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

我的研究重点是最优运输理论在动力学中的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.最优运输理论在动力学中的应用。空间非齐次动力学方程。这些方程描述了粒子在周围溶液中进行扩散、输运和碰撞的运动。它们被用作颗粒介质、激光中的电子或超离子导体中的离子的时间演化模型。数学方程是动力学的Fokker-Planck方程（如Vlasov-Poisson-Fokker-Planck系统，Boltzmann方程）和颗粒介质方程。来自最优输运理论的技术有望使我们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全局存在性、唯一性和大时间行为，特别是在存在平衡点的情况下，推导出这些解收敛到平衡点的-sharp-速率。2. 抛物扩散方程。通常，这些是与上面提到的相关的空间齐次方程（例如，Fokker-Planck型方程，p- laplace方程，双重退化方程）。我们计划研究关于最优输运微分结构的p-拉普拉斯方程和双重退化方程是否为梯度流（如已知的空间齐次Fokker-Planck型方程），并找到这些方程解的平衡点的-尖锐-收敛率。3. 几何不等式。几何不等式（如对数-索博列夫不等式）中的最佳常数和极值可以导致进化方程（如线性Fokker-Planck方程）的急剧衰减到平衡。我们打算用最优输运理论找出所有Gagliardo-Nirenberg不等式的最佳常数和极值，并将它们应用于演化方程的大时间渐近性研究。

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.