Optimal transport, geometry, kinetic and partial differential equations

拟议的研究涉及的几何最佳运输（OT），其数值分辨率，其应用几何不等式和偏微分方程（PDE）所产生的扩散过程和动力学模型。研究的基本主题可以分为四个部分。 1.动力学模型。科学中的各种现象可以通过动力学方程来建模;例如，描述鸟类群集/群集的动力学方程、颗粒介质方程、Vlasov-Poisson系统、（相对论）Vlasov-Maxwell系统及其达尔文近似。OT中的技术预计是有用的，在研究这些方程中，已经注意到的情况下的Vlasov-Poisson系统。我们计划研究颗粒介质方程解的整体存在性和大时间行为，并利用OT中的工具重点研究相对论Vlasov-Maxwell和相对论Vlasov-Darwin系统弱解的唯一性。还将探索一种改进的鸟类群集模型。 2.最佳运输几何学。OT问题定义了一个距离--瓦瑟斯坦距离--它取决于运输的基本成本。我们将探讨的几何结构的Wasserstein空间，作为流形，一般凸成本及其应用到抛物扩散方程。特别是，我们将研究相应的Wasserstein距离的收缩性质。 3. OT和PDE的数值分辨率。我们将解决一般凸成本在更高的维度的OT问题的数值解决方案，然后将其应用于数值求解抛物扩散方程，这是已知的“梯度流”在Wasserstein空间与这些成本。 4.几何不等式与偏微分方程的渐近分析。已知OT导致一些拟线性偏微分方程和线性/非线性Fokker-Planck型方程之间的对偶。我们计划使用这种对偶来研究这些拟线性偏微分方程。这个想法是成功的，在获得唯一性结果的基态准线性偏微分方程。我们期望用它来得到相应的发展方程的稳定性结果。

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.