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

我的研究主要集中在最优运输理论在动力学偏微分方程和几何不等式中的应用，主要包括以下三个部分：1。空间非齐次动力学方程。这些方程描述了颗粒在周围熔池中进行扩散、输运和碰撞的运动。它们被用来作为模型的时间演变的粒状介质，电子在激光，或离子在超离子导体。数学方程是动力学Fokker-Planck方程（例如Vlasov-Poisson-Fokker-Planck系统，Boltzmann方程）和颗粒介质方程。从最优运输理论的技术，预计将导致更好地理解这些方程。本研究的目的是利用最优运输理论的工具来研究这些方程解的整体存在性、唯一性和大时间行为，特别是，如果存在一个平衡点，则导出这些解收敛到平衡点的锐速率。2.抛物扩散方程通常，这些是与上述方程相关联的空间齐次方程（例如，福克-普朗克型方程，p-Laplacian方程，双退化方程）。我们计划调查是否的p-Laplacian和双退化方程的梯度流相对于一个最佳的运输微分结构（如已知的空间齐次的福克-普朗克型方程），并找到这些方程的解的平衡收敛的尖锐率。3.几何不等式几何不等式（如对数-Sobolev不等式）中的最佳常数和极值会导致演化方程（如线性Fokker-Planck方程）快速衰减到平衡态。本文利用最优运输理论寻找所有Gagliardo-Nirenberg不等式的最佳常数和极值，并将其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.