Applications of optical mass transport theory to partial differential equations and to geometric inequalities

本文主要研究最优输运理论在运动理论中的偏微分方程组和几何不等式中的应用，主要包括以下三个部分：1.空间非齐次运动方程。这些方程描述了粒子在周围熔池中的扩散、传输和碰撞的运动。它们被用作颗粒介质、激光中的电子或快离子导体中的离子的时间演化的模型。建立的数学方程是基本的动力学Fokker-Planck方程(如Vlasov-Poisson-Fokker-Planck系统、Boltzmann方程)和颗粒介质方程。来自最优运输理论的技术有望导致对这些方程的更好理解。本研究的目的是利用最优运输理论中的工具来研究这些方程解的整体存在性、唯一性和大时间性态，特别地，如果存在平衡点，则得到这些解收敛到平衡点的-锐率。2.抛物型扩散方程。通常，这些方程是与上述方程相关的空间齐次方程(例如，福克-普朗克型方程、p-拉普拉斯方程、双重退化方程)。我们计划研究p-Laplace方程和双重退化方程是否是关于最优传输微分结构的梯度流(就像已知的空间齐次Fokker-Planck型方程)，并找到这些方程解的收敛到平衡点的-锐收敛速度。3.几何不等式。几何不等式(如对数-索布列夫不等式)中的最佳常量和极值会导致演化方程(如线性福克-普朗克方程)的急剧衰减到平衡点。我们打算利用最优传输理论来寻找所有Gagliardo-Nirenberg不等式的最佳常数和极值，并将它们应用于研究发展方程的大时间渐近性。

My research focuses on applications of Optimal transportation theory to partial differential equations arising in kinetic theory, and to geometric inequalities, and it consists of the following three parts: 1. Spatially inhomogeneous kinetic equations. These equations describe the motion of particles undergoing diffusion, transport and collision in a surrounding bath. They are used as models for the time evolution of granular media, of electrons in a laser light, or ions in a superionic conductor. The mathematical equations are the  kinetic Fokker-Planck equations (e.g. Vlasov-Poisson-Fokker-Planck system, Boltzmann equation), and the granular media equations. Techniques from Optimal transportion theory are expected to lead to a better understanding of these equations. The purpose of this research is to use tools from Optimal transportation theory to study the global existence, uniqueness and large time behavior of solutions of these equations, and in particular, derive the -sharp- rates at which these solutions converge to an equilibrium, in case there is one. 2. Parabolic diffusion equations. Typically, these are the spatially homogeneous equations associated to the ones mentioned above (e.g. Fokker-Planck type equations, p-Laplacian equations, doubly degenerate equations). We plan to investigate whether the p-Laplacian and the doubly degenerate equations are gradient flows with respect to an optimal transportation differential structure (as known already for the spatially homogeneous Fokker-Planck type equations), and to find the -sharp- rates of convergence to equilibria of the solutions of these equations. 3. Geometric inequalities. Best constants and extremals in geometric inequalities (e.g. Logarithmic-Sobolev inequality) can lead to sharp rates of decay to equilibria for evolution equations (e.g. linear Fokker-Planck equation). We intend to use Optimal transportion theory to find the best constants and extremals of all the Gagliardo-Nirenberg inequalities, and apply them to study large time asymptotics of evolution equations.