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 == Abstracts ==   == Abstracts == 
   
−  ===Julian LopezGomez===  +  === === 
   
−  Title: The theorem of characterization of the Strong Maximum Principle  +  Title: 
   
−  Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, MolinaMeyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.  +  Abstract: 
−   
−  ===Hiroyoshi Mitake===
 
−  Title: On approximation of timefractional fully nonlinear equations
 
−   
−  Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider timefractional fully nonlinear equations. GigaNamba (2017) recently has established the wellposedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)
 
−   
−   
−   
−  ===Changyou Wang===
 
−   
−  Title: Some recent results on mathematical analysis of EricksenLeslie System
 
−   
−  Abstract: The EricksenLeslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the NavierStokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.
 
−   
−  ===Matthew Schrecker===
 
−   
−  Title: Finite energy methods for the 1D isentropic Euler equations
 
−   
−  Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible NavierStokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.
 
−   
−  ===Anna Mazzucato===
 
−   
−  Title: On the vanishing viscosity limit in incompressible flows
 
−   
−  Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the NavierStokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under noslip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseentype equation (linearization around a steady Euler flow) in general smooth domains.
 
−   
−  ===Lei Wu===
 
−   
−  Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
 
−   
−  Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with inflow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.
 
−   
−   
−  ===Annalaura Stingo===
 
−   
−  Title: Global existence of small solutions to a model waveKleinGordon system in 2D
 
−   
−  Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled waveKleinGordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasilinear nonlinearity, that expresses in terms of « null forms » .
 
−  Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasilinear equations, in their paradifferential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general nonlinearities.
 
−   
−  ===YeonEung Kim===
 
−   
−  Title: Construction of solutions to a HamiltonJacobi equation with a maximum constraint and some uniqueness properties
 
−   
−  A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and nonuniqueness properties.
 
−   
−  ===Albert Ai===
 
−   
−  Title: Low Regularity Solutions for Gravity Water Waves
 
−   
−  Abstract: We consider the local wellposedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for wellposedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of AlazardBurqZuily and low regularity Strichartz estimates, we apply this idea to the wellposedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.
 
−   
−  ===Trevor Leslie===
 
−   
−  Title: Flocking Models with Singular Interaction Kernels
 
−   
−  Abstract: Many biological systems exhibit the property of selforganization, the defining feature of which is coherent, largescale motion arising from underlying shortrange interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the socalled flocking phenomenon. Within the family of models that we consider (of which the CuckerSmale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and longtime dynamics of the Euler Alignment model and the ShvydkoyTadmor model.
 
−   
−  ===Serena Federico===
 
−   
−  Title: Sufficient conditions for local solvability of some degenerate partial differential operators
 
−   
−  Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.
 
−   
−  ===Max Engelstein===
 
−   
−  Title: The role of Energy in Regularity
 
−   
−  Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the EulerLagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
 
−   
−  However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.
 
−   
−  We will then turn the tables, and examine PDEs which look like they should be an EulerLagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.
 
−   
−   
−  ===RuYu Lai===
 
−  Title: Inverse transport theory and related applications.
 
−   
−  Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.
 
−   
−  ===Seokbae Yun===
 
−  Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation
 
−   
−  Abstract: In this talk, we consider the propagation of the uniform upper bounds
 
−  for the spatially homogenous relativistic Boltzmann equation. For this, we establish two
 
−  types of estimates for the the gain part of the collision operator: namely, a potential
 
−  type estimate and a relativistic hypersurface integral estimate. We then combine them
 
−  using the relativistic counterpart of the Carlemann representation to derive a uniform
 
−  control of the gain part, which gives the desired propagation of the uniform bounds of
 
−  the solution. Some applications of the results are also considered. This is a joint work
 
−  with Jin Woo Jang and Robert M. Strain.
 
−   
−   
−   
−  ===Daniel Tataru===
 
−   
−  Title: A Morawetz inequality for water waves.
 
−   
−  Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.
 
−   
−  Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit.
 
−   
−   
−  ===Wenjia Jing===
 
−   
−  Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
 
−   
−  Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the holecell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of holecell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.
 
−   
−   
−  ===Xiaoqin Guo===
 
−   
−  Title: Quantitative homogenization in a balanced random environment
 
−   
−  Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss nondivergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic nondivergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UWMadison).
 
−   
−  ===Sverak===
 
−   
−  Title: PDE aspects of the NavierStokes equations and simpler models
 
−   
−  Abstract: Does the NavierStokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.
 
−   
−  ===Jonathan Luk===
 
−   
−  Title: Stability of vacuum for the Landau equation with moderately soft potentials
 
−   
−  Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique globalintime smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a longrange interaction.
 
−   
−   
−  ===Jiaxin Jin===
 
−   
−  Title: Convergence to the complex balanced equilibrium for some reactiondiffusion systems with boundary equilibria.
 
−   
−  Abstract: We first analyze a threespecies system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general twospecies reversible reactiondiffusion network with boundary equilibria.
 
−   
−  ===Jingrui Cheng===
 
−   
−  Title: Gradient estimate for complex MongeAmpere equations
 
−   
−  Abstract: We consider complex MongeAmpere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.
 
−   
−   
−  ===Yao Yao===
 
−   
−  Title: Radial symmetry of stationary and uniformlyrotating solutions in 2D incompressible fluid equations
 
−   
−  Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformlyrotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformlyrotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier GómezSerrano, Jaemin Park and Jia Shi.
 
−   
−  ===Jessica Lin===
 
−   
−  Title: Speeds and Homogenization for ReactionDiffusion Equations in Random Media
 
−   
−  Abstract:
 
−  The study of spreadings speeds, front speeds, and homogenization for reactiondiffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.
 
−   
−   
−   
−  ===Beomjun Choi===
 
−  In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex noncompact solution will be discussed.
 
−   
−  The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.K. Hung concerning the evolution of singular hypersurfaces.
 