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 =Abstract=   =Abstract= 
   
−  ===Hung Tran===  +  === === 
−   
−  Some inverse problems in periodic homogenization of HamiltonJacobi equations.
 
−   
−  Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.
 
−   
−   
−  ===Eric Baer===
 
−   
−  Optimal function spaces for continuity of the Hessian determinant as a distribution.
 
−   
−  Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(22/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(22/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(22/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.
 
−   
−  ===Donghyun Lee===
 
−   
−  FLUIDS WITH FREESURFACE AND VANISHING VISCOSITY LIMIT.
 
−   
−  Abstract : Freeboundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, nonsurface tension), we will see how to get local regularity of incompressible freeboundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.
 
−  If possible, we also talk about applying the similar technique to the freeboundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.
 
−   
−  ===HyungJu Hwang===
 
−   
−  The FokkerPlanck equation in bounded domains
 
−   
−  abstract: In this talk, we consider the initialboundary value problem for the FokkerPlanck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of wellposedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.
 
−   
−  ===MinhBinh Tran===
 
−   
−  Nonlinear approximation theory for kinetic equations
 
−   
−  Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I
 
−  n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose nonphysics conditions on the equation.
 
−   
−  ===Bob Jensen===
 
−   
−  CrandallLions Viscosity Solutions of Uniformly Elliptic PDEs
 
−   
−  Abstract: I will discuss CL viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the CL viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.
 
−   
−  ===Luis Silvestre===
 
−   
−  A priori estimates for integral equations and the Boltzmann equation.
 
−   
−  Abstract: We will review some results on the regularity of general parabolic integrodifferential equations. We will see how these results can be applied in order to obtain a priori estimates for the Boltzmann equation (without cutoff) modelling the evolution of particle density in a dilute gas. We derive a bound in L^infinity for the full Boltzmann equation, and Holder continuity estimates in the space homogeneous case.
 
−   
−  ===Connor Mooney===
 
−   
−  Counterexamples to Sobolev regularity for degenerate MongeAmpere equations
 
−   
−  Abstract: W^{2,1} estimates for the MongeAmpere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.
 
−   
−  ===Javier Gomez Serrano===
 
−   
−  Existence and regularity of rotating global solutions for active scalars
 
−   
−  A particular kind of weak solutions for a 2D active scalar equation are the so called patches, i.e., solutions for which the scalar is a step function taking one value inside a moving region and another in the complement. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a nonlocal one dimensional equation for the contour. In this talk we will discuss the existence and regularity of uniformly rotating solutions for the vortex patch and generalized surface quasigeostrophic (gSQG) patch equation. We will also outline the proof for the smooth (non patch) SQG case. Joint work with Angel Castro and Diego Cordoba.
 
−   
−  ===Yifeng Yu===
 
−   
−  Gequation in the modeling of flame propagation.
 
−   
−  Abstract: Gequation is a well known model in turbulent combustion. In
 
−  this talk, I will present joint works with Jack Xin about how the
 
−  effective burning velocity (turbulent flame speed) depends on the strength
 
−  of the ambient fluid (e.g. the speed of the wind) under various Gequation
 
−  model.
 
−   
−  ===Nam Le===
 
−   
−  Global smoothness of the MongeAmpere eigenfunctions
 
−   
−  Abstract:
 
−  In this talk, I will discuss global smoothness of the eigenfunctions of the MongeAmpere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate MongeAmpere equations. This is joint work with Ovidiu Savin.
 
−   
−  ===Qin Li===
 
−   
−  Kineticfluid coupling: transition from the Boltzmann to the Euler
 
−   
−  Abstract: Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime coexist, how to couple the two systems remains an open problem. The key is to understand the halfspace problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the wellposedness of a class of halfspace equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.
 
−   
−  ===Lu Wang===
 
−   
−  Asymptotic Geometry of Selfshrinkers
 
−   
−  Abstract: In this talk, we will discuss some recent progress towards the conjectural asymptotic behaviors of twodimensional selfshrinkers of mean curvature flow.
 
−   
−  ===Christophe Lacave===
 
−   
−  Wellposedness for 2D Euler in nonsmooth domains
 
−   
−  The wellposedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. In a first part, we will establish the existence of global weak solutions of the 2D incompressible Euler equations for a large class of nonsmooth open sets. Existence of weak solutions with $L^p$ vorticity is deduced from an approximation argument, that relates to the socalled $\gamma$convergence of domains. In a second part, we will prove the uniqueness if the open set is the interior or the exterior of a simply connected domain, where the boundary has a finite number of corners. Although the velocity blows up near these corners, we will get a similar theorem to the Yudovich's result. Theses works are in collaboration with David GerardVaret, Evelyne Miot and Chao Wang.
 