Regularity of Vortex Sheets [email protected] with Surface Tension We prove well-posedness of vortex sheets with surface ten- Lu Lu sion in the 3D incompressible Euler equations with vor- Wachovia Investments ticity. This is a moving boundary-value problem for the New York, NY motion of two incompressible perfect fluids separated by a [email protected] surface of discontinuity, wherein the motion of the fluid interacts with the motion and regularity of the vortex sheet at highest-order. MS35 The Bifurcation and Stability Analysis of Double- Steve Shkoller Diffusive Convection Univiversity of California, Davis Department of Mathematics In this talk, we present a bifurcation and stability analy- [email protected] sis on the double-diffusive convection. Both steady state 66 PD07 Abstracts

MS36 chanics. When the viscosity is small, very complex phe- Incompressible Fluids in Thin Domains with nomena can occur near turning points, which are not yet Navier Friction Boundary Conditions well understood. A model problem, corresponding to a linear convection-diffusion equation (e.g., suitable lineariza- This study is motivated by problems arising in oceanic tion of the Navier-Stokes or Benard convection) equation dynamics. Our focus is the Navier-Stokes equations in a is considered. Our analysis shows the diversity and com- three-dimensional domain Ωε, whose thickness is of order plexity of behaviors and boundary or interior layers which 0(ε)asε → 0, having non-trivial topography. The velocity already appear for our equations simpler than the Navier- vector field is subject to the Navier friction boundary con- Stokes or Benard convection equations. Of course the di- ditions on the bottom and the top boundaries of Ωε with versity and complexity of these structures will have to be the friction coefficients γ0,ε and γ1,ε, respectively. Assume taken into consideration for the study of the nonlinear that γ0,ε and γ1,ε are of order 0(1), and |γ0,ε − γ1,ε| is of problems. In our case, at this stage, the full theoretical order 0(3/4) as ε → 0. It is shown that if the initial veloc- asymptotic analysis is provided. ity field and the body force belong to ”large sets”, then the strong solution of the Navier-Stokes equations exists for all Chang-yeol Jung time. Our proofs rely on the study of the dependence of the Indiana University Stokes operator on ε, and the non-linear estimate in which [email protected] the contributions of the boundary integrals are non-trivial. Roger Temam Luan T. Hoang Inst. f. Scientific Comput. and Appl. Math. SCHOOL OF MATHEMATICS Indiana University UNIVERSITY OF MINNESOTA [email protected] [email protected]

MS37 MS36 Thin-Film Coarsening Driven by Surface Relax- Averaged Bounds on Moments for the 2D Navier- ation and the Ehrlich-Schwoebel Effect Stokes Equation The surface of an epitaxially growing thin film often ex- Upper and lower bounds for the ensemble averages of en- hibits a mound-like structure with its characteristic lateral ergy, enstrophy, and palinstrophy (the first three moments size increasing in time. In many material systems, such of the 2D, periodic, Navier-Stokes equations) are derived a coarsening process is driven mainly by two competing both in the general case, and in the case where the en- mechanisms. One is the surface relaxation described by ergy spectrum for fully developed turbulence holds. In the high-order gradients of the surface profile, and the other turbulent case, the bounds are sharp, up to a logarithm, the Ehrlich-Schwoebel effect which is the upper-lower ter- and provide a new lower bound on the Landau-Lifschitz race asymmetry in the adatom attachment and detachment degrees of freedom. A key Sobolev-type estimate of the to and from atomic steps. In this talk, we present an analy- inertial term is shown to be sharp on a significant portion sis of a class of partial differential equation models that are of the global attractor. mathematically gradient-flows of some effective free-energy functionals describing these mechanisms. Our analysis con- Michael S. Jolly sists of two parts: (1) variational properties of the energies, Indiana University such as “ground states” and their large-system-size asymp- Department of Mathematics totics, showing the unboundedness of surface slope and re- [email protected] vealing the relation between some of the models; (2) rigorous bounds for the scaling law of the roughness, the rate of Ciprian Foias increase of surface slope, and the rate of energy dissipation, TexasA&MUniv all of which characterize the coarsening process. [email protected] Bo Li Radu Dascaliuc Department of Mathematics, UC San Diego Indiana University [email protected] Department of Mathematics [email protected] MS37 Modeling and Analysis of Stepped Crystal Surfaces MS36 Discrete in Time Coupling and Synchronization in The morphological evolution of crystal surfaces is stud- the 2D Navier-Stokes Equations ied as a prototypical case of modeling across the scales. At the nanoscale, the governing discrete schemes describe Abstract not available at time of publication. the motion of line defects (“steps”) of atomic height. At the macroscale, the resulting free-boundary problems for Eric Olson PDEs retain a novel microstructure. I will focus on re- UNR cent progress and challenges in answering the question: For ejolson what continuum theory is step motion a consistent numerical solution scheme ? I will present the continuum laws and free-boundary problems that result from step motion. MS36 I will also explore the validity of the continuum theory from Singularly Perturbed Convection-Diffusion Equa- the perspective of kinetic BBGKY-type hierarchies for step tions with a Turning Point correlation functions.

Turning points occur in many circumstances in ﬂuid me- Dionisios Margetis PD07 Abstracts 67

University of Maryland, College Park dynamics naturally extend to probability distributions on [email protected] the half-line with zero and infinity appended, represent- ing populations of clusters of zero and infinite size. The “scaling attractor” (set of subsequential limits) is compact MS37 and has a Levy-Khintchine-type representation that lin- Epitaxial Growth, Atomistic Simulations on Diffu- earizes the dynamics and allows one to establish several sive Time Scales signatures of chaos. In particular, for any given solution trajectory, there is a dense family of initial distributions We apply a phase field crystal model to epitaxial growth (with the same initial tail) that yield scaling trajectories and study equilibrium shapes of islands and mounds. The that shadow the given one for all time. model is derived from classical density functional theory (DFT) and leads to a higher order parabolic PDE for the Govind Menon atom density. The derivation from DFT as well as a coarse Applied Mathematics, Brown University graining which leads to classical phase field models will be [email protected] discussed. Furthermore efficient numerical methods based on finite element discretizatioins will be shown. Robert Pego Carnegie Mellon University Andreas Raetz [email protected] University of Dresden [email protected] MS38 MS37 Aggregation of Floating Particles by Surface Ten- sion Forces: From Nano- to Millimeter Scales. A Continuum Model for the Long-Range Elastic Effect on Epitaxial Surfaces We derive an evolution equations for self-assembly of floating particles. First, we present results of numerical simu- Below the roughening transition, the surface of an epitaxial lations showing that energy density of a realistic particle film consists of steps whose distribution and motion deter- clump of an arbitrary shape is described by a surprisingly mine the surface morphology. The elastic effects on the simple formula, and that the continuum approximation epitaxial surface provide a driving force for the motion of may be used for clumps with as few as five particles. Next, these surface steps. The stress in the epitaxial film, which we deduce an evolution equation for density evolution for may be due to the misfit between the lattice constants in in the case of central potential and describe analytically the film and in the substrate, generates a long-range elas- stationary and spatio-temporal solutions forming the base tic effect on the epitaxial surface. We present a continuum of the dynamics. Finally, we suggest an evolution equations model for this long-range elastic effect, incorporating the for particles with interaction potential dependent not only discrete features of the stepped surface. The model is ob- on their density, but also on the mutual orientation. In the tained by taking continuum limit of the discrete model for process, we also derive the analogue of Darcy’s law and cor- the elastic interaction between surface steps. responding equation of motion for an arbitrary geometric Yang Xiang quantity. Hong Kong University of Science and Technology Darryl Holm [email protected] Imperial College London and Los Alamos National Laboratory MS38 [email protected] A Variational Theory for Point Defects in Patterns Vakhtang Putkaradze We derive a rigorous scaling law for minimizers in a natural Colorado State University version of the regularized Cross-Newell model for pattern [email protected] formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physi- MS38 cal systems and in numerical simulations of the microscopic Rigorous Scaling Laws and Crossover Behavior in equations that describe these systems. Elastic Ridges Shankar C. Venkataramani Abstract not available at time of publication. University of Arizona Department of Mathematics Shankar C. Venkataramani [email protected] University of Arizona Department of Mathematics [email protected] Nick Ercolani University of Arizon [email protected] MS39 Controllability of a Cochlea and Related Fluid- MS38 Elastic Systems Scaling Dynamics for Smoluchowski’s Coagulation Standard 2-dimensional cochlea model consists of one- Equations with Dust and Gel dimensional elastic structure surrounded by incompressible fluid within cochlear cavity. Dynamics are typically driven We study limiting behavior of rescaled size distributions by pressure differential across the basilar membrane trans- that evolve by Smoluchowski’s rate equations for coagula- mitted through round and oval windows. Idealized basilar tion, with rate kernel K=2, x+y or xy. We find that the 68 PD07 Abstracts

membrane is modeled as inﬁnite array of oscillators and MS40 ﬂuid is described by Laplace’s equation. We show approx- On the Spectrum Associated with Periodic Waves imate controllability with control acting on arbitrary open for Generalized KdV Equations set of membrane. If membrane has longitudinal elasticity, then exact controllability can be proved. Abstract not available at time of publication. Scott Hansen Todd Kapitula Iowa State University Department of Mathematics Department of Mathematics Calvin College [email protected] [email protected]

MS39 MS40 Optimal Static Control of Groundwater Semi-Strong N-Pulse Interaction in a Hypberbolic- Parabolic System Control theory has been less developed for elliptic partial differential equations than it has for hyperbolic and We consider the semi-strong interaction of optical pulses parabolic variety. In this talk we indicate how control can in a thermally detuned optical parametric oscillator. The be applied to study certain problems for elliptic systems. pulses evolve under a parametrically forced nonlinear We use Darcy’s equation for groundwater flow together Schrodinger equation coupled to a thermal profile gener- with mathematically idealized representation of control in- ated by the optical heating. The thermal profile decays tervention by means of pumping and transport as an ex- at a much slower spatial rate and provides for long-range ample. We will indicate the relevant theoretical questions interaction in what is effectively a three species, hyperbolic- and present the results of some computational simulations. parabolic, singularly perturbed system. We show that by controlling the commutator of a scaling operator and David Russell a spectral projection, that the associated semi-group is Virginia Tech strongly contractive and use the RG machinary to derive [email protected] interaction laws for N pulses comprised of optical pulse and long-range thermal envelope. MS39 Keith Promislow Multiscale Modeling of Preferential Flow Michigan State University [email protected] We discuss new multi-scale models of fully saturated flow coupled to transport through highly heterogeneous media. These are systems of partial differential equations that in- MS40 clude the usual diffusion effects from various spatial scales A Simple Technique for Solving Partial Differential and additionally model the effects of local advective trans- Equations on Surfaces port through fast flow regions. Abstract not available at time of publication. Malgorzata Peszynska, Ralph Showalter Department of Mathematics Steve Ruuth Oregon State University Simon Fraser University [email protected], Department of Mathematics [email protected] [email protected]

MS39 MS40 Fluid-Structure Interaction Problems HydroGel Swelling: Models and Analysis

We consider parabolic-hyperbolic fluid-structure interac- Abstract not available at time of publication. tion PDE model (dimensions 2 and 3): structure of arbitrary shape, modeled by the system of dynamic elasticity, Hang Zhang is immersed in a fluid modeled by Navier-Stokes equation, Michigan State University with coupling at the interface. In linear case, semigroup [email protected] well-posedness in the energy space is asserted. Spectral properties of the generator are described. Strong stability is analyzed. To obtain uniform stability, the interface con- MS41 dition is modified to become dissipative. Backward unique- Shock Reflection, Transonic Flow, and Free Bound- ness result is presented. uniqueness. ary Problems

George Avalos In this talk we will start with various shock reflection phe- University of Nebraska-Lincoln nomena and their fundamental scientific issues. Then we Department of Mathematics and Statistics will describe how the shock reflection problems can be for- [email protected] mulated into free boundary problems for nonlinear partial differential equations of mixed-composite type. Finally Roberto Triggiani we will discuss some recent developments in the study of University of Virginia the shock reflection problems. In particular, we will dis- [email protected] cuss our recent results on the global theory for solutions to shock reflection by wedges for potential flow. The approach includes techniques to handle transonic shocks via free boundary techniques, sonic curves via degenerate ellip- PD07 Abstracts 69

tic techniques, and corner singularities when sonic curves MS41 meet transonic shocks that connect the subsonic phase with Instability Criteria in a Model for Shear Bands the supersonic phase. Further remarks, trends, and open problems in this direction will be also addressed. This talk We consider a system of hyperbolic-parabolic equations de- is based mainly on our joint work with Mikhail Feldman. scribing the material instability mechanism associated to the formation of shear bands at high strain-rates. We de- Gui-Qiang G. Chen rive a quantitative criterion for the onset of instability: Department of Mathematics, Northwestern University Using ideas from the theory of relaxation systems we de- USA rive equations that describe the effective behavior of the [email protected] system. The effective equation turns out to be a forward- backward parabolic equation regularized by fourth order term. MS41 Symmetric Stationary Inviscid and Viscous Profiles Athanasios Tzavaras in Multi-D University of Maryland [email protected] We construct stationary solutions with spherical or cylindrical symmetry for the 2D and 3D compressible Euler and Navier-Stokes equations. The solutions are defined in the MS42 region between two concentric spheres or cylinders and sat- A Model-Based Inversion Algorithm for the isfy Dirichlet boundary conditions at the inner and outer Controlled-Source Electromagnetic Data boundaries. We first build stationary solutions with a single shock for the Euler equations (both isentropic and full We present the parametric inversion algorithm (PIA), system). These are then used in the construction of cor- which uses a priori information on the geometry to re- responding solutions of the Navier-Stokes equations. In duce the number of unknown parameters and improve the particular, the viscous solutions converge to the inviscid quality of the reconstructed conductivity image. This PIA shock solutions as the viscosities tend to zero. approach can be also used to refine the conductivity image that we obtained using the pixel-based inversion (PBI) Mark Williams algorithm. The PIA adopts the Gauss-Newton minimiza- Department of Mathematics tion method, with nonlinear constraints and regularization UNC Chapel Hill for the unknown parameters. It also employs a line search [email protected] approach to guarantee the reduction of the cost function after each iteration. The forward modeling simulation is Kris Jenssen a two-and-half dimensional finite-difference solver, and the Pennsylvania State University parameters that govern the location and the shape of a [email protected] reservoir include the depth and the location of the user- defined nodes for the boundary of the region. The un- Erik Endres known parameter that describes the physical property of Pennsylvania State University the region is the electrical conductivity. We will show some Department of Mathematics numerical examples to illustrate the advantageous of using [email protected] this PIA approach. Tarek Habashy, Yan Zhang, Maokun Li MS41 Schlumberger-Doll Research Shocks, Rarefactions and Triple Points in Multidi- [email protected], yzhang37@ridgefield.oilfield.slb.com, mensional Conservation Laws [email protected]

Following the initial work of Tesdall and Hunter [SIAP, Andrea Zerilli 2003] which found a new shock reflection pattern in the Schlumberger unsteady transonic small disturbance equations, we have [email protected]field.slb.com now exhibited this Guderley Mach reflection in numerical simulations of a number of equations – specifically the non- Aria Abubakar linear wave system and the adiabatic gas dynamics equa- Schlumberger Doll Research tions. Within this complicated pattern, the details of how [email protected] the rarefaction wave interacts with the sonic line form a mathematically appealing subproblem. We present some numerical and analytical results on this problem. MS42 Title Not Available at Time of Publication Barbara L. Keyfitz Fields Institute and University of Houston Abstract not available at time of publication. bkeyfitz@fields.utoronto.ca or keyfi[email protected] Aria Abubakar Allen Tesdall Schlumberger-Doll research Southern Methodist University [email protected]field.slb.com [email protected] MS42 Mary Chern Fields Institute Optimal Reduced Models for Dispersive Multi- cher [email protected] Conductor Transmission Line Systems Through the use of non-uniform grids generated by Gaus- 70 PD07 Abstracts

sian spectral rules we develop reduced models for disper- MS43 sive, multi-conductor transmission line systems. The syn- Title Not Available at Time of Publication thesized multi-port is represented as the concatenation of sections of lumped circuits; hence, it is compatible with Abstract not available at time of publication. circuit simulators such as SPICE. The passivity of the reduced model is guaranteed through the use of passive cir- Vladislav Panferov cuit representations of the frequency-dependent, per-unit- McMaster University length series impedance and shunt admittance matrices of Department of Mathematics and Statistics the transmission line system. [email protected] Aravind Ramachandran, Andreas Cangellaris University of Illinois at Urbana-Champaign MS43 [email protected], [email protected] Recent Results on Transport in Random Media Abstract not available at time of publication. MS42 Spectrally Matched Grids for Anistropic Problems Alexis Vasseur University of Texas at Austin Spectral convergence of grids targeted for the Neumann to [email protected] Dirichlet map is observed for anisotropic problems, despite the fact that the grids were designed for isotropic ones. We explain why this happens. For elliptic problems, the MS44 gridding algorithm is reduced to a Stieltjes rational ap- Existence and Stability of Ginzburg-Landau Vor- proximation on an interval of a line in the complex plane tices in Arbitrary Two Dimensional Domains instead of the real axis as in the isotropic case. We show rigorously why this occurs for a semi-infinite and bounded We consider Ginzburg-Landau equation with Neumann or interval. We then extend the gridding algorithm to hyper- Dirichlet boundary conditions in a bounded domain. We bolic problems on bounded domains. For the propagative show that whenever the renormalized energy W has critical modes, the problem is reduced to a rational approxima- points (not necessary nondegenerate) then there exist so- tion on an interval of the negative real semiaxis, similarly lution to Ginzburg-Landau equation with vortices located to in the isotropic case. For the wave problem we present near the set of critical points of W . We also show that the numerical examples in 2-D anisotropic media. Morse index of these solutions is equal to the number of the negative eigenvalues of the hessian of the renormalized Shari Moskow energy. University of Florida Department of Mathematics Manuel Del Pino, Michal Kowalczyk [email protected]fl.edu Departamento de Ingeniera Matemática Universidad de Chile [email protected], [email protected] MS43 From Kinetic Fokker-Planck Traffic Models to Sec- Monica Musso ond Order Conservation Laws Facultad de Matemáticas Pontificia Universidad Católica de Chile We revisit multilane kinetic traffic models of Fokker-Planck [email protected] type and show that the Aw-Rascle second order model of conservation type can be derived from such models. Fur- thermore, we present a linear stability analysis of the equi- MS44 libria associated with the (multivalued) fundamental dia- A Gradient Flow Approach to a Mean Field Model gram. of Superconductivity Reinhard Illner In a joint work with Luigi Ambrosio we studied an The University of Victoria, Canada evolution problem proposed by Chapman-Rubinstein- [email protected] Schatzman as a mean-field model for the evolution of the vortex density in a superconductor. We consider the case of a bounded domain where vortices can exit or enter the do- MS43 main. We show that the equation can be derived rigorously Analysis of a Fluid-Kinetic System of Equations as a gradient-flow of a specific energy for the Riemmanian structure induced by the Wasserstein distance on probabil- We study a coupled system of kinetic and fluid equa- ity measures. This leads to some existence and uniqueness tions modeling fluid-particles interactions arising in sprays, results and some energy-dissipation identities. We also ex- aerosols or sedimentation problems. More precisely, we hibit some ”entropies” which decrease along the flow and consider a Vlasov-Fokker-Planck equation coupled to com- allow to get regularity results. pressible Navier-Stokes equation via a drag force. We establish the existence of solutions and we rigorously derive Sylvia Serfaty the asymptotic regime corresponding to a strong drag force Courant Institute, NYU and a strong Brownian motion. New York, NY [email protected] Antoine Mellet University of British Columbia [email protected] MS45 Optimal Lower Bounds on the Stress Inside Ran- PD07 Abstracts 71

dom Two-Phase Composites this property, and in fact, minimizers of the relaxed energy must be in SBV . I will describe work in this direction, A prescribed uniform stress is applied to the boundary which is joint with G. Dal Maso and A. Garroni. of a composite material made from two perfectly bonded isotropic linear elastic components. We present lower Christopher Larsen bounds on the maximum stress generated inside the com- Worcester Polytechnic Institute posite that apply to all mixtures made from two elastic Math Dept materials in fixed volume fractions. We present several sit- [email protected] uations in which these lower bounds are attained by simple configurations of the two materials. MS45 Bacim Alali On the Eshelby Conjecture and a Symmetry Result Mathematics Department for Overdetermined Elliptic Problems Louisiana State University [email protected] In this paper we present solutions to the Eshelby conjecture. We first discuss the exact meanings of the original Robert P. Lipton Eshelby conjecture. We then prove one version of the Es- Department of Mathematics helby conjectures and construct a counterexample to the Louisiana State University other by exploring a related variational inequality problem. [email protected] In particular we construct multiply-connected regions in all dimensions and nonellipsoidal connected regions in three and higher dimensions that share the special property of MS45 an ellipsoid that some uniform eigenstress of the region in- Gradient Estimates for the Perfect Conductivity duces uninform strain on the region. We also show that one Problem version of the Eshelby conjecture is equivalent to asserting that some overdetermined elliptic problem has a solution if We consider the optimal bound on the stress occurring be- and only if the region is an ellipsoid. We therefore obtain tween a pair of closely spaced fibers in a bounded domain. a symmetry result for this overdetermined elliptic problem With Dirichlet boundary condition, the conductivity prob- by studying the Eshelby conjecture. lem can be described as follows Liping Liu div [1 + χ(D1 ∪ D2)(k − 1)]∇uk = 0 in Ω, Mechanical Engineering Department California Institute of Technology uk = ϕ [email protected]∂Ω, (1) where Ω contains two inclusions D1 and D2 which are ε- apart, and ∇uk represents the stresses.We study the per- MS46 fect conductivity problem, where k =+∞. We provide Stable Numerical Schemes for Constitutive Equa- an optimal blow-up rate of the gradient estimate as the tions of Complex Fluids separation distance ε approaches 0. It is shown that the blow-up rate depends not only on the dimension, but also Modelling of polymeric fluids can be done using two ap- on the shape of the inclusions. proaches: macro-macro formulations or micro-macro formulations using kinetic equations to model the evolution ShiTing Bao, YanYan Li, Biao Yin of the microstructures in the fluid. Macro-macro models Mathematics Department are cheaper to discretize than micro-macro models, but Rutgers University are known to blow up when a parameter, the Weissenberg [email protected], [email protected], number, increases. Many studies (e.g. [1]) have identified [email protected] possible reasons for that so-called High Weissenberg Num- ber Problem (HWNP), but have not led yet to a complete understanding of the numerical instabilities. Micro-macro MS45 models often yield more information and better under- On Cohesive Fracture Without Relaxation standing of the coupling between scales than macro-macro models. The key role of the Free energy in long-time be- Cohesive models of fracture involve energies of the form haviour was first understood thanks to the equivalence of the Oldroyd-B model with a kinetic model [2]. We will n−1 E(u):= W (∇u)dx + φ([u])dH , discuss the stability of finite element methods to discretize Ω K macro-macro models, using Free energy estimates. [1] R. Fattal and R. Kupferman, Time-dependent simulation of where u is a deformation with crack set (i.e., discontinuity viscoelastic flows at high Weissenberg number using the set) K and [u] is the jump in u across K. What makes the → → log-conformation representation, J. Non- Newtonian Fluid model cohesive is that φ(x) 0asx 0, so that there Mechanics, 126 (1) (2005) 23-37. [2] B. Jourdain and C. Le are forces across K if [u] is small enough. If φ is concave Bris and T. Lelievre and F. Otto, Long-Time Asymptotics with φ(0) = ∞, then existence of a minimizer is straight- ∞ of a Multiscale Model for Polymeric Fluid Flows , Archive forward. However, if φ (0) < , there at first seems to for Rational Mechanics and Analysis, 181 (2006) 97-148. be a fatal issue in that one needs to relax the energy to functions u that have singular behavior besides disconti- Sebastien Boyaval nuities (i.e., relaxed from SBV to BV ), resulting in the CERMICS addition of a new term to the energy. Precisely, a sequence Ecole Nationale des Ponts et Chaussees {ui} in SBV with bounded energy might converge to a [email protected] function u/∈ SBV , which appears fatal to applying the direct method of the calculus of variations. However, there Tony Lelievre is reason to believe that minimizing sequences cannot have 72 PD07 Abstracts

ENPC consider the coupling between the bulk fluid and the mem- [email protected] brane dynamics. Dynamic structure factors are computed in varies cases. These new systems of equations allow us to carry out stable and robust numerical simulations for the MS46 dynamics of the membranes. Global Orientation Dynamics for Liquid Crys- talline Polymers Pingwen Zhang Peking University We study global orientation dynamics of the Doi- [email protected] Smoluchowski equation with the Maier-Saupe potential on the sphere, which arises in the modeling of rigid rod-like molecules of polymers. Using the orientation tensor we MS47 first reconfirm the structure and number of equilibrium so- Asymptotics of the Semiclassical Sine-Gordon lutions established in [Comm. Math. Sci. 3 (2), (2005), Equation 201-218] by H. Liu, H. Zhang and P.-W. Zhang. We then examine global orientation dynamics in terms of eigenval- The study of the sine-Gordon equation with a fixed ini- ues of the orientation tensor via the Doi closure approxi- tial condition and dispersion tending to zero is motivated mation. It is shown that for small intensity 0 <α<4, all by the modeling of magnetic flux propagation in Josephson states will evolve into the isotropic phase; for large inten- junctions. We discuss the recent discovery of two families of sity α>4.5, all states will evolve into the nematic prolate initial data, with topological charge zero and one, respec- phase; and for the intermediate intensity 4 <α<4.5, an tively, for which the sine-Gordon equation can be solved initial state will evolve into either the isotropic phase or explicitly for arbitrarily small dispersion. Plots of the solu- the nematic prolate phase, depending on whether such an tions for small dispersion reveal, depending on the choice of initial configuration crosses a critical threshold. Moreover, other parameters, regions of pure librational and rotational the uniaxial symmetry structure is shown to be preserved motion, as well as regions of multi-phase waves separated in time. by primary and secondary nonlinear caustics. We present current progress on the asymptotic analysis of these solu- Hailiang Liu tions for small dispersion. Iowa State University USA Robert J. Buckingham [email protected] University of Michgan [email protected]

MS46 On the New Multiscale Rod-Like Model of Poly- MS47 meric Fluids Universality Classes for Semiclassical Eigenvalue Problems It is concerned with the well-posedness for the new rigid rod-like model in a polymeric fluid and the structure of We will discuss some generalized eigenvalue problems (in some solutions in special case. The constitutive relations which the eigenvalue does not necessarily enter linearly) in considered in this work are motivated by the kinetic theory. a semiclassical scaling (where derivatives are multiplied by The micro equation has five spatial freedom variables, two small coefficients). Such problems arise frequently in the of them are in the configuration domain and the others are asymptotic analysis of nonlinear problems solvable by an in the macro flow domain. inverse-scattering transform. We will show how the asymptotics of the discrete spectrum leads to the idea of univer- Hui Zhang sality classes of potentials and describe how this idea can Beijing Normal University be used to approximate the discrete spectrum with quan- [email protected] titative error estimates. Peter D. Miller MS46 University of Michigan Dynamical Modeling and Simulation of Fluid Bio- [email protected] Membranes

This work provides a set of equations for the dynamics of MS47 evolving fluid membranes, such as cell membranes, in the Semiclassical Limit Scattering Map (Direct and In- presence of bulk fluids. We model the membrane as a sur- verse ) for the Focusing Nonlinear Schroedinger face endowed with a director field, which describes the local Equation. average orientation of the molecules on the membrane. A model for the elastic energy of a surface endowed with a We discuss the progress in finding the semiclassical (zero director field is derived using liquid crystal theory. This dispersion) limit of the scattering map for the focusing elastic energy reduces to the well-known Helfrich energy NLS, that put into correspondence the initial potential in the limit when the directors are constrained to be nor- q(x, ) and the scattering data. In the solitonless case, the mal to the surface. We then derive the full dynamic equa- scattering data consists of the reflection coefficient r(z,), tions for the membrane that incorporate both the elastic where z is the spectral and is the semiclassical parameter and viscous effects, with and without the presence of the respectively. So, in this case, we are considering the → 0 bulk fluids. We also consider the effect of local sponta- limit of the maps q → r and r → q. neous curvature, arising from the presence of membrane proteins. Overall the new systems of equations allow us Stephanos Venakides to carry out stable, accurate and robust numerical model- Duke University ing for the dynamics of the membranes. In addition, we Department of Mathematics [email protected] PD07 Abstracts 73

Alexander Tovbis isfy alternative, equivalent forms of the Harnack inequality University of Central Florida Department of Mathematics Emmanuele DiBenedetto [email protected] Department of Mathematics Vanderbilt University Xin Zhou [email protected] Duke University [email protected] MS48 Continuity of the Saturation in the Flow of Two MS47 Immiscible Fluids Through a Porous Medium Persistence of Modulated Post-Break NLS Wave- We consider a weakly coupled, highly degenerate system forms in the Presence of Solitons. of an elliptic equation and a parabolic equation, arising in We analyze the persistence of modulated 2-phase NLS the theory of flow of immiscible fluids in a porous medium. waves developed after the first break (first nonlinear caus- The unknown functions in the system are u and v, and tic line in space-time). In previous work, we have shown represent pressure and saturation respectively. By using (within an initial wave structure) global time-persistence of suitable DeGiorgi’s techniques, we prove that the satura- these waves when the initial data are solitonless. We now tion is a continuous function in the space–time domain of show that, for initial data containing solitons, the same is definition. Here the main technical point is that no as- true for all space points whose distance from the origin is sumptions at all are made on the nature of the degeneracy less than a calculated distance d. For spatial points closer of the system. | | to the origin, we calculate an asymptotic formula (as x Vincenzo Vespri increases to d) for time t = t(x) up to which the waveform Department of Mathematics is guaranteed not to undergo a second break. University of Florence Stephanos Venakides [email protected]fi.it Duke University Department of Mathematics Ugo P. Gianazza [email protected] Department of Mathematics University of Pavia Alexander Tovbis [email protected] University of Central Florida Department of Mathematics Emmanuele DiBenedetto [email protected] Department of Mathematics Vanderbilt University, Nashville Xin Zhou, Sergey Belov [email protected] Duke University [email protected], [email protected] MS48 Beyond the Laplacian MS48 In this talk we discuss some applications of boundary Har- Wiener’s Criterion for the Regularity of the Point nack inequalities for positive p-harmonic functions (re- at Infinity and its Measure-Theoretical Counter- cently proved by Lewis and Nystrom) vanishing on the part boundary of a Lipschitz or Reifenberg flat domain. Pos- We introduce a notion of regularity (or irregularity) of the sible topics include the dimension of p-harmonic measure point at infinity for the unbounded open set concerning the for Wolff snowflake, the construction of p-harmonic qua- heat equation, according as whether the parabolic measure sispheres, and codimension ¿ 1 Harnack - free boundary of the point at infinity is zero or positive. We prove the regularity. necessary and sufficient condition for the existence of a John Lewis unique bounded solution to the parabolic Dirichlet problem University of Kentucky in an arbitrary unbounded open set expressed in terms of [email protected] the Wiener’s criterion for the regularity of the point at infinity. MS48 Ugur G. Abdulla Department of Mathematical Sciences On the Regularity of the Free Boundary for the Florida Institute of Technology Classical Stefan Problem abdulla@fit.edu The regularity of the temperature and the free boundary for the Stefan problem has attracted much attention over MS48 the last 3 decades or so. In this talk I will consider the classical one- or two-phase Stefan problem, where the free Connecting Non–Negative Solutions to Quasi– boundary is represented as the graph of a function. Un- Linear Degenerate Equations, Sub-potentials der mild regularity assumptions on the initial data it will Non–negative weak solutions of quasilinear degenerate be shown that the problem admits a local solution that parabolic equations of p–Laplacian type are shown to be lo- is analytic in space and time. The approach is based on cally bounded below by sub–potentials. As a consequence optimal regularity results for a linearized problem, and on non–negative solutions expand their positivity set and sat- a scaling-translation argument in conjunction with the im- 74 PD07 Abstracts

plicit function theorem. The University of North Carolina at Chapel Hill [email protected] Gieri Simonett Vanderbilt University [email protected] MS49 Propagation of a KPP-Type Fronts in a Model of Jan Prüss Pressure Driven Flames Martin-Luther University Halle [email protected] We study the model of pressure driven-flames in porous media introduced by G. Sivashinsky et. al. Under the Jürgen Saal assumption that the chemical kinetics is described by the University of Konstanz, Germany KPP-type nonlinearity, we show that the problem admits a [email protected] family of positive traveling solutions. Moreover, it is shown that the long time behavior of the model is prescribed by the rate of decay of initial data at infinity. This is a joint MS49 work with A. Ghazaryan. On Long-Time Peter Gordon Dynamics for Competition-Diffusion Systems with New Jersey Institute of Technology Inhomogeneous Dirichlet Boundary Conditions [email protected] We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous MS49 Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solu- Blow Up and Regularity for Burgers Equation with tion converges to that of a freeboundary problem. If all sta- Fractional Dissipation tionary solutions of this limit problem are non-degenerate I will discuss recent results on regularity and blow up of and if a certain linear combination of the boundary data solutions of Burgers equation with fractional dissipation. does not identically vanish, then for sufficiently large in- Fluid mechanics models with fractional dissipation, such as terspecific competition, all non-negative solutions of the surface quasi-geostrophic equation, have recently attracted competition-diffusion system converge to stationary states significant attention. The Burgers equation is perhaps the as time tends to infinity. Such dynamics are much simpler simplest model for studying interaction between nonlinear- than those found for the corresponding system with either ity and fractional viscosity. We establish a sharp transition homogeneous Neumann or homogeneous Dirichlet bound- between singular front formation and smooth evolution as ary conditions. the fractional dissipation exponent passes 1/2. We also Norman Dancer prove analyticity of the solutions in the subcritical case, School of Mathematics and Statistics and prove existence of solutions for rough initial data. University of Sydney Alexander Kiselev [email protected] University of Wisconsin, Madison [email protected] Danielle Hilhorst Laboratoire de Mathématiques Université de Paris-Sud MS50 [email protected] Global Well-Posedness of the Three-Dimensional Geostrophic Turbulence Mixing Model Elaine Crooks Oxford University The three-dimensional viscous Boussinesq equations under [email protected] hydrostatic balance govern large scale dynamics of atmo- sphere and oceanic motion. To overcame the turbulence mixing a vertical diffusion is added. In this paper we prove MS49 the global existence and uniqueness (regularity) of strong High Lewis Number Combustion Fronts: a Geo- solutions to this model. metric Singular Perturbation Theory Point of View Edriss Titi I will discuss combustion wavefronts that arise in a math- Department of Mathematics ematical model for high Lewis number combustion pro- University of California, Irvine cesses. An efficient method for the proof of the existence [email protected] and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported Chongsheng Cao by the model with very large Lewis numbers are small per- Department of Mathematics turbations of the front supported by the model with infinite Florida International University Lewis number. The question of stability of fronts is more caoc@fiu.edu complicated. I will describe the issues arising in the stability analysis and up to date known results, in particular, how a geometric approach which involves construction of MS50 the augmented unstable bundles can be used to relate the Wave Turbulence: A Story Far from Over spectral stability of the wavefront with high Lewis number to the spectral stability in the case of infinite Lewis Wave turbulence, the study of the long time statistical number. This is a joint work with C.K.R.T. Jones behavior of solutions of nonlinear field equations describing weakly nonlinear dispersive waves in the presence of Anna Ghazaryan sources and sinks, is a subject far from over. Among the PD07 Abstracts 75

remaining challenges are: 1. Since wave turbulence theory MS51 almost never holds at all scales, one has to understand how Some Numerical Results on Rudin-Osher-Fatemi the wave turbulence stationary states coexist with fully Smoothing nonlinear states. 2. Understanding how the Kolmogorov- Zakharov spectra are realized in cases where these solutions We present some numerical algorithms, their analyses, and have finite capacity. The results of 2. may very well have results for ROF-style variational image smoothing. relevance for turbulence in general. I will briefly discuss these challenges and suggest some resolutions. Bradley Lucier Department of Mathematics Alan Newell Purdue University University of Arizona [email protected] Department of Mathematics [email protected] MS51 Some Properties of a Perturbed Version of the MS50 Perona-Malik Equation Daynimic Transitions in Geophysical Fluid Dynam- ics The one-dimensional version of Perona-Malik equation, originally proposed for image selective smoothing, is con- First we shall present a new dynamic transition theory for sidered. This is an ill-posed parabolic equation for which nonlinear dynamical systems, both finite and infinite di- stable numerical schemes exist. We investigate a perturbed mensional. The theory will be used to 1) classify dynamic version of the equation such that the perturbation does transitions of different flow regimes of rotating Boussinesq not change the order of the differential operator. The per- equations, a basic model in geophysical fluid dynamics, and turbed problem is well posed and the asymptotic behaviour 2) to provide a new theory for ENSO. of its solutions is analyzed when a suitable perturbation parameter tends to zero. Tian Ma Sichuan University Riccardo March [email protected] Istituto per le Applicazioni del Calcolo, CNR [email protected] Shouhong Wang Department of Mathematics Mario Rosati, Andrea Schiaffino Indiana University Universita’ Roma II [email protected] [email protected], schiaffi@mat.uniroma2.it

MS50 MS51 The Surface Quasi-Geostrophic Equation Curvature of Level Sets of BV Functions

This talk reports several recent results concerning solutions Abstract not available at time of publication. of the inviscid quasi-geostrophic (QG) equation and of the dissipative QG equation with supercritical dissipation. Murali Rao University of Florida Jiahong Wu [email protected]fl.edu Oklahoma State University [email protected] MS51 (Phi,Phi*) Image Decomposition Models and Min- MS51 imization Algorithms Diffuse Interface Methods for Image Reconstruc- tion We review minimization algorithms for image restoration and decomposition using dual functionals and dual norms. We consider some recent diffuse interface methods for im- In order to extract a clean image u from a degraded version age reconstruction. Applications include image inpainting, f=Ku+n, where f is the observation, K is a blurring opera- in which missing information must be recovered in some re- tor and n represents additive noise, we impose a standard gion, and deconvolution, in which an image is blurred due regularization penalty on u, of the form Phi(u), depending to the process of obtaining the data. Real life applications on the gradient of u; however, on the residual f-Ku, we im- include document explotation, following roads in sattelite pose a weaker dual penalty Phi*(f-Ku), instead of the more imagery, and bar code scanners. We present some new standard L2 data fidelity term. In particular, when Phi(u) methods with fast solvers based on diffuse interface equa- is the total variation of u, we recover the (BV,BV*) decom- tions from materials science. One class of methods involves position of the data f, as suggested by Y. Meyer. Practical the Cahn-Hilliard equation for binary mixtures. Another minimization methods are presented, together with exper- class of methods involves a nonlinear wavelet-based func- imental results and comparisons to illustrate the validity tional related by scaling to the Ginzburg-Landau energy of the proposed models. from diffuse interface methods. We consider both binary and greyscale applications with possible extension to high Luminita A. Vese dimensional pixel data. This is joint work with Julia Do- University of California, Los Angeles brosotskaya, Selim Esedoglu, and Alan Gillette. Department of Mathematics [email protected] Andrea L. Bertozzi UCLA Department of Mathematics Triet M. Le [email protected] 76 PD07 Abstracts

Yale University MS52 [email protected] Lorentz Space Estimates for Ginzburg-Landau Vor- tices Linh M. Lieu University of California, Davis We discuss the Ginzburg-Landau model of superconductiv- [email protected] ity in two dimensions. We present a technical improvement of the “vortex balls construction” that allows for the ex- traction of a new positive term in the energy lower bounds MS52 in the vortex balls. This term is then estimated using the 2,∞ A Variational Model for Bent-Core Liquid Crystals Lorentz space L , which is critical for the expected vortex proﬁles. From this we can estimate the total number We present a nonlinear variational model for bent-core of vortices and prove improved convergence results. (banana-shaped) liquid crystals which contains energy terms including polarization, smectic aned chiral eﬀects, Ian Tice elasticity, and surface tension. We analyze solutions in a Courant Institute, NYU physically realistic regime involving a free boundary prob- [email protected] lem for the energy and the existence of a Gamma limit. Daniel Phillips MS53 Department of Mathematics, Purdue University Monge’s Mass Transportation in the Heisenberg [email protected] Group

Patricia Bauman Abstract not available at time of publication Purdue University Robert Berry [email protected] University of Pittsburgh. [email protected] MS52 Equilibrium Configurations of Epitaxially Strained MS53 Crystalline Films: Existence and Regularity Re- Title Not Available at Time of Publication sults Abstract not available at time of publication. Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elastic- Nicola Garofalo ity. The total free energy of the system is assumed to be Purdue University the sum of the energy of the free surface of the film and [email protected] the strain energy. Due to the lattice mismatch between film and substrate, flat configurations are, in general, energetically unfavorable and a corrugated or islanded morphology MS53 is the preferred growth mode of the strained film. New reg- Regularity of Lipschitz Free Boundaries in Two- ularity results for volume constrained local minimizers of Phase Problems for the p-Laplace Operator the total free energy are established, leading to a rigorous proof of the zero contact-angle condition between islands In this paper we study the regularity of the free bound- and wetting layers. ary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lips- Irene Fonseca chitz free boundaries are C1,γ -smooth for some γ ∈ (0, 1). Carnegie Mellon University As part of our argument, and which is of independent in- Center for Nonlinear Analysis terest, we establish the Hopf boundary principle for non- [email protected] negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain. MS52 Kaj Nystrom A Gamma-Convergence Approach to the Cahn- Umea University, Sweden Hilliard Equation [email protected]

We study the asymptotic dynamics of the Cahn-Hilliard John Lewis equation via the “Gamma convergence” of gradient flows University of Kentucky scheme initiated by Sandier and Serfaty. This gives rise [email protected] to an H1-version of a conjecture by De Giorgi, namely, the slope of the Allen-Cahn functional with respect to −1 the H -structure Gamma-converges to a homogeneous MS53 Sobolev norm of the scalar mean curvature of the limit- P-Harmonic Measure on Wolff Snowflakes. ing interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, un- We study the dimension of p-harmonic measure on Wolff der suitable conditions for which the conjecture is true, snowflakes using some of the new results on boundary Har- we prove that the limiting dynamics for the Cahn-Hilliard nack inequalities of Lewis and Nystrom. equation is motion by Mullins-Sekerka law. Kaj Nystrom Nam Le Umea University, Sweden Courant Institute, NYU [email protected] [email protected] PD07 Abstracts 77

John Lewis MS54 University of Kentucky Darcy-Brinkman Models of Fast Channel Flow at [email protected] an Interface

Andrew Vogel We show that preferential flow or channeling effects at the Syracuse University boundary of a porous medium can be effectively modeled [email protected] by a Darcy-Brinkman interface system. Fernando Morales Bjorn Bennewitz Oregon State University University of Jyvaskyla [email protected] [email protected].fi Ralph Showalter MS54 Department of Mathematics Rheology of Bacterials Suspensions Oregon State University [email protected] Modeling of bacterial suspensions and, more generally, of suspensions of active microparticles has recently become an increasingly active area of research. The focus of this work MS55 is on the development and analysis a mathematical PDE Mathematical Problems Arising in Near-Shore model for a multiscale problem of bacterial suspensions. Zone Sediment Trasport We start from a comparison of bacterial suspensions (active suspensions) and suspensions of solid particles (passive sus- A wave - sediment interaction model is put forward and pensions). The ultimate goal is to develop a model which validated. Some mathematical issues that arise as a conse- describes large scale pattern formationand rhelogical prop- quence will be described. erties of bacterial suspensions. We discuss recent results Jerry Bona on effective viscosity of dulite bacterial suspensions (with University of Illinois at Chicago Aronson, Haines and Karpeev) and asymptotic analysis of [email protected] swimming patterns of bacteria (with Aronson, Gyrya and Karpeev). MS55 Leonid Berlyand Penn State University Water Waves Over Bottom Topography Eberly College of Science We present a Hamiltonian formulation for water waves over [email protected] bottom topography based on potential flow theory. In this formulation, the problem is reduced to a lower-dimensional MS54 system involving boundary variables alone. This is ac- complished by introducing the Dirichlet-Neumann opera- New Exact Bounds for Effective Conductivity of tor which expresses the normal fluid velocity at the free Multiphase Composites and Optimal Microstruc- surface in terms of the velocity potential there, and in tures terms of the surface and bottom variations which deter- Abstract not available at time of publication. mine the fluid domain. A Taylor series expansion of the Dirichlet-Neumann operator in powers of the surface and Andrej V. Cherkaev bottom variations is proposed. This formulation has im- Department of Mathematics plications for the convenience of perturbation calculations University of Utah and numerical simulations of the full Euler equations. We [email protected] develop an efficient and accurate spectral method based on the fast Fourier transform to evaluate numerically the Dirichlet-Neumann operator and solve the full Euler equa- MS54 tions. Tests and validation will be shown. This is joint Homogenization and Field Concentrations in Het- work with W. Craig (McMaster University), D. P. Nicholls erogeneous Media (UIC) and C. Sulem (University of Toronto).

A method for upscaling the local field concentrations inside Philippe Guyenne random composite and polycrystalline media is presented. University of Delaware The talk focuses on gradient or strain fields associated with [email protected] solutions of second order elliptic PDE used in the description of thermal transport and elasticity inside random media. We develop a method for assessing the Lp integrability MS55 of gradient and strain fields inside microstructured media. Long Nonlinear Internal Waves Models The results are described in terms of the pth order moments of the solution of two-scale corrector problems. Examples We derive systematically and justify rigorously asymptotic are provided that illustrate the theory and its application. models for the propagation of long nonlinear internal waves in 2+1 dimensions. This is a joint work with Jerry Bona Robert P. Lipton and David Lannes. Department of Mathematics Louisiana State University Jean-Claude Saut [email protected] Paris-Sud France [email protected] 78 PD07 Abstracts

MS56 MS56 On a Variational Approach for Stokes Conjecture Pulsating Wave for Motion by Mean Curvature in on Water Waves Inhomogeneous Medium

We study the behaviour of a two-dimensional, inviscid, ir- We prove the existence and uniqueness of pulsating waves rotational fluid in a domain of finite depth and in the pres- for the motion by mean curvature of a hypersurface in an ence of gravity. More precisely, we investigate the mini- inhomogeneous medium given by periodic forcing. The mizers of the functional main difficulty is caused by the degeneracy of the equation and the fact the inhomgeneity is allowed to change its sign. Under the assumption of weak forcing, we obtain uni- 2 form oscillation and gradient bounds so that the evolving Jλ (u)= |∇u| +(λ − y)+ χ{u>0} dx dy Ω surface can be represented as a graph over a hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the with Dirichlet boundary condition at the bottom. We show Lipshitz continuity of the speed with respect to the nor- that for large values of the paremeter λ, minimizers of Jλ mal and various stability properties of the pulsating wave. have a regular (analytic) free boundary while for small val- Some connection with homogenization theory will also be ues of λ, minimizers are ”nonphysical”. This leads to a discussed. This is joint work with N. Dirr and G. Karali. critical value of λ related to Stokes wave of greatest height. Aaron Yip Danut Arama Purdue University Carnegie Mellon University Department of Mathematics Pittsburgh, PA 15213 [email protected] [email protected] MS57 MS56 Ginzburg-Landau Vortices Concentrating on A Variational Formulation for Level Set Represen- Curves tation of Multiphase Flow. We study a two-dimensional Ginzburg-Landau functional, We discuss variational formulations for level set represen- which describes superconductors in an externally applied tations of various multiphase flows that involve curvature. magnetic field. We are interested in describing the en- Our formulations lead to practical algorithms for comput- ergy minimizers at the critical value of the magnetic field ing these flows accurately. Joint work with Peter Smereka. for which vortices first appear in the superconductor (the ”lower critical field”.) The vortices are quantized singu- Selim Esedoglu larities, and we are interested in their number and their Department of Mathematics distribution in the sample nearby the critical field. I will University of Michigan describe recent results with L. Bronsard and V. Millot in [email protected] which we study the number and distribution of these vortices which concentrate along a curve. Their distribution is determined by a classical problem from potential theory. MS56 Phase Transitions in Thin Films Stanley Alama McMaster University In this talk, I will present some models for solid-solid phase Department of Mathematics and Statistics transitions in thin films obtained by Γ-convergence. The [email protected] different models are determined by the asymptotic ratio between the characteristic length scale of the phase transition Vincent Millot and the thickness of the film. Depending on the regime, I Carnegie Mellon University will show that a separation of scales holds or not and how Pittsburgh, PA 15213 this later case is related to some rigidity results. This is a [email protected] joint work with V. Millot. Bernardo Galvao-Sousa Lia Bronsard Carnegie Mellon University McMaster University [email protected] [email protected]

MS56 MS57 Optimal Transport for the System of Isentropic Eu- Global Minimizers of the LawrenceDoniach Model ler Equations with Oblique External Field

In this talk we report on a variational time discretization We study periodic minimizers of the Lawrence-Doniach for the system of isentropic Euler equations. The scheme model for layered superconductors, in various limiting is inspired by the theory of abstract gradient flows on the regimes. We are particularly interested in determining the space of probability measures, and by Dafermos’ entropy direction of the internal magnetic field (and vortex lattice) rate criterion: The total energy should be decreased at as a function of the applied external magnetic strength and maximal rate. This is joint work with Wilfrid Gangbo. its orientation with respect to the superconducting planes. We identify the corresponding lower critical fields, and Michael Westdickenberg compare the Lawrence-Doniach and anisotropic Ginzburg- Georgia Institute of Technology Landau minimizers in the periodic setting. This talk rep- [email protected] PD07 Abstracts 79

resents joint work with S. Alama and E. Sandier. MS58 Propagation of Fisher Information for Inelastic Stanley Alama Collisions McMaster University Department of Mathematics and Statistics We prove bounds on the Fisher information for solutions of [email protected] the spatially homogeneus Boltzmann equation with inelastic collisions. The bounds diverge as time tends to inﬁnity, Etienne Sandier as they must, due to the dirac mass limit, but nonetheless, Univ. Paris 12 the sharp bound obtained here plays a key role in a proof Creteil, France that when the solutions are rescaled to preserve the second [email protected] moment, the result is regular uniformly in time. This is joint work with E. Carlen and J. Carillo. Lia Bronsard McMaster University Maria Carvalho Department of Mathematics University of Lisbon and Georgia Tech [email protected] [email protected]

MS57 MS58 Analysis of Heterogeneous Superconducting Sys- Kinetic Dynamics of Multilinear Stochastic Inter- tems Via the Ginzburg-Landau Approach. actions

Generalizations of the Ginzburg-Landau energy functional Abstract not available at time of publication. have been introduced in the physics literature to describe Irene M. Gamba composite superconducting bodies. For example, the pres- Department of Mathematics and ICES ence of a metal or semiconductor that surrounds or coats a University of Texas superconductor can be modeled by a suitable extension of [email protected] the energy to the non-superconducting components or by the addition of surface integrals. We will present analytical results which illustrate how eﬀective these models can be MS58 in describing the physical response of the systems under Spectral - Lagrangian Based Deterministic Method examination. for Space Inhomogenous Boltzmann Equation.

Tiziana Giorgi A new spectral Lagrangian based deterministic solver for New Mexico State Univeristy the non-linear Boltzmann Transport Equation for Variable Department of Mathematics Hard Potential VHP) modeling collision kernels has been tgiorgi@nmsu proposed for both conservative and non-conservative binary interactions. The method is based on the symmetries MS57 of the Fourier transform of the collision integral, where the complexity in computing it is reduced to a separate inte- Vortex Lines in Bose-Einstein Condensates gral over the unit sphere S2. In addition the conservation In this talk I will present some results that give a prelimi- of moments is enforced by Lagragian constrains. The re- nary description of the vortex lines exhibited by Bose Ein- sulting scheme is very versatile and adjust to in a very stein condensates, as well as super-conductors, subject to simple manner to several that involved energy dissipation a stirring in an appropriate range. What one finds is that due to local micro-reversibility (inelastic interactions) or these vortex lines can be described by means of an ODE elastic model of slowing down process. The issue of con- that represents the equation of the geodesics in an appro- vergence has also been addressed. Since the equation is priate manifold. This is joint work with Ben Stephens. solved for explicit distribution function, the high velocity tail behavior can be analyzed explicitly. Such an analysis Alberto Montero is crucial in understanding the physical processes involved University of Toronto in certain problems such as boundary and strong shock Department of Mathematics layer formation and sheared flows in classical elastic di- [email protected] luted gases or rapid granular flows or a mixture of gases where classical fluid dynamical models fail.The proposed numerical method has been used to solve the space inho- MS58 mogenous Boltzmann equation for Riemann problem and Strong Convergence to Homogeneous Cooling Couette flow problems. States Sri Harsha Tharkabhushanam We study solutions of the spatially homogeneus Boltzmann The University of Texas at Austin equation with inelastic collisions with rescaling to fix the [email protected] temperatures. We prove regularity properties for these solutions that are valid uniformly in time, and use thses to quantify the rate of convergence to the homogeneous cool- MS59 ing state in the strong L1 norm. This is joint work with Some Mathematical Models in Emerging Biomedi- M.C. Carvalho and J. Carrillo. cal Imaging Eric A. Carlen Abstract not available at time of publication. Georgia Tech [email protected] Habib Ammari CNRS & Ecole Polytechnique, France 80 PD07 Abstracts

[email protected] is frequently exact. Jared Bronski MS59 University of Illinois Urbana-Champain Modeling and Design of Coated Stents in Interven- Department of Mathematics tional Cardiology [email protected]

Stents are used in interventional cardiology to keep a dis- eased vessel open. New stents are coated with a medicinal MS60 agent to prevent early reclosure due to the proliferation of Dispersion Relations for Subwavelength Photonic smooth muscle cells. It is recognized that it is the dose of Crystals the agent that effectively controls the growth. This paper focusses on the asymptotic behavior of the dose for gen- We examine wave propagation in sub-wavelength dielec- eral families of coated stents under a fixed ratio between tric/metal photonic crystals near the high plasma fre- the coated area of the stent and the targeted area of the quency limit. We characterize the band structure for this vessel and set therapeutic bounds on the dose. It gener- case. Examples are presented for layered media and wave alizes the results of Delfour, Garon, and Longo (SIAM J. guides. Appl Math) for stents made of a sequence of thin equally spaced rings to stents with an arbitrary pattern. It gives Stephen P. Shipman, Robert P. Lipton, Santiago Fortes the equation of the asymptotic dose for a normal tiling of Department of Mathematics the target region. Louisiana State University [email protected], [email protected], Michel C. Delfour [email protected] Centre de recherches mathematiques Université de Montréal, Canada [email protected] MS60 Fano Resonance in Photonic Crystal Slabs

MS59 Spectrally embedded guided modes in photonic crystal Optimization and Control Techniques for Medical slabs are nonrobust under perturbations of the structure Image Registration or the wave vector. When the eigenvalue for the nonrobust mode dissolves into the continuous spectrum, it leaves a Abstract not available at time of publication. spectral region of anomalous transmission in its wake. This phenomenon has its analog in the Fano resonances in quan- G¨unter Leugering tum mechanics. Our rigorous asymptotic formula for the University Erlangen-Nuremberg transmission anomalies for periodic slabs is a generalization Institute of Applied Math. of the Fano formula for line shapes in physical chemistry. [email protected] Joint work with S. Venakides. Stephen P. Shipman MS59 Department of Mathematics Drag Minimization for Stationary, Compressible Louisiana State University Navier-Stokes Equations [email protected]

Abstract not available at time of publication. Stephanos Venakides Duke University Jan Sokolowski Department of Mathematics Institut Elie Cartan [email protected] Universite Henri Poincare Nancy I [email protected] MS61 Pavel Plotnikov Schroedinger Maps (Landau-Lifschitz equation) Lavryentyev Institute of Hydrodynamics Novosibirsk, Russia Abstract not available at time of publication. [email protected] Stephen Gustafson Department of Mathematics MS60 University of British Columbia Defect Eigenvalues and Diophantine Conditions [email protected] Via the Evans Function.

We consider Schrodinger eigenvalue problem in one dimen- MS61 sion with a potential consisting of a compactly supported On the Lifetime of Quasi-Stationary States in Non- ”defect” potential plus a periodic part. This is a model for Relativistic QED similar structures in optics. We give a result which allows one to count the number of point eigenvalues which are cre- We consider resonances in the Pauli-Fierz model of non- ated in a gap in the periodic spectrum. This count can be relativistic QED. We use and slightly modify the analysis expressed either in terms of an associated purely periodic developed by V. Bach, J. Froehlich, and I.M. Sigal to ob- spectral problem or as the Maslov index of a certain curve tain an upper and lower bound on the lifetime of quasi- in the symplectic group. This count is always within one of stationary states. This is joint work with I. Herbst and M. the actual number of eigenvalues (in either direction) and PD07 Abstracts 81

Huber. David Hasler University of Virginia [email protected]

MS61 Fractional Nonlinear Schroedinger Equations

This talk reviews recent results on fractional nonlinear Schroedinger equations (NLS). This novel class of dispersive PDEs contains model equations for various physical applications, ranging from relativistic astrophysics to molecular dynamics. I will present results concerning well- posedness of the initial-value problem, ﬁnite-time blowup, and solitary wave solutions for fractional NLS. As a guid- ing example, I will discuss the so-called pseudo-relativistic Hartree equation which arises as an eﬀective model equation for relativistic, self-gravitating matter in astrophysics. Enno Lenzmann Mathematics MIT [email protected]

MS61 Mathematical Aspects of Bose-Einstein Condensa- tion

The Bose-Einstein condensation (BEC) was predicted in 1924 and discovered experimentally in trapped gases in 1995. It oﬀers challenging and beautiful mathematical problems in Quantum Many-Body Theory and in nonlinear PDEs. In the latter case the main questions deal with behavior of solitons in the nonlinear Schrdinger equation with an external potential (the Gross-Pitaevskii equation). In this talk I review some recent results and open problems in the theory of BEC and, speciﬁcally, on the dynamics of solitons in the Gross-Pitaevskii equations. Michael Sigal University of Toronto Toronto, Canada [email protected]