PD09 Abstracts 45

IP1 functions. Free Boundary Regularity Yann Brenier We discuss level surfaces of solutions u to singular semilin- CNRS ear elliptic equations such as Δu = δ(u). The level surface Universitéde Nice Sophia-Antipolis u = 0 can be interpreted as a solution to a free boundary [email protected] problem, the problem of finding the optimal shape of in- sulating material. A variant of this equation, the Prandtl- Batchelor equation, describes the wake created by a boat. IP4 We will prove that stable free boundaries are smooth in The Importance of PDEs for Modeling and Simu- three-dimensional space. Numerical methods are needed lation on Peta and Exascale Systems to explore counterexamples in higher dimensions and to establish a more robust regularity theory with numerically As scientists and engineers tackle more complex problems effective bounds. involving multiphysics and multiscales and then seek to solve these problems through computation, the partial dif- David Jerison ferential equation formulations which incorporate and more Massachusetts Institute of Technology closely approximate the complex physical or biological phe- [email protected] nomena being modeled will also be important to unlocking the approach taken on Peta and Exascale systems. Scien- tists and engineers working on problems in such areas as IP2 aerospace, biology, climate modeling, energy and other ar- Fluid Biomembranes: Modeling and Computation eas demand ever increasing compute power for their problems. For Petascale and Exascale systems to be useful mas- We study two models for biomembranes. The first one is sive parallelism at the chip level is not sufficient. I will de- purely geometric since the equilibrium shapes are the min- scribe some of the challenges that will need to be considered imizers of the Willmore energy under area and volume con- in designing Petascale and eventually Exascale systems. straints. We present a novel method based on ideas from Through the combination of High Performance Comput- shape differential calculus. The second model incorporates ing (HPC) hardware coupled with novel mathematical and the effect of the inside (bulk) viscous incompressible fluid algorithmic approaches emerging from the original PDE and leads to more physical dynamics. We use a parametric formulations some efforts toward breakthroughs in science approach, which gives rise to fourth order highly nonlinear and engineering are described. While progress is being PDEs on surfaces and involves large domain deformations. made, there remain many challenges for the mathematical We discretize these PDEs in space with an adaptive fi- and computational science community to apply ultra-scale, nite element method (AFEM), with either piecewise linear multi-core systems to Big science problems with impact on or quadratic polynomials, and a semi-implicit time step- society. In conclusion, some discussion not only on the ping scheme. We employ the Taylor-Hood element for the most obvious way to use ultra-scale, multi-core HPC sys- Navier-Stokes equations together with iso-parametric ele- tems will be given but also some thoughts on incorporat- ments, the latter being crucial for the correct approxima- ing more physics in the algorithms derived from the PDE tion of curvature. We discuss several computational tools models which might allow us to better use such systems to such as space-time adaptivity and mesh smoothing. We tackle previously intractable problems. also discuss a method to execute refinement, coarsening, and smoothing of meshes on manifolds with incomplete in- Kirk E. Jordan formation about their geometry and yet preserve position IBM T.J. Watson Research and curvature accuracy. This is a new paradigm in adap- [email protected] tivity. Ricardo Nochetto IP5 Department of Mathematics Singular Limits in Thermodynamics of Viscous Flu- University of Maryland, College Park ids [email protected] We discuss some recent results concerning singular limits of the full Navier-Stokes-Fourier system, where one or several IP3 characteristic numbers become small or tend to infinity. Rearrangement, Convection and Competition The main ingredients of our approach read as follows:

Rearrangement theory is about reorganizing a given func- • a general existence theory of global-in-time weak so- tion (or map) in some specific order (monotonicity, cycle lutions for any finite energy data; monotonicity etc...). This is somewhat similar to the convection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid • uniform bounds independent of the value of the sin- at bottom and light fluid at top). This can also be related gular parameters based on total dissipation balance to some competition models in economy, where agents act (Second law of thermodynamics); according to their rank. In our talk, we make these analogies more precise by analysing the Navier-Stokes equa- • analysis of propagation of acoustic waves, in particu- tions with Boussinesq approximation of the buoyancy force lar on unbounded or “large’ spatial domains. and two distinct approximations of this model (Darcy and Boussinesq). We will see how these approximations are related to the concept, well known in optimal transport theory, of rearrangement of maps as gradient of convex Eduard Feireisl Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1 Czech Republic [email protected] 46 PD09 Abstracts

IP6 berger and Rosinger but it is actually based on topological Q-curvature in Conformal Geometry processes known as convergence structures.

In this talk , I will survey some analytic results concerned Roumen Anguelov, Jan Harm Van Der Walt with the top order Q-curvature equation in conformal ge- University of Pretoria ometry. Q-curvature is the natural generalization of the [email protected], [email protected] Gauss curvature to even dimensional manifolds. Its close relation to the Pfaffian, the integrand in the Gauss-Bonnet formula, provides a direct relation between curvature and topology. The notion of Q-curvature arises naturally in CP1 conformal geometry in the context of conformally covari- Existence of a Unique Solution to Equations Mod- ant operators. In 1983, Paneitz gave the first construction eling Fluid Flow with Time-Dependent Density of the fourth order conformally covariant Paneitz opera- Data tor in the context of Lorentzian geometry in dimension four. The ambient metric construction, introduced by Fef- We consider a system of nonlinear equations modeling the ferman and Graham , provides a systematic construction flow of a barotropic, compressible fluid. The equations in general of conformally covariant operators. Each such consist of a hyperbolic equation for the velocity, an alge- operator gives rise to a semi-linear elliptic equation anal- braic equation (the equation of state) for the pressure, and ogous to the Yamabe equations which we shall call the a modified version of the conservation of mass equation. Q-curvature equation. These equations share a number of We prove the existence of a unique solution to the model’s common features. Among these we mention the following: equations with time-dependent density data and spatially- (i) the lack of compactness: the nonlinearity always oc- dependent velocity data, under periodic boundary condi- cur at the critical exponent, for which the Sobolev imbed- tions. ding is not compact; (ii) the lack of maximum principle: Diane Denny for example, it is not known whether the solution of the Texas A&M University - Corpus Christi fourth order Q-curvature equation on manifolds of dimen- [email protected] sions greater than four may touch zero. In spite of these difficulty, there has been significant progress on questions of existence, regularity and classification of entire solutions CP1 for these equations in the recent literature. In the talk, I Localization of Analytic Regularity Criteria on the will give a brief survey of the subject with emphasize on Vorticity and Balance Between the Vorticity Mag- applications to problems in conformal geometry and the nitude and Coherence of the Vorticity Direction in connection between the a class of conformal covariant op- the 3D NSE erators to that of the fractional Laplacian opeators. 2q 2q−3 1 Sun-Yung Alice Chang DaVaiga has shown that Duq ∈ L (0,T) is a regu- Princeton University larity class for the Navier-Stokes Equations for any 3 ≤ [email protected] q<∞. Beale-Kato-Majda proved the regularity for vorticity in time-space L1L∞-norm. Geometric conditions on the vorticity direction improve the regularity and the local- IP8 ized versions have been recently obtained by Grujic. The p q Title Not Available at Time of Publication scaling-invariant regularity class of weighted L L -type for vorticity magnitude with coherence factor as a weight will Abstract not available at time of publication. be demonstrated in a localized setting. Felix Otto Rafaela Guberovic University of Bonn Seminar for Applied Mathematics, ETH, Zurich, Germany Switzerland [email protected] University of Virginia, Charlottesville, USA [email protected]

IP9 Zoran Grujic Title Not Available at Time of Publication Department of Mathematics University of Virginia Abstract not available at time of publication. [email protected] George C. Papanicolaou Stanford University CP1 Department of Mathematics [email protected] Fully-Coupled Aeroelastic Computations of High Reynolds Number Flows

CP1 Strong (two-way) coupling of fluid and structure presents interest to vary engineering applications, particularly when Solving Nonlinear Systems of Pdes Via Real Set- the flow is turbulent and sensitive to structural motions. A Valued Maps CFD based algorithm, using large-eddy simulation (LES), The existence and regularity of solutions of nonlinear PDEs is proposed for the numerical investigation of strong aeroe- resulting from constitutive relations of fluids are problems lastic coupling. The aeroelastic response of a cranked dou- with acknowledged difficulty. We prove that the solutions ble delta wing to forced flap motion is subject of investi- of very large classes of systems of nonlinear PDE can be gation. The results show that the flapping frequency in- assimilated with real set-valued maps. Our approach is fluences the wing deformation, as well the flow field in the similar to the order completion method of Oberguggen- PD09 Abstracts 47

wake of the wing. [email protected] Marcel Ilie University of Central Florida CP2 Department of Mechanical, Material and Aerospace Analysis of An Epidemic Model with Age Depen- Engineering dency: The Case of 2009 H1N1 Influenza Outbreak [email protected] It is well known that epidemic models concerned with the infection-age are represented by PDE. In resent years, a CP1 few researchers have investigated models take account of Mixed Bottom-Friction-Kelvin-Helmholtz Destabi- not only infection-age but chronological age, and this as- lization of Source-Driven Abyssal Overflows in the sumption seems to appropriate for the case of 2009 H1N1 Ocean influenza outbreak which more likely to affect young peo- ple. Our purpose of this presentation is to develop an age- Source-driven ocean currents that flow over topographic structured model of this epidemic and to study the math- sills are important initiation sites for the abyssal compo- ematical character. nent of the thermohaline circulation. These overflows exhibit vigorous space and time variability over many scales Toshikazu Kuniya as they progress from a predominately gravity-driven down Waseda university slope flow to a geostrophic along slope current. Observa- Graduate school of fundamental science and engineering tions show that in the immediate vicinity of a sill, grounded [email protected] abyssal ocean overflows can possess current speeds greater than the local long internal gravity wave speed with bottom friction and down slope gravitational acceleration dominat- CP2 ing the flow evolution. It is shown that these dynamics lead Mathematical Analysis of a Free Boundary Com- to the mixed frictionally-induced and Kelvin-Helmholtz in- bustion Model stability of grounded abyssal overflows. Within the overflow, the linearized instabilities correspond to bottom- This study concerns mathematical analysis of a free- intensified baroclinic roll waves and in the overlying wa- boundary model of solid combustion. Nonlinear transi- ter column amplifying internal gravity waves are gener- tion behaviors of small disturbances of front propagation ated. The stability characteristics are described as a func- and temperature as they evolve in time, as well as the tion of the bottom drag coefficient and slope, Froude, bulk large-time quasi-steady oscillations of these disturbances, Richardson and Reynolds numbers associated with the are studied. Previous asymptotic solutions with some dom- overflow and the fractional thickness of the abyssal current inant modes have significant errors for almost all cases. In compared to the mean depth of the overlying water col- this study, a special asymptotic expansion method is used umn. The marginal stability boundary and the boundary to improve the accuracy of the approximation. separating the parameter regimes where the most unstable mode has a finite or infinite wavenumber are determined. Jun Yu, Yi Yang When it exists, the high wavenumber cut-off is obtained. University of Vermont Conditions for the possible development of an ultra-violet [email protected], [email protected] catastrophe are determined. In the infinite Reynolds number limit, an exact solution is obtained which fully includes Jianping Cai, Meili Lin the effects of mean depth variations in the overlying water Zhangzhou Normal University, P. R. China column associated with a sloping bottom. For parameter [email protected], [email protected] values characteristic of the Denmark Strait overflow, the most unstable mode has wavelength of about 19 km, a geostationary period of about 14 hours, an e-folding am- CP2 plification time of about 2 hours and a down slope phase Stability and Bifurcation Analysis to Dissipative speed of about 74 cm/s. Cavity Soliton of Lugiato-Lefever Equation in One Dimensional Bounded Interval Gordon E. Swaters Department of Mathematical and Statistical Sciences We show a mathematically rigorous analysis for bifurca- University of Alberta tion structure of a spatially uniform equilibrium in non- [email protected] linear Schrödingerequations with dissipation, driving and detuning terms. Numerically, it has been reported that the “ snake bifurcation occurs. We ensure that the pitchfork CP1 bifurcation happens by using the bifurcation theory with On Conserved Quantities of Some Fluid Mechanical the symmetry, and make a much finer analysis at the codi- Equations mension two bifurcation point to prove the existence of fold bifurcation of nontrivial equilibrium around there. The study of conserved quantities for PDEs related to fluid mechanics is significant from both theoretical and practical Tomoyuki Miyaji points of views. For example, conservation of energy of the Dept. of Math. and Life Sci., Euler equations is closely related to turbulence and thus Hiroshima University plays an important role in its weak solution theory. In this [email protected] talk I will present recent results giving necessary and sufficient conditions for the conservation of various quantities Isamu Ohnishi of the Euler, MHD and SQG equations. Hiroshima University isamu [email protected] Xinwei Yu University of Alberta Yoshio Tsutsumi 48 PD09 Abstracts

Department of Mathematics, Graduate School of Science The upper and lower solutions are verified through inte- Kyoto University gral equations and only required to be continuous. Appli- [email protected] cations of the reaction-diffusion equations include several important models in biological invasion and disease spread. CP2 Mathematical Analysis to Coupled Oscillators Sys- Haiyan Wang tem with a Conservation Law Arizona State University [email protected] We consider the 3-component reaction-diffusion system of a coupled oscillator with conservation law on an interval O = [0, 1] with homogeneous Neumann boundary condi- CP3 tion. We can prove mathematically rigorously that the A New Glimm Functional and Convergence Rate of wave instability can occur under natural and appropriate Glimm Scheme for General Systems of Hyperbolic conditions for this system. We especially notice that this Conservation Laws system has a preferable cluster size of synchronization of oscillations, which tends to smaller and smaller as e goes In this work, we introduce a new Glimm functional for to 0. It may be interesting that, if the effect by which the general systems of hyperbolic conservation laws. This new synchronized oscillation occurs is too much, then the syn- functional is consistent with the classical Glimm functional chronized cluster is vanishing and a kind of homogenization for the case when each characteristic field is either gen- happens. uinely nonlinear or linearly degenerate, so that it can be viewed as “optimal’ in some sense. With this new func- Isamu Ohnishi tional, the consistency of the Glimm scheme is proved Hiroshima University clearly for general systems. Moreover, the convergence rate isamu [email protected] of the Glimm scheme is shown to be the same as the one obtained for systems with each characteristic field being Tomoyuki Miyaji genuinely nonlinear or linearly degenerate. Dept. of Math. and Life Sci., Hiroshima University Zaihong Jiang [email protected] Joint Advanced Research Center of University of Science and Technology of China and City University of Hong Kong Ryo Kobayashi [email protected] Dept. of Math. and Life Sci.s, Hiroshima University [email protected] CP3 Stability of a New Difference Scheme With Un- Atsuko Takamatsu bounded Operator Coefficients in Hilbert Space Department of Electrical Engineering and Bioscience 2 Waseda University d u(t) ≤ The initial-value problem dt2 + A(t)u(t)=f(t)(0 atsuko [email protected] t ≤ T ),u(0) = ϕ, u(0) = ψ for hyperbolic equations in a Hilbert space H with the self-adjoint positive definite CP2 operator A(t) is considered. A new second-order accurate absolutely stable difference scheme generated by integer Bifurcation in Two-Dimensional Electrohydrody- powers of A(t) for approximately solving this problem is namics developed. The stability estimates for the solution of this Experimental work in annular electroconvection in a smec- difference scheme are established. Theoretical results are tic film reveals the birth of convection-like vortices in the supported for solving one-dimensional hyperbolic partial plane as the electric field intensity is increased. Modeling differential equation with initial conditions. this phenomenon involves two-dimensional Navier-Stokes Mehmet Emir Koksal equations coupled with simplified Maxwell equations. The Florida Institute of Technology system satisfies the prerequisites for application of O(2)- mkoksal@fit.edu equivariant bifurcation theory. A center manifold reduction with coefficients computed numerically unveils interesting dynamics, including stationary and spatio-temporal Allaberen Ashyralyev patterns. Results are compared with those from previous Fatih University work in two-dimensional thermoconvection. aashyr@@fatih.edu.tr Dan Rusu Indiana University-Purdue University Columbus CP3 [email protected] Analytical and Numerical Methods in the Problem of Extinguishing of Vibration for Petrovsky Hyper- bolic Equations CP2 Traveling Wave Solutions for a Class of Reaction- The problem of damping vibration described by the Petro- Diffusion Equations vsky hyperbolic equation

Combining upper and lower solutions, monotone itera- m−1 2 m (−1) ytt = a Dxxy + g(t, x), 0

small vicinities of fixed points, is considered. For example conduction may not be suitable for hyperbolic heat transfer. In this paper, we present an anti-diffusive method t to solve the hyperbolic heat transfer equation. The solution is compared with the analytical one as well as the one g(t, x)=u(t)δ(x − x0 − v(τ) dτ), obtained from a high-order TVD (Total Variation Dimin- 0 ishing) scheme. The order of accuracy of the anti-diffusive method is also investigated numerically. where δ – delta-function of Dirac, x0 – fixed point, u(t) and v(t) are two control functions. Note, that m = 1 cor- Wensheng Shen responds to the string or membrane vibration, and m =2 SUNY Brockport those ones of beam or plate. The solution of the problem of Department of Computer Science damping vibrations requires a study on the special problem [email protected] of moments, the description of the set of damper movement optimal trajectories and the solution of the minimization Leigh J. Little problem of functional arising on the trajectories. Department of Computational Science State University of New York, Brockport Igor Mikhailov [email protected] Russian Academy of Sciences mikh [email protected] CP4 Leonid A. Muravey Boundary Integral Solution of the Time Fractional Russian State Aviation Technology University Diffusion Equation in C∞ Domains [email protected] Recently, fractional derivatives have found new applications in engineering, physics, finance and hydrology. CP3 In physics, fractional diffusion-type equations describe Singular Specific Heat and Second Order Phase anomalous diffusion on heterogeneous media. Transitions Our interest is to study the boundary integral solution of the fractional diffusion equation Motivated by the presence of second order phase transitions α in materials such as liquid helium, we examine specific heat ∂t Φ − ΔxΦ=0, in QT =Ω× (0,T) (1) functions containing an integrable singularity at a critical temperature. Our system is non-strictly hyperbolic, with the Dirichlet boundary condition Φ = g on ΣT = damped and may fail to be genuinely nonlinear. We will Γ × (0,T) and the zero initial condition Φ(x, 0)=0,x∈ Ω α discuss breakdown of solutions, results which vary with the and where ∂t is the fractional Caputo time derivative of specific heat assumptions. A connection between steady- order 0 <α≤ 1. Boundary integral approach allows us state solutions and dipole solutions for the porous media to prove the solvability of (1) in the scale of anisotropic equation will be presented. Sobolev spaces. Katarzyna Saxton Jukka Kemppainen Loyola University, New Orleans University of Oulu [email protected] [email protected].fi

CP3 CP4 What Went Wrong? Cauchy Problems for Semilinear Parabolic Equa- tions Viscosity approximations are constructed for a class of two-dimensional Riemann problems for the two-dimensinal In this paper, we consider solvability of nonlinear evolu- ”p-system” also called the nonlinear wave system. Under tion equations in Banach spaces. In particular, we study seemingly mild additional assumptions, a subsequence con- Cauchy problems for semilinear parabolic equations on un- verges and produces a global weak solution in the limit of bounded domains using fixed point methods. vanishing viscosity. However, such weak solutions provably do not exist in the generality considered. Nickolai Kosmatov University of Arkansas at Little Rock Michael Sever [email protected] Einstein Institute of Mathematics Hebrew University of Jerusalem [email protected] CP4 On Solvability of Initial-Boundary Value Problems for Multi-Weighted Parabolic Systems CP3 Anti-Diffusive Method for Hyperbolic Heat Trans- I.G. Petrovskii 1938, T. Shirota 1955 and V. Solonnikov fer 1965 introduced successively more and more general defi- nitions of parabolic systems. In spite of the fact that the The hyperbolic heat transfer equation is a model used to entries of matrix operator have different orders with respect replace the Fourier heat conduction for heat transfer of ex- to the spacial variables, the t-differentiation has a constant tremely short duration or at very low temperature. Unlike weight 2b with some natural b. Therefore all these sys- the Fourier heat conduction, in which heat energy is trans- tems can be called single-weighted. In the present work, fered by diffusion, thermal energy is transfered as wave a more general definition of multi-weighted parabolic sys- propagation at finite speed in the hyperbolic heat trans- tems is formulated and solvability of the initial-boundary fer model. Therefore methods accurate for Fourier heat value problems for the multi-weighted systems is proved in 50 PD09 Abstracts

appropriate Sobolev-type spaces. ical Quantum Dots with Finite Confinement Bar- riers Alexander Kozhevnikov Professor, Department of Mathematics, University of By employing a novel model for the dielectric interface be- Haifa, 31905, Israel tween a spherical quantum dot and the surrounding matrix, [email protected] an analytical solution of the self-polarization energy of a spherical quantum dot with a finite confinement barrier is first presented. A numerical procedure is then proposed to CP4 calculate the self-polarization energy of a spherical quan- Asysmptotic Behavior of the Solutions for a Cou- tum dot with arbitrary continuous model for the dielectric pled Nonlinear Schrodinger Equations interface, which can fully recover the exact solution of the self-polarization energy. We study the solutions of a coupled nonlinear Schrödinger equation which has been derived in many areas of physical Shaozhong Deng modeling: University of North Carolina at Charlotte 2 2 2 2 [email protected] iut +Δu − (a|u| + b|v| )u =0,ivt +Δv − (c|u| + d|v| )v =0 where the spatial dimension n is 3. Assume that (1) a>0 CP5 (2) b = c>0 (3) d>0 are real constants. We show that the local energy of a solution is integrable in time t and A Study of the Level Sets with Point Correspon- the local L2 norm of the solution approaches zero as time dence Approach to the Stress-Strain Analysis of t approaches the infinity. Deformable Thin Elastic Membranes Jeng-Eng Lin One of the most challenging problems in the stress-strain George Mason University analysis of deformable membranes is building accurate nu- [email protected] merical discretizations for the membranes’ complex geometries. We propose to use the level set method with point correspondence to generate a simplified geometry of a lin- CP4 ear elastic thin membrane, perform the stress-strain anal- Periodic Solutions to a p−Calogero Class of Evolu- ysis on this new configuration which is easier to handle tion Equations numerically, and use the point correspondence mapping to predict the stress-strain distributions in the original geom- We consider spatially periodic solutions to the class of evo- etry. lution equations Jason S. Fritz 1 2 2 The Pennsylvania State University utx+uuxx = pux−(p+1) uxdx, 0

Semiconductor Gaz Sensor Boltzmann Equation. where the number of band-gaps can be made arbitrarily large. Finally, we present some numerical examples by us- The aim of this work is to derive, from a kinetic description, ing the finite element method. a drift-diffusion model for electron transport in semiconductor gaz sensor, by using diffusion and homogenization Keijo M. Ruotsalainen approximation. A homogenized absorption term appears Professor, University of Oulu in the fluid model. In the presence of a self-consistent spa- [email protected].fi tially oscillating electrostatic potential, the right hand side of the Boltzmann equation presents, an absorption term Sergey Nazarov added to the electron-phonon operator collision. This ab- Russian Academy of Sciences, IPME sorption term has the periodicity of the oscillating electro- St. Petersburg static potential. [email protected] Hechmi Hattab ENIT-LAMSIN - Tunis TUNISIA Jari Taskinen [email protected] Department of Mathematics University of Helsinki [email protected].fi CP5 Effect of Anisotropic Surface Energy in An Epitax- CP6 ial Growth Model On the Numerical Solution of Differential- Mullins’ general diffusion equation can be used for model- Algebraic Equations (daes) By Using Legendre ing the epitaxial growth of quantum dots. Several effects Polynomials Approximation have to be considered when defining a chemical potential. Stresses appear due to misfit in the crystal’s grid spacings, The numerical solution of differentialalgebraic equations a thin layer between the dots is apparent because of in- (DAEs) using the Legendre polynomials approximation is termolecular interactions, and the inherent anisotropy of considered in this paper. By using the theories and meth- crystals results in pyramidal shapes of the islands. In self- ods of mathematical analysis and computer algebra, a algo- assembly systems such as Ge/Si, flat films are unstable rithm of Legendre polynomials approximation method for after they excess a critical height. Nano-dots evolve and solving differential-algebraic equation systems was estab- coarsen with time. Their small slopes allow to apply a thin- lished and a Maple procedures mainproc was established. film reduction. We extend a recent model by letting the Two different problems are solved using the Legendre poly- surface energy depend on orientation. Simulations with a nomials approximation and the solutions are compared pseudospectral method show that one additional nonlinear with the exact solutions. First, we calculate the power se- term in the final evolution equation is suitable to guaran- ries of a given equation system and then transform it into tee faceting of the structures. A linear stability analysis Legendre polynomials approximation form, which gives an explains how anisotropy destabilizes flat films. The criti- arbitrary order for solving the DAE numerically. cal height is reduced while the most unstable wave number Mutlu Akar grows with increasing anisotropy parameter. Yildiz Technical University Maciek D. Korzec [email protected] WIAS Berlin [email protected] Ercan elik Ataturk University Peter L. Evans [email protected] Humboldt University in Berlin [email protected] CP6 The Discontinuous Galerkin Method for Two Di- CP5 mensional Hyperbolic Problems Motion of a Viscous Fluid Around a Rigid Body We investigate the superconvergence properties of the dis- The lecture will contain the results about the problem of continuous Galerkin method applied hyperbolic problems on triangle meshes. We show that the discontinuous finite a motion of a viscous fluid around a rotating rigid body p+2 in cases when velocity of fluid at infinity is or not parallel element solution is O(h ) superconvergent at the Legen- with rotation of body. dre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite p+2 Sarka Necasova element error is O(h ) superconvergent at the end points Mathematical Institute of Academy of Sciences of the inflow edge. We use these results to construct sim- Czech Republic ple, efficient and asymptotically correct a posteriori error [email protected] estimates for hyperbolic problems on unstructured meshes. Mahboub Baccouch CP5 Department of Mathematics Computation of the Band-Gap Structure of the University of Nebraska at Omaha Elasticity Operator on Periodic Waveguide [email protected]

In this paper we shall study the spectral problem in the linearized elasticity. It will be shown that an inﬁnite waveg- CP6 uide with periodically positioned cells, the essential spec- Discrete Compactness for the p-Version of Discrete trum contains gaps. Moreover, we construct examples 52 PD09 Abstracts

Differential Forms Pamukkale University, Engineering Faculty Mechanical Engineering Department The aim of this paper is to prove the discrete compactness [email protected] property for a wide class of p finite element approximations of the Maxwell eigenvalue problem in two and three space dimensions. In a very general setting, we find sufficient CP6 conditions for the p-version of the generalized discrete com- Space-Time Finite Volume Differencing Method for pactness property which is formulated in the framework of Effective Higher Order Accurate Schemes for Gen- discrete differential forms of order on a polyhedral do- eral Diffusion Equations main in Rd (0 <

Vyacheslav Trofimov Faculty of Computational Mathematics Paramjeet Singh, Kapil Sharma Moscow State University Department of Mathematics [email protected] Panjab University Chandigarh, India [email protected], [email protected] I.A. Shirokov Moscow State University CP7 [email protected] On Schauder’s Fixed Point Theorem Applications to Nonlinear Variational Inequality with One Or CP6 Two Obstacle Application of the Generalized Iterative Differen- We use Schauder’s fixed point theorem to obtain existence tial Quadrature Method for Solving Lid-Driven of a solution of the elliptic unilateral problem. We consider 1,p Cavity Problem a compact convex subspace S of the space W0 (Ω). Then a mapping J is defined such that it is a continuous map in In this work, Lid-driven cavity problem was solved by using itself. Then J will have a fixed point. The nonlinear op- Generalized differential quadrature method and results are erator is of the form A(u)+F (x, u, Du) in an unbounded given by figure and tables. Good agreements are seemed domain Ω of Rn. The operator is defined from the space compared with other works. 1,p −1,p W0 (Ω) into the space W (Ω). F has growth of order Zekeriya Girgin p with respect to variable Du satisfying the sign condition. PD09 Abstracts 53

The obstacle is a measurable function with variable in R¯. viscosity solutions are allowed. We propose a finite dif- A and F satisfy Carathéodory conditions. Similar solu- ference numerical scheme to approximate the solution of tions can be obtained for a Parabolic Inequality. The last the extended class of equations and analyze a set of initial- section gives an idea about numerical approximation of the boundary value problems for Hamilton-Jacobi equations unilateral problem. arising as the convective component of nonlinear flux lim- iter diffusion equations. Kumud S. Altmayer University of Arkansas Susana Serna kumud [email protected] Department of Mathematics Universitat Autonoma de Barcelona [email protected] CP7 Asymptotic and Numerical Solutions to Laplace Antonio Marquina Young Capillary Equation at Singularity. University of Valencia, Spain [email protected] The Laplace Young Equation is a nonlinear elliptic PDE models capillary surfaces (static liquid surfaces). An asymptotic sequences motivated by the geometry of do- CP7 main made it possible to construct an asymptotic solution Multitime Dynamic Programming in the non-power series cusp region. Also an exact-solution of an approximated equation has allowed us to obtain an This paper introduces a new type of dynamic programming accurate asymptotic solution in the circular cusp domain. PDEs for optimal control problems with performance crite- At last, a fast converging finite volume element method ria involving multiple integrals or curvilinear integrals. The was developed using asymptotic solutions. main novel feature of the multitime dynamic programming PDEs, relative to the standard Hamilton-Jacobi-Bellman Yasunori Aoki PDEs, is that they are connected to the multitime max- University of Waterloo imum principle. In other words, we present an interest- [email protected] ing and useful connection between the multitime maximum principle and the multitime dynamic programming, CP7 characterizing the optimal control by means of a multitime Hamilton-Jacobi-Bellman PDE (or system) that may The Dirichlet Problem for Willmore Surfaces of be viewed as a feedback law. In the case of performance Revolution criteria involving either multiple integrals or curvilinear in- Boundary value problems for the “Willmore equation’ are tegrals, with quadratic integrands, the new equations lead quasilinear and, in particular, frame invarint counterparts to multitime variants of the Riccati equation. Section 1 of the fourth-order linear clamped plate equation. They do introduces the multitime Hamilton-Jacobi PDEs from ge- not enjoy any form of a general comparison principle. In ometrical point of view splitting the discussions after the contrast to closed Willmore surfaces, only a few results are type of actions (multiple integrals or curvilinear integrals). known for boundary value problems. In the talk, a partic- Section 2 shows how multi-time control dynamics, based ularly symmetric situation is studied. Here, one succeeds on curvilinear integral action or on multiple integral ac- in finding strong a-priori-estimates on suitable minimising tion, determines the multitime Hamilton-Jacobi-Bellman sequences and obtains a-priori bounded classical solutions. PDEs via the maximum value function or its generating vector field. Section 3 describes the connection between Hans-Christoph Grunau multitime dynamic programming and the multitime max- Institut fur Analysis und Numerik imum principle. In the general case, the HJB PDEs do Otto-von-Guericke-Universitat, Magdeburg - Germany not have classical (smooth) solutions, but we can introduce [email protected] multitime viscosity solutions (extending the theory of P.- L. Lions and M. Crandall) or multitime minmax solutions Anna Dall’Acqua, Klaus Deckelnick (extending the theory of I. Subbotin). Otto von Guericke-Universitaet Magdeburg Constantin Udriste [email protected], [email protected] Bucharest, RO 060042, Romania [email protected] Steffen Froehlich Freie Universitaet Berlin [email protected] CP7 Over-Determined Systems of Partial Differential Friedhelm Schieweck Equations, Equilibrium Behavior of Two-Phase Otto von Guericke-Universitaet Magdeburg Granular Mixtures and Equations of Monge- [email protected] Ampere Type

The equilibrium behavior of two-phase granular mixtures CP7 is described via an over-determined, quasi-linear system of Numerical Approximation of Viscosity Solutions partial diﬀerential equations. In the ﬁrst part of the paper with Shocks we perform a compatibility analysis and various existential and regularity results are also presented. In the second We consider a class of Hamilton-Jacobi equations that part of the paper we demonstrate the connection of our admits a unique viscosity solution containing shocks, [Y. system to equations of Monge-Ampere type and discuss Giga, Viscosity solutions with shocks, Comm. Pure Appl. further implications. Math., vol. LV, 2002], as an extension of the classical theory by Crandall and Lions (1983) where only continuous Christos Varsakelis, Miltiadis Papalexandris 54 PD09 Abstracts

Department of Mechanics Michael Weinstein Universite catholique de Louvain Columbia University [email protected], [email protected] [email protected]

CP8 CP8 Non-Existence and Existence of Localized Solitary Generalized Analytic Functions in Axially Sym- Waves for the Two-Dimensional Long Wave-Short metric Oseen Flows Wave Interaction Equations A class of generalized analytic functions arising from the In this study we consider the two-dimensional long wave- Oseen equations in the axially symmetric case has been short wave interaction (LSI) equations which describe identified. For this class of functions, the generalized the interaction between high-frequency and low-frequency Cauchy integral formula has been obtained, and a series waves near the long wave-short wave resonance. We es- representation for the region exterior to a sphere has been tablish non-existence and existence results for the localized constructed. The velocity field has been represented in solitary waves of the LSI equations. Both the non-existence terms of two generalized analytic functions, and it has been and the existence results are based on Pohozaev-type iden- shown that for an exterior Oseen flow problem, those func- tities. We prove the existence of solitary waves by showing tions are uniquely determined provided that they both van- that the solitary waves are the minimizers of an associated ish at infinity. Also the pressure and vorticity have been variational problem. determined as real and imaginary parts of the two functions representing the velocity field, and the drag exerted Handan Borluk, Husnu Ata Erbay, Saadet Erbay on a solid body of revolution in the axially symmetric Os- Department of Mathematics, Isik Universty een flow has been expressed in terms of one of the involved [email protected], [email protected], generalized analytic functions. The problem of the axially [email protected] symmetric Oseen flow past a solid body of revolution has been reduced to an integral equation based on the generalized Cauchy integral formula. The developed framework of CP8 generalized analytic functions has been illustrated in solv- Dispersion in 1-D Variational Boussinesq Model ing the problem of the Oseen flow past a solid sphere and solid bispheres. The Variational Boussinesq Model for waves over ideal fluid conserves mass, momentum, energy, and contains de- Michael Zabarankin creased dimensionality compared to the full problem. It Stevens Institute of Technology is derived from the Hamiltonian formulation via an ap- [email protected] proximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive prop- CP9 erties of the 1-D linear model over flat bottom and, using Eigenfunctions and Eigenvalues for High Energy the finite element and spectral numerical codes, investigate Resonance for Non-Standard Sturm-Liouville Op- its robustness. erators with a Perturbation of the Coulomb Poten- tial. Ivan Lakhturov, Brenny Van Groesen University of Twente Resonances for the Hamiltonian operator result from the [email protected], separated radial Schrödingerequation. Perturbations e(r) [email protected] of the Coulomb potential can produce a sequence of resonances. The high-energy resonance contributing equations could be the three dimensional Schrödingeroperator CP8 with Coulomb potential or a non-standard Sturm-Liouville Scattering Resonances and the Dynamics of De- eigenproblem with interior singularities. We re-format the forming Bubbles in a Compressible Fluid radial Schrödingeroperator in R, which happens to be a non-standard Sturm-Liouville operator, and by specifying We consider the dynamics of a nearly spherical gas bub- boundary conditions that retain the resonance poles the ble in an inviscid, compressible fluid with surface tension. eigenfunctions solutions are obtained and subsequently the The time-decay of symmetric (volume) and asymmetric eigenvalues are derived. These solutions are obtained with (shape) deformation modes is governed by scattering res- minimal restriction on a non-constant perturbation e(r) of onances. These are eigenvalues of a non-self-adjoint spec- the Coulomb potential. tral problem associated with the linearization about the spherical equilibrium. We study the locations of scatter- Thomas Acho ing resonance energies and prove resonance expansions for University of the Free State, Bloemfontein, South Africa the bubble surface perturbation and fluid velocity poten- [email protected] tial. These perturbations decay exponentially with ad- vancing time. While the lifetime of decaying radial oscil- −1 lations, Tradial ∼ ( denotes the Mach number), there CP9 are asymmetric shape modes that decay much more slowly. Improved Convergence of the Allen-Cahn Equation In particular, the slowest decaying mode has a lifetime 2 to Generalized Motion by Mean Curvature Tshape ∼ exp(κ/ ), κ>0, thus occurring beyond all orders in . It is known that, as ε → 0, the Allen-Cahn equation ε ε −2 ε ut =Δu + ε f(u ) converges — for all times— to the Abby Shaw-Krauss generalized motion by mean curvature defined via the level- Columbia University set approach and viscosity solutions. In this talk, given a Dept of Applied Physics and Applied Mathematics rather general initial data, we include the transition lay- [email protected] PD09 Abstracts 55

ers of the solutions uε in sets moving by mean curvature. be more intricate. In some special cases, this enables to obtain an O(ε| ln ε|) estimate of the thickness (measured in space-time) of the Galimzyan G. Islamov transition layers. Udmurt State University AMS Matthieu Alfaro, Jerome Droniou [email protected] Universite Montpellier 2 [email protected], [email protected] CP9 Min-Max Game Theory and Non-Standard Differ- ential Riccati Equations under Singular Estimates Hiroshi Matano At At University of Tokyo for e B and e G in the Absence of Analyticity [email protected] We consider an abstract dynamical system, which is characterized by ”singular estimates”, as it arises from many CP9 concrete hyperbolic/parabolic coupled PDE models, that Existence and Asymptotic Behavior of Solutions to are subject to boundary/point control and deterministic Quasilinear Thermoelastic Systems Without Me- disturbance. For such a system, we study a min-max game chanical Dissipation theory problem with quadratic cost functional over a finite horizon. The problem is fully solved in feedback form via a We consider a class of quasilinear thermoelastic systems. Riccati operator which satisfies a non-standard differential The model considered is characterized by supercritical non- Riccati equation. Several PDE-illustrations are presented. linearity and accounts for large displacements and large strains. Existence of finite energy (weak)global solutions and exponential decay rates for the energy of weak solu- Jing Zhang, Roberto Triggiani tions will be shown. The methods of proof will involve com- University of Virginia pensated compactness along with the application of special [email protected], [email protected] PDO non-local multipliers that have been developed in the context of the proof. CP10 Albert Altarovici, Irena Lasiecka Numerical Valuation of Multi-Currency Interest University of Virginia Rate Derivatives Using a Partial Differential Equa- [email protected], [email protected] tion Approach We consider the 3D PDE arising from a multi-currency CP9 model with foreign exchange (FX) skew. Finite differences Analysis on a Reaction-Diffusion System with Non- are used to discretize the space variables. Both GMRES linear Rate of Growth with pre-conditioner solved by FFT techniques and the Alternating-Direction Implicit (ADI) methods are consid- In this talk, we present some recently qualitative results ered. We analyze the impacts of the FX skew on long- on a nonlinear reaction-diffusion system which is used as a dated exotic interest rate derivatives with cancelable and model for animal and insect dispersal. We find that under knockout features, and study the effects of the boundary certain parametric conditions the system admits one pa- conditions on the accuracy of the numerical methods. rameter Lie groups of transformations, which lead to two independent first integrals. Duy Minh Dang University of Toronto Zhaosheng Feng Computer Science Department University of Texas-Pan American [email protected] Department of Mathematics [email protected] Christina Christara University of Toronto [email protected] CP9 Pseudo-Wave Functions. Ken Jackson Dept. of Computer Science ccording to J. von Neumann a set of all having a prop- University of Toronto erty G elements of Hilbert space B is a closed subspace [email protected] PB. Here via P is denoted an orthogonal projector on this subspace. We extended this original method to finite dimensional projectors P of general form. An evolu- CP10 tion of quantum mechanical system is described by self- Explicit Solutions to the Kolmogorov Backward adjoint energy operator H. We study finite dimensional Equation solutions K of equation PH − HP = K(I − P ) with minimal rank. Any solution ϕ(t) of perturbed equation To obtain the mean time to failure of a device, a model i¯hϕ (t)=(H − K)ϕ(t),t>0 is called a pseudo-wave func- involving a system of two stochastic differential equations tion if a quadratic form ((H − K)ϕ(t),ϕ(t)) is real for all is proposed. This leads to a partial differential equation t>0. Efficient Einstein-Broglie formula E =¯hν con- known as Kolmogorov’s backward equation. This equa- nects energy and frequency spectrums of object described tion is solved explicitly in important particular cases. The by wave function. For object described by pseudo-wave method of similarity solutions is used. function a correlation between these types of spectrum may Mario Lefebvre 56 PD09 Abstracts

Ecole Polytechnique Montreal tions for the p(x)-Laplacian with Rough Exponents Dept of Mathematics and Industrial Engineering [email protected] In this talk, properties of local ﬁelds inside mixtures of two nonlinear power law materials are studied. This constitu- Djilali Ait Aoudia tive model is frequently used to describe several phenomena Department of Mathematics and Statistics ranging from plasticity to optical nonlinearities in dielec- University of Montreal tric media. We develop new multiscale tools to bound the [email protected] local singularity strength inside microstructured media in terms of the macroscopic applied ﬁelds.

CP10 Silvia Jimenez, Robert P. Lipton Department of Mathematics On Existence of Optimal Relaxed Controls for Dis- Louisiana State University sipative Stochastic PDEs in Banach Spaces [email protected], [email protected] We study existence of optimal relaxed controls for semilinear stochastic PDEs in UMD type-2 Banach spaces with CP11 multiplicative noise driven by a cylindrical Wiener process. The state equation is controlled through the drift Some Applications of Minimizing Variational Prin- coefficient which satisfies a dissipative-type condition with ciples for Helmholtz Equation respect to the state-variable. The main tools are the fac- We use the variational principles of Milton et al. for the torization method for stochastic convolutions in Banach complex Helmholtz/acoustic equation to formulate a finite spaces, and certain compactness properties of the factor- element method for solving these equations that is based ization operator and the class of relaxed control policies on entirely on minimization. Various boundary conditions are Suslin metrisable control sets. addressed. Also, we apply these finite element methods to Rafael Serrano the tomography problem and use the variational principles PhD Student, University of York, UK to derive some elementary bounds on effective properties. [email protected] Russell Richins, David C. Dobson University of Utah Zdzislaw Brzezniak [email protected], [email protected] University of York, UK [email protected] Graeme W. Milton University of Utah CP11 Department of Mathematics [email protected] Homogenization of a Heat Transfer Problem with Robin Dynamic Boundary Condition MS0 In this paper, we deal with the asymptotic behaviour of the heat equation with highly oscillating coefficients in a Some Non Linear Problems Involving Integral Dif- -periodic two-constituent composite with dynamic heat fusions - Part II transfer interfacial formulation. Problems involving integral diffusion appear in questions Abdelhamid Ainouz of continuum mechanics ( in variational , or ”divergence University of Sciences and Technology Houari form” ) and in probability and optimal control, from Levi [email protected] processes, in ” non divergence” form. In the first case, to develop a non linear theory ( similar to quasilinear equations in the calculus of variations, it is necessary to develop CP11 the equivalent of the De Giorgi,Nash, Moser theory for ” Geometrically Nonlinear Models in Crystal Plas- bounded measurable kernels coming from non local varia- ticity with Penalization of Elastic Deformations tional integrals in energy spaces of fractional derivatives . In the second, parallel to the theory of optimal control,the In this talk we consider models for crystal plasticity in theory parallels the Krylov Safanov Harnack inequality and the framework of the modern calculus of variations. As a the Evans Krylov theorem. We will present the main ideas particular example we study the case of one slip system and arguments of the corresponding non local theorems. in two dimensions with rigid elasticity and we investigate whether these models arise as Gamma limits of models in Luis Caffarelli which elastic deformations are penalized. Surprisingly, the University of Texas at Austin answer depends strongly on the growth conditions satisfied Department of Mathematics by the elastic and the plastic contributions to the energy. caff[email protected] In the natural setting of quadratic potentials the compact- ess of sequences with finite energy is obtained by using a MS0 version of the classical div-curl lemma. Some Non Linear Problems Involving Integral Dif- Georg Dolzmann fusions Universitaet Regensburg [email protected] Problems involving integral diffusion appear in questions of continuum mechanics ( in variational , or ”divergence form” ) and in probability and optimal control, from Levi CP11 processes, in ” non divergence” form. In the first case, to Correctors, Homogenization, and Field Fluctua- develop a non linear theory ( similar to quasilinear equations in the calculus of variations, it is necessary to develop PD09 Abstracts 57

the equivalent of the De Giorgi,Nash, Moser theory for ” Istituto per le Applicazioni del Calcolo, CNR bounded measurable kernels coming from non local varia- [email protected] tional integrals in energy spaces of fractional derivatives . In the second, parallel to the theory of optimal control,the theory parallels the Krylov Safanov Harnack inequality and MS1 the Evans Krylov theorem. We will present the main ideas Convergence of Perturbed Allen Cahn Equations and arguments of the corresponding non local theorems. to Forced Mean Curvature Flow

Luis Caffarelli We study perturbations of the Allen–Cahn equation and University of Texas at Austin prove the convergence to forced mean curvature flow in the Department of Mathematics sharp interface limit. We allow for perturbations that are caff[email protected] square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for MS1 the Allen–Cahn action functional. Finally we discuss some Wavelet Analogue of the Ginzburg-Landau Energy applications. in Variational Models of Image Processing Luca Mugnai We discuss a wavelet analogue of the classical Ginzburg- Max Planck Institute for Mathematics in the Sciences Landau energy (WGL). Functionals of this type retain cer- [email protected] tain diffuse interface features and converge, in the Γ-sense, to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets. However, for the MS1 same values of the diffuse interface parameter, minimiz- Existence of Weak Mean Curvature Flow with ers of WGL have sharper transitions between equillibria Non-smooth Forcing Term than those of the classical Ginzburg-Landau functional, thus making the WGL energy an advantageous tool for We prove the existence of weak solution for the surface image processing. We design a variational model for multi- evolution problem where the velocity of hypersurface is purpose image reconstruction, where WGL is used as the the sum of the mean curvature vector and a given vec- regularizing energy part and the fidelity terms are defined tor field. The vector field is in a suitable Sobolev space so by specifics of each application (such as image inpainting, the evolving hypersurface satisfies good measure-theoretic blind deconvolution, superresolution and more). properties in addition to the Brakke’s evolution law. Julia Dobrosotskaya Yoshihiro Tonegawa Department of Mathematics, UMD Hokkaido University [email protected] [email protected]

Andrea L. Bertozzi MS2 UCLA Department of Mathematics [email protected] Homogenization of Multiple Integrals Under Dif- ferential Constraints

Two-scale techniques are developed for sequences of maps MS1 p M {uk}k∈N ⊂ L (Ω; R ) satisfying a linear differential con- Variational Properties of the Mumford Shah Func- straint Auk = 0. These, together with Γ-convergence ar- tional with Singular Operators guments and using the unfolding operator, provide a homogenization result for energies of the type We consider functionals of the Mumford and Shah type with singular operators. In such functionals the fidelity x p M Fε := f x, ,u(x) dx with u ∈ L (Ω; R ), Au =0 term of the Mumford and Shah functional is replaced by ε a functional which penalizes the L2 distance between the Ω given image and a version of the unknown function which that generalizes current results in the case where A = curl. is filtered by means of a non-invertible linear operator. De- This is joint work with Stefan Krömer. pending on the type of the involved operator the resulting variational problem may have several applications in the Irene Fonseca, Stefan Kroemer field of image reconstruction: for instance image deconvo- Carnegie Mellon University lution preserving discontinuities in the case of convolution [email protected], [email protected] operators, inpainting problems in the case of suitable local operators. We study the existence of minimizers of functionals of this type in suitable classes of discontinuous MS2 functions. The discontinuity set of such functions belongs Models for Dynamic Fracture to a class of compact sets introduced by L. Rondi to study problems of crack formation in solid structures. We will We introduce mathematical models for “sharp interface” consider the cases of local operators and convolution oper- dynamic fracture that make no assumption about crack ators. geometry, etc. Existing models for static/quasi-static fracture, and phase-field models for dynamic brittle fracture, Massimo Fornasier are based on variational principles that cannot be directly Johann Radon Institute for Computational and Applied transferred to the dynamic setting. We show how these Mathematics (RICAM) principles can be reformulated, so that they can be incorpo- [email protected] rated in dynamics. Time permitting, progress on existence will also be discussed. Riccardo March Christopher Larsen 58 PD09 Abstracts

Worcester Polytechnic Institute ancient solutions are King-Rosenau solutions. Math Dept [email protected] Panagiota Daskalopoulos Columbia University [email protected] MS2 The Time-dependent von Krmn Plate Equation as a Limit of 3d Nonlinear Elasticity MS3 Monge-Ampere Type Equations and Geometric In this talk we will discuss the asymptotic behaviour of Optics solutions of the hyperbolic equation of nonlinear elastody- namics in a three-dimensional thin domain. More precisely, This talk will describe recent work on problems in geomet- we show that when the initial data and loadings are suf- ric optics related to Monge-Ampere type equations. They ficiently small in terms of the thickness, the limit dynam- mostly concern with the construction of surfaces that re- ics corresponds to the time-dependent von Kármánplate flect and/or refract radiation in a prescribed manner. equation. Cristian Gutierrez Helmut Abels Department of Mathematics Max Planck Institute Temple University Leipzig [email protected] [email protected] MS3 Maria Giovanna Mora Sissa, Trieste, Italy Title Not Available at Time of Publication [email protected] Abstract not available at time of publication.

Stefan Mueller YanYan Li Hausdorﬀ Center for Mathematics Mathematics Department Bonn, Germany Rutgers University [email protected] [email protected]

MS2 MS3 Simulating 2-D crack propagation with XFEM and On the Levi Monge Ampre Equation a hybrid mesh We give a survey about some notions of curvatures asso- We will present a technique for simulating quasistatic crack ciated with pseudoconvexity and the Levi form the way propagation in 2-D which combines the extended finite ele- the classical Gauss and Mean curvatures are related to the ment method (XFEM) with a general algorithm for cutting convexity and to the Hessian matrix. We shall first show triangulated domains, and introduce a simple yet general that these curvature equations contain information about and flexible quadrature rule based on the same geometric the geometric feature of a closed hypersurface. Then, we algorithm. The combination of these methods gives several shall show that the curvature operators lead to a new class advantages. First, the cutting algorithm provides a flexi- of second order fully nonlinear equations which are not ble and systematic way of determining material connectiv- elliptic at any point. However, they have the following re- ity, which is required by the XFEM enrichment functions. deeming feature: the missing ellipticity direction can be Also, our integration scheme is straightfoward to imple- recovered by suitable commutation relations. We shall use ment and accurate, without requiring a triangulation that this property to study the Dirichlet problem for graphs incorporates the new crack edges or the addition of new with prescribed Levi curvature. degrees of freedom to the system. Annamaria Montanari Casey Richardson Universita’ di Bologna, Italy UCLA [email protected] Los Angeles, CA (USA) [email protected] MS4 Jan Hegemann Critical Thresholds in Hyperbolic Relaxations University of M × ”unster Critical threshold phenomena in one dimensional 2 2 [email protected] quasi-linear hyperbolic relaxation systems are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the Eftychios Sifakis, Jeffrey Hellrung, Joseph Teran critical threshold phenomena associated with the under- UCLA lying relaxation systems. Our results apply to the well- [email protected], [email protected], known isentropic Euler system with damping. This is a [email protected] joint work with Hailiang Liu.

Tong Li MS3 Department of Mathematics Ancient Solutions to the Ricci Flow on Surfaces University of Iowa [email protected] We provide a classiﬁcation of ancient solutions to the Ricci ﬂow on compact surfaces. We show that the only type II PD09 Abstracts 59

Hailiang Liu duced by Liu and Yu for the study of Boltzmann proﬁles, Iowa State University but applied to the stationary rather than the time- evolu- USA tionary equations. This yields a simple proof by contrac- [email protected] tion mapping in weighted Hs spaces. Kevin Zumbrun MS4 Indiana University Global Well-Posedness for the Microscopic Fene USA Model with a Sharp Boundary Condition [email protected]

We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well- MS5 posedness of the FENE model that consists of the incom- Kinetic Theory of Synchronization pressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the nat- A kinetic theory for phase synchronization through binary ural flux boundary condition. Recently it was illustrated interactions will be discussed. This theory describes the in [C. Liu and H. Liu, Boundary Conditions for the Mi- nonequilibrium phase transition from a nematic, ordered croscopic FENE Models, 2008, SIAM J. Appl. Math., phase to an isotropic, disordered phase that occurs in a 68(5):1304–1315] that any preassigned boundary value of system of interacting rods, that are also subjected to ran- a weighted distribution will become redundant once the dom forcing. The steady-state solution for the nonlinear non-dimensional parameter b>2. In this talk, we present and nonlocal theory is obtained by expressing the Fourier that for the well-posedness of the microscopic FENE model transform of the orientation distribution as a function of (b>2) the least boundary requirement is that the distribu- the order parameter, which in turn, is obtained in terms tion near boundary needs to approach zero faster than the of the driving strength. This exact solution is obtained distance function. This condition is strictly weaker than using iterated partitions of the integer numbers. This ki- the natural flux boundary condition. Under this condition netic theory described alignment of vibrated granular rods it is shown that there exists a unique weak solution in a or chains, as well as biological systems such as molecular weighted Sobolev space. The sharpness of this boundary motors and microtubules. requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as Eli Ben-Naim the distance function. Los Alamos National Laboratory [email protected] Jaemin Shin Department of Mathematics, Iowa State University [email protected] MS5 Kinetic Models for Swarming

Hailiang Liu I will present a kinetic theory for swarming systems of in- Iowa State University teracting, self-propelled discrete particles. Starting from USA the the particle model, one can construct solutions to a ki- [email protected] netic equation for the single particle probability distribution function using distances between measures. Moreover, MS4 I will introduce related macroscopic hydrodynamic equations. General solutions include flocks of constant den- Weak Solutions and Critical Thresholds for the sity and fixed velocity and other non-trivial morphologies One-dimensional Vlasov- Poisson Equation such as compactly supported rotating mills. The kinetic We give an explicit weak solution of the one-component theory approach leads us to the identification of macro- Vlasov-Poisson equations in a single space dimension with scopic structures otherwise not recognized as solutions of smooth electron sheet initial data. Moreover, we give sharp the hydrodynamic equations, such as double mills of two conditions on whether the moments of this weak solution superimposed flows. I will also present and analyse the will blow up or not. It depends on whether or not the initial asymptotic behavior of solutions of the continuous kinetic configuration of the density and velocity crosses a critical version of flocking by Cucker and Smale, which describes threshold. Beyond the critical time, we first construct a the collective behavior of an ensemble of organisms, ani- multi-valued solution to the Euler-Poisson equation, then mals or devices. This kinetic version introduced in Ha and we give a weak solution to the Vlasov-Poisson equation. Tadmor is obtained from a particle model. The large-time behavior of the distribution in phase space is subsequently Dongming Wei studied by means of particle approximations and a stabil- Dept. of Math. ity property in distances between measures. A continuous University of Wisconsin at Madison analogue of the theorems of Cucker-Smale will be shown to [email protected] hold for the solutions on the kinetic model. More precisely, the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a MS4 translational flocking solution. Existence of Semilinear Relaxation Shocks Jose Carrillo We establish existence with sharp rates of decay and dis- ICREA tance from the Chapman–Enskog approximation of small- Universitat Autnoma de Barcelona amplitude shock profiles of a class of semilinear relaxation [email protected] systems including discrete velocity models obtained from Boltzmann and other kinetic equations. Our method of analysis is based on the macro–micro decomposition intro- 60 PD09 Abstracts

MS5 which describes initiation of combustion process in inert Self-similarity for Ballistic Aggregation Equation porous media. In the framework of the considered model initiation is associated with blow up whereas global exis- We consider ballistic aggregation equation for gases in tence of solution suggests quenching. I will present neces- which each particle is identiﬁed either by its mass and im- sary conditions for global existence of solution and condi- pulsion or by its sole impulsion. For the constant aggrega- tions ensuring blow up. tion rate we prove existence of self-similar solutions as well as convergence to the self-similarity for generic solutions. Peter Gordon For some classes of mass and/or impulsion dependent rates Department of Math Sci we are also able to estimate the large time decay of some NJIT moments of generic solutions. [email protected] Stephane Mischler CEREMADE, Universite Paris Dauphine MS6 [email protected] Uniqueness of Steady States for a Chemotaxis Model

Miguel Escobedo 2 Departamento de Matemáticas,Universidad del Pais We consider in a disc of R a class of parameter-dependent, Vasco nonlocal elliptic boundary value problems that describe the Apartado 644, Bilbao. España steady states of some chemotaxis systems. If the appearing [email protected] parameter is less than an explicit critical value, we establish several uniqueness results for solutions that are invariant under a group of rotations. Furthermore, we discuss the as- MS5 sociated consequences for the time asymptotic behavior of Kinetic Formulations and Regularizing Effects in the solutions to the corresponding time dependent chemo- Degenerate Quasi-linear PDEs taxis systems. Our results also provide optimal constants in some Moser-Trudinger type inequalities. We quantify the regularizing effects in a general family of quasi-linear scalar PDEs, using velocity averaging of Marcello Lucia their underlying kinetic formulations. The PDEs are first Department of Mathematics and second order equations which involve nonlinear trans- CUNY, College of Staten Island port and (possibly degenerate) diffusion. In particular, [email protected] we use the Chen-Perthame characterization of solutions to degenerate problems to derive new regularity results for MS6 convection-diffusion and elliptic equations driven by degenerate, non-isotropic diffusion. Boundary Blowup-type Solutions to Elliptic Equa- tions with Hardy Potentials Eitan Tadmor University of Maryland Semilinear elliptic equations which give rise to solutions USA blowing up at the boundary are perturbed by a Hardy po- [email protected] tential involving distance to the boundary. The size of this potential affects the existence of a certain type of solutions (large solutions): if the potential is too small, then Terence Tao no large solution exists. The presence of the Hardy poten- Department of Mathematics, UCLA tial requires a new definition of large solutions, following [email protected] the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be MS6 given. Considerations are based on a Phragmen-Lindeloef type theorem which enables to classify the solutions and A Constrained Capacitive System Arising in sub-solutions according to their behaviour near the bound- MEMS ary. A model describing and electrostatic deflections of an elas- Vitaly Moroz tic membrane with a volume constraint, arising in MEMS, Department of Mathematics is considered. Existence and nonexistence of a solution is Swansea University explored as well as the structure of the solution set for the [email protected] governing equation. Time-of-fight analysis, construction of limiting singular solutions, and bifurcation diagrams, based on the variational formulation of the model, are used to MS7 determine the structure of the solution set and stability of Liouville Type Theorems in the Fluid Mechanics solutions found. In this talk we discuss Liouville type theorems in the fluid Regan Beckham mechanics. For incompressible fluids suitable decay con- Department of Mathematics dition near infinity combined with sign conditions for the University of Texas at Tyler pressure integrals leads to triviality of the solution. For sta- Regan [email protected] tionary compressible fluids we can omit the conditions on the pressure to lead to similar conclusion. For time depen- MS6 dent viscous compressiblle fluids we need to assume energy inequality for weak solution to get trivilaity of solution. Ignition in Porous Media as a Blowup Problem Dongho Chae We consider a system of two reaction diffusion equations Sungkyunkwan University PD09 Abstracts 61

[email protected] [email protected]

MS7 MS8 Blow-up Problems in Fluid Mechanics: a Case Sharp Estimates on Minimum Traveling Wave Study Speed of Reaction Diﬀusion Systems

We study a family of 2D fluid equations, including 2D Euler Abstract not available at time of publication. and surface quasi-geostrophic (SQG) equations. In hypo- viscous dissipative cases, an oscillatory damping in long- Yuanwei Qi time evolution is observed, both below/above the critical- University of Central Florida ity. This suggests all-time regularity even for supercritical [email protected] cases, but with a complicated transient. For inviscid cases, the growth rates of scalar gradient in Lp are compared within the family; in particular 2D Euler grows faster than MS8 SQG. This order is reversed in L∞. Generalized Traveling Wave Solutions and Their Propagating Speeds in Almost Periodic Media Koji Ohkitani The University of Sheffield This talk is concerned with the existence, stability, and K.Ohkitani@sheffield.ac.uk propagating speeds of generalized traveling wave solutions in time dependent, specially, in time almost periodic and unique ergodic bistable and monostable reaction diffusion MS7 equations, Questions on Polymeric Equations Wenxian Shen I will consider questions related to existence and decay Department of Mathematics and Statistics of solutions to kinetic models of incompressible polymeric Auburn University flow, in the case when the space of elongations is bounded, [email protected] and the spatial domain of the polymer is either a bounded domain Ω ⊂,n=2, 3 or it is the whole space R3. Consid- eration will also be given to solutions where the probability MS8 density function is radial in the admissible elongation vec- An Introduction to Fronts in Heterogeneous Media tors. Abstract not available at time of publication. Maria E. Schonbek Univ. of California at Santa Cruz Jack Xin Dept. of Math. Department of Mathematics [email protected] UCI [email protected]

MS7 MS8 Loss of Smoothness and Energy Conserving Rough Weak Solutions for the 3d Euler Equations Pulsating Front Speed-up and Diffusion Enhance- ment in Flows A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions Abstract not avialable at time of publication. of the Euler equations of incompressible ideal fluids. In par- Andrej Zlatos ticular, they proved by means of this example that weak University of Chicago limit of solutions of Euler equations may, in some cases, fail [email protected] to be a solution of Euler equations. We use this example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional 1,α MS9 Euler equations, for initial data that do not belong to C . Mortar Discretization for Stokes-Darcy Systems In addition, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler We consider the coupling across an interface of fluid and equations that conserve the energy, an issue which related porous media flows with Beavers-Joseph-Saffman transmis- to the Onsager conjecture. Moreover, we show by means of sion conditions. Under an adequate choice of Lagrange this shear flow example that, unlike to the two-dimensional multipliers on the interface we analyze inf-sup conditions case, the minimal regularity in the three-dimensional vor- and optimal a priori error estimates associated with the tex sheet Kelvin-Helmholtz problem need not to be the continuous and discrete formulations of this Stokes-Darcy class of real analytic solutions. system. We allow the meshes of the two regions to be non- matching across the interface. Using mortar finite element Edriss S. Titi analysis and appropriate scaled norms we show that the University of California, Irvine and constants that appear on the a priori error bounds do not Weizmann Institute of Science, Israel depend on the viscosity, permeability and ratio of mesh [email protected], [email protected] parameters. Numerical experiments are presented.

Claude Bardos Juan Galvis Laboratory J.L. Lions, Universit Texas A&M University ’{e} Pierre et Marie Curie [email protected] Paris, 75013, France 62 PD09 Abstracts

MS9 MS9 Mixed Finite Element Methods for a Nonlinear Domain Decomposition for Strokes-Darcy Systems Stokes-Darcy Coupled Problem In this talk we couple deterministic and stochastic Darcy’s In this paper we analyze mixed finite element methods pressure equations with log-normal permeability. We first for a nonlinear model of the coupling of fluid flow with formulate the problem as an infinity-dimensional system porous media flow. More precisely, flows are governed by and then show well-posedness using inf-sup techniques. the Stokes equations in the fluid and a nonlinear Darcy Then we formulate the finite dimensional Galerkin scheme equation in the porous medium, and the corresponding and establish well-posedness and a priori error estimates. transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman Marcus Sarkis law. The pseudostress and the velocity in the fluid, to- IMPA (Brazil) and WPI (USA) gether with the velocity, the pressure, and the gradient of [email protected] pressure in the porous medium, constitute the main unknowns of the model. In addition, the transmission condi- Juan Galvis tions become essential, which leads to the introduction of TexasA&M the traces of the porous media pressure and the fluid veloc- [email protected] ity as the associated Lagrange multipliers. We show that the resulting variational formulation has a twofold saddle- point structure and then we prove the unique solvability MS10 of it. Also, we establish suitable hypotheses on the finite Constructing Markov Models for Barrier Options element subspaces ensuring that the associated Galerkin scheme becomes well posed. Finally, a reliable and effi- In this talk we will present an approach for modeling bar- cient residual-based a posteriori error estimator is derived. rier options which is analogous to the ”local volatility” approach for modeling European options. In place of the forward equation, we employ a recent weak existence re- Gabriel Gatica sult which can handle the path dependence in the option Universidad de Concepcion payoff. We then use a minor variation on the martingale Departamento de Ingeniera Matematica problem of Stroock and Varadhan to conclude that the re- [email protected] sulting model is Markov.

Ricardo Oyarzua Gerard Brunick Universidad de Concepcion Department of Mathematics [email protected] The University of Texas at Austin [email protected] Francisco Sayas Department of Mathematics MS10 University of Minnesota Portfolios and Risk Premia for the Long Run [email protected] This talk develops a method to derive optimal portfolios and risk-premia explicitly in a general diffusion model, MS9 for an investor with power utility in the limit of a long On the Coupled Problem of Navier-Stokes, Darcy horizon. The market has several risky assets and is po- and Transport Equations tentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables following The study of a coupled flow and transport system in ad- an autonomous diffusion. The framework nests models of jacent surface and subsurface regions is of interest for the stochastic interest rates, return predictability, stochastic environmental problem of contaminated aquifers through volatility and correlation risk. For each relative risk aver- rivers. First, two models describing the coupled Navier- sion parameter value, the long-run portfolio, its implied Stokes and Darcy equations are presented. The two mod- risk-premia, the long-run pricing measure, and their per- els differ only by the interface condition for the balance formance on finite horizons are obtained. In the case of of forces. Weak solutions are constructed using a Galerkin a single state variable, a candidate solution is derived for approach. The numerical discretization of the coupled flow the finite horizon value function, and hence optimal port- problem uses continuous elements in the surface region folio and pricing measure. An application to a single state and discontinuous finite elements in the subsurface. Nu- variable, potentially non affine, model concludes. merical solutions obtained with the two models are compared as the input data (permeability, viscosity) varies. Scott Robertson Second, a transport equation is coupled with the Navier- Department of Mathematical Sciences Stokes/Darcy flow. We prove existence and uniqueness of Carnegie Mellon University a weak solution. The transport equation is solved by a [email protected] discontinuous Galerkin method. Error estimates for the coupled problem are obtained. Numerical simulations are shown for heterogeneous porous media. MS10 Optimal Investment with Random Discrete Order Beatrice Riviere Flow in Illiquid Markets Rice University Houston, Texas, USA We consider an optimal investment problem where a risky [email protected] asset is traded and observed only at a sequence of arrival times whose intensity approaches infinity as we get closer to maturity. We solve this stochastic control problem us- PD09 Abstracts 63

ing dynamic programming, by ﬁrst solving the integral model oscillations. The idea is that a family of the homo- Hamilton-Jacobi-Equation and then going trough a veri- geneous Sobolev spaces could provide us with the distin- ﬁcation argument. The presentation is based on joint work guishable patterns when it comes to texture and noise in with Paul Gassiat and Huyen Pham. images. Mihai Sirbu Yunho Kim University of Texas at Austin Department of Mathematics [email protected] University of California, Irvine [email protected]

MS10 Luminita A. Vese On the Optimal Stopping Problems for Levy Pro- University of California, Los Angeles cesses Department of Mathematics [email protected] Value functions of optimal stopping problems for processes with Levy jumps are known to be generalized solutions of variational inequalities. Assuming the diffusion compo- John Garnett nent of the process is nondegenerate, in my talk I will show Department of Mathematics, UCLA that the value function is a classical solution of the vari- [email protected] ational inequality in the continuation region for problems with either finite or infinite variation jumps. Moreover, MS11 the smooth-fit property is shown via the global regularity of the value function. Our global regularity results gen- Exact Reconstruction of Piecewise Constant Color eralize the results of Bensoussan and Lions (1984) which Images by a Total Variation Based Model were developed for bounded domains. On the other hand, 2,1 In this work we address the following colorization problem: until now the value function was known to be C only for How can a color image be recovered when the underly- the optimal stopping problems when the jumps have finite ing gray level function is known everywhere but only small activity. This is a joint work with my Ph.D. student Hao patches of color are available? We consider the Red-Green- Xing. Blue (RGB) total variation model proposed by Fornasier Erhan Bayraktar and subsequently studied by Fornasier and March, and we University of Michigan seek to study the faithfulness of the reconstructions it pro- Department of Mathematics vides, with particular emphasis on the possible creation of [email protected] new, spurious contours in the restored image. In the main result of the paper, using calibrations techniques, we show that the reconstruction is faithfull in the case of sufficiently Hao Xing regular piecewise constant images, if the exact information Boston University on the colors is known over a (possibly) very small but uni- [email protected] formly distributed area. This is joint work with I. Fonseca, G. Leoni, and F. Maggi. MS11 Massimiliano Morini A Variational Approach to Hyperspectral Image SISSA Fusion Trieste,Italy [email protected] There has been significant research on pan-sharpening mul- tispectral imagery with a high resolution image, but there has been little work extending the procedure to high dimen- Irene Fonseca sional hyperspectral imagery. We present a wavelet-based Carnegie-Mellon University variational method for fusing a high resolution image and Department of Mathematical Scienc a hyperspectral image with an arbitrary number of bands. [email protected] To ensure that the fused image can be used for tasks such as classication and detection, we explicitly enforce spectral Giovanni Leoni coherence in the fusion process. This procedure produces Carnegie Mellon University images with both high spatial and spectral quality. We [email protected] demonstrate this procedure on several AVIRIS and HY- DICE images. This is joint work with Michael Moeller and Francesco Maggi Todd Wittman. University of Florence, Italy [email protected]fi.it Andrea L. Bertozzi UCLA Department of Mathematics [email protected] MS11 Bidirectional Inverse Consistent Deformable Image Registration MS11 Texture - Noise Separation Model This paper presents a novel variational model for inverse consistent deformable image registration. This model de- The BV space is an excellent space to capture piecewise forms the source and target image simultaneously, and smooth components in an image. However, we still do not aligns the deformed source and deformed target images in know what differentiates texture from noise in corrupted the way that the both transformations are inverse consis- images. We will pursue a method to answer this ques- tent. The model does not computes the inverse transforms tion using homogeneous Sobolev spaces, which are good to explicitly, alternatively it finds two more deformation fields 64 PD09 Abstracts

satisfying the invertibility constraints. Moreover, to im- in time of the propagation along a ”smooth” path. Thus, prove the robustness of the model to noises and the choice if time continuity is viewed as essential, the classical view of parameters, the dissimilarity measure is derived from of crack kinking along a single crack branch seems incor- the likelihood estimation of the residue image. The pro- rect and a change in crack direction is necessarily a more posed model is formulated as an energy minimization prob- complex process. lem, which involves the regularization for four deformation fields, the dissimilarity measure of the deformed source and Antonin Chambolle deformed target images, and the invertibility constraints. Centre de MathématiquesAppliquées The experimental results on clinical data indicate the ef- Ecole´ Polytechnique, France ficiency of the proposed method, and improvements in ro- [email protected] bustness, accuracy and inverse consistency. Gilles Francfort Xiaojing Ye Universitéde Paris Nord University of Florida Villetaneuse, France Department of Mathematics [email protected] xye@ufl.edu Jean-Jacques Marigo Yunmei Chen UniversitéP. et M. Curie University of Florida Paris, France [email protected]fl.edu [email protected]

MS12 MS12 Quasistatic Evolution for Cam-Clay Plasticity Symmetry of Locally Minimizing Solutions for the 3D Ginzburg-Landau System Cam-Clay nonassociative plasticity exhibits both harden- ing and softening behavior, depending on the loading. For We shall present a result classifying nonconstant entire many initial data the classical formulation of the qua- local minimizers of the standard Ginzburg-Landau func- sistatic evolution problem has no smooth solution. We tional for maps from R3 into R3 satisfying a natural energy propose here a notion of generalized solution, based on a bound. Up to translations and rotations, such solutions of viscoplastic approximation. To study the limit of the vis- the Ginzburg-Landau system are given by an explicit solu- coplastic evolutions we rescale time, in such a way that tion equivariant under the action of the orthogonal group. the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized Vincent Millot through an energy-dissipation balance, that can be writ- Universit´ede Paris VII ten in a natural way using the rescaled time. It turns out Paris, France that the proposed solution may be discontinuous with re- [email protected] spect to the original time, and our formulation allows to compute the amount of viscous dissipation occurring in- Adriano Pisante stantaneously at each discontinuity time. University of Rome La Sapienza Roma, Italy Gianni Dal Maso [email protected] SISSA, Trieste, Italy [email protected] MS12 Antonio DeSimone Onsager Theory and Singularities in Nematic Liq- SISSA uid Crystals 34014 Trieste, ITALY [email protected] We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct ap- Francesco Solombrino proaches based on the director ﬁeld (Oseen and Frank), SISSA order parameter tensor (Landau and de Gennes), and ori- Trieste, Italy entation probability density function (Onsager). [email protected] Ibrahim Fatkullin Department of Mathematics MS12 The University of Arizona Crack Kinking in Brittle Fracture [email protected]

Smooth crack propagation in an isotropic 2d brittle mate- Valeriy Slastikov rial is widely viewed as the interplay between two separate University of Bristol criteria. Griffith’s cap on the energy release rate along Bristol, UK the crack path decides when the crack propagates, while [email protected] the Principle of Local Symmetry decides in which direction that crack propagates (the kinking direction). Other com- peting criteria yield different and even contradictory kink- MS13 ing angles. We revisit crack path in the light of energetic High Field and Diffusion Asymptotics for the meta-stability of the current state among suitable compet- Boltzmann Equation ing crack states. We then demonstrate that 2d crack kinking in an isotropic setting is incompatible with continuity The high field asymptotics of the fermionic Boltzmann PD09 Abstracts 65

equation is sconsidered. It consists in a singular perturba- n. tion where the driving forces balance the collision effects. The limiting equation is a nonlinear conservation law for Hailiang Liu the particle density. We show the convergence to entropy Iowa State University solutions of the limiting conservation law. Under symme- USA try conditions on the collission cross section, the nonlinear [email protected] limiting flux is parallel to the force field. We therefore investigate the orthogonal direction to the electric field, and prove after a suitable rescaling that the behaviour is gov- MS13 erned by the original conservation law with an additional A Bloch Band Based Level Set Method for Com- nonlinear diffusion in the orthogonal (to the force field) di- puting the Semiclassical Limit of Schrödinger rection. The main tool used in the convergence proof is a Equations new estimate for the dissipation of a family of kinetic entropies which converge to convex entropies of the limiting A novel Bloch band based level set method is proposed for conservation law. While the entropy dissipation is usually computing the semiclassical limit of Schrödingerequations in periodic media. For the underlying equation subject to estimated by quantities of the type dist(f, Feq) representing the distance of the distribution function to the equi- a highly oscillatory initial data, a hybrid of the WKB ap- librium set, the new estimate involves a quantity of the proximation and homogenization leads to the Bloch eigenvalue problem and an associated Hamilton-Jacobi system form dist(f, Feq(u))dμ(u), where the macroscopic equilibria depend on a velocity variable u and μ is a probability for the phase in each Bloch band, with the Bloch eigen- measure. This estimate allows to control high velocities, value be part of the Hamiltonian. We formulate a level set pass to the limit in the diffusion current and shows the description to capture multi-valued solutions to the band convergence to the entropy solution of the limiting equa- WKB system, and then evaluate total homogenized density tion. over a sample set of bands. A superposition of band densities is established over all bands and solution branches Naoufel Ben Abdallah when away from caustic points. The numerical approach University of Toulouse splits the solution process into several parts: i) initialize [email protected] the level set function from the band decomposition of the initial data; ii) solve the Bloch eigenvalue problem to com- Hedia Chaker pute Bloch waves; iii) evolve the band level set equation to Ecole Nationale d’Ingenieurs de Tunis compute multi-valued velocity and density on each Bloch Tunisia band; iv) evaluate the total position density over a sample [email protected] set of bands using Bloch waves and band densities obtained in step ii) and iii), respectively. Numerical examples with different number of bands are provided to demonstrate the MS13 good quality of the method. Global Existence of a Free Boundary Problem with Zhongming Wang Non-Standard Sources Dept. of Math. We present a kinetic model with free boundary in price UCSD formation. The model consists on high nonlinear parabolic [email protected] equations with nonstandard sources.Global existence results and regularity for the free boundary are given. MS14 Maria Gualdani Analysis of Coupled Systems Describing Sprays Dept. of Math. Dynamics UCLA [email protected] This joint work with A. Moussa, L. He and P. Zhang is concerned with a system that couples the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equa- MS13 tion. Such a system arises in the modeling of sprays, where Global Recovery of High Frequency Wave Fields a dense phase interacts with a disperse phase. The coupling arises from the Stokes drag force exerted by a phase Computation of high frequency solutions to wave equations on the other. We establish the global-in-time existence of is important in many applications, and notoriously difficult classical solutions for data close to an equilibrium and we in resolving wave oscillations. Gaussian beams are asymp- investigate their long time behavior. The proofs use energy totically valid high frequency solutions concentrated on a estimates and the hypoelliptic structure of the system. single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate Thierry Goudon more general high frequency solutions to PDEs. An alter- INRIA, Universite Lille 1 native way to compute Gaussian beam components such as [email protected] phase, amplitude and Hessian of the phase, is to capture them in phase space by solving Liouville type equations Ayman Moussa on uniform grids. In this work we review and extend re- CMLA, ENS Cachan, France cent constructions of asymptotic high frequency wave fields [email protected] from computations in phase space. We give a new level set method of computing the Hessian and higher derivatives Lingbing He of the phase. Moreover, we prove that the kth order phase Department of Mathematical Sciences, Tsinghua space based Gaussian beam superposition converges to the University 2 k − n original wave field in L at the rate of 2 4 in dimension Beijing, China [email protected] 66 PD09 Abstracts

Ping Zhang ties in constructing such solution in three dimensions. Beijing Academy of Science [email protected] Changfeng Gui University of Connecticut [email protected] MS14 Large Time Behavior of Collisionless Plasma MS15 The motion of a collisionless plasma, a high-temperature, Limiting Problems for Ginzburg-Landau Models of low-density, ionized gas, is described by the Vlasov- Superconductivity in 3D Maxwell equations. In the presence of large velocities, relativistic corrections are meaningful, and when magnetic In this paper we consider the asymptotic behavior of the effects are neglected this formally becomes the relativistic Ginzburg- Landau energy in 3-d, in various energy regimes. Vlasov-Poisson system. Similarly, if one takes the classical We derive applications to superconductivity and Bose- limit as the speed of light tends to infinity, one obtains the Einstein condensation: a general expression for the first classical Vlasov-Poisson system. We study the long-time critical field, and the curvature equation for the vortex dynamics of these systems of PDE and contrast the behav- lines. ior when considering the cases of classical versus relativistic velocities and the monocharged (i.e., single species of ion) Giandomenico Orlandi versus neutral plasma situations. Department of Computer Science University of Verona Stephen Pankavich [email protected] Univertity of Texas, Arlington [email protected] MS15 Global Solutions of Phase Transitions Equations MS14 Recent Global Results for the Relativistic Boltz- We study some geometric properties of global solutions of mann Equation Allen-Cahn-type equations. For homogeneous equations, we deal with rigidity and symmetry properties. For meso- We will discuss several recent results regarding the rel- scopic equations, we constuct solutions with interface close ativistic Boltzmann equation. The talk will start with to a plane or to a sphere and we discuss their multibump a broad overview of relativistic Kinetic theory for non- features. specialists. New results to be discussed include the previously open problem of stability of the Maxwellian equi- Matteo Novaga librium for the relativistic Boltzmann equation with soft University of Padova (Italy) interactions. The soft potentials are important for parti- [email protected] cles moving at relativistic speeds. We can also prove for the first time the global validity of the Newtonian Limit Enrico Valdinoci in the near Vacuum regime. Additionally we can estab- Department of Mathematics lish the rigorous connection between the Boltzmann equa- University of Rome II, Tor Vergata tion and Relativistic Euler via a Hilbert Expansion, this is [email protected] joint work with Jared Speck. Furthermore, if time permits, we will discuss the relativistic Boltzmann equation coupled with it’s internally generated electric and magnetic forces. MS15 Despite its importance, no global in time solutions have Singular Limit of a Reaction-diffusion System De- been established so far for this Lorentz invariant model. scribing Combustion in Porous Media We prove existence of the first global in time classical solutions. This project is joint work with Yan Guo. Abstract not available at time of publication. Robert Strain Georg S. Weiss University of Pennsylvania Graduate School of Mathematical Sciences [email protected] University of Tokyo [email protected]

MS15 MS16 Quadruple Junction Solutions in the Entire Three Dimensional Space A Couple of Regularity Problems for the Mhd Equations In this talk, I will discuss the quadruple junction solutions in the entire three dimensional space to a vector-valued First, we present our work on the global regularity of clas- Allen-Cahn equation which models multiple phase separa- sical solutions of the 2D MHD equations with mixed dis- tion. The solution is the basic profile of the local structure sipation and diffusivity. Then, we present two regularity near a quadruple junction in three dimensional crystalline criteria for the 3D incompressible HMD equations. One material under the generalized Allen-Cahn model, and is is in terms of the derivative of the velocity field in one- the three dimensional counterpart of triple junction solu- direction and the other requires suitable boundedness of tion which is two dimensional. I will start with one di- the derivative of the pressure in one-direction. mensional heteroclinic solutions, and describe how we can Chongsheng Cao construct higher dimensional solutions from the lower di- Department of Mathematics mensional ones, and explain the complications and difficul- Florida International University caoc@fiu.edu PD09 Abstracts 67

MS16 being a part of the evolution problem to check that such Motion of Viscous Fluid Around a Rotating Rigid property prevails for a short time, as well as it does the Body Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to Abstract not available at time of publication. the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from Sarka Nekasova the mere fact that we are dealing with weak solutions of Academy of Sciences of the Czech Republic Euler’s equation in the whole space. [email protected] Francisco Gancebo The University of Chicago MS16 [email protected] The Quintic NLS as the Mean Field Limit of a Bo- son Gas with Three-body Interactions MS17 In this talk we will discuss the dynamics of a boson gas with Dispersive Properties of Surface Water Waves three-body interactions in dimensions d=1,2. We prove that in the limit as the particle number N tends to infinity, I will speak on the dispersive character of waves on the the BBGKY hierarchy of k-particle marginals converges interface between vacuum and water under the influence to a limiting Gross-Pitaevskii (GP) hierarchy for which of surface tension and possibly gravity. I will begin by we prove existence and uniqueness of solutions. For fac- giving a precise account of the formulation of the surface torized initial data, the solutions of the GP hierarchy are water-wave problem and discussion of its defining features. shown to be determined by solutions of a quintic nonlinear I will describe the local smoothing effect and the gain of Schrödingerequation. Time permitting, we will briefly de- regularity for the nonlinear water-wave problem with some scribe our new approach for studying well-posedness of the detail of the proofs. I will examine the frequency-localized Cauchy problem for focusing and defocusing GP hierarchy. and global dispersion of the linear system and indicate how the microlocal dispersion estimates gives the Strichartz estimates for the nonlinear problem. If time permits, I will Thomas Chen, Natasa Pavlovic speak on the application of global dispersion on the long- University of Texas at Austin time existence. This work is partly joint with Hans Chris- [email protected], [email protected] tianson and Gigliola Staffilani (MIT). Vera Mikyoung Hur MS16 University of Illinois at Urbana-Champaign Inverse Problems for Some Transport Equations [email protected] Transport equations are used often to model various types of biophysical phenomena, including the propagation of MS17 light in tissues. In one limiting case one has the traditional How Gravity Can Stabilize Kelvin-Helmholtz In- X-ray tomography, which has been the subject of tremen- stabilities and Applications dous attention both for its applications and for its theoretical interest. The inverse problem, in the X-ray tomogra- The mathematical properties of the motion of the inter- phy case, is a linear one. Yet it is ill-posed and presents face between two layers of fluid differ dramatically from many computational challenges. In this talk discuss nonlin- what is observed for water-waves (which is a particular ear inverse problems related to transport equations within case of the two fluids problem with an upper fluid of zero the frame-work of the so called diffuse tomography mod- density). While the water-waves equations are well-posed, els. The work presented here originated in joint research with or without surface tension, on time scales that are with F.A. Grunbaum (UCB-Mathematics) and J.R. Singer pertinent for the observation of oceanographic phenomena, (UCB-EECS). Despite many recent progresses the inverse the ”interfacial waves” equations are known to be ill-posed problem associated to diffuse tomography still presents a without surface tension, due to Kelvin-Helmotz instabili- number interesting open problems. ties. It has been shown that adding surface tension allows the solution to exist locally in time. However, the exis- Jorge P. Zubelli tence time vanishes as the surface tension goes to zero and IMPA since the physical value of surface tension is very small, [email protected] this wellposedness result does not explain why we can observe many stable manifestions of interfacial waves. In this talk, we will show that it is possible to control the growth MS17 of Kelvin-Helmoltz instabilities by a careful analysis of the Interface Evolution: Water Waves in 2-D role played by gravity and by the physical characteristics (depths of the layers, amplitude, etc.) of the flow. In We study the free boundary evolution between two irro- particular, we exhibit a new condition for the existence tational, incompressible and inviscid fluids in 2-D with- of ”stable” solutions to the ”interfacial waves” equations, out surface tension. We prove local-existence in Sobolev that can be seen as a generalization of both the Rayleigh- spaces when, initially, the difference of the gradients of Taylor criterion for the stability of surface waves and the the pressure in the normal direction has the proper sign, Chandrasekhar law on the instable modes in shear flows. an assumption which is also known as the Rayleigh-Taylor high-level commands. Do not include references or cita- condition. The well-posedness of the full water wave prob- tions separately at the end of the abstract. Instead, all lem was first obtained by Wu. The methods introduced citations must be in text in the general form [Authorname, in this paper allows us to consider multiple cases: with or Title, etc] without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It David Lannes is assumed that the initial interface does not touch itself, 68 PD09 Abstracts

Ecole Normale Superieure MS18 France Non Conforming Space-time Domain Decomposi- [email protected] tion Algorithms for Coupling Heterogeneous Prob- lems

MS17 In this talk we present non conforming space-time domain Almost Global Wellposedness of the 2-D full Water decomposition algorithms for solving evolution problems Wave Equation with discontinuous coefficients. The objective is long time computations in highly discontinuous media such as nu- We consider the problem of global in time existence and clear waste disposal simulations, or climate modeling. The uniqueness of solutions of the 2-D infinite depth full water strategy is to split the time interval into time windows wave equation. It is known that this equation has a solution and to perform, in each window, few iterations of Opti- for a time period [0,T/ ] for initial data of type Φ, where mized Schwarz Waveform Relaxation methods [D. Ben- T depends only on Φ. We show that for such data there nequin, M.J. Gander, and L. Halpern., A homographic T/ exists a unique solution for a time period [0,e ]. This best approximation problem with application to optimized is achieved by better understandings of the nature of the Schwarz waveform relaxation, Math. of Comp., pages nonlinearity of the water wave equation. 259-266, 2009]. It has been proposed in [L. Halpern and C. Japhet, Discontinuous Galerkin and Nonconforming in Sijue Wu Time Optimized Schwarz Waveform Relaxation for Het- University of Michigan at Ann Arbor erogeneous Problems, In U. Langer, M. Discacciati, D.E. [email protected] Keyes, O.B. Widlund, and W. Zulehner, editors, Decom- position Methods in Science and Engineering XVII, volume 60 of Lecture Notes in Computational Science and Engi- MS18 neering, pages 211-219. Springer, 2008] to use a discon- Domain Decomposition for Heterogeneous, Multi- tinuous Galerkin method in time as a subdomain solver scale Problems in order to have a high degree of accuracy, time-stepping approaches, and ultimately adaptive control of the time Multiscale methods are used to solve flow problems in step, see [C. Johnson, K. Eriksson, and V. Thomee, Time porous media with heterogeneous permeability. We de- discretization of parabolic problems by the discontinuous velop multiscale mortar mixed finite element discretiza- Galerkin method, RAIRO Modl. Math. Anal. Numr., 19, tions for second order elliptic equations. The continuity 1985], [Ch. Makridakis and R. Nochetto, A posteriori error of flux is imposed via a mortar finite element space on a analysis for higher order dissipative methods for evolution coarse grid scale, while the equations in the coarse elements problems, Numerische Mathematik, 2007]. We present the (or subdomains) are discretized on a fine grid scale. The analysis of the method and numerical results to illustrate mortar approximation space is based on homogenization, the performances of the method. and the method is shown to converge with respect to this microstructure. Laurence halpern UniversitéParis 13 Todd Arbogast [email protected] Dept of Math; C1200 University of Texas, Austin [email protected] Caroline Japhet Paris 13 university, France [email protected] MS18 On the Coupling of Hyperbolic and Elliptic Equa- JérémieSzeftel tions: Comparison Between Virtual Control Meth- Princeton University and Bordeaux 1 University ods and Extended Variational Formulations [email protected]

We address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring MS18 boundary layers. We consider both non-overlapping and Filtration Into Deforming Porous Media overlapping domain decompositions and we face up the heterogeneous problem by different approaches. Starting from We review the theory of flow through deformable porous a rigorous variational analysis, the aim of this research is media. Then we describe more recent extensions to com- to investigate if other approaches may be considered to posite media, partially-saturated flow, flow through rather formulate this heterogeneous problem, for a possible ex- general poro–inelastic media, and coupling to the Stokes tension to more general couplings. In particular we will flow in an adjacent open channel. The problems will be consider the virtual control approach, based on the opti- posed and resolved in appropriate variational formulations. mal control theory, and an extend variational formulation recently proposed for geometric multiscale problems. Ralph Showalter Paola Gervasio Department of Mathematics Dipartimento di Matematica,Facoltàdi Ingegneria Oregon State University Universitàdegli Studi di Brescia, Brescia, Italy [email protected] [email protected] MS19 Alfio Quarteroni Ecole Pol. Fed. de Lausanne Convergence to Homogenized and Stochastic PDEs Alfio.Quarteroni@epfl.ch Equations with small scale structures abound in applied sciences. Many such structures cannot be modeled at a mi- PD09 Abstracts 69

croscopic level and thus require that one understand their MS19 macroscopic influence. I will consider the situation of par- Homogenization of Hamilton-Jacobi Equations in tial differential equations with random, highly oscillatory, Time-Space Random Environments potential. One is then interested in the behavior of the solutions to that equation as the frequency of oscillations We present results on the stochastic homogenization of in the micro-structure tends to infinity. Depending on spa- Hamilton-Jacobi equations in time-space random environ- tial dimension and the decorrelation properties of the ran- ments. Previous results of Souganidis, 1999, and simul- dom potential, I will show that the limit is the solution taneously by Rezakhanlou and Tarver, 2000, had been in to either a deterministic, homogenized (effective medium) the context of random fluctuations in space only. Special equation or a stochastic equation with multiplicative noise attention will be placed on the interplay between the use that should be understood in the sense of a Stratonovich of the Subadditive Ergodic Theorem and continuity esti- product. More precisely, there is a critical spatial dimen- mates for solutions of these equations that depend only on sion above which we observe convergence to a determinis- the growth of the Hamiltonian. tic solution and below which we observe convergence to a stochastic solution. In the former case, a theory of correc- Russell Schwab tors to homogenization allows one to capture the remaining Carnegie Mellon University randomness in the solution to the equation with the small [email protected] scale structure. Once properly rescaled, this corrector is shown to solve a stochastic equation with additive noise. MS20 Guillaume Bal Generalized Newton-type Methods for Energy For- Columbia University mulations in Image Processing Department of Applied Physics & Applied Mathematics [email protected] Many problems in image processing are addressed via the minimization of a cost functional. The most prominently used optimization technique is gradient-descent, often used MS19 due to its simplicity and applicability where other tech- SPDE Limits of the Random Walk Metropolis Al- niques, e.g., those coming from discrete optimization, can gorithm in High Dimensions not be applied. Yet, gradient-descent suffers from slow convergence, and often to just local minima which highly de- Diffusion limits of MCMC methods in high dimensions pro- pend on the initialization and the condition number of the vide a useful theoretical tool for studying efficiency. In functional Hessian. Newton-type methods, on the other particular they facilitate precise estimates of the number hand, are known to have a faster, quadratic, convergence. of steps required to explore the target measure, in station- In its classical form, the Newton method relies on the L2- arity, as a function of the dimension of the state space. type norm to define the descent direction. In this work, we However, to date such results have only been proved for generalize and reformulate this very important optimiza- target measures with a product structure, severely limit- tion method by introducing Newton-type methods based ing their applicability to real applications. I will present on more general norms. Such norms are introduced both a study diffusion limits for a class of naturally occuring in the descent computation (Newton step), and in the high dimensional measures and their associated MCMC corresponding stabilizing trust-region. This generalization algorithm. The diffusion limit to an infinite dimensional opens up new possibilities in the extraction of the New- Hilbert space valued SDE (or SPDE) will be proved. ton step, including benefits such as mathematical stability and the incorporation of smoothness constraints. We first Jonathan C. Mattingly present the derivation of the modified Newton step in the Duke University calculus of variation framework needed for image process- [email protected] ing. Then, we demonstrate the method with two common objective functionals: variational image deblurring and geometric active contours for image segmentation. We show MS19 that in addition to the fast convergence, norms adapted to Unbiased Random Perturbations of Navier-Stokes the problem at hand yield different and superior results. Equation. Leah Bar A random perturbation of a deterministic Navier-Stokes University of Minnesota equation is considered in the form of an Stochastic PDE [email protected] with Wick product in the nonlinear term. The equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos ex- MS20 pansion. The generalized solution is obtained as an inverse From TV-L1 Model to Continuous Graph-Cuts: of solutions to corresponding quantized equations. An in- Global & Fast Optimization Algorithms for Image teresting feature of this type of perturbation is that it pre- & Geometry Processing serves the mean dynamics: the expectation of the solution of the perturbed equation solves the underlying determin- In this talk, I will review some recent developments in istic Navier-Stokes equation. From the stand point of a global continuous optimization methods for image and ge- statistician it means that the perturbed model is unbiased. ometry processing. Starting from the TV-L1 model for The talk is based on a joint work with R. Mikulevicius. image denoising, we will show that this model has opened the path to a new paradigm to define convex formulations Boris Rozovsky for e.g. image segmentation and geometry reconstruction. Brown University Indeed, unlike state-of-the-art methods, the proposed con- Boris [email protected] vex energy minimization approach allows to solve geometry problems independently of initial condition. Besides, this approach provides fast optimization algorithms, bor- 70 PD09 Abstracts

rowed from operator splitting techniques, to solve energy and alternating direction methods. Recent results on the minimization problems related with non-linear hyperbolic Bregman methods are included. Applied to 1-type regu- PDEs. We will show some results in segmentation and sur- larization, the original Bregman method has an exact error face reconstruction. Joint work with T.F. Chan, S. Osher, cancellation property that leads to machine-precision solu- T. Goldstein (UCLA). tions. In addition, we briefly show an exact penalty property of the linearized Bregman method. These properties Xavier Bresson explain the great performance of the Bregman methods. UCLA The analysis and discussions broadly apply to polyhedral Department of Mathematics regularization functions such the (weighted) 1-norm, total [email protected] variation, piece-wise linear functions, as well as the matrix nuclear norm. MS20 Wotao Yin Coarsening in High Order, Discrete, Ill-posed Dif- Rice University fusion Equations [email protected] We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order MS21 (n = 2) version of these equations constitutes our main mo- Radial Symmetry of Positive Solutions to Nonlin- tivation, as it appears prominently in image processing and ear Polyharmonic Dirichlet Problems computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining We extend the celebrated symmetry result of Gidas-Ni- sharp object boundaries (edges). The second order equa- Nirenberg to semilinear polyharmonic Dirichlet problems tion (n = 1) corresponds to another famous model from in the unit ball. In the proof we develop a new variant of image processing, namely Perona and Maliks anisotropic the method of moving planes relying on fine estimates for diffusion, and was studied previously. The equations pre- the Green function of the polyharmonic operator. We also sented in this talk are high order analogues of the Perona- consider minimizers for subcritical higher order Sobolev Malik equation, and like the second order model, their con- embeddings. For embeddings into weighted spaces with tinuum versions violate parabolicity and hence lack well- a radially symmetric weight function, we show that the posedness theory. As in the previous work, we follow a minimizers are at least axially symmetric. This result is recent technique by Kohn and Otto to establish rigorous sharp since we exhibit examples of minimizers which do upper bounds on the coarsening rate of the discrete ver- not have full radial symmetry. The lecture is based on joint sions of these high order equations, in any space dimension work with E. Berchio (Milan) and T. Weth (Frankfurt). and for a large class of diffusivities. However, the high order nature of our equations requires different arguments Filippo Gazzola and constructions from before. Dipartimento di Matematica - Politecnico di Milano fi[email protected] Catherine Kublik University of Michigan Tobias Weth [email protected] Institut fürMathematik Johann Wolfgang Goethe-UniversitätFrankfurt - Selim Esedoglu Germany Department of Mathematics [email protected] University of Michigan [email protected] Elvise Berchio Dipartimento di Matematica MS20 Politecnico di Milano - Italy [email protected] Geodesics in Level-set Based Shape Space

The equations for geodesic movement (with respect to MS21 an appropriately constructed metric) of a level-set function representing a shape variable are derived. The re- ”Almost Positivity”’ in the Fourth Order Clamped lated equations for parallel transport are also constructed. Plate Equation Geodesic movement and parallel transport are used to re- A classical example for a fourth order problem in mechan- alize algoritmic concepts for non-linear optimization in the ics is the linear clamped plate equation. It is known that non-linear level-set based shape space. it does not satisfy any form of a general maximum or pos- Wolfgang Ring itivity preserving (comparison) principle. However, there Institut fuer Mathematik are positivity issues as e.g. “almost positivity’ to be dis- Universitaet Graz cussed. In particular, it is shown that the negative part [email protected] of the corresponding Green function obeys much sharper bounds than the singular positive part. For example, if n>4, the negative part is bounded while the positive part 4−n MS20 displays the singular behaviour of |x − y| . A Review of the Bregman Methods Hans-Christoph Grunau We review four Bregman methods that arise in applica- Institut fur Analysis und Numerik tions of imaging, compressed sensing, and other inverse Otto-von-Guericke-Universitat, Magdeburg - Germany problems. We explain their numerical properties and high- [email protected] light their relationships with the augmented Lagrangian Frederic Robert PD09 Abstracts 71

Laboratoire J.-A.Dieudonne Laws Universite de Nice-Sophia Antipolis [email protected] Abstract not available at time of publication. Constantine Dafermos Guido Sweers Brown Univ. Mathematisches Institut constantine [email protected] Universitat zu Koln - Germany [email protected] MS22 MS21 New Models of Chemotaxis: Analysis and Numer- ics Q-curvature Equation in Conformal and CR Ge- ometry Patlak-Keller-Segel (PKS) system is a classical PDE model of the chemotaxis. The system admits solutions that de- In this talk, I plan to outline the role this 4th order equa- velop delta-type singularities within a ﬁnite time. Even tion plays in the conformal geometry of four dimension, as though such blowing up solutions model a concentration well as the CR geometry in 3-dimension. In CR geometry, phenomenon, they are not realistic since biological cells there is a natural 4th order operator which plays an impor- do not converge to one point (while the cell density grows tant rolein the analogue of the Bochner formula governing sharply, it must remain bounded at all times). I will present the kernel of the sub-Laplacian. a new chemotaxis model, which can be viewed as a regu- Paul Yang larized PKS system. The proposed regularization is based Department of Mathematics on a basic physical principle: boundedness of the chemo- Princeton University, USA tactic convective ﬂux, which should depend on the gradi- [email protected] ent of the chemoattractant concentration in a nonlinear way. Solutions of the new system may develop spiky structures that model the concentration phenomenon. However, MS21 both cell density and chemotattractant concentration re- Fourth Order Elliptic Problems Involving Singular main bounded as supported by both our analytical results and Critical Nonlinearities and extensive numerical experiments.

We study a class problems for the biharmonic operator Alina Chertock with Dirichlet boundary conditions in dimension four in- North Carolina State University volving critical growth corresponding to a Trudinger-Moser Department of Mathematics embedding. [email protected] Joao Marcos do O Alexander Kurganov Universidade Federal da Paraba Tulane University [email protected] Department of Mathematics [email protected]

MS22 Xuefeng Wang On the Classical Solutions of Two Dimensional In- Tulane University viscid Rotating Shallow Water System [email protected] We prove the global existence and asymptotic behavior of the classical solutions for two dimensional inviscid Rotat- MS22 ing Shallow Water system with small initial data subject A Low Mach Number Limit from a Dispersive to the zero-relative-vorticity constraint. One of the key Navier-Stokes System ¿ to a Ghost-Effect System steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, whose existence of the A low Mach number limit for classical solutions of a dis- classical solutions is then proved with combination of the persive system beyond the full Navier-Stokes equations is vector field approach and the normal forms, adapting ideas investigated. We prove the convergence of this dispersive developed in [1]. The symmetric form of the system facili- system to a ghost effect system studied extensively by Sone tates the closure of energy estimates for the problem. We in the whole space case. The combined effects of large also probe the case of general initial data and reveal a lower temperature variations, thermal conduction as well as the bound for the lifespan that is almost inversely proportional dispersive structure are taken into account. to the size of initial relative vorticity. [1] Tohru Ozawa, Kimitoshi Tsutaya, and Yoshio Tsutsumi, Global existence Weiran Sun and asymptotic behavior of solutions for the Klein-Gordon Department of Mathematics equations with quadratic nonlinearity in two space dimen- University of Chicago sions, Math. Z. 222 (1996), no. 3, 341362. [email protected]

Bin Cheng, Chunjing Xie University of Michigan MS23 [email protected], [email protected] Well-posedness and Mean-ﬁeld Limit for Kinetic Models of Collective Motion

MS22 There has been a recent interest in models for the com- Continuous Solutions of Hyperbolic Conservation plex behavior of large groups of animals, such as ﬂocks of birds, swarms, or schools of ﬁsh. Kinetic equations, which 72 PD09 Abstracts

involve the time evolution of the distribution function of lutions. An interesting feature of the model is that the individuals depending on their position and velocity, are ”trivial” steady state characterized by the uniform density tentative models for these phenomena, and raise many in- remains locally stable in the subcritical region while for teresting mathematical questions. In this work, we give sufficiently large perturbations the dynamics prefers other, a well-posedness theory for kinetic PDE models for the spatially nonuniform stady states. collective behavior of a group of individuals that interact through a pair potential and/or try to adapt their velocity Vladislav Panferov to the average velocity of their neighbours, possibly includ- California State University, Northridge ing also self-propulsion effects. The aim is to give a solid Department of Mathematics mathematical basis for the study of these models and the [email protected] range of phenomena they display. Jose Alfredo Canizo MS24 Universitat Autonoma de Barcelona Entire Solutions of the Allen Cahn Equation in the [email protected] Hyperbolic Space Qualitative properties of entire solutions of equations with MS23 double well potential in the Hyperbolic space will be dis- Accuracy and Consistency of Spectral-Lagrangian cussed, both in relation with conditions at the asymptotic Approximating Methods for the Boltzmann Equa- boundary and with conditions of stability or minimality. tion Differences and similarities with respect to the euclidean case will be discussed. We study the accuracy and consistency properties of the recent Lagrangian based spectral method to produced an ap- Isabeau Birindelli proximating scheme to solutions of the Boltzmann equation Sapienza Universitàdi Roma [1,2]. We develop bounds on the optimization correction in [email protected] terms of spectral accuracy and also prove consistency and conservation properties of the scheme. We also obtain error estimate for the conservation routine implemented as an MS24 extended isoperimetric problem with the moment conser- A Moving Boundary Problem Involving Surface vation properties as the constraints. Combining Sobolev Tension estimates of the collisional integral with projection error and conservation correction estimates yields a Lagrange- Abstract not available at time of publication. spectral error result. Nestor Guillen Irene M. Gamba Department of Mathematics University of Texas at Austin Department of Mathematics and ICES [email protected] University of Texas [email protected] MS24 Stability in the Stefan Problem with Surface Ten- Sri Harsha Tharkabhushanam sion University of Texas at Austin [email protected] Abstract not available at time of publication. Mahir Hadzic MS23 Brown University Fractional Diffusion Limits for Kinetic Equations [email protected] We will discuss some diffusion limits for linear Boltzmann equations when the equilibrium distribution function is not MS24 a Maxwellian distribution. In particular, we will show that Domain Walls and Triple Junctions for an appropriate time scale, the small mean free path limit gives rise to fractional diffusion equations. The studies of domain walls and triple junctions arise in many applications. Their appearences , such as honeycomb Antoine Mellet structures in 2D, are deceptively simple. If the underlying University of Maryland variational energy driven such a structure is giving by the [email protected] Surface Area, it is then possible to describe some geometric and analytic properties of the structure via the classical minimal surfaces theory. In many applications, however, MS23 such a structure is driven by the so-called stress, that is, Phase Transitions for the McKean-Vlasov Equation the Bulk energy, then not much were known before. In in Finite Volume this lecture, I shall describe some recent progress on the Bulk-energy driven domain walls and multiple junctions. We consider a nonlinear parabolic equation that describes the mean-field limit for a system of N diffusions in a Fang-Hua Lin bounded domain interacting through a finite-range poten- Courant Institute of Mathematical Sciences tial V. We show that for all potentials that are not of pos- [email protected] itive type (characterized by the positivity of the Fourier transform of V) the system generically has a first-order phase transition at the critical temperature, which is man- MS25 ifested by the nonuniqueness/instability of the steady so- Global Bv Solutions to Heat Flows of Linear PD09 Abstracts 73

Growth Maps into Spheres Free Boundary

In this talk I shall first present some recent developments We study evolution of simple planar curves driven by the for p-harmonic map heat flows into spheres, in particular, weighted mean curvature flow (wmc) for 1-harmonic map heat flow. The highlight of the developments is the BV (bounded variation) solution theory for βV = κγ + σ the flow, which will be reviewed in detail. I shall then discuss the extension of these results to heat flows for a class when the anisotropy function is piecewise linear and con- of linear growth maps. Finally, applications (and their nu- vex. The diffusion in certain direction is so strong that merical approximations) of these geometric flows to color facets are created. At the same time in other directions image denoising based on the chromaticity and brightness the equation degenerates to a first order equation. In or- decomposition will also be discussed. The results presented der to prove existence and uniqueness of solutions to the in this talk are based on recent joint works with John Bar- resulting problem we develop a theory of viscosity solutions rett of Imperial College (UK), Andreas Prohl of University to Hamilton-Jacobi equation with discontinuous Hamilto- of Tuebengin (Germany), and Minun Yoon of University of nian. In particular, we show a comparison principle and a Tennessee. stability theorem. Xiaobing H. Feng Piotr Rybka The University of Tennessee Warsaw University [email protected] Mathematics Department [email protected]

MS25 Yoshikazu Giga Local and Nonlocal Weighted p-Laplacian Evolu- Graduate School of Mathematical Sciences tion Equations with Neumann Boundary Condi- University of Tokyo, Japan tions [email protected]

In this talk we study existence and uniqueness for solutions Przemys law G´orka of the nonlocal diﬀusion equation with Neumann boundary Instituto de Matem conditions { } ’ a tica y F ’{i}sica x + y p−2 ut(t, x)= J(x−y)g |u(t, y)−u(t, x)| (u(t, y)−u(Universidadt, x)) dy de Talca Ω 2 [email protected] in ]0,T[×Ω, and for solutions of its local counterpart p−2 MS26 ut = div (g|∇u| ∇u) in]0,T[×Ω, p−2 Expanding the Prodi-Serrin Critera for Regularity |∇ | ∇ · × p g u u η in]0,T[ ∂Ω. to Weak-L Spaces

We consider 1 ≤ p<∞ and g ≥ 0. We pay special atten- In this talk we discuss weakening the Prodi-Serrin criteria tion to the case in which g vanishes somewhere in Ω even for regularity of Leray-Hopf solutions to the 3d Navier- in a set of positive measure. Joint work with: F. Andreu, Stokes equation. In particular, the criteria is expanded to J. Rossi and J. Toledo include solutions which satisfy the following bound:

t p Jose Mazon u(s) q,∞ 3 L (R ) ∞ Universitat de Valencia ds < Departamento de An´alisisMatem´atico 0 e + log(e + u(s) L∞(R3)) [email protected] 3 2 Here p and q satisfy the usual relation 1 = q + p . The result is based on bounding the local energy in terms of MS25 weak Lp norms, a scaling argument, and an application of Regularity of the Total Variation Flow the De Giorgi method.

I would like to investigate phenomena appearing in analy- Clayton Bjorland sis of a class of systems represented by the total variation University of Texas flow. The basic gaol is to create a new point of view at the [email protected] regularity and the notion of solution when the nonlinear diffusion is so strong that it causes nonlocal effects. Our Alexis F. Vasseur approach will allow us to call the obtained solutions al- University of Texas at Austin most classical, although the classical theory sees this type [email protected] of equations as meaningless. The talk will be based on joint results with Piotr Rybka. MS26 Piotr Mucha Stochastic Particle Systems for the Navier-Stokes Warsaw University and Burgers Equations Mathematics Department [email protected] I will introduce an exact stochastic representation for the 3D-Navier-Stokes equations based on noisy Lagrangian paths, and use this to construct a (stochastic) particle MS25 system. On any fixed time interval, this particle system A Singular Weighted Mean Curvature Flow in the converges to the Navier-Stokes equations as the number Plane Hamilton-Jacobi Equation with an Unusual of particles goes to infinity. Curiously, a similar system 74 PD09 Abstracts

for the (viscous) Burgers equations shocks in finite time, techniques. and solutions can not be continued past these shocks using classical methods. I will describe a resetting procedure by Mohammed B. Ziane which these shocks can (surprisingly!) be avoided. University of Southern California MAthematics Department, DRB141 Peter Constantin [email protected] Department of Mathematics University of Chicago [email protected] MS27 Fully Nonlinear Hamiltonian Equations for for the Gautam Iyer Dynamics of Fluid Films Stanford University Department of Mathematics We discuss the exact nonlinear equations for the dynam- [email protected] ics of fluid films, modeled as a two dimensional manifold. Our main goal is to illustrate the differences and similarities between the fluid film equations and Euler’s equations, Jonathan C. Mattingly their classical three dimensional counterpart. Since the Duke University geometry of fluid films is fundamentally different – three [email protected] dimensional velocity field on a two dimensional support with a time varying Riemannian metric – all classical the- Alexei Novikov orems must be properly modified. We offer adaptations of Penn State University the following theorems: conservation of mass and energy, Mathematics pointwise conservation of vorticity and Kelvin’s circulation [email protected] theorem. We present proofs of these theorems by employing the calculus of moving surfaces. It is of great interest to develop a simplified model that captures normal deforma- MS26 tions of fluid films by assuming tangential velocities vanish Exterior Problems while preserving the exact nonlinear nature of the full system. This cannot be accomplished simply by neglecting the We develop some techniques to deal with exterior problems tangential components, for such an attempt leads to inter- for equations of interest to fluid dynamics, such as the two nal contradictions. Instead, we alter the initial formulation dimensional dissipative quasi-geostrophic equation and present a modified variational approach that leads to a simplified system of equations capable of capturing a broad

∂θ α range of deeply nonlinear effects. + u ·∇θ + κ(−Δ) θ =0, 0 <α<1. ∂t Pavel Grinfeld Drexel University The emphasis is in obtaining decay of solutions of these [email protected] equations extending the Fourier splitting methods used in the full space situation. MS27 Tomas Schonbek Stability of Periodic Water Waves Florida Atlantic University Department of Mathematics Periodic traveling waves exist in water waves and lots of [email protected] dispersive models, such as Stokes waves of deep water (Stokes, 1847) and Cnoidal waves of KDV equation. I will discuss an unified approach to study the stability and insta- MS26 bility of periodic waves of water waves and a large class of Higher Derivatives Estimates for the 3D Navier- dispersive models, under perturbations of the same period. Stokes Equation The results include a sharp stability criterion for KDV and BBM type models, and a proof of the existence of unstable We will present, in this talk, new applications of De Stokes waves under some natural assumptions. Giorgi’s methods and blow-up techniques to fluid mechanics problems. We introduce a new nonlinear family of Zhiwu Lin spaces allowing to control higher derivatives of solutions Georgia Institute of Technology to the 3D Navier-Stokes equation. Finally, we show a reg- School of Mathematics ularity result for a reaction-diffusion system which has al- [email protected] most the same supercriticality than the 3D Navier-Stokes equation. MS27 Alexis F. Vasseur Spectral Stability of Traveling Water Waves University of Texas at Austin [email protected] The water wave problem arises in a number of problems of practical interest, consequently its reliable and accurate numerical simulation is of central importance in many ap- MS26 plications. Recently, an efficient, stable, and high-order Oscillations and Regularity of the Navier-Stokes Boundary Perturbation scheme for simulating traveling wa- Equations ter waves (due to the author and F. Reitich) has been extended to address the equally important topic of their We shall give some global regularity results for the 3D dynamic (spectral) stability. In this talk we will discuss Navier-Stokes equations when the initial data are oscil- this algorithm and present new results on the ”motion” of lating in some sense. The proofs rely on thin domains the spectrum of the linearized water wave equations as the PD09 Abstracts 75

traveling waveform is varied. In particular, we will focus and 2D/3D Navier-Stokes equations. The potentialities upon the radius of convergence of a Taylor series expansion and the performance of the strategy are shown through of the spectral data and its possible connection to spectral several examples involving transient ﬂows. instability of traveling water waves. Pablo J. Blanco David P. Nicholls Laboratorio Nacional de Computacao Cientiﬁca University of Illinois at Chicago Brasil [email protected] [email protected]

Jorge Leiva MS27 Instituto Balseiro Spectral Stability of Solitary Waves on Water [email protected] The talk will discuss spectral stability of two-dimensional solitary waves on water. It is assumed that the fluid is Gustavo Buscaglia bounded by a free surface and a rigid horizontal bottom. Instituto de Ciencias Matematicas e Computacao The solitary wave is moving under the gravity and the sur- Brasil face tension is ignored. It was known that the fully nonlin- [email protected] ear Euler equations have a solitary-wave solution. In this talk, we will show that the linear operator arising from MS28 linearizing the Euler equations around the solitary-wave solution has no spectrum points lying on the right half of Fluid-Structure Interaction in Blood Flow the complex plane, which implies the spectral stability of The speaker will give an overview of the basic issues and the solitary waves. This is a joint work with R. Pego. difficulties associated with the study of fluid-structure in- Shu-Ming Sun teraction in blood flow. A list of open problems and the re- Virginia Tech lated results obtained by the hemodynamics group in Hous- Department of Mathematics ton will be presented. They include an existence result for [email protected] a nonlinear hyperbolic-parabolic effective moving boundary problem, a design of a novel, stable, loosely-coupled computational scheme for the full 2D/3D problem, and a MS28 novel model for the mechanical behavior of endovascular Physiological Models for Evaluating Biocompatibil- stents. Movies showing application to the medically rele- ity of Heart Assist Devices vant problems will be presented.

Several challenges and methods in blood pump design are Suncica Canic described. The objective often involves the unique behav- Department of Mathemtics ior of blood as the flowing medium, necessitating, e.g., ac- University of Houston curate modeling of thrombosis and hemolysis. The fluid [email protected] constitutive behavior, in particular shear-thinning, may affect the outcome of shape optimization more than it affects MS28 direct flow analysis. Finally, target applications often involve intricate time-varying geometry, and thus, realistic Data Assimilation in Fluid-structure Problems. solutions can be only obtained on high-performance paral- Application to blood flows. lel computers. We present a method to estimate the state and the pa- Marek Behr rameters in systems involving the mechanical interaction RWTH Aachen University of a viscous incompressible fluid and an elastic structure. Chair for Computational Analysis of Technical Systems We consider a sequential approach inspired by Kalman fil- [email protected] tering strategies assuming observations on structural displacements. The proposed method is illustrated through examples inspired from vascular biomechanics. MS28 Cristobal Bertoglio, Miguel Fernandez Partitioned Strategies for Strong INRIA Coupling of Dimensionally-heterogeneous Models [email protected], [email protected] for the Navier-Stokes Equations

In this work an iterative procedure is developed to Jean-Frederic Gerbeau tackle the problem of strong coupling of dimensionally- Stanford university & INRIA Paris-Rocquencourt heterogeneous models in fluid mechanics. The procedure [email protected] proposed here makes use of a reinterpretation of the problem as a partitioned problem yielding a system of non- Philippe Moireau linear equations in terms of interface variables, for which INRIA classical non-linear solvers are applied. The goal is to cou- [email protected] ple different mathematical models, which can be treated as black boxes, through the imposition of proper boundary information at the coupling interfaces. The main application MS29 for which this strategy is envisaged arises when addressing On Some Averaging Results for SPDE’s the interaction between hydraulic components which aim The behavior of solutions of infinite dimensional systems at retrieving information from different geometrical scales −1 in complex hydraulic systems. The specific examples are on time intervals of order was not very well understood, provided in the context of coupling 0D lumped equations even if applied mathematicians believed that the averaging 76 PD09 Abstracts

principle holds and usually approximate of the slow motion fidelity terms both from a continuous and a discrete point by the averaged motion, also with n = ∞. In my talk I of view. Two algorithms are presented: a) a pure combina- will present some results on multiscaling limits for systems torial algorithm relying on the parametric maximum-flow of SPDE’s, obtained also in collaboration with M. Freidlin. in a network. and b) an approximation algorithm that I will be mainly concerned with the problem of the valid- allows an extremely efficient parallel programming imple- ity of an averaging principle for general systems of fully mentation. Several applications such as crystalline/mean coupled SPDE’s of reaction diffusion type and I will also curvature flow, deconvolution and compressive sensing re- study normal deviations from the averaged motion for a construction are presented. particular class of systems. Jerome Darbon Sandra Cerrai Mathematics University of Maryland UCLA [email protected] [email protected]

MS29 MS30 Singular Perturbations of Stochastic PDEs Existence and Regularity of Minimizers for a Mul- tiphase Segmentation Variational Model We study of a class of parabolic PDEs depending on a small parameter with the following very interesting fea- We consider a new functional for image segmentation which ture. As → 0, the solutions to the PDEs converge to the has been proposed by Sandberg, Kang and Chan. The solutions to the heat equation. However, when we add to functional is a modified version of the Mumford and Shah these equations additive space-time white noise (of order functional for the partition problem. In the new functional one), their solutions converge to the solutions to a stochas- the length term of the Mumford and Shah functional is tic PDE that is different from the stochastic heat equa- multiplied by the sum of the ratios perimeter/area of the tion. The additional term in the limiting equation can be sets of the partition. We study the regularity of bound- interpreted as a spatial analogue to the Itô-Stratonowich aries of sets which constitute an optimal partition by using correction term appearing in the usual theory of stochastic an existence result of minimizers for a weak version of the differential equations. functional defined on the class of sets with finite perimeter. Techniques for the present problem are adapted from Martin Hairer the analysis of the Mumford and Shah functional in the New York University piecewise constant case. [email protected] Sung Ha Kang Georgia Inst. of Technology MS29 Mathematics On the Fluctuations of Fronts in Random Media [email protected]

I will describe some asymptotic properties of reaction- Riccardo March diﬀusion fronts propagating in random media, including Istituto per le Applicazioni del Calcolo, CNR the ﬂuctuations of the front position around the large-time [email protected] mean behaviour.

James Nolen MS30 Duke University Mathematics Department Analysis of a Total Variation-Based Model for Im- [email protected] age Restoration with Higher Order Regularization The higher order total variation-based model for image MS29 restoration proposed by Chan, Marquina, and Mulet is analyzed in one dimension. A suitable functional framework Waves and Particles in Slowly Decorrelating Ran- in which the minimization problem is well posed is being dom Media proposed, and it is proved analytically that the higher order Slow decay of correlations in random media leads to frac- regularizing term prevents the occurrence of the staircase tional limits for waves and particles in the large distance eﬀect. The generalized version of the model considered limit. I will describe some results in this direction concern- here includes, as particular cases, some curvature depen- ing the phase of a wave in a random medium. dent functionals. Lenya Ryzhik Gianni Dal Maso Department of Mathematics, University of Chicago SISSA, Trieste, Italy Chicago, IL 60637 [email protected] [email protected] Irene Fonseca Carnegie-Mellon University MS30 Department of Mathematical Scienc Combinatorial and Parallel Programming Point of [email protected] Views for Markovian Energies Minimization Giovanni Leoni Many image processing processing problems can be formu- Carnegie Mellon University lated as the minimization of a Markovian energy. In this [email protected] talk, a combinatorial point of view is considered. I focus on the minimization of the Total Variation with convex data Massimiliano Morini PD09 Abstracts 77

S.I.S.S.A., Trieste, Italy biharmonic equation Δ2u = u|u|p−1. [email protected] Paul G. Schmidt Auburn University, USA MS30 [email protected] An Extension of the Wasserstein Distance Monica Lazzo The Wasserstein distance has been used in a variety of ap- University of Bari, Italy plications as a measure of likeness of two images. One issue [email protected] in such applications is that the Wasserstein metric only measures distances between measures of the same total ”mass”. So the images need to be normalized, which leads MS31 to a loss of potentially useful information. Here we extend Biharmonic Equations and Domains with Corners the Wasserstein distance to nonnegative measures of potentially different total mass. In particular we show from For one of the first boundary value problem in pde, namely a large class of possible extensions, only a one-parameter Δu = f in Ω with Dirichlet boundary conditions on ∂Ω one family satisfies certain requirements which are natural for knows that for most nice approximations Ωn of Ω, the cor- the image-processing applications. responding solutions un on Ωn converge to u. For the biharmonic problem this changes drastically. We will present Dejan Slepcev, Gustavo K. Rohde some of the problems that occur when the boundary con- Carnegie Mellon University tains corners. [email protected], [email protected] Guido Sweers Mathematisches Institut MS31 Universitat zu Koln - Germany Entire Solutions for Biharmonic Equations with [email protected] Power Nonlinearities

It is well known that the biharmonic equation Δ2u = MS32 p−1 n u|u| with p ∈ (1, ∞) has positive solutions on R if A Discontinuous Galerkin Solver for Full-Band and only if the growth of the nonlinearity is critical or Boltzmann-Poisson Models supercritical. We complement this result by proving the existence and uniqueness, up to scaling and symmetry, of We present new results of a discontinuous Galerkin (DG) oscillatory radial solutions on Rn in the subcritical case scheme applied to deterministic computations of the tran- and analyzing their nodal properties. sients for the Boltzmann-Poisson (BP) system describing electron transport in semiconductor devices. In recent Monica Lazzo years, results for one and two dimensional devices were ob- University of Bari, Italy tained in the case of silicon semiconductor assuming the [email protected] non-parabolic band approximation. Here, more general band structures are considered. Preliminary benchmark Paul G. Schmidt numerical tests on Kane and Brunetti et al. band models Auburn University, USA are reported. [email protected] Yingda Cheng Dept. of Math and ICES MS31 University of Texas at Austin Nonlinear Waves in Suspension Bridges and Re- [email protected] lated Problems

I will review a series of results on travelling waves in nonlin- MS32 early suspended beams. The equation will be a nonlinearly The Discontinuous Galerkin Method for the perturbed fourth order problem on R1. I will cover some Wigner-Fokker-Planck Equation important open questions on the interactions and ﬁssion of these waves. Several open problems will be adressed at the The Wigner-Fokker-Planck equation is a model of an open end of the talk. quantum system coupled to a heat bath. We have developed a discontinuous Galerkin method for computing time Joseph McKenna dependent solutions given a wide range of external poten- University of Connecticut tial functions. The method does not depend on a polyno- [email protected] mial approximation space. I will discuss the method and an application to calculate the eﬀect of grain boundaries on resistivity in small wires. MS31 Qualitative Behavior and Asymptotics of Radial R. Sharp Solutions for Polyharmonic Equations with Power- Dept. of Math. type Nonlinearities Carnegie Mellon University [email protected] We are interested in polyharmonic equations of the form Δmu = f(u), where m is a positive integer and f a p- homogeneous function with p ∈ (1, ∞). A wealth of in- MS32 formation about radial solutions can be inferred from the A New Discontinuous Galerkin Method for dynamics of the corresponding ODE u(2m) = f(u). We describe the technique and present detailed results for the 78 PD09 Abstracts

Hamilton-Jacobi Equations Joel Smoller University of Michigan Different to Hu-Shu’s paper in 1999, we develop a local [email protected] discontinuous Galerkin method to directly solve Hamilton- Jacobi equations. For the linear case, the method is equivalent to the discontinuous Galerkin method for conserva- MS33 tion laws. Thus, stability and error analysis are valid. For On the Two-dimensional Pressureless Equations both convex and nonconvex Hamiltonian, optimal (k+1)-th order of accuracy for smooth solutions are obtained with We discuss the existence of weak solutions of the 2D non- piecewise k-th polynomial approximations. The schemes linear pressureless equations. We develop an L1 theoretical are numerically tested on a variety of one and two dimen- framework for dual solutions of such equations. Their ex- sional problems. The method works well to capture sharp istence is realized as vanishing viscosity solutions, which corners(discontinuous derivatives) and converges to the vis- follows from spectral dynamics. cosity solution. Eitan Tadmor Jue Yan University of Maryland Dept. of Math. USA Iowa State University [email protected] [email protected] Dongming Wei Dept. of Math. MS33 University of Wisconsin at Madison Hyperbolic Conservation Laws on a Spacetime [email protected] We consider nonlinear hyperbolic conservation laws posed on a differential (n+1)-manifold with boundary (referred MS33 to as a spacetime) for which the “flux’ is defined as a field Weak Stability and Hydrodynamic Limit of Vlasov- of n-forms depending on a parameter (the unknown vari- Maxwell-Boltzmann Equations able). Under a global hyperbolicity condition which implies no topological restriction on the manifold, we estab- Weak stability and hydrodynamic limit of renormalized so- lish the existence and uniqueness of an entropy solution lutions to the Vlasov-Maxwell-Boltzmann equations will be to the initial and boundary value problem. No foliation is discussed. assumed on the spacetime and a new formulation of the finite volume method based on ‘total flux’ functions is in- Dehua Wang troduced. References: P. Amorim, P.G. LeFloch, and B. University of Pittsburgh Okutmustur, Finite volume schemes on Lorentzian mani- USA folds, Comm. Math. Sc. 6 (2008), 1059–1086. M. Ben- [email protected] Artzi, J. Falcovitz, and P.G. LeFloch, Hyperbolic conservation laws on the sphere. A geometry-compatible finite Xianpeng Hu volume scheme, J. Comput. Phys. (2009). J. Giesselman University of Pittsburgh and P.G. LeFloch, Hyperbolic conservation laws on space- [email protected] times with boundary, preprint 2009. P.G. LeFloch, W. Neves, and B. Okutmustur, Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes, MS34 Acta Math. Sinica (2009). Estimates for the Boltzmann Collision Kernel via Analysis of an N-particle Stochastic System Philippe G. LeFloch University of Paris 6 In 1956, Mark Kac proposed a novel approach to the Laboratoire Jacques-Louis Lions study of the Boltzmann equation via the large N limit of lefl[email protected] a stochastic system of N particle undergoing binary col- lisions. In the 1960’s, Henry McKean and his students made many significant contributions to this program, par- MS33 ticularly with regard to the problem of propagation of Stability of White Dwarfs Stars chaos. However, analysis of the rate of equilibriation for this model remained an open problem for many years, We prove existence of rotating star solutions which are and progress on this front was much more recent. Until steady-state solutions of the compressible isentropic Euler- now, this progress has been made for what corresponds to Poisson (EP) equations in 3 spatial dimensions, with pre- ”Maxwellian molecules”. Recent work of myself, Carvalho scribed angular momentum and total mass. This problem and Loss extends this progress to the physically signifi- can be formulated as a variational problem of finding a cant hard-sphere case. This talk will explain this result, minimizer of an energy functional in a broader class of but starting from the beginning, assuming no knowledge functions having less symmetry than those functions con- of Kac’s program. sidered in the classical Auchmuty-Beals paper. We prove the nonlinear dynamical stability of these solutions with Eric Carlen perturbations having the same total mass and symmetry Rutgers University as the rotating star solution. These results apply to white [email protected] dwarf stars when the total mass is below a critical mass.

Tao Luo MS34 Georgetown University On Strong Convergence to Equilibrium for the [email protected] PD09 Abstracts 79

Boltzmann Equation with Soft Potentials introduced which are characterized by a non-local depen-

1 dence in the diffusivity via fractional derivatives. The new The paper concerns L -convergence to equilibrium for weak models are well-posed but still allow for non-trivial dynam- solutions of the spatially homogeneous Boltzmann Equa- ical behavior as desired in their application to performing tion for soft potentials (œB!]œ(B4œ(C!Bœ(Bγ < 0), with image processing tasks. Theoretical and experimental re- and without angular cutoff. We prove the time-averaged L sults will be presented. 1-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to Patrick Guidotti order greater than 2 + |γ|. For the usual L1-convergence University of California at Irvine we prove that the convergence rate can be controlled from [email protected] below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cut- MS35 off on the collision kernel with œB!]œ(B1œ(C!Bœ(Bγ < 0, The Matching and Density Properties of Infinites- there are algebraic upper and lower bounds on the rate of imal Isometries of Convex Surfaces, with Applica- L1-convergence to equilibrium. Our methods of proof are tions to Nonlinear Elasticity of Shells. based on entropy inequalities and moment estimates. This is joint work with E. A. Carlen and Xuguang Lu. We shall discuss the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isome- Maria Carvalho tries, and matching smooth infinitesimal isometries with Rutgers University exact isometric immersions on smooth elliptic surfaces. We [email protected] shall point to our major application, which is deriving the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, un- MS34 der the assumption that the elastic energy of deformations Vanishing Diffusion Limit for the Coagulation scales like hβ with β>2. Namely, contrary to the case Equation of plates, for the given scaling regime, the limiting theory reduces to the linear pure bending. Abstract not available at time of publication. Marta Lewicka Pavel B. Dubovski University of Minnesota Stevens Institute of Technology [email protected] [email protected]

MS35 MS34 Crystal Surface Diffusion: Numerical Simulation Global Small Solutions of a Micro-macro Models and Homogenization for Polymeric Fluids Near Local Equilibrium Models of surface diffusion on crystals mainly invoke: (i) Abstract not available at time of publication. a singular interfacial energy E; (ii) Fick’s law relating flux and the variational derivative of E; and (iii) a conservation Ning Jiang law for the graph. In the first part of my talk, I will present Courant Institute of Mathematical Sciences, NYU progress and puzzles in simulating surface evolution by the [email protected] Finite Element Method in 2+1 dimensions. In the second part, I will address the derivation of (ii) from microscopic MS35 models in the case of a surface composite. The Stefan Problem with Anisotropic Gibbs Thom- Dionisios Margetis son Law: Analysis and Numerical Approximation University of Maryland, College Park [email protected] The Stefan problem with Gibbs-Thomson law models so- lidification phenomena such as dendritic growth. In my talk I will present a global existence result in situations MS36 where the Gibbs-Thomson law is anisotropic. In a second Trapping of Quasigeostrophic Waves in the Equa- part of my talk I will discuss a novel stable finite element torial Shallow Water Equations discretization of the Stefan problem with anisotropic Gibbs Thomson law. Finally I will show several numerical com- In this talk we shall discuss a joint work with Ch. Cheverry, putations including dendritic growth and facetted growth. Th. Paul and L. Saint-Raymond showing that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large Harald Garcke times closed trajectories due to the propagation of Rossby Department of Mathematics waves, while the Poincaré waves are shown to disperse. The University of Regensburg techniques used to prove that result involve semi-classical [email protected] analysis and spectral methods.

Isabelle Gallagher MS35 Universit´eParis-Diderot Some New Nonlocal Diﬀusions with Application to [email protected] Image Processing

After a brief historical perspective on the use of nonlinear diﬀusion in image processing, some novel equations will be 80 PD09 Abstracts

MS36 follows by a standard argument. ‘Stability’ must however Title Not Available at Time of Publication be understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves. Abstract not available at time of publication. Mark D. Groves Theodore Tachim-Medjo Universitätdes Saarlandes Florida International University [email protected] tachimt@fiu.edu MS37 MS36 The Pressure Beneath a Steady Water Wave A One Parameter Family of Expanding Wave So- lutions of the Einstein Equations that Induces an A Stokes wave is a irrotational incompressible periodic 2D Anomalous Acceleraton into the Standard Model steady water wave under the influence of gravity. It is well- of Cosmology known that there is a one-parameter family of such waves. I will prove that the pressure in the fluid strictly increases I introduce a new family of general relativistic expand- both (i) with depth and (ii) horizontally toward the crest ing wave perturbations of the Standard Model of Cosmol- line. Numerical evidence shows that this is not necessarily ogy, and explore the possibility that they can account for true in the presence of vorticity. the Anomalous Acceleration of the galaxies within classical General Relativity, without the need for Dark Energy or Walter Strauss the Cosmological Constant. [Joint work with Joel Smoller.] Brown University [email protected] Blake Temple Department of Mathematics University of California, Davis MS37 [email protected] Existence and Stability of Solitary Water Waves with Weak Surface Tension

MS36 We prove that two-dimensional solitary water waves with Stable Directions for Degenerate Excited States of weak surface tension can be constructed by minimising the Nonlinear SchrödingerEquations energy subject to the constraint of fixed momentum. The proof relies on the concentration-compactness method and We consider the nonlinear Schrödingerquations i∂tψ = the main difficulty is to prove that the infimum of the en- 2 3 H0ψ + λ|ψ| ψ ∈ R × [0, ∞) where H0 =Δ+V and λ = ergy is a strictly sub-additive function of the momentum. ±1. Our potential V, decays sufficiently fast at infinity. This is done by a careful analysis of a certain minimising The linear Hamiltonian H0 has two discrete eigenvalues, sequence. The resulting solutions are periodic wave trains one simple and one of multiplicity three. We show that modulated by exponentially decaying envelopes and the there exist three branches of nonlinear excited states and fact that they are constrained minimisers guarantees some for certain finite codimension subset in the space of initial kind of stability. This is a joint work with Mark Groves data, we construct solutions converging to these excited (Saarbrcken). states in both non-resonant and resonant cases. Erik Wahlen Tuoc Van Phan Lund University University of British Columbia Sweden [email protected] [email protected]

MS37 MS37 Existence and Stability of Fully Localised Three- Stratified and Steady Water Waves dimensional Gravity-capillary Solitary Water Waves We will discuss two-dimensional, periodic, stratified, traveling water waves propagating over an impermeable flat A solitary wave of the type advertised in the title is a crit- bed and with a free surface. The wave’s motion is assumed ical point of the Hamiltonian, which is given in dimension- to be driven by surface tension on the upper boundary less coordinates by and a gravitational force acting on the body of the fluid. Such waves are commonly seen to form when, for example, 1 1 2 2 2 a wind blows over a quiescent body of water. We shall H(η, ξ)= ξG(η)ξ + η + β 1+ηx + ηz − β , R2 2 2 present some new results on the existence of global con- tinua of classical solutions of this type. In the process, subject to the constraint that the impulse we shall also answer some open questions for the constant density case.

I(η, ξ)= ηxξ Samuel Walsh R2 Brown University is ﬁxed. Here η(x, z) is the free-surface elevation, ξ is the Division of Applied Mathematics trace of the velocity potential on the free surface, G(η)is Samuel [email protected] a Dirichlet-Neumann operator and β>1/3 is the Bond number. In this talk I show that there exists a minimiser of H subject to the constraint I =2μ, where 0 <μ MS38 1. The existence of a solitary wave is thus assured, and Eddy Current Problems: Diﬀerent Choices of the since H and I are both conserved quantities its stability PD09 Abstracts 81

Main Unknown in the Conductor and the Insulator clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and The typical setting for an eddy current model distinguishes steady-state properties of crime hotspots. This talk focuses between a bounded conducting region and its complement, on a two-dimensional lattice model for residential burglary, the air region. A fundamental difference of the eddy cur- where each site is characterized by a dynamic attractive- rent model and the full Maxwell’s is the broken symmetry ness variable, and where each criminal is represented as between the electric and the magnetic field in the air re- a random walker. The dynamics of criminals and of the gion. We will discuss the different variational problems attractiveness field are coupled to each other via specific that emerge by retaining either E or H as principal un- biasing and feedback mechanisms. Depending on parame- known, and their finite element approximation. ter choices, the model has several regimes of aggregation, including hotspots of high criminal activity. I also discuss Ana M. Alonso Rodriguez the model and related models in the context of real data Department of Mathematics from the Los Angeles Police Department and Long Beach University of Trento (Italy) Police Department. [email protected] Andrea L. Bertozzi UCLA Department of Mathematics MS38 [email protected] Automatic Coupling and Finite Element Dis- cretization of the Navier-Stokes and Heat Equa- tions MS39 Stochastic Shell Models and Their Statistical Prop- Abstract not available at time of publication. erties

Christine Bernardi We will define some stochastic shell models and their re- UniversitéParis VI lationship with the 3D Navier-Stokes equations. We will Labo J-L Lions prove the well posedness of the models and study their [email protected] longtime behavior such as invariant measures. Moreover, we will introduce a linear version of the Shell model and MS38 study its statistical properties and relationship with the nonlinear model. Finite Elements for the Immersed Boundary Method and Added-mass Effect Hakima Bessaih University of Wyoming We review a finite element version for the Immersed Bound- [email protected] ary Method, designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems. The finite element version offers interesting fea- MS39 tures for the analysis of the problem, the robustness and Existence, Uniqueness and Statistical Theory of flexibility of the numerical scheme. We present a stability the Stochastic Navier-Stokes Equation in Three Di- analysis showing that the time-step and the discretization mensions. parameters are linked by a CFL condition, independently of the ratio between the densities of fluid and solid. We will discuss the existence of unique rough solutions of the Navier-Stokes equation in three dimensions. These so- Lucia Gastaldi lutions are the result of noise that the equation produces Dipartimento di Matematica at high Reynolds numbers. They also give a unique invari- Universitàdi Brescia ant measure that permits the development of Kolmogorovs [email protected] statistical theory of turbulence. Bjorn Birnir MS38 University of California Coupling of Eddy Current and Circuit Problems Department of Mathematics [email protected] Eddy current equations are a reduced model used in elec- tromagnetism when the frequency is small. It can be seen as a coupled problem, where coupling takes place through MS39 the interface between a conductor and its complement. We Renormalization and Rescaling of Elliptic SPDEs present a new formulation of the eddy current equations in with Gaussian Coefficients terms of the electric field in the conductor and the magnetic field in the insulator, and we use it for facing the In many SPDEs with stochastic coefficients, modeling ran- problem of coupling with circuits. domness by spatial white noise may lead to ill-posed problems. We will consider an elliptic problem with spatial Alberto Valli Gaussian coefficients and present a methodology that re- Dipartimento di Matematica solves this issue. It is based on stochastic convolution im- Universita‘ di Trento plementation via generalized Malliavin operators in con- [email protected] juncture with weighted Wiener Chaos expansions that re- solves the ellipticity issue. The talk is based on joint work MS39 with S. Lototsky and X. Wan Mathematical Models for Urban Crime Boris Rozovsky Division of Applied Mathematics Motivated by empirical observations of spatio-temporal adamBrown University Providence, RI 02912 82 PD09 Abstracts

[email protected] lent agreement with the simulation as well as experimental results. We discuss the implications of these results for understanding the migration of general grain boundaries. MS40 Kinetic Approaches in Mesoscale Modeling of Poly- David Srolovitz crystals The Institute for High Performance Computing in Singapore Polycrystalline materials are important in many techno- [email protected] logical applications, yet there are still many challenges they present for mathematical modeling and analysis. One Adele Lim such challenge lies in understanding how statistical distri- Mechanical & Aerospace Engineering butions develop in the process of coarsening of materials Princeton University microstructure and how these distributions in turn relate [email protected] to materials properties. In this talk, we will discuss and compare several recent continuum level models resulting in Mikko Haataja nonlinear evolution equations. Special focus will be placed Princeton Materials Institute on newly discovered features of interface dynamics that Princeton University connect this problem to the theory of nonhomogeneous [email protected] Poisson processes and Boltzmann equations. Numerical and analytical characteristics of the solutions will be discussed and compared against the results produced by ex- MS40 periments and large-scale simulations. Kinetic Theories for Complex Fluids

Maria Emelianenko Kinetic theory is a powerful tool to formulate hydrody- George Mason University namic models for complex fluids and soft matter mate- [email protected] rial systems. It can provide much needed insight into the model at multiple length scales and even time scales. In MS40 this talk, I will present a framework for multicomponent complex fluid mixtures to study polymer blends, polymer- Bounds on Rate of Coarsening in Grain-boundary particulate nanocomposites, and biofilm flows. Spatial in- Networks homogeneity of the complex fluid systems can be modeled Boundaries between grains in polycrystalline materials can effectively through translational diffusion and nonlocal and evolve to reduce the interfacial energy. For example, if long range molecular interactions. Analytical analysis in the energy is isotropic the grain boundary network moves limiting cases and numerical examples will be presented to with velocity equal to the mean curvature (with appropri- show the effectiveness of the kinetic theory modeling. ate conditions at interface junctions). The evolution leads Qi Wang to coarsening of the network: the number of grains de- University of South Carolina creases, while the average grain size grows. I will present [email protected] mathematical results on the rate at which the coarsening occurs. In particular, an upper-bound on coarsening rate which is rigorous under some structural assumptions on MS41 the network. The results will also be presented for a sim- A Variational Model for Denoising Surfaces in plified vertex model. Numerical experiments that support Computer Graphics Applications the conjecture that the upper bound is optimal will also be shown. The talk is based on joint work with Eva Eggeling Geometry denoising, also known as surface fairing, is an and Shlomo Ta’asan. important problem in computer graphics. It is needed whenever 3D objects are digitized and represented as tri- Dejan Slepcev angulated surfaces, typically by sampling points from the Carnegie Mellon University surface of an object using a 3D scanner. Measurement er- [email protected] rors (noise) in coordinates of the sampled points inevitably lead to oscillations in the digitized surface, giving it a rough MS40 and unnatural appearance. The goal of surface fairing is to smooth out the oscillations that are due to noise without Low-Angle Grain Boundary Mobility in a Disloca- also rounding out ”legitimate” singularities such as creases tion Dynamics Framework and corners that are comonly found on man-made objects We investigated the migration of a symmetric tilt, low- (such as machine parts). We will describe a new varia- angle grain boundaries (LAGB) under an applied shear tional model for surface fairing that is the natural geomet- stress in the presence of extrinsic dislocations within a dis- ric analogue of an image processing model due to Rudin, location dynamics model. The results demonstrate that Osher, and Fatemi. Gradient descent for the energy that there is a threshold stress for the LAGB to depin from ex- we propose, which is defined on surfaces, leads to an intrinsic dislocations. Below the threshold stress, the LAGB teresting new geometric motion that removes the noise but remains immobile at zero dislocation climb mobility, while keeps creases and corners intact as it evolves a surface. Its for finite climb mobilities, it migrates at a velocity that is numerical implementation (discretization) using triangu- directly proportional to the applied stress, with a propor- lated surfaces turns out to be quite tricky; to accomplish tionality factor that is a function of misorientation, disloca- this task, we rely on important previous work of geometers tion climb mobility and extrinsic dislocation density. We who were interested in extending certain classical theorems derive analytical expressions for the LAGB mobility and from smooth to polyhedral surfaces. threshold stress for depinning from extrinsic dislocations. Matt Elsey The analytical prediction of the LAGB mobility is in excel- University of Michigan PD09 Abstracts 83

[email protected] currently available in the literature. In this talk, we show that one cannot hope for better results in the general case, Selim Esedoglu showing that free-discontinuity problems are NP hard. We Department of Mathematics nevertheless introduce an iterative thresholding algorithm University of Michigan that is guaranteed to converge to local minimizers of such [email protected] functionals; through this algorithm, we shed light on certain properties of minimizers, including a “gap” property that corroborates the cartoon- like nature of images recon- MS41 structed using free-discontinuity type minimization. Fi- Geodesics in Shape Space via Variational Time Dis- nally, we explain how free-discontinuity problems can be cretization viewed as a relaxation of the sparse recovery problem in compressed sensing. In the talk a variational approach to defining geodesics in the space of implicitly described shapes is introduced. Boris Alexeev The proposed framework is based on the time discretiza- Princeton University tion of a geodesic path as a sequence of pairwise matching [email protected] problems, which is strictly invariant with respect to rigid body motions and ensures a 1-1 property of the induced Massimo Fornasier flow in shape space. For decreasing time step size, the Johann Radon Institute for Computational and Applied proposed model leads to the minimization of the actual Mathematics (RICAM) geodesic length, where the Hessian of the pairwise match- [email protected] ing energy reflects the chosen Riemannian metric on the shape space. Considering shapes as boundary contours, Rachel Ward the proposed shape metric is identical to a physical dissi- Princeton University pation in a viscous fluid model of optimal transportation. [email protected] If the pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, for decreasing time step size one obtains an additional opti- MS41 cal flow term controlling the transport of the shape by the Variational Methods in Hyperspectral Image Pro- underlying motion field. The implementation is based on cessing a level set representation of shapes, which allows topological transitions along the geodesic path. For the spatial Variational and PDE methods have been successfully ap- discretization a finite element approximation is employed plied to a wide range of problems for processing color im- both for the pairwise deformations and for the level set ages, but there has been little work extending these meth- representation. The numerical relaxation of the energy is ods to higher dimensional imagery. A hyperspectral image performed via an efficient multi–scale procedure in space typically contains hundreds of image bands, each corre- and time. Examples for 2D and 3D shapes underline the sponding to a precise wavelength of light. Each pixel in effectiveness and robustness of the proposed approach. the image is a signal which can be used to identify the materials in the image, often with much greater accuracy than Benedikt Wirth a color image could provide. Due to the nature of hyper- Rheinische Friedrich-Wilhelms-UniversitätBonn spectral data, their usage is much different from standard [email protected] color images and this gives rise to a different set of image processing problems. We will present recent work applying Leah Bar variational methods to the problems of spectral unmixing University of Minnesota and image fusion, with an emphasis on the Split Bregman [email protected] iterative method used for fast numerical minimization.

Martin Rumpf Andrea L. Bertozzi Rheinische Friedrich-Wilhelms-Universit¨atBonn UCLA Department of Mathematics [email protected] [email protected]

Guillermo Sapiro Stanley J. Osher University of Minnesota University of California [email protected] Department of Mathematics [email protected]

MS41 Todd Wittman Inverse Free-Discontinuity Problems and Iterative UCLA Thresholding Algorithms [email protected]

The minimization of non-convex functionals having quadratic data-fidelity term and truncated quadratic regu- MS42 larization have been used frequently in inverse problems Layered Superconductors in Nearly Parallel Mag- related to image processing, most notably in the areas netic Fields of inpainting and image segmentation. The so- called Mumford-Shah functional is perhaps the most well-known We consider minimizers or stable solutions of the Lawrence- of this type; such problems in general are known as free- Doniach energy for three-dimensional layered (anisotropic) discontinuity problems. Despite several successful numeri- superconductors in parallel or nearly parallel magnetic cal results, no rigorous results on the existence of minimiz- fields used to model a large class of high-temperature su- ers, let alone on the convergence to such minimizers, are perconducors. We find that the lower critical field becomes 84 PD09 Abstracts

arbitrarily large as the field becomes nearly parallel to the Bernardo Galvao-Sousa layers, and we obtain upper and lower bounds on its value. Department of Mathematics & Statistics We analyze the qualitative behavior of stable minimizers McMaster University in nearly parallel magnetic fields in the regime of small [email protected] Josephson constants, and compare with that of nonlay- ered superconductors given by minimizers of the three- dimensional anisotropic Ginzburg-Landau model. MS42 Vortex Density Models in 3 Dimensions Patricia Bauman Purdue University In this paper we consider the asymptotic behavior of the [email protected] Ginzburg-Landau functional in 3-d, and we derive a limiting functional, in the sense of Gamma-convergence, in Zhenqiu Xie various energy regimes. We derive applications to super- Dept. of Math conductivity and Bose-Einstein condensation: a general ex- Purdue University pression for the first critical field (for superconductivity) [email protected] and the first critical rotational forcing (for Bose-Einstein condensation), and the curvature equation for the vortex lines. MS42 Small Volume Fraction Gamma-limits for a Nonlo- Sisto Baldo cal Cahn-Hilliard Functional University of Verona [email protected] We discuss the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase Robert Jerrard separation of diblock copolymers. In particular, we address University of Toronto the limit in which the volume fraction tends to zero but the Department of Mathematics number of minority phases (called particles) remains O(1). [email protected] Using the language of Γ-convergence, we focus on two levels of this convergence, and derive first and second order Giandomenico Orlandi effective energies, whose energy landscapes are simpler and Department of Computer Science more transparent. These limiting energies are only finite University of Verona on weighted sums of delta functions, corresponding to the [email protected] concentration of mass into ‘point particles’. At the high- est level, the effective energy is entirely local and contains H. M. Soner information about the structure of each particle but no Sabanci University information about their spatial distribution. At the next [email protected] level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions. MS43 Rustum Choksi Sharp Energy Estimates for Fractional Diffusion Department of Mathematics Equations Simon Fraser University In this talk, I will mainly describe recent results in col- [email protected] laboration with Eleonora Cinti (Barcelona and Bologna) concerning elliptic nonlinear equations with fractional dif- MS42 fusion. In a work in collaboration with Y. Sire (Mar- seille), we studied the equation (−Δ)αu = f(u)inRn with Thin Film Limits for Ginzburg-Landau with Strong α ∈ (0, 1). Crucial to our analysis is a result of Caffarelli- Applied Magnetic Fields Silvestre which allows to realize this nonlocal equation as a n+1 degenerate elliptic equation in R+ together with a non- We study thin-film limits of the full three-dimensional n n+1 Ginzburg-Landau model for a superconductor in an ap- linear Neumann boundary condition on R = ∂R+ .We plied magnetic field oriented obliquely to the film surface. characterized the nonlinearities f for which there exists a We obtain Γ−convergence results in several regimes, deter- layer solution —meaning, essentially, a solution increasing mined by the asymptotic ratio between the magnitude of in one direction. We also established several properties of these solutions, such as their uniqueness and decay at in- the parallel applied magnetic field and the thickness of the n 2 film. Depending on the regime, we show that there may finity in R, minimality in R , and 1D symmetry in R .In be a decrease in the density of Cooper pairs. We also show a more recent work in collaboration with E. Cinti, we es- that in the case of variable thickness of the film, its ge- tablish the optimal energy estimates for minimizers, layer solutions, and also other solutions such as saddle-shaped ometry will affect the effective applied magnetic field, thus 3 influencing the position of vortices. solutions. These estimates allow to prove in R the 1D symmetry result of De Giorgi type for the nonlocal equa- Stan Alama tion, previously only known in R2. McMaster University Department of Mathematics and Statistics Xavier Cabre [email protected] ICREA and Universitat Politecnica de Catalunya Barcelona, Spain Lia Bronsard [email protected] McMaster University Department of Mathematics Eleonora Cinti [email protected] Universitat Politecnica de Catalunya PD09 Abstracts 85

Barcelona, Spain to be the Deborah number. The cases of infinite Reynolds [email protected] number and infinite Weissenberg number are in fact identical to the pure elasticities. Moreover, there are many physical and rheological situations can be modeled/formulated MS43 in large/high Weissenberg number problems. From PDE Fractional Laplacian in Conformal Geometry point of view, the system will demonstrate more properties of hyperbolic wave equation in cases of large/infinite Weis- The fractional Laplacian a non-local, fractional order oper- senberg number. This present major challenges and diffi- ator. We explore the relations with conformal geometry, in culties in both analysis and numerical simulations. Most particular, scattering theory of conformally compact Ein- well-established work in mathematics have been in the area stein manifolds. We consider a fractional order Paneitz- of fluid dominate region (with small Weissenberg number). type operator and its associated curvature, and try to find High Weissenberg number problems are also present in their geometrical implications. wide range of applications, such as the benchmark problems of flow through contraction, flow past a cylinder, Maria Del Mar Gonzalez micromixing, as well as in MHD, where in many cases Universitat Politecnica de Catalunya (such as the modeling of the Sun), the magnetic diffu- [email protected] sion can be mostly neglected. The problem is closely related/conjectured to the elastic turbulence phenomena. In MS43 the situations with large/infite Weissenberg number, the total system is still dissipative (due to the fluid viscos- Regularity Results for Elliptic Nonlocal Equations ity). Mathematically, this is a partial dissipative system, Abstract not available at time of publication. where most of the established analysis and numerical tools fail. In fact, even the global existence of the energy solu- Luis Silvestre tions (Leray’s solution) is still an open problem. Recently, University of Chicago we succeed to prove the global existence of the classical [email protected] solution with small initial data. The proof relied heavily on the energetic variational structure of the system, in particular, the coupling between the induced elastic stress MS43 and the kinematic transport of the elastic variable. From Non-local Phase Transitions the mathematical point of view, the system can be derived from the energetic variational approach framework. En- I will describe some symmetry results for solutions of equa- ergetically, there is the competition between the kinetic tions involving the fractional laplacian. As an important energy and the elastic energy. The competition is through step, I will explain how to construct the one-dimensional the kinematic transport, which (after the various varia- profile and study its properties such as uniqueness, asymp- tional procedures) gives the induced elastic stresses, such totic behaviour. as the Maxwell stress in MHD and the elastic stresses in the viscoelatic fluids. All analysis and the numerics need to Yannick Sire preserve these structures (the kinematic transport, as well Universite Paul Cezanne as the variational structure), especially when the solutions Marseille, France involve singularities or pattern/structures. [email protected] Chun Liu Dept of Math MS44 IMA/Penn St University Strong Solutions to Viscoelastic Fluids and Liquid [email protected] Crystals

The existence and uniqueness of the strong solutions with MS44 small data to the three-dimensional viscoelastic ﬂuids and Hyperbolic Balance Laws liquid crystals will be discussed. Hyperbolic balance laws arise naturally in continuum Xianpeng Hu physics, elasticity and other applied ﬁelds. In this talk University of Pittsburgh recent developments in hyperbolic balance laws are pre- [email protected] sented. Sharp decay estimates are derived for the positive waves in an entropy weak solution to a class of hyperbolic Dehua Wang balance laws. The result is obtained by introducing a par- University of Pittsburgh tial ordering within the family of Radon measures and a USA comparison with a solution of Burgers’s equation with im- [email protected] pulsive sources. An application and further consequences will be discussed.

MS44 Konstantina Trivisa Title Not Available at Time of Publication University of Maryland [email protected] The complex fluids posses the properties of both the fluids and solids. The relative strength of the each effects is re- Cleopatra Christoforou flected through the ”relative” magnitude of the Reynolds University of Houston number and the Weissenberg number. While the Reynolds [email protected] number represent the ”viscous” dissipation of the fluids, the Weissenberg number represents the elastic dissipation (relaxation). Sometime, the constant is identified/confused 86 PD09 Abstracts

MS44 ler equations. For the Navier-Stokes equations, the sup- Nonlinear Stability of Non-isentropic Conical norm of solutions can not be controlled since there exist Flows Past a Perturbed Cone with Large Angle no natural invariant regions for the equations with the physical viscosity term and, furthermore, convex entropy- In [Wen-Ching Lien, Tai-ping Liu, Nonlinear Stability of a entropy flux pairs may not produce a signed entropy dis- Self-Similar 3-Dimensional Gas Flow], the authors proved sipation measures. To overcome these difficulties, we first that the 3-d supersonic gas flow past an infinite cone with develop uniform energy-type estimates with respect to the very small vertex angle was globally stable. We are in- viscosity coefficient for the solutions to the Navier-Stokes terested in the case when the vertex angle was large so equations and establish the existence of measure-valued so- that the strong shock issuing from the tip of the cone is no lutions to the isentropic Euler equations generated by the longer negligible, we analyze, in particular the wave inter- Navier-Stokes equations. Based on the uniform energy- action between the strong shock and local weak waves, and type estimates and special features of the isentropic Euler constructed a globally decreasing Glimm-type functional to equations, we establish that the entropy dissipation mea- finally establish the stability of the strong shock structure. sures of the Euler equations for the solutions to the Navier- Stokes equations for weak entropy-entropy flux pairs are Dianwen Zhu contained in a compact set in H−1 which leads to the div- University of California-Davis curl commutator relation for the corresponding measure- [email protected] valued solutions, determined by the vanishing viscosity sequence of solutions. A careful characterization of the un- Gui-Qiang G. Chen bounded support of the measure-valued solutions yields the Department of Mathematics, Northwestern University reduction of the measure-valued solutions to a Delta mass, USA which lead to the convergence of the finite-energy solutions [email protected] of the Navier-Stokes equations to a finite-energy entropy solution to the isentropic Euler equations. Yongqian Zhang Fudan University Mikhail Perepelitsa [email protected] Vanderbilt University [email protected]

MS45 Gui Qiang Chen Sharp-Interface Models for Fluidic Particles Northwestern University [email protected] Sharp-interface models for two-phase flows allow different levels of physico-chemical interface properties, starting from a purely geometric dividing interface to the case when MS45 the interface is a phase for itself with surface viscosity and On the Rayleigh-Taylor Instability for the Two- variable surface tension - the so-called Boussinesq-Scriven phase Navier-Stokes Equations surface fluid. For the different levels of interfacial properties, the corresponding models together with main an- We consider the motion of two superposed immiscible, vis- alytical results are outlined. Some results of VOF-based cous, incompressible, capillary fluids that are separated by numerical approaches are also included. a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, Dieter Bothe we prove local well-posedness of the problem. In particu- Technical University Darmstadt (Germany) lar, we obtain well-posedness for the case where the heavy [email protected] fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously, MS45 and we give a rigorous proof of the Rayleigh-Taylor insta- Fingering Patterns for the Muskat problem bility.

Of concern is the study of a free boundary problem de- Gieri Simonett scribing the flow of a two-phase flow in a Hele-Shaw cell. Vanderbilt University The less dense fluid is located in the bottom part of the [email protected] cell. The free boundary separating the fluids moves under the influence of surface tension and gravity. The existence Jan Pruess of nontrivial stationary fingering patterns is proved for a Martin-Luther University Halle-Wittenberg null sequence of surface tension coefficients. These fingers [email protected] appear as global bifurcation branches from the trivial flat stationary solution MS46 Joachim Escher Steady-state Navier-Stokes Flows with Rough Ex- Lebiniz Universitat Hannover (Germany) ternal Forces [email protected] We address the existence, the asymptotic behavior and sta- p p,∞ 3 ≤∞ bility in L and L , 2

lutions can be large. We also give non-existence results wave speed 1

Gabriela Planas C. Eugene Wayne UNICAMP Boston University Campinas, Brazil Department of Mathematics [email protected] [email protected]

MS47 MS48 Stability of Co-propagating N-solitons of FPU Lat- How to Best Approximate a Fully Viscous Solution tices in the KdV Limit with Inviscid Approximations in Subregions

Solitary waves of the FPU lattices cannot be characterized In many applications, viscous terms are not important in as a critical point of conservation due to the lack of in- large parts of the computational domain, and one would ﬁnitesimal invariance in the spatial variable. I prove an ex- like to solve a cheaper inviscid problem there instead. We ponential stability property of the linearized FPU equation show in this talk that for the model problem of advection around an N-soliton in a weighted space which is biased reaction diﬀusion equations, there are coupling conditions in the direction of motion and prove stability of N-soliton based on the factorization of the operator, which lead to solutions in the energy space. approximate solutions much closer to the fully viscous one than with classical coupling conditions from the literature. Tetsu Mizumachi Faculty of Mathematics Martin J. Gander Kyushu University, Japan University of Geneva [email protected] [email protected]

MS47 MS48 On Spectral Stability for Small Solitary Water Weak Coupling Between Scales and Models: A Fre- Waves quency Oriented Approach for Transferring Infor- mation in Heterogeneous Models I will discuss a draft proof (with Shu-Ming Sun) of spectral stability of small solitary waves for the 2D Euler equations The coupling of models on different scales (e.g. molecu- for water of finite depth without surface tension. lar dynamics/continuum mechanics) is a source for hetero- geneities in (discrete) models. The information transfer Robert L. Pego between the models requires some ”filtering’ of the passed Carnegie Mellon University information: only the information usable in both interfac- Department of Mathematical Sciences ing models should be transferred, thus avoiding spurious [email protected] effects. In this talk, we present a new frequency oriented coupling concept for concurrent multiscale simulations and will discuss its relation to existing coupling concepts for fi- MS47 nite element discretizations. Nonlinear Instability of Periodic Travelling Waves for a 1D Boussinesq System Rolf Krause Universita’ della Svizzera Italiana We build periodic travelling wave solutions of the non- [email protected] linear one-dimensional Boussinesq system qt = rx, rt = −1 2 2 B (Aqx − 2rqqx − 2q qt), where A = I − a∂x and B = 2 I − b∂x. We discuss orbital instability of these solutions MS49 and we provide numerical evidence on a type of instabil- Crossover in Coarsening Rates for the Monopole ity arising when perturbing with small amplitude distur- Approximation of the Mullins-Sekerka Model with bances. Orbital instability of periodic travelling waves for 88 PD09 Abstracts

Kinetic Drag MS49 Kinetic Descriptions of Evolution of Crystalline The Mullins-Sekerka sharp-interface model for phase Surfaces transitions interpolates between attachment-limited and diffusion-limited kinetics if kinetic drag is included in the The evolution of crystalline surfaces is often described Gibbs-Thomson interface condition. Heuristics suggest via particle-type equations for line defects (steps) at the that the typical length scale of patterns may exhibit a nanoscale. In this talk, I will focus on challenges encoun- 1/2 crossover in coarsening rate from l(t) ∼ t at short times tered and recent progress made in extracting mesoscale and to l(t) ∼ t1/3 at long times. We establish rigorous, univer- macroscale descriptions of surface evolution by methods of sal one-sided bounds on energy decay that partially justify kinetic theory. I will discuss the role of step correlation this understanding in the monopole approximation and in functions and their hierarchies in stochastic settings. The the associated LSW mean-field model. Numerical simula- kinetic approach reveals useful analogies with other physi- tions for the LSW model illustrate the crossover behavior. cal systems such as non-uniform liquids. Dionisios Margetis Shibin Dai University of Maryland, College Park Worcester Polytechnic Institute [email protected] [email protected] MS50 Barbara Niethammer Mathematical Institute Optimal Estimates for Semilinear Elliptic Equa- University of Oxford tions [email protected] In this work we study the distribution function of the solutions to the Dirichlet problem Robert L. Pego Carnegie Mellon University Department of Mathematical Sciences −Δpu = f(u)inΩ [email protected] u>0inΩ (2) u =0 on∂Ω, MS49 Aspects of Coarsening in Microstructural Evolu- where Ω is an open bounded set of Rn and f is a non- tion negative Lipschitz function with a suitable growth. Our main concern is to compare the distribution function of a Coarsening during microstructural evolution has many fea- solution associated to Ω with the maximal one associated tures which appear to be driven by different mechanisms. to the ball B with same measure, obtaining results similar Some of these are transport related while others are more to some Talenti estimates. We get also some estimates for easily described as entropic. What sort of models can we general domains. We apply these results to estimate the derive to present the array of phenomena and what math- maximum of an eigenfunction by its L2 norm and by the ematical questions do they pose? corresponding eigenvalue. David Kinderlehrer Leonardo Bonorino Carnegie Mellon University Departamento de Matematica Pura e Aplicada, [email protected] Federal University of Rio Grande do Sul, Brazil [email protected] MS49 A Continuum Model for Selectivity Mechanism in MS50 Ion Channels: An Energetic Variational Approach Remarks on Aleksandrov-Bakelman-Pucci Maxi- mum Principle, Weak Harnack Inequality and The interactions of ions flowing through biological systems Their Consequences has been a central topic in biology for more than 100 years. Flows of ions produce signaling in the nervous system, ini- Aleksandrov-Bakelman-Pucci (ABP) maximum principle tiation of contraction in muscle, coordinating the pump- and Harnack inequality are classical tools in the theory ing of the heart and regulating the flow of water through of elliptic PDE and in the last two decades these topics kidney and intestine. Ion concentrations inside cells are have been revisited for fully nonlinear uniformly elliptic controlled by ion channels through the lipid membrane. In equations from the point of view of viscosity solutions. We this talk, I will propose a continuum model that is derived will discuss old and new results and show when the ABP from the energetic variational approach which include the maximum principle is true for equations with superlinear coupling between the electrostatic forces, the hydrodynam- growth in the gradient (in which case it has been known ics, diffusion and crowding (due to the finite size effects). to fail in general). Moreover we will present a small recent The model provides some basic understanding of one of the improvement in weak Harnack inequality for fully nonlin- most important properties of proteins, the ion selectivity. ear equations. As a consequence we will show how these This is a joint work with Yunkyong Hyon (IMA), Taichia techniques allow to obtain new results about solvability Lin (National Taiwan University) and Robert Eisenberg of nonlinear PDE, in particular of certain Pucci extremal (Rush Medical School). equations. Chun Liu Andrzej J. Swiech Penn. State University Georgia Tech University Park, PA Department of Mathematics [email protected] [email protected] PD09 Abstracts 89

MS50 University of Michigan Estimates the Solutions of Free Boundary Prob- [email protected] lems

This is a joint work with Diego Moreira, where we will MS51 estimate the Hausdorﬀ measure of the free boundary of Existence, Stability, and Regularity of Global So- general fully nonlinear elliptic equations. lutions to Shock Reﬂection Problems, Part II

Lihe Wang In these two talks, we will discuss our recent results on Iowa University shock reflection problems for the potential flow equation. Department of Mathematics We will start with discussion of shock reflection phenom- [email protected] ena. Then we will describe the results on the existence and stability of global solutions to regular shock reflection for all wedge angles up to the sonic angle, as well as on MS50 the regularity near the sonic arc (joint work with M. Bae). Some Results About the Uniqueness of Infinity The approach is to reduce the shock reflection problem to Ground States a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the It was proved by Juutinen, Lindqvist and Manfredi that sonic arc). We will discuss techniques to handle such free asymptotic limits of principle eigenfunctions of p-Laplacian boundary problems and degenerate elliptic equations. Fur- operators are viscosity solutions of a free boundary prob- thermore, some related multidimensional shock problems lem, i.e, the so called ?nfinity ground states? A very in- and further developments in this direction will be also ad- teresting question is whether the infinity ground state is dressed. simple. In this talk, we will discuss some results related to this uniqueness issue. Gui-Qiang G. Chen Department of Mathematics, Northwestern University Yifeng Yu USA University of California [email protected] Irvine [email protected] Mikhail Feldman University of Wisconsin-Madison MS51 [email protected] Existence, Stability, and Regularity of Global So- lutions to Shock Reflection Problems, Part I MS51 In these two talks, we will discuss our recent results on Entropies for Hyperbolic Systems with Prescribed shock reflection problems for the potential flow equation. Eigenfields We will start with discussion of shock reflection phenom- Consider the following problem: given n linearly indepen- ena. Then we will describe the results on the existence dent vector fields on an open subset of Rn - when is it and stability of global solutions to regular shock reflection possible to find a hyperbolic system of n conservation laws for all wedge angles up to the sonic angle, as well as on ut + f(u)x = 0 such that the flux f has eigenfields equal the regularity near the sonic arc (joint work with M. Bae). to the given vector fields? This problem has been analyzed The approach is to reduce the shock reflection problem to by the authors in a previous work. We now add the con- a free boundary problem for a nonlinear elliptic equation, straint that the resulting system should be equipped with with ellipticity degenerate near a part of the boundary (the a non-trivial entropy. We provide a self-contained formu- sonic arc). We will discuss techniques to handle such free lation of this problem (i.e. entirely in terms of the given boundary problems and degenerate elliptic equations. Fur- vector fields) where the unknowns are the eigenvalues of thermore, some related multidimensional shock problems the Hessian of the entropy. The resulting overdetermined and further developments in this direction will be also ad- system for these eigenvalues will be derived and we discuss dressed. its solvability. Various examples will be considered. Gui-Qiang G. Chen Kristian Jenssen Department of Mathematics, Northwestern University Penn State University USA [email protected] [email protected] Irina Kogan Mikhail Feldman North Carolina State University University of Wisconsin-Madison [email protected] [email protected]

MS52 MS51 Blow-up in Multidimensional Aggregation Equa- Non-uniqueness of Inviscid Flows tions with Mildly Singular Interaction Kernels

We report on several recent results on non-uniqueness of I will discuss recent results for the multidimensional aggre- the Cauchy problem for the incompressible and compress- gation equation ut −∇·(u∇K ∗u) = 0 in which the radially ible Euler equations. The consequences for theory and nu- symmetric attractive interaction kernel has a mild singu- merics, and the relation to physically observed phenomena, larity at the origin (Lipschitz or better). Under mild mono- will be discussed in detail. tonicity assumptions on the kernel K, the Osgood condition Volker W. Elling for well-posedness of the ODE characteristics is a necessary 90 PD09 Abstracts

and sufficient condition for global in time well-posedness of work with J.J. Manfredi and J.D. Rossi. the PDE with compactly supported bounded nonnegative initial data in all dimensions. I also discuss the case of data Juan J. Manfredi in Lp for which the same condition also determines local Department of Mathematics vs. global-in time well-posedness for p larger than a criti- University of Pittsburgh cal value, depending on the potential. I also present some [email protected] numerical computations illustrating that the Lp problem is important for finite time blowup and some examples that Mikko Parviainen illustrate that the equation is not locally well-posed in Lp University of Pittsburgh for subcritical p. This is joint work with Thomas Laurent, mikko.parviainen@tkk.fi Jose Carrillo, Jesus Rosado, and Yanghong Huang. Julio D. Rossi Andrea L. Bertozzi Departamento de Matematica UCLA Department of Mathematics Universidad de Buenos Aires [email protected] [email protected]

MS52 MS53 A Harnack-type Estimate for Subcritical Singular Classical Solutions and Stability Results for the Parabolic Equations Boltzmann Equation

For non-negative solutions of quasilinear singular parabolic This talk focuses on the study of existence and unique- equations of p-laplacian type a novel intrinsic Harnack es- ness of distributional and classical solutions to the Cauchy timate is established and its relation to Holder continuity Boltzmann problem for the soft potential case assuming is traced. Sn−1 integrability of the angular part of the collision kernel Emmanuele DiBenedetto (Grad cut-off assumption). We study the propagation of Department of Mathematics regularity using a recent estimate for the positive collision Vanderbilt University operator given by E. Carneiro and the authors, that per- [email protected] mits to study such propagation without additional conditions on the collision kernel. Finally, an Lp-stability result (with 1 ≤ p ≤∞) is presented assuming the aforemen- Ugo P. Gianazza tioned condition. Department of Mathematics University of Pavia Ricardo J. Alonso [email protected] UCLA, IPAM Rice University Vincenzo Vespri [email protected] Department of Mathematics University of Florence Irene M. Gamba [email protected]fi.it Department of Mathematics and ICES University of Texas [email protected] MS52 Symmetry Properties of Positive Solutions to Parabolic Problems MS53 The Dynamics of Viral Entry Symmetry properties of non-negative bounded solutions of fully nonlinear parabolic equations on bounded reflec- Successful viral infection of a healthy cell requires com- tionally symmetric domains with Dirichlet boundary con- plex host-pathogen interactions. In this talk we focus on ditions will be studied. A sufficient conditions on the equa- the dynamics specific to the HIV virus. We model viral tion and domain, which guarantee asymptotic symmetry of entry as a stochastic engagement of receptors and corecep- solutions will be given. The basic techniques include the tors on the cell surface. We also consider the transport of method of moving hyperplanes, maximum principles and virus material to the cell nucleus by coupling microtubular Harnack inequalities. motion to the concurrent biochemical transformations that render the viral material competent for nuclear entry. We Juraj Foldes discuss both mathematical and biological consequences of University of Minnesota our model, such as the formulation of an effective integrod- [email protected] ifferential boundary condition embodying a memory kernel and optimal timing in maximizing viral probabilities.

MS52 Maria Rita D’Orsogna Mean Value Theorems for Nonlinear PDEs California State University, Northridge [email protected] In this talk we present a characterization of viscosity solutions to a class of nonlinear parabolic equations in terms of an asymptotic mean value property. The correspond- MS53 ing elliptic problem was considered in [J.J. Manfredi, M. Kinetic Models for Polymer with Inertial Eﬀects Parviainen, and J.D. Rossi. An asymptotic mean value characterization for p-harmonic functions]. This is a joint Novel kinetic models for both Dumbbell-like and rigid-rod like polymers are derived, based on the probability distribution function f(t, x, n, n) for a polymer molecule posi- PD09 Abstracts 91

tioned at x to be oriented along direction n while embedded tion Problems in a n environment created by inertial effects. It is shown that the probability distribution function of the extended We study the vanishing viscosity limit for certain Taylor- model, when converging, will lead to well accepted kinetic Couette flows in pipes and channels. We establish conver- models when inertial effects are ignored such as the Doi gence of the Navier-Stokes solution to the corresponding models for rod like polymers, and the Finitely Extensible Euler solution as viscosity vanishes in various norms. In the Non-linear Elastic (FENE) models for Dumbbell like poly- process we obtain a detailed analysis of the small-diffusion mers. limit for a heat equation with drift, using a parametrix construction. Hailiang Liu Iowa State University Anna L. Mazzucato [email protected] Department of Mathematics Pennsylvania State University [email protected] MS53 Transport In Narrow Geometries Under Strong Milton C. Lopes Confinement IMECC, Unicamp, Brazil [email protected] Kinetic transport in thin plates or tubes, involving scattering of particles with a background, is modeled by sub- Helena J Nussenzveig Lopes band type macroscopic equations for thedensity of parti- IMECC, UniCamp, Brazil cles. The result is a diffusion equation with the projection [email protected] of the (asymptotically conserved) energy tensor on the con- fined direction(s) as an additional free variable, on large time scales. Classical transport of ions through protein Michael E. Taylor channels and quantum transport of electrons in thin (SOI- UNC-Chapel Hill type) semiconductor devices are discussed as examples of [email protected] the application of this methodology. (Joint work with N. BenAbdallah and C. Heitzinger) MS54 Christian Ringhofer Vortex Sheets in Domains with Boundary Arizona State University [email protected] Vortex sheets in 2D correspond to flows for which there is a discontinuity in the tangential component of velocity along a curve, while the normal component is continuous. MS54 In 1991 J.-M. Delort established the existence of a weak On the Uniqueness of Weak Solutions to the 2D solution for the incompressible 2D Euler equations with Euler Equations vortex sheet initial data, under the hypothesis that the vorticity be of distinguished sign. Delort considered three flow An interesting class of initial data for solutions to the Euler contexts: full-space flow, flow on a two-dimensional com- equations in 2D are those for which uniqueness of solutions pact manifold, and flow in a smooth, bounded domain. In in the natural energy space exist, but for which the flow this talk we will re-examine Delort’s analysis for the case of map can be highly irregular, indicating that perhaps such bounded domain flows; while Delort considered the velocity classes of initial data are near minimal for insuring unique- formulation of the incompressible Euler equations we will ness. I will discuss three such classes, two due to Yudovich discuss the vorticity formulation. We will discuss the pos- and one to Vishik, and some properties of their flow maps. sibility that mass of vorticity accumulates on the boundary of the domain and we study the dynamics of this vorticity defect. We extend Delort’s analysis, as well as ours, to James Kelliher smooth, possibly unbounded, domains with boundary. We UC Riverside note that our results are consistent with the expected be- [email protected] havior of the vanishing visocity limit of the incompressible Navier-Stokes equations. MS54 Dragos Iftimie On Partial Regularity for Solutions of the Navier- Universite de Lyon I Stokes System [email protected]

A classical result of Caﬀarelli, Kohn, and Nirenberg states Milton C. Lopes Filho that the one dimensional Hausdorﬀ measure of singularities IMECC-UNICAMP of a suitable weak solution of the Navier-Stokes system is [email protected] zero. We present a short proof of the partial regularity result which allows the force to belong to a singular Morrey Helena J. Nussenzveig Lopes space. State University of Campinas, Brazil Igor Kukavica [email protected] USC [email protected] Franck Sueur Universite de Paris VI [email protected] MS54 Vanishing Viscosity Limits and Singular Perturba- 92 PD09 Abstracts

MS55 from Interior Measurements Faber-Krahn Type Inequalities in Inverse Scatter- ing Theory This talk concerns a non-smooth optimization problem occurring in conductivity imaging. The magnitude of the We first consider the scattering of time harmonic plane current density field while maintaining a prescribed volt- waves by a perfectly conducting infinite cylinder of cross age is assumed known. Local stability of the minimizer is section D. We observe that the Dirichlet eigenvalues for the shown to hold nearby an admissible interior data. Laplacian in D can be determined from the far field pattern of the scattered wave and hence from the Faber-Krahn Alexandru Tamasan inequality we can obtain a lower bound for the area of D. University of Central Florida We then consider the corresponding problem for a dielec- [email protected] tric cylinder. Here we observe that a relatively new type of spectra called transmission eigenvalues can be determined MS56 from the far field pattern of the scattered wave and show that transmission eigenvalues exist and form a discrete set. Traveling Wave Solutions to Reaction-diffusion We then obtain a Faber-Krahn type inequality for trans- Systems mission eigenvalues which,if D is known, provide a lower We will present some recent results on traveling waves in bound on the index of refraction n(x). Of special inter- nonlinear reaction-diffusion equations and systems. We est is the case when cavities may be present,i.e. regions will pay particular attention to the generalized Fisher equa- where n(x)=1.We consider both isotropic and anisotropic tion, which has applications in biology and chemistry. materials. Zhaosheng Feng Fioralba Cakoni University of Texas-Pan American University of Delaware [email protected] [email protected]

MS56 MS55 Coupled Chemotaxis-Fluid Models Transformation Optics and Active Acoustic Cloak- ing Devices We consider coupled chemotaxis-fluid models aimed to describe swimming bacteria, which show bio-convective flow In this talk, we shall describe our recent progress on the patterns on length scales much larger than the bacteria use of transformation optics to the design of active cloaking size. This behaviour can be modelled by a system consist- devices. The cloaking device makes the interior medium to- ing of the chemotaxis equations coupled with the viscous gether with a source/sink in the cloaked region invisible to incompressible fluid equations through transport and ex- boundary/exterior wave detection. Finite energy solutions ternal forcing. We give global-in-time existence results for for these acoustic cloaking devices are studied in weighted weak potentials, for small initial concentration of the chem- Sobolev spaces with singular weights. We analyze the be- ical and for nonlinear diffussion of the cells. havior of the finite energy solution in the cloaking medium, in the cloaked region, and at the interface between the two Alexander Lorz regions. A novel finite element discretization for the corre- Department of Applied Mathematics and Theoretical sponding singular PDEs shall also be discussed. Physics University of Cambridge Hongyu Liu [email protected] University of Washington Department of Mathematics [email protected] Peter Markowich, Klemens Fellner University of Cambridge [email protected], MS55 [email protected] Inverse Obstacle Problems with Nonlinear Terms in the Underlying Equation Renjun Duan University of Linz This talk will be concerned with determining hidden in- [email protected] clusions or obstacles from (Cauchy) data measurements on an accessible, external boundary. The underlying equation Marco di Francesco will be of second order elliptic type but may contain non- L’Aquila, Italy linear terms in either the equation itself or the boundary [email protected] condition. Further, these terms may themselves be unknown. The aim is to determine the minimal amount of data required in order to obtain uniqueness and to examine MS56 some reconstruction algorithms. Stationary Patterns in Prey-predator Models with Stage Structure William Rundell Texas A&M University We look at a diffusive prey-predator model with stage Department of Mathematics structure for the predator, which is in the form of a [email protected] reaction-diffusion system with the homogeneous Neumann boundary condition, with the presence of a cross-diffusion term. We show that stationary patterns, i.e., nonconstant MS55 positive stationary solutions, emerge for this system as a On a Minimizing Problem in Conductivity Imaging PD09 Abstracts 93

result of the presence of cross diffusion. range and lengths in micron range. Double-walled CNTs are two concentrically nested CNTs with different char- Peter Y. Pang acteristics. The individual tubes are not bonded to- National University of Singapore gether; they only interact through non-bonded Van-der- [email protected] Waals forces. They remain free to slide and rotate independently with small resistive forces. It seems natural that the system of a nanometer size should be described MS56 as a quantum system and studied by using an appropri- Traveling Fronts to Auto-Catalytic System with ate Schrodinger operator. In fact, this method is the core Two Orders of Reactions of molecular simulations. However, at present quantum- mechanical methods have been used for analysis of single- In this talk, we study the existence and non-existence of walled CNT and no attempts have been made to deal with travelling wave to parabolic system of the form double-walled CNT. In our research, we use an alterna- at = axx − af(b); bt = Dbxx + af(b), tive approach based on traditional continuum mechanics models such as beams, shells, and membranes. So, CNT with f a degenerate nonlinearity. Using a complete new are viewed as a continuous vibrating structures and mod- approach, we are able to derive sharp bounds on the min- eled by PDEs. The main object of our analysis is a math- imum speed with interesting dependance relation on D, ematical model of a double-walled CNT. We consider a when f(b)=bm + kbn, where m, n are two different posi- system of two beams interacting through distributed Van- tive exponents and k a positive constant. der-Waals forces. There are the most used beam models: Euler-Bernoulli beam and Timoshenko beam models. The Yuanwei Qi second model is more complete since it takes into account University of Central Florida not only transverse displacement of a vibrating beam but [email protected] also torsional angle between the originally parallel cross- sections. At the beginning, double-walled CNTs have been modeled by two coupled E/B beams, then by E/B and T- MS57 beams. However, an agreement between numerical data Control and Stabilization of the Multilayer Rao- provided by mathematical model and experimental data Nakra Beam System was not satisfactory. The latest set of models is given by two T-beams coupled through distributed Van-der-Waals We consider the multi-layer Rao-Nakra beam system forces. It is precisely our model. The model is governed by with combinations of passive internal damping and active a system of four hyperbolic PDEs. It is really complicated boundary damping. With passive internal damping alone, and probably for this reason the results available in the we can prove that the system is exponentially stable except literature are all computational. Our goal is to give purely in two exceptional cases: when all the wave speeds are the theoretical analysis. In particular, we obtained explicit same, or when all wave speeds except one are the same and asymptotic formulas for the vibrational frequencies. Here the common wave speeds match one of a sequence of criti- we would like to add the following. In macroscopic vibrat- cal numbers. In either case, a single feedback on one layer ing systems high frequency vibrations are heavily damped is sufficient to achieve exponential stability. Under some and their amplitudes decrease in time very rapidly. That is restrictions on parameters, we also show that boundary why high frequency modes are not of primary interest for feedback alone is sufficient for exponential stability. engineers. But for nanostructures, high frequency modes Scott Hansen are almost not damped. So, our asymptotic analysis of Iowa State University high frequency vibrations (in terahertz frequency range) Department of Mathematics could be very important. [email protected] Marianna Shubov University of New Hampshire MS57 Department of Mathematics and Statistics [email protected] Swimming Models and Their Controllability

We consider a simplified model of an abstract object, MS57 formed by narrow interconnected rectangles, which propels itself in a fishlike fashion in a fluid. The results include an Uniform Null Controllability of a Parabolic Equa- asymptotic formula for “small” motions of this obect which tion with Rapidly Oscillating Periodic Coefficients is then applied to the study of its global controllability We consider a parabolic equation with fast oscillating pe- properties. riodic coefficients, and an interior control in a bounded do- Alexander Y. Khapalov main. First, we prove sharp convergence estimates depend- Washington State University ing explicitly on the initial data for the corresponding un- [email protected] controlled equation; these estimates are new in a bounded domain, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove MS57 that the original equation is uniformly null controllable, Asymptotic and Spectral Properties of Double- provided a carefully chosen extra vanishing interior control walled Carbon Nanotube Model is added to that equation. This uniform null controllability result is the first in the multidimensional setting for We present the results on rigorous mathematical analysis parabolic equations with oscillating coefficients. Finally, of a model of a double-walled carbon nanotube (CNT). As we prove that the sequence of null controls converges to is well-known, nanoscience is an extremely fast-developing the optimal null control of the homogenized equation when area with numerous experimental and numerical results. CNT are elongated tubes with diameters in nanometer 94 PD09 Abstracts

the period tends to zero. novel features of the coupled problem, not observed in the thin structure case. The derivation of the model and its Louis Tebou novel features will be discussed. This is a joint work with Department of Mathematics Andro Mikelic of the University of Lyon1, and Giovanna Florida International University Guidoboni of the University of Houston. teboul@fiu.edu Suncica Canic Department of Mathematics MS57 University of Houston Stabilization of Structure-Acoustics Interactions [email protected] for a Reissner-Mindlin Plate by Localized Nonlin- ear Boundary Feedbacks MS58 This work addresses a system comprised of a wave equa- Boundary Layers for the Primitive Equations tion coupled with a Reissner-Mindlin plate. The acoustic damping is restricted to the flexible boundary and only a We present in this lecture some convergence results related portion of the rigid wall; the plate is likewise damped on a to the Linearized Primitives Equations (LPEs) as the vis- segment of its boundary. The nonlinearities in both feed- cosities go to zero. The (full nonlinear) Primitives Equa- backs may include sub- and super-linear growth at infinity. tions read: ⎧ We establish stabilization estimates and energy decay rates 2 for this model. ⎪ ∂v ·∇ ∂v × 1 ∇− − ∂ v ⎪ +(v )v + w + fk v + p μvΔv νv 2 = F ⎪ ∂t ∂z ρ0 ∂z ⎪ Daniel Toundykov ⎪ ⎪ ∂p − University of Nebraska-Lincoln ⎪ = ρg, [email protected] ⎨ ∂z (PEs) ∂w ⎪ ∇v + =0, George Avalos ⎪ ∂z ⎪ University of Nebraska-Lincoln ⎪ 2 ⎪ ∂T ∂T ∂ T Department of Mathematics and Statistics ⎪ ·∇ − − ⎪ +(v )T + w μTΔT νT 2 = QT, [email protected] ⎪ ∂t ∂z ∂z ⎩ ρ = ρ0(1 − α(T − T0)).

MS58 One of our aims, among others, is to give the limit solution NonLinear Possio Integral Equation and AeroElas- associated with the Linearized system of (PEs) that we ob- tic Flutter Limit Cycle Oscillation tain by dropping the nonlinear terms. However, a difficulty for the limit LPEs system (that is we set the viscosities to With a zero thickness wing structure model (Goland) that be equal to zero) lies in the fact that no set of local bound- is linear but aerodynamics that is nolinear- inviscid isen- ary conditions ensures its well-posedness. Several choices tropic flow characterised by the Euler full potential equa- of nonlocal boundary conditions are possible. Hence, in tion with Kutta-Joukowsky conditions - we show that Flut- view of the uniqueness, our aim is to give an asymptotic ter is an LCO. The speed is a Hopf Bifurcation point, de- expansion of the solution of the LPEs at small viscosities termined by the linearised model and so is the period. The confirming thus our choice for the boundary conditions. key relation of the pressure jump to the wing normal velocity is now given by a 2D nonlinear time domain extension Makram Hamouda of the linear Possio Integral equation. As for the disturbed Indiana University flow itself, we show that it can be decomposed as the sum [email protected] of two parts, one part that produces the lift determined by nonlinear Possio equation and does not depend on the value of the ratio of specific heats; and can be linearised; MS58 and the other part which produces no lift but may con- Limit Cycle Oscillations of Very Flexible High- tain discontinuities but not across the wing and cannot be Aspect-Ratio Wings linearized. An aeroelastic analysis is performed for a flexible high- A V. Balakrishnan aspect-ratio wing representative of a high altitude long en- University of California durance (HALE) aircraft. A number of features relevant to Dept of Electrical Engineering the aeroelastic response of HALE aircraft are highlighted, [email protected] including the sensitivity of the computed flutter boundary and limit cycle oscillations to aerodynamic data from computational fluid dynamics (CFD) simulations and wind MS58 tunnel tests, and the role of stall in the dynamic stability Fluid-stucture Interaction Between Blood Flow of high-aspect-ratio wings. and Thick Arterial Walls Justin Jaworski A new effective 3D axially symmetric nonlinear hyperbolic- Duke University parabolic model of fluid-structure interaction between an [email protected] incompressible, viscous fluid and a 3D (thick) elastic or viscoelastic structure will be presented. The model was motivated by the study of blood in muscular arteries. The MS58 resulting model is a generalization to the thick structure Flutter Analysis of a Wedge Shaped Airfoil in Po- case of an effective model obtained earlier by the speaker tential Flow and the co-authors. The new effective model reveals some We study the effect of nonzero camber thickness on flut- PD09 Abstracts 95

ter speed in potential flow using continuum models. We related to the classical /Hodgkin//Huxley/ models. solve the Possio integral equation from which the Lift and Moment are computed. We examine specifically the cases Chun Liu M=0 and M=1 . Our main result is that the flutter speed Penn. State University increases with the thickness. University Park, PA [email protected] Amjad Tuffaha Department of Mathematics University of Southern California MS59 tuff[email protected] A Variational Approach to Bar Code Reconstruc- tion

MS59 When a bar code is read by a reader, the reader produces Review of New Phase Field Models in Image Anal- a blurred noisy signal that lacks the original binary char- ysis and Microfluidics acter. One method for recovering the code is minimization of the Rudin-Osher-Fatemi phase field functional, which I will review some recent results using phase field mod- involves a fidelity and total variation term. The binary els in image analysis and microfluidics. The talk will in- structure of the original image allows for analytical results clude an overview of (a) Cahn-Hilliard methods for im- stating when this method will give back the original code. age inpainting, (b) a wavelet-Allen-Cahn method for im- The method also allows for generalizations to two dimen- age reconstruction, and (c) a diffuse interface model for sions. the EWOD (electrowetting on dieletric) device. Yves van Gennip Andrea L. Bertozzi Department of Mathematics UCLA Department of Mathematics Simon FRaser University [email protected] [email protected]

MS59 MS60 Variational Methods in Image Processing Minimizers of the Lawrence-Doniach Functional in Parallel Applied Fields Deblurring, denoising, inpainting and recolorization of images are fundamental problems in image processing and We consider minimizers of the Lawrence-Doniach func- have given rise in the past few years to a vast variety of tional, which models highly anisotropic superconductors techniques and methods touching different fields of math- with layered structure, in the simultaneous limit as the ematics. Among them, variational methods based on the layer thickness tends to zero and the Ginzburg-Landau pa- minimization of certain energy functionals have been suc- rameter tends to infinity. In particular, we consider the cessfully employed to treat a fairly general class of im- properties of minimizers when the system is subjected to age restoration problems. The underlying theoretical chal- an external magnetic field applied either tangentially or lenges are common to the variational formulation of prob- normally to the superconducting planes. For normally ap- lems in other areas (e.g. materials science). Here first plied fields, our results show that the resulting “pancake’ order RGB variational problems for recolorization will be vortices will be vertically aligned. In horizontal fields we analyzed, and the use of second order variational problems show that there are two parameter regimes in which mini- to eliminate the staircasing effect will be validated. mizers exhibit very different characteristics. The low-field regime resembles the Ginzburg-Landau model, while the Irene Fonseca high-field limit gives a “transparent state’ described in the Carnegie-Mellon University physical literature. Department of Mathematical Scienc [email protected] Stan Alama McMaster University Giovanni Leoni Department of Mathematics and Statistics Carnegie Mellon University [email protected] [email protected] Lia Bronsard Francesco Maggi McMaster University University of Florence, Italy Department of Mathematics [email protected]fi.it [email protected]

Massimiliano Morini Etienne Sandier S.I.S.S.A., Trieste, Italy Univ. Paris 12 [email protected] Creteil, France [email protected]

MS59 On Ionic Fluids in Ion Channels MS60 Ginzburg-Landau-type Vortices in a Model for Liq- We will study the dynamics of general ionic ﬂuids, in par- uid ticular those associated with ion channels. We will emphasize on the selectivities of ion channels as well as those Abstract not available at time of publication. Dmitry Golovaty 96 PD09 Abstracts

Department of Mathematics ear elasticity form quasilinear hyperbolic systems, which University of Akron are subject to shocks. The introduction of a strong dissi- [email protected] pative mechanism converts these equations to quasilinear parabolic-hyperbolic systems (for strain-rate viscoelasticity). This lecture describes the subtle and critical roles MS60 of such dissipative mechanisms and corresponding invari- Existence Results for Liquid Crystal Energies ance requirements for the global existence of solutions, for blowup of solutions, for time-periodic solutions, and We examine the problem of minimizing the Chen-Lubensky for the construction of accurate numerical shock-capturing liquid crystal energy. In the case that the energy has a schemes. SmC ground state we show that anchoring conditions on the smectic layering at the boundary are needed in order Stuart Antman for minimumizers to exist. We further give examples of Mathematics Department & IPST strong and weak anchoring conditions that suﬃce. UMCP [email protected] Dan Phillips Department of Mathematics, Purdue University [email protected] MS61 On the Necessary Conditions for Stability in Non- Patricia Bauman linear Elasticity Purdue University [email protected] Elastic equilibrium in solids capable of phase transformations can be modeled as a strong local minimizer of the Jinhae Park energy functional. The necessary conditions for such local University of Minnesota minimizers are well-known but poorly understood. I will [email protected] discuss these conditions and their relative strengths. Many simple-sounding problems still remain open.

MS60 Yury Grabovsky Temple University Disc Droplets, Ring Droplets, and Oval Droplets in [email protected] Some Morphogenesis and Morphology Problems

The Gierer-Meinhardt system for morphogenesis in devel- MS61 opment and the Ohta-Kawasaki theory for block copolymer morphology give rise to one nonlocal geometric problem of Existence via the Inverse Formulation in 2-d Non- finding a (often disconnected) subset of a given domain. linear Elasticity The subset satisfies an equation that involves the curvature Abstract not available at time of publication. of the boundary of the subset and the inverse Laplacian of the characteristic function of the subset. Three solutions Phoebus Rosakis are found: a subset of many small discs, a subset of many University of Crete (Greece) small rings, and a small oval shaped subset. A resonance [email protected] diagram determines the existence and stability of the first two solutions. It reveals a complicated landscape of the free energy functional. An analysis near a resonance point MS61 yields the third solution with unexpected properties. On the Symmetry of Energy Minimising Deforma- tions in Nonlinear Elasticity Xiaosong Kang Wuhan University Consider a homogeneous, isotropic, hyperelastic body that [email protected] occupies a three-dimensional, thick spherical shell in its ref- erence state and is subject to radially symmetric displace- Jieun Lee ment boundary conditions on its inner or outer boundary. George Washington University We show that for a large class of polyconvex stored-energy [email protected] functions the radial minimiser of this problem is an abso- lute minimiser of the elastic energy. The key ingredient is a Xiaofeng Ren new radial-symmetrisation procedure that yields a one-to- Department of Mathematics one map. Such a one-to-one symmetrisation is important George Washington University in Continuum Mechanics where interpenetration of matter [email protected] must be prohibited. J. Sivaloganathan Juncheng Wei Department of Mathematics University of Bath (UK) Chinese University of Hong Kong [email protected] [email protected] scott spector Southern Illinois University MS61 [email protected] Dynamical Problems of Nonlinear (Visco) Elastic- ity

The dynamical equations of (standard models of) nonlin- PD09 Abstracts 97

MS62 Flow Nonlinear Subelliptic Equations We study the existence of a solution to a three-dimensional In this paper, we study the higher order regularity for axially symmetric, moving boundary problem arising in weak solutions of a class of quasilinear subelliptic equa- modeling blood flow. The model equations form a tions. We introduce the notion of ν-closed system vector hyperbolic-parabolic system of partial differential equa- fields, which includes all the previously studied nilpotent tions of Biot type with degenerate diffusion. Degenerate systems and extends them to some non-nilpotent systems diffusion is a consequence of the fact that the effects of of vector fields, including those generating the Lie Alge- the fluid viscosity in the axial direction of a long and slen- bra of the rotation group SO(n) and other non-compact der tube are small in comparison with the effects of the semisimple and solvable Lie groups. fluid viscosity in the radial direction. Degenerate fluid diffusion and the hyperbolic features of the problem cause Juan J. Manfredi lower regularity of a weak solution and are a source of the Department of Mathematics main difficulties associated with the existence proof. Cru- University of Pittsburgh cial for the existence proof is the viscoelasticity of vessel [email protected] walls which provides the main smoothing mechanisms in the energy estimates. Using the Implicit Function Theorem AndrásDomokos we obtain existence of a solution which is a perturbation California State University Sacramento of the famous Womersley flow in a tube with fixed walls [email protected] and periodic forcing, thereby providing the existence of a solution which is physiologically relevant. This result con- trasts other related results in this field which primarily deal MS62 with small perturbations of solutions corresponding to the Estimates of Hausdoff Measure of Free Boundaries zero-velocity scenarios. Our estimates show that the viscoelasticity of vessel walls is crucial in smoothing out the Here we will talk about estimates of the measure of the free sharp fronts which are generated by the heart’ steep pres- boundary problems for equations the could be nonhomoge- sure pulse in human large-to-medium arteries. Numerical neous p-laplacian problems. The strong nondegeneracy of solution of the resulting nonlinear problem shows excellent the solution is the technical contribution for this problems. agreement with experimental data. Diego Moreira Suncica Canic University of Iowa Department of Mathemtics Italy University of Houston [email protected] [email protected]

Taebeom Kim MS62 Department of Mathematics The Structure of the Free Boundary for Lower Di- University of Houston mensional Obstacle Problems [email protected] Abstract not available at tinme of publication. Giovanna Guidoboni Sandro Salsa University of Houston Politecnico di Milano Department of Mathematics Germany [email protected] [email protected] Andro Mikelic University of Lyon 1 MS62 Department of Mathematics A Bifurcation Phenomenon in Two-Phase Free [email protected] Boundary Problems

In an ellipitc free boundary problem, the uniqueness of a MS63 viscosity solution of the boundary value-problem and of a A Variational Approach to Nonlinear Elasticity of minimizer of the variational problem in general fails. In Non-Euclidean Plates. the counter-examples we constructed, a bifurcation phenomenon presents. A critical situation exists so that the This talk is motivated by studying elasticity of thin struc- uniqueness problem has different answers to the different tures which show non-zero strain at free equilibria (non- sides of the critical situation. The corresponding evalution- Euclidean plates). Many growing tissues (leaves, flowers or ary problem explains this phenomenon. marine invertebrates) exhibit complicated configurations during their free growth and one would like to have them Peirong Wang reproduced with man-made means. Recall that a smooth Wayne State University Riemannian metric on a simply connected domain can be [email protected] realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, MS63 one seeks a deformation yielding the closest metric realiza- Existence of a Solution to a Three-Dimensional Ax- tion. We set up a variational formulation of this problem ially Symmetric Moving-Boundary Biot Problem by introducing the non-Euclidean version of the standard Arising in Modeling Blood Flow: the Womersly nonlinear elasticity energy functional, and establish its Γ- limit under a proper scaling. As a corollary, we obtain new 98 PD09 Abstracts

necessary and sufficient conditions for existence of a W 2,2 Christian Pfrang isometric immmersion of a given 2d metric, into R3. Division of Applied Mathematics Brown University Marta Lewicka [email protected] University of Minnesota [email protected] Ravi Srinivasan Brown University Reza Pakzad [email protected] University of Pittsburgh [email protected] MS64 Instability and Mixing in Driven and Active Com- MS63 plex Fluids Cauchy Problem for Relativistic Landau-Maxwell System Complex fluids are fluids whose suspended microstructure feeds back and influences the macroscopic flow. A well- We prove the global existence of classical solutions for known example is a polymer suspension and a novel one is the relativistic Landau-Maxwell system near the global a bacterial bath wherein many swimming micro-organisms Maxwellian with generic initial data. This is a joint work interact with each other through the surrounding fluid. In with Hongjun Yu. either case, these systems can display very rich dynamics even at system sizes where inertia is negligible, and both Ronghua Pan systems have important applications to micro-fluidic mix- Georgia Tech. ing and transport. I will discuss examples of each where [email protected] hydrodynamic instabilities drive these systems into dynamics characterized by the emergence of coherent structures, MS64 multiple time-scales, and strong fluid mixing. A Kinetic Model for the Sedimentation of Rod-like Michael J. Shelley Particles New York University Courant Inst of Math Sciences The sedimentation of suspensions of rod-like particles [email protected] shows an interesting pattern formation that has been studied by several authors in theoretical, numerical and experimental work. Here we try to understand these phe- MS64 nomena by considering a kinetic model which couples a Kinetic Models for Dilute Suspensions and Shear microscopic Smoluchowski equation (for the rod orienta- Bands in Viscoelastic Flows tion) to the macroscopic Stokes equation. By looking at special flow configurations and scalings, we derive simpler In this talk we will present a class of models introduced by models which describe concentration. We show results of Doi and describing suspensions of rod–like molecules in a numerical simulations and present algorithms for a detailed solvent fluid. Such kinetic models couple a microscopic to a simulation of the microscopic equation. macroscopic equation. The macroscopic one is the Stokes equation for the fluid velocity, the microscopic equation Christiane Helzel is a Fokker–Planck (Smoluchowski) equation for the prob- Ruhr-University Bochum ability distribution of rod orientations in every point of Germany physical space. For certain parameter values, the velocity [email protected] gradient vs. stress relation defined by the stationary and homogeneous flow is not rank–one monotone. This induces Felix Otto that steady states can have discontinuous solutions anal- University of Bonn ogous to the ones studied in the context for macroscopic Germany viscoelastic models (e.g. for Oldroyd-B models) and spurt [email protected] phenomena or shear bands in that context. (joint work with Ch. Helzel, Univ. of Bochum and F. Otto, Univ. of Athanasios Tzavaras Bonn) University of Maryland [email protected] Athanasios Tzavaras University of Maryland [email protected] MS64 Kinetic Theory and Lax equations for shock clustering MS65 An Example of Global Classical Solution for the We derive kinetic equations that describe the clustering Perona-Malik Equation of shocks in scalar conservation laws with random initial The Perona–Malik equation data. These equations have the structure of a Lax pair, and admit remarkable exact solutions for Burgers equations. ∇u There are strong hints that the kinetic equations form a |∇ | ut = div ϕ ( u ) |∇ | 2+1-dimensional completely integrable system. u Govind Menon is a typical example of forward-backward diffusion process. Brown University It is well known that in dimension n = 1 the Cauchy [email protected] problem does not admit global-in-time classical solutions (namely C2,1 or even C1 solutions) if the initial condition PD09 Abstracts 99

u0 is transcritical, namely |∇u0(x)|−1 is a sign changing time according to the corresponding dynamic. The frame- funcion. On the contrary, we show that in dimension n ≥ 2 work of stochastic analysis can be used to construct so- there are examples of global-in-time solutions of class C2,1 lutions. In our presentation, we follow the second point with transcritical initial datum. of view to present some results on the 2D Euler equation with periodic boundary conditions. We construct surface Massimo Gobbino type measures on the level sets of the renormalized energy Universita di Pisa and establish the existence of weak solutions living on such [email protected] level sets. We also prove the existence of weak solutions for the forward and backward transport equations associ- Marina Ghisi ated with the 2D Euler equation. Such solutions can be Dipartimento di Matematica interpreted, respectively, as a statistical Lagrangean and Universitàdi Pisa statistical Eulerian description of the motion of the fluid. [email protected] Fernanda Cipriano GFM - UL and FCT - New University of Lisbon MS65 [email protected] Anistropic Diffusion Models Old and New

Since the pioneering work of Perona and Malik, so-called MS66 anisotropic diﬀusion has found a fertile ground of applica- Existence for the 2D Vortex-wave System tion in Image processing. A survey of models will be given with particular emphasis on second and fourth order diﬀu- The 2D vortex-wave system is obtained by coupling the sions and their application to the resolution of important 2D vorticity equation with the equation for the evolution basic image processing tasks. Analytical results will be of point vortices. Our objective in this talk is to present complemented with substantiating numerical experiments. recent results for this system, including existence of a weak solution with initial vorticity in Lp, p>2. Patrick Guidotti University of California at Irvine Milton C. Lopes Filho [email protected] IMECC-UNICAMP [email protected]

MS65 Evelyne Miot A Semidiscrete Scheme for the Perona-Malik Equa- Univ. Paris VI tion [email protected]

I present a semidiscrete scheme for the Perona-Malik Equa- Helena J. Nussenzveig Lopes tion in one space dimensione, rigorously proving the con- State University of Campinas, Brazil vergence for a special class of initial data. Particular em- [email protected] phasis will be given to the different behavior of the solutions in the different time-scales, and the long-time behavior will be discussed in detail. MS66 Matteo Novaga A Generalization of the Helmholtz-Kirchhoff Model. University of Padova (Italy) [email protected] The two-dimensional Navier-Stokes equations are rewrit- ten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the MS66 moments of an expansion of the vorticity with respect to The Free Boundary Value Problem of the Navier- Hermite functions and of the centers of vorticity concen- Stokes Equations with Surface Tension trations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit We study the free boundary value problem of the Navier we formally recover the Helmholtz-Kirchhoff model for the Stokes equations on a moving domain of finite depth, evolution of point-vortices. The present expansion system- bounded above by a free surface and bounded below by atically incorporates the effects of both viscosity and finite a solid flat domain. We show that there exists a unique, vortex core size. This is joint work with Ray Nagem, Guido global-in-time solution for small initial data in energy Sandri and David Uminsky of Boston University. spaces. Gene Wayne Hantaek Bae Boston University CSCAMM - University of Maryland Department of Mathematics and Statistics [email protected] [email protected]

MS66 MS67 Statistical solutions for the 2D Euler equation Sensitivity Analysis in Imaging

In the study of Euler and Navier-Stokes equations we can An imaging procedure is an inverse problem. While gen- consider two diﬀerent approaches. The most classical one erally ill-posed, such problems are solvable on an adequate consist in the study of the equations with speciﬁque ini- subspace. In particular, the small perturbation case can tial and boundary conditions. Another approach, the so- be rigorously studied. Considering the scalar Helmholtz called statistical approach, consist in the construction of equation and the elasticity model, we perform asymptotic suitable probability measures and study its evolution in 100 PD09 Abstracts

analysis of the perturbed fields when a small inclusion is ded in heavy cluttered media. To make coherent imaging added in an homogeneous background. Using this pertur- possible in such media we propose to filter the unwanted bation analysis we derive imaging algorithms and discuss echoes due to the cluttered medium prior to imaging. The their resolution limits. main idea is to identify windows in the time-direction of arrival plane that contain the coherent echoes from the Habib Ammari reflectors, by doing a spectral decomposition of the local CNRS & Ecole Polytechnique, France cosine transform of the response matrix recorded at the [email protected] array.

Pierre Garapon Chrysoula Tsogka Laboratoire Ondes et Acoustique University of Crete and FORTH-IACM ESPCI [email protected] [email protected] Liliana Borcea Franois Jouve Rice University Universit´eParis VII - Denis Diderot Dept Computational & Appl Math [email protected] [email protected]

George C. Papanicolaou MS67 Stanford University Imaging by Cross Correlation of Noisy Signals Department of Mathematics [email protected] We introduce a self-consistent theoretical framework for the analysis of interferometric imaging techniques. We describe the conditions required to observe the emergence of MS68 the Green’s function between two sensors from the cross Power-law Variational Principles and Aronsson correlation of ambient noise recorded by the sensors. We Equations discuss passive sensor imaging by migration of the cross correlations. We show that reflectors can be imaged in a Γ-convergence results for power-law functionals acting on scattering medium with passive sensor networks by migrat- fields which satisfy constant rank differential constraints ing suitable fourth-order cross correlations. will be presented, together with a discussion of selected Aronsson equations associated to minimization problems Josselin Garnier for the limiting supremal functionals. Universite Paris 7 [email protected] Marian Bocea NDSU George C. Papanicolaou [email protected] Stanford University Department of Mathematics [email protected] MS68 Stability Analysis of Generalized Forchheimer Knut Solna Flows in Porous Media. University of California at Irvine This presentation focuses on the stability of non-linear [email protected] flows of slightly compressible fluids in porous media not adequately described by Darcy’s law. We study a class of MS67 generalized nonlinear momentum equations which covers three well-known Forchheimer equations, the so-called two- Array Imaging of Sparse Scenes with L1 Minimiza- term, power, and three-term laws. The generalized Forch- tion heimer equation is inverted to a non-linear Darcy equation Most array imaging is done with variants of an L2 mini- with implicit permeability tensor depending on the pres- mization algorithm, including travel time migration which sure gradient. This results in a degenerate parabolic equa- is a simplified form of it for wide area imaging. With the tion for the pressure. Two classes of boundary conditions exception of SVD-based algorithms, L2 minimization is not are considered, given pressure and given total flux. The efficient for sparse image scenes. We will describe an L1 uniqueness, Lyapunov and asymptotic stability of the so- based imaging algorithm that works well when there is lutions, and their continuous dependence on the boundary sparsity, analyze it using ideas from compressed sensing, data are analyzed. and show results of numerical simulations (work jointly Luan Hoang, Aulisa Eugenio, Lidia Bloshanskya with A. Chai). Texas Tech University George C. Papanicolaou Department of Mathematics and Statistics Stanford University [email protected], [email protected], Department of Mathematics [email protected] [email protected] Akif Ibragimov Department of Mathematics and Statistics. MS67 Texas Tech University Data Filtering for Coherent Array Imaging in [email protected] Heavy Clutter

We consider the problem of imaging small reﬂectors embed- PD09 Abstracts 101

MS68 Department of Mathematics Asymptotic Analysis of Phase Field Formulations New Mexico State University of Bending Elasticity Models [email protected]

In this talk, we give the asymptotic analysis of sharp inter- Robert Smits face analysis of the phase field function in some phase field New Mexico State University models for Willmores problem or equilibrium lipid bilayer [email protected] cell membrane problems. We derive the explicit expression of the asymptotic expansion of the phase field functions minimizing the Willmore energy. Based on the structure of MS69 the phase field functions obtained via the asymptotic anal- Ginzburg-Landau Vortices Driven by the Landau- ysis, we can then demonstrate the consistency of phase field Lifschitz-Gilbert Equations models and the sharp interface models. Also some error estimates of energy and Euler number formulae are further A simplified model for the energy of the magnetization of a analyzed. Some numerical experiments are performed to thin ferromagnetic film gives rise to a version of the theory verify our assumptions and results. The results of this pa- of Ginzburg-Landau vortices for sphere-valued maps. In per lead to better understanding of the structure of the particular we have the development of vortices as a cer- phase field functions in the phase field models for Will- tain parameter tends to 0. The dynamics of the mag- mores problem and the equilibriums configurations of the netization is ruled by the Landau-Lifshitz-Gilbert equa- lipid vesicle membranes. tion, which combines characteristic properties of a nonlinear Schrödingerequation and a gradient flow. We study the Xiaoqiang Wang motion of the vortex centers under this evolution equation. Florida State University [email protected] Matthias Kurzke Institute for Applied Mathematics, University of Bonn MS69 Bonn, Germany Superconductivity Near the Normal State in the [email protected] presence of current Christof Melcher We consider the linearization of the time-dependent RWTH Aachen University Ginzburg-Landau near the normal state. We assume that [email protected] an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the nor- Roger Moser mal state is always stable. Using Fourier analysis the de- University of Bath tailed behaviour of solutions is obtained as well. Relying on [email protected] semi-group theory we then obtain the spectral properties of the steady-state elliptic operator. We shal also consider Daniel Spirn the spectral properties of the same elliptic operator near University of Minnesota a flat wall, and obtain the critical current in the limit of [email protected] small and large normal conductivity Yaniv Almog MS69 Lousiana State University Time-dependent Ginzburg-Landau with an Ap- Department of Mathematics plied Voltage [email protected] We study a Ginzburg-Landau model for the response of a Bernard Helffer thin superconducting wire to an applied voltage difference. University of Paris, Orsay In this model, the voltage difference leads to time-periodic bernard.helff[email protected] boundary conditions. Different asymptotic regimes will be discussed including large/small voltage, and long/short wires. Physically and numerically observed phenomena in- Xingbin Pan clude period doubling, chaotic behavior and boundary layer East China Normal University effects. [email protected] Peter Sternberg Indiana University MS69 Department of Mathematics Gauge Uniqueness of Solutions to the Ginzburg- [email protected] Landau System for Small Bi-dimensional Domains

We study the problem of type I behavior and gauge unique- Jacob Rubinstein ness of solutions to the two-dimensional Ginzburg-Landau Department of Mathematics system, in the presence of an applied field when the sample Indiana Univ. and Technion-Israel Institute of Technology is sufficiently small. We show that there are only two pos- [email protected] sible solutions: one normal and one vortex-free, with the normal solution being the only solution above a critical Junghwa Kim field and the vortex-free being the global minimizer below. Indiana University [email protected]

Tiziana Giorgi 102 PD09 Abstracts

MS70 MS70 Diﬀusion Generated Motion for Evolving Interfaces Statistical Analysis of Shapes based on Nonlinear in Imaging and Vision Elasticity and Phase Field represented Geometries

We describe efficient and accurate algorithms for gener- The talk deals with the covariance analysis of a set of of ating a variety of interfacial motions of interest in image given shapes. These shapes are interpreted as boundary processing and computer vision. They reduce geometric contours of elastic objects and algorithmically treated as motions to essentially two simple operations: Redistanc- phase fields. Based on the notion of nonlinear elastic de- ing and convolution with a kernel, both of which have fast, formations from one shape to another, a suitable lineariza- classical algorithms. Our methods can be seen as a vari- tion of geometric shape variations is introduced. Once such ant of Merriman, Bence, and Osher’s threshold dynamics a linearization is available, a principal component analy- which itself stems from operator splitting on well-known sis can be investigated. This requires the definition of a phase field formulations of certain geometric motions. covariance metric—an inner product on linearized shape variations. The resulting covariance operator robustly cap- Selim Esedoglu tures strongly nonlinear geometric variations in a physi- Department of Mathematics cally meaningful way and allows to extract the dominant University of Michigan modes of shape variation. The underlying elasticity con- [email protected] cept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L2-type covariance metric with a metric based on the Hessian of the MS70 nonlinear elastic energy. Furthermore, we explore the de- On the Convergence of the Ohta-Kawasaki Equa- pendence of the principal component analysis on the type tion to Motion by Nonlocal Mullins-Sekerka Law of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is em- We rigorously justify the nonlocal Mullins-Sekerka dy- ployed which allows for a non-exact matching. Shape con- namics from the Ohta-Kawasaki equation on any smooth ≤ tours are approximated by single well phase fields, which domain in space dimensions N 3. These equations enables an extension of the method to a covariance anal- arise in modeling microphase separation in diblock copoly- ysis of image morphologies. The model is implemented mers. We establish convergence results for the case of well- with multilinear finite elements embedded in a multi-scale preparedness of the initial data and smoothness of the lim- approach. The characteristics of the approach are demon- iting interface. Our method makes use of the “Gamma- strated on a number of illustrative and real world exam- convergence’ of gradient flows scheme initiated by Sandier ples in 2D and 3D. The talk is based on joint work with and Serfaty and the constancy of multiplicity of the limit- Benedikt Wirth. ing interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give Martin Rumpf a new and short proof of Henry’s result for all space di- Rheinische Friedrich-Wilhelms-UniversitätBonn mensions. Finally, we establish transport estimates for so- [email protected] lutions of the Ohta-Kawasaki equation characterizing their transport mechanism MS71 Nam Le Weak Solutions in Second-gradient Nonlinear Elas- Department of Mathematics ticity Columbia University [email protected] We consider a class of second-gradient elasticity models for which the internal potential energy is the sum of a convex function of the second gradient of the deformation MS70 and a general function of the first gradient. In consonance Smoke-rings and Toroidal Tubes in Some with classical nonlinear elasticity, the latter is assumed to Activator-inhibitor grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is rou- Ring shaped objects in space have been observed in many tine, the existence of weak solutions is not. We focus on physical and biological systems. We find them in the that question here. We provide a general setting in which Gierer-Meinhardt system for morphogenesis in cell develop- the determinant of any admissible deformation correspond- ment. This system is a minimal model that provides a the- ing to bounded energy is strictly positive on the closure of oretical bridge between observations on the one hand and the domain. In particular, the total potential energy is the deduction of the underlying molecular-genetic mecha- readily shown to be Gateaux differentiable at an energy nisms on the other hand. In the non-saturation case we minimizer. We obtain weak solutions for a wide variety of show the existence of a smoke-ring type solution in space. mixed boundary conditions on Lipshitz domains. The activator component of the solution vanishes outside the smoke-ring and remain large on the smoke ring. In the Tim Healey saturation case the GM system is reduced to a nonlocal Cornell University geometric problem where a solution is a set in space that Universidad de Los Andes satisfies an equation that links the mean curvature of the Email [email protected] boundary of the set to the Newtonian potential of the set. A torus shaped, tube like solution is found. MS71 Xiaofeng Ren n-Harmonic Deformations of Annuli, The Art of The George Washington University Integration Of Free Lgrangians Department of Mathematics [email protected] We study homeomorphisms h : X −→ Y between annuli in Rn. The aim is to minimize the associated n-harmonic PD09 Abstracts 103

energy integral. Because of conformal invariance such an of the heat equation. The scaling invariance property of alternative to the classical Dirichlet energy has drawn the the self-similar solution depends on the nonlinear opera- attention of researchers in n-dimensional Geometric Func- tor, and is in general different from that of the heat kernel. tion Theory. We adopt the interpretations and ideas of We will see that this difference has an interesting interpre- nonlinear elasticity where the applied aspects of our re- tation in terms of controlled diffusion processes. This work sults originated. The underlying integration of nonlinear is joint with M. Trokhimtchouk. differential forms, called free Lagrangians, comes into play. Scott Armstrong Tadeusz Iwaniec Louisiana State University Syracuse University USA [email protected] [email protected]

MS71 MS72 Violation of the Complementing Condition and Lo- Title Not Available at Time of Publication cal Bifurcation in Nonlinear Elasticity Abstract not available at time of publication. In this talk we review several examples of boundary value problems in nonlinear elasticity where the linearization Ivan Blank about a suitable trivial solution (depending on a param- Kansas State University eter) fails to satisfy the complementing condition (CC). [email protected] The CC is an algebraic compatibility requirement between the principal part of a linear elliptic differential operator and the principal part of the corresponding boundary op- MS72 erators. The examples we review suggest that failure of Asymptotic Energy Concentration in the Phase the CC implies the existence of bifurcating branches of so- Space of the Weak Solutions to the Navier-Stokes lutions accumulating at the point where the CC fails. We Equations will discuss the essential features of this relationship in a more general context. We consider the asymptotic behavior of the weak solution to the Navier-Stokes equations in whole space Rn. More Errol Montes precisely, we consider the asymptotic behavior of the lower Mathematics frequency part of solutions to the Navier-Stokes equations. University of Puerto Rico - Cayey In fact, the kinetic energy of the lower frequency part of [email protected] solutions dominates the total energy of the solution asymptotically, if the initial values have some lower frequency. We Pablo Negron characterize the set of initial data which causes the energy University of Puerto Rico - Humacao concentration to the lower frequency part of the solution. [email protected] Takahiro Okabe Tohoku University MS71 Japan [email protected] A Degree Theory for Proper Fredholm Maps in Continuum Mechanics MS72 We present a topological degree theory for proper Fred- holm maps that applies to the equilibrium equations of Montonocity Formulas for Free Boundary Prob- nonlinear continuum mechanics. The equations are of lems on Manifolds. quasilinear elliptic type in a bounded domain subject to For free boundary problems on Euclidean spaces, the displacement/traction boundary conditions. Applications monotonicity formulas of Alt-Caffarelli-Friedman and are made directly to some compressible and incompressible Caffarelli-Jerison-Kenig are cornerstones for the regularity elasticity and steady state Navier-Stokes equations with theory. I will present the analogs of these results for the global continuation and existence results. Laplace-Beltrami operator on Riemannian manifolds. As Henry Simpson an application I will show that the new monotonicity theo- Mathematics rems can be employed to prove the Lipschitz continuity for University of Tennessee the solutions of a general class of two-phase free boundary [email protected] problems. This is a joint work with E. Teixeira. Lei Zhang MS72 University of Florida [email protected] Self-similar Solutions and Long-time Asymptotics for Fully Nonlinear Parabolic Equations MS73 I will present results on the existence and uniqueness of a self-similar solution of a fully nonlinear, parabolic equa- Title Not Available at Time of Publication tion (an example of which include the Bellman-Isaacs equa- Abstract not available at time of Publication. tion arising in the theory of stochastic optimal control and stochastic differential game theory). As an application, John Hunter we are able to describe the long-time behavior of solu- Department of Mathematics tions to the Cauchy problem, and derive a conservation UC Davis law which generalizes the conservation of mass in the case [email protected] 104 PD09 Abstracts

MS73 MS74 Rarefaction Wave Interaction for the Unsteady Variational Characterization of Solutions of Non- Transonic Disturbance Equations linear Parabolic Equations

We study a Riemann problem for the unsteady transonic This talk will describe some variational characterizations small disturbance equations that results in a diverging rar- of the solutions of initial boundary value problems for efaction problem. The self-similar reduction leads to a parabolic equations and systems of parabolic type. These boundary value problem with equations that change type characterizations enable some different proofs of existence (hyperbolic-elliptic) and a sonic line that is a free bound- uniqueness and other results for these equations. ary. We summarize the principal ideas and present the main features of the problem. The flow in the hyperbolic Giles Auchmuty part can be described as a solution of a degenerate Goursat Department of Mathematics boundary problem, the interaction of the rarefaction wave University of Houston with the subsonic region is illustrated and the subsonic [email protected] flow is shown to satisfy a second order degenerate elliptic boundary problem with mixed boundary conditions. MS74 Katarina Jegdic Smoothing Properties and Lack of Compactness for University of Houston-Downtown a Coupled Fluid-Structure Semigroup [email protected] In this talk we shall derive certain qualitative properties for Jun Chen, Cleopatra Christoforou a partial differential equation (PDE) system which com- University of Houston prises (parabolic) Stokes fluid flow and a (hyperbolic) elas- [email protected], [email protected] tic structure equation. The appearance of such coupled PDE models in the literature is well-established, inasmuch as they mathematically govern many physical phenomena; MS73 e.g., the immersion of an elastic structure within a fluid. A Two-phase Stefan Problem for the Unsteady The coupling between the distinct hyperbolic and parabolic Transonic Small Dis- turbance Equations dynamics occurs at the boundary interface between the media. In previous work, we have established semigroup A complex structure is found in numerical solutions of weak wellposedness for such dynamics, in part through a non- shock reflection problems for the unsteady transonic small standard elimination of the associated pressure variable. disturbance equations, the nonlinear wave system, and However, one problem with this fluid-structure semigroup the full Euler equations at a set of parameter values for setup is that, due to the definition of the domain of the which regular reflection is impossible. The solutions con- generator, there is no immediate implication of smooth- tain a sequence of triple points and tiny supersonic patches ing in all the fluid-structure variables; viz., the resolvent embedded in the subsonic flow directly behind a leading of the generator is not compact on the finite energy space. triple point. The supersonic patches are separated from Consequently, one is presented with the basic question of the subsonic flow by a sonic line of complex shape that whether smooth initial data will give rise to higher regular- contains embedded shock waves. This sonic line can be ity of the solutions. Accordingly, one main result described considered a free boundary, and the resulting free bound- here states that the mechanical, fluid, and pressure vari- ary problem for a system of equations of mixed type results ables do in fact enjoy a greater regularity if an extra unit in a new type of problem that has not previously been for- of Sobolev smoothness is imposed upon the initial struc- mulated or analyzed. As a step in this direction, we pose tural component (only). A second problem of the model is a much simpler problem for the unsteady transonic small the inherent lack of long time stability. In this connection, disturbance equations that contains some of the same fea- a second result described here provides for uniform stabi- tures: a free boundary consisting of a sonic line containing lization of the fluid-structure dynamics, by means of the an embedded shock, and transonic coupling between the insertion of a damping term at the interface between the subsonic and supersonic regions. We linearize the problem two media. and solve it exactly. We also obtain numerical solutions of the nonlinear problem. George Avalos University of Nebraska-Lincoln Allen Tesdall Department of Mathematics and Statistics College of Staten Island [email protected] City University of New York [email protected] MS74 Barbara Keyfitz Stability of Inverse Problems for The Wave Equa- Ohio State University tion bkeyfi[email protected] Consider the inverse problem of determining the coefficients from the Dirichlet to Neumann map on the boundary MS73 for the wave equation with known initial conditions. In this talk, the speaker will discuss some recent stability study of Title Not Available at Time of Publication. the nonlinear inverse problem and the implications on in- Abstract not available at time of publication. verse scattering problems. Recent results on related inverse problems will also be highlighted. Kevin Zumbrun Indiana University Gang Bao USA Michigan State University [email protected] [email protected] PD09 Abstracts 105

MS74 tivistic heat equation, Uniform Stabilization of the System of Dynamic Elasticity with Nonlinear Dissipation in the Dirich- ⎧ ⎛ ⎞ let Boundary Conditions ⎪ ⎪ ∂u ⎝ uDu ⎠ × ⎪ = ν div in QT =(0,T) Ω We consider the system of dynamic elasticity defined on ⎪ ∂t 2 ν2 2 ⎨ u + 2 |Du| a bounded domain, with a nonlinear, nonlocal, dissipative c term in the Dirichlet boundary conditions; and we estab- ⎪ 2 −1 ⎪ lish its uniform stability in the space L × H of optimal ⎪ u = g on ST =(0,T) × ∂Ω, ⎪ regularity. Two approaches will be noted: one is a direct ⎩⎪ approach; the other is instead based on the very strong 0 ∗ 2 u(0,x)=u (x)inΩ property that B L for the linear model is bounded, L on (4) the boundary in time and space. In both cases, a pseu- where Ω is an open bounded subset of RN with Lipschitz dodifferential analysis is needed. This is joint work with ∞ ∞ boundary ∂Ω, 0 ≤ u0 ∈ L (Ω) and 0 ≤ g ∈ L (∂Ω). I. Lasiecka. It is a companion version of similar results We prove existence and uniqueness of entropy solutions for originally established by the authors for wave equations the Dirichlet problem. This problem is however more deli- with nonlinear, nonlocal dissipative terms in the Dirichlet cate than the Cauchy problem in RN or the homogeneous boundary conditions. Neumann one. In particular, difficulties arise because of Roberto Triggiani the fact that the boundary condition is in general not at- University of Virginia tained and this condition has to be weakened to become rt7u@virginia,edu an obstacle condition on the boundary. Jose Mazon MS74 Universitat de Valencia Departamento de AnálisisMatemático Stabilty of Moving Structures [email protected] The stability of moving wings and moving structures is an open problem. Several destructions of unmanned flying ob- MS75 jects occur during the shape dynamical changes. Indeed, in this period, no classical eigenmodes analysis is avail- Afem for Shape Optimization Problems able. We extend the stability criteria introduced in Optimal We examine shape optimization problems in the context Morphing Control (by A.V. Balakrishnan and J.-P. Zolesio, of inexact sequential quadratic programming. Inexactness in Sixth International Conference on Mathematical Prob- is a consequence of using adaptive finite element methods lems in Engineering and Aerospace Sciences, Cambridge, (AFEM) to approximate the state equation, update the 2007), and present new results concerning moving bound- boundary, and compute the geometric functional. Numer- aries with Neumann boundary condition (including some ical simulations illustrating the method are presented for sharp regularity and tube’s derivative). the design of a drag minimizing object and a cantilever Jean-Paul Zolesio minimizing compliance. CNRS and INRIA Miguel S. Pauletti Sophia Antipolis, France University of Maryland [email protected] [email protected]

MS75 Pedro Morin The Dirichlet Problem for the Relativistic Heat Universidad Nacional del Litoral and IMAL CONICET Equation Santa Fe, Argentina [email protected] To correct the infinite speed of propagation of the classical linear diffusion equation Ph. Rosenau proposed the Ricardo Nochetto tempered diffusion equation Department of Mathematics University of Maryland, College Park ⎛ ⎞ [email protected] | | ∂u ⎝ u Du ⎠ = ν div , (3) MS75 ∂t 2 ν2 | |2 u + c2 Du Shape Optimization of Peristaltic Pumping

We present a variational method for optimizing peristaltic where ν is a constant representing a kinematic viscosity pumping in a two dimensional periodic channel with mov- and c the speed of light. This equation was derived by Y. ing walls to pump fluid (peristalsis is common in biology). Brenier by means of Monge-Kantorovich’s mass transport No a priori assumption is made on the wall motion, except theory and he named it as the relativistic heat equation. In that the shape is static in a moving wave frame. Thus, we pose an infinite dimensional optimization problem and solve it with finite elements. L2-type projections are used a a series of papers we have studied existence and unique- to compute quantities such as curvature and boundary ness of entropy solutions for the Cauchy problem and for stresses. the homogeneous Neumann problem associated to a class of quasi-linear parabolic equations which include as a par- Shawn Walker ticular case the relativistic heat equation. Here we are New York University, Courant Institute interested in the Dirichlet problem associated to the rela- [email protected] 106 PD09 Abstracts

MS76 IFREMER Spurious Caustics of Dispersion Relation Preserv- [email protected] ing schemes

We presently determine classes of traveling solitary wave MS77 solutions for a differential approximation of a dispersion- Imaging in Random Waveguides relation preserving scheme by means of an hyperbolic ansatz. The occurance of such spurious solitary waves re- I will present a new method for imaging sources in random sults in a non-vanishing numerical error for arbitrary time waveguides, from the time traces of the signals recorded at in unbounded numerical domains, leading to the structural remote arrays of passive sensors. By random waveguides instability of the scheme, since the space of solutions en- we mean that the wave speed has rapid fluctuations, that compasses types of solutions (solitary waves) that are not cause wave scattering and a significant loss of coherence of solution of the original continuous equations. the signals at the array. Our imaging method is based on a special form of transport equations in random waveguides, Claire David and it can determine in a statistically stable manner the UniversitéPierre et Marie Curie-Paris location of sources in the waveguide and statistical infor- Institut Jean Le Rond dAlembert mation about the wave speed fluctuations. [email protected] Liliana Borcea Rice University MS76 Dept Computational & Appl Math Traveling Wave Phenomena to the Modified Ko- [email protected] rteweg de Vries-Burgers Equation

In this talk, our goal is to study a modified Korteweg-de MS77 Vries-Burgers equation with two higher–order nonlineari- Active Exterior Cloaking ties. An asymptotic analysis of traveling wave solutions is established by means of the qualitative theory of differ- We present a new method for cloaking an object from a ential equations. An approximate solution with arbitrary known incident wave. This method is based on active de- velocity is obtained by using the decomposition method, vices that create a region, outside but near the devices, which agrees well with the phase plane analysis. with small wave amplitudes and while not significantly ra- diating waves. An object in this region is for all practical Zhaosheng Feng purposes invisible. We will discuss how this method relates University of Texas-Pan American to array imaging. [email protected] Fernando Guevara Vasquez University of Utah MS76 [email protected] Regularity of Attractor for 3D Ginzburg-Landau equation Graeme W. Milton University of Utah A three dimensional Ginzburg-Landau type equation with Department of Mathematics periodic initial value condition is considered. Firstly, the [email protected] smoothing property of the solution is obtained by a uniform priori estimates; then, the existence of the global i Daniel Onofrei attractors, i ⊂ Hp(Ω) (i =2, 3, ···), of the semi-group (i) Department of Mathematics {S (t)}t≥0 of operators generated by the equation is pre- University of Utah sented by using the compactness principle; finally, the reg- [email protected] ularity of the global attractors is proved by the decomposition of semi-group. MS77 Shujuan Lu Limit of Fluctuations of Solutions of the Wigner School of Science Equation for White-noise Potentials Beihang University [email protected] Large distance wave energy propagation in random media can often be described by kinetic equations. I will describe some recent results on the correctors to that limit when the MS76 random potential is time-dependent. This is a joint work Numerical Study of a Boussinesq System with T. Komorowski. We presently study the propagation, as well as the colli- Lenya Ryzhik sion, of solitary waves. A Boussinesq system for two-way Department of Mathematics, University of Chicago propagation of interfacial waves in a rigid lid configuration Chicago, IL 60637 is derived. If, in most cases, the nonlinearity is quadratic, [email protected] when the square of the depth ratio gets close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. MS77 Multi-Scale Approach to Seismic Inverse Scatter- ing and Partial Reconstruction via Microdiffraction Hai Yen Nguyen Laboratoire de Physique des Océans PD09 Abstracts 107

Tomography

We present a framework for inverse scattering via imaging and partial reconstruction in connection with limited boundary acquisition geometry. We ﬁrst discuss a microlocal analysis viewpoint using the generalized Radon transform. Then we introduce a construction using downward data continuation leading to the introduction of microd- iﬀraction tomography. To carry out the analysis we make use of higher-dimensional curvelets (Joint research with H. Smith, G. Uhlmann, S. Wang and B. Ursin.) Maarten de Hoop Center for Computational & Applied Mathematics Purdue University [email protected]