Optimal transport, geometry, kinetic and partial differential equations

提出的研究涉及最优输运（OT）的几何，其数值分辨率，及其在扩散过程和动力学模型中产生的几何不等式和偏微分方程（PDE）中的应用。本研究的基本主题可分为四个部分。

The proposed research concerns the geometry of optimal transport (OT), its numerical resolution, and its applications to geometric inequalities and to partial differential equations (PDE) arising in diffusion processes and in kinetic models. The fundamental themes of the research can be organized in four sections. 1. Kinetic models. Various phenomena in sciences can be modeled by kinetic equations; e.g., kinetic equations describing the flocking/swarming of birds, granular media equations, the Vlasov-Poisson system, the (relativistic) Vlasov-Maxwell system and its Darwin approximation. Techniques in OT are expected to be useful in studying these equations, as already noticed in the case of the Vlasov-Poisson system. We plan to investigate the global existence and large time behavior of solutions to the granular media equation, and to focus on the uniqueness of weak solutions for the relativistic Vlasov-Maxwell and relativistic Vlasov-Darwin systems using tools in OT. Also to be explored is an improved model of flocking and swarming of birds. 2. Geometry of Optimal transport. The OT problem defines a distance - the Wasserstein distance - which depends on the basic cost of transportation. We will explore the geometric structure of the Wasserstein spaces, as manifolds, for general convex costs and their applications to parabolic diffusion equations. In particular, we will investigate the contraction properties of the corresponding Wasserstein distances. 3. Numerical resolution of OT and PDEs. We will address the numerical resolution of the OT problem for general convex costs in higher dimensions, then apply it to numerically solve parabolic diffusion equations which are known to be "gradient flows" in the Wasserstein spaces associated to these costs. 4. Geometric inequalities & Asymptotic analysis for PDEs. OT is known to leads to a duality between some quasilinear PDEs and linear/nonlinear Fokker-Planck type equations at equilibrium. We plan to use this duality to study these quasilinear PDEs. This idea was successful in obtaining uniqueness results on ground states for quasilinear PDEs. We expect to use it to derive stability results on their corresponding evolution equations.