Optimal transport, geometry, kinetic and partial differential equations

这项研究涉及最优输运(OT)的几何、数值解，以及它在几何不等式、扩散过程和动力学模型中产生的偏微分方程(PDE)中的应用。研究的基本主题可分为四个部分。 1.动力学模型。科学中的各种现象可以用动力学方程来模拟；例如，描述鸟类聚集/聚集的动力学方程、颗粒介质方程、弗拉索夫-泊松系统、(相对论)弗拉索夫-麦克斯韦系统及其达尔文近似。正如在Vlasov-Poisson系统中已经注意到的那样，OT中的技巧有望在研究这些方程时有用。我们计划研究颗粒介质方程解的整体存在性和大时间性态，并利用OT中的工具研究相对论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成群和成群模型。 2.最优运输几何。加班问题定义了一个距离--沃瑟斯坦距离--这取决于运输的基本成本。我们将探索Wasserstein空间的几何结构，作为流形，对于一般的凸成本及其在抛物扩散方程中的应用。特别地，我们将研究相应的Wasserstein距离的压缩性质。 3.OT和PDEs的数值解。我们将解决高维一般凸成本的OT问题的数值解，然后将其应用于数值求解抛物型扩散方程，这些抛物型扩散方程被认为是与这些成本相关的Wasserstein空间中的“梯度流”。 4.偏微分方程解的几何不等式及渐近分析。已知OT导致某些拟线性偏微分方程组与线性/非线性Fokker-Planck型方程在平衡点的对偶性。我们计划利用这种对偶性来研究这些拟线性偏微分方程组。这一思想成功地获得了拟线性偏微分方程组基态的唯一性结果。我们期望用它来推导出它们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.