[00418] Nonlinear PDE: beyond the well-posedness theory
Session Time & Room : 2C (Aug.22, 13:20-15:00) @F402
Type : Proposal of Minisymposium
Abstract : The theory of nonlinear partial differential equations (PDEs) is of fundamental importance in mathematical analysis, and through recent developments, it has reached a stage where some difficult and important questions beyond the well-posedness theory can be fruitfully addressed. The aim of this session focuses on a large class of nonlinear PDEs particularly related to Hamilton-Jacobi equations, level-set mean curvature flow equations, mean field games, and reaction diffusion equations, and brings experts to give a constructive and inspiring reflection on the state of the literature surrounding such equations, which will boost some further research in related areas.
[01572] Hessian Riemannian flows in mean-field games
Format : Talk at Waseda University
Author(s) :
Diogo Gomes (KAUST)
Abstract : Hessian Riemannian flows are a powerful tool for the construction of numerical schemes for monotone mean-field games that have their origin in constrained optimization problems. In this talk, we discuss the general construction of these flows for monotone mean-field games, their existence and regularity properties, and their asymptotic convergence.
[02413] Homogenization of Reactions in Random Media
Format : Talk at Waseda University
Author(s) :
Yuming Paul Zhang (Auburn University)
Andrej Zlatos (University of California, San Diego)
Abstract : Homogenization is a general phenomenon when physical processes in periodic or random environments exhibit homogeneous long time dynamic due to large space averaging of the variations in the environment. While this area of Mathematics saw a slew of remarkable developments in the last 20 years, the progress in the case of reaction-diffusion equations has been somewhat limited due to the homogenized dynamic involving discontinuous solutions to different (first-order) equations. In this talk I will discuss stochastic homogenization for reaction-diffusion equations in several spatial dimensions. These include the cases of both time-independent and time-dependent reactions, with the later proof employing a new subadditive ergodic theorem for time-dependent environments. This talk is based on joint works with Andrej Zlatoš.
[02535] Continuum limit of dislocations with annihilation in one dimension
Format : Talk at Waseda University
Author(s) :
Norbert Pozar (Kanazawa University)
Abstract : In this talk I discuss the many-particle limit for a system of particles in one dimension. The particles carry a signed charge, interact via a Newtonian potential and when two particles with opposite charges meet, they annihilate and are removed from the system. This serves as a simplified model of dislocation dynamics in a crystalline lattice. This talk is based on joint work with Mark Peletier and Patrick van Meurs.
[04586] Quantitative periodic homogenization of a front propagation model in dynamic environments
Format : Talk at Waseda University
Author(s) :
Wenjia Jing (Tsinghua University)
Abstract : In this talk we review the developments of homogenization theory for a front propagation model. It is described by a first order Hamilton-Jacobi equation with a Hamiltonian that grows linearly with respect to the absolute value of the momentum variable. We focus on the case of dynamic environment where the Hamiltonian has highly oscillations in time as well as in space. We present some key steps in the proof of the qualitative homogenization theory and in the quantification of the convergence rate in the periodic setting. The talk is based on several joint works with Souganidis, Tran and Yu.
