Gradient flows

Abstract

In this chapter we present one of the most spectacular applications of optimal transport and Wasserstein distances to PDEs. We will see that several evolution PDEs can be interpreted as a steepest descent movement in the space W2. This includes the Heat equation, the Fokker-Planck equation, and many others. We will present the main ideas, provide a rigorous analysis of the Fokker-Planck case, and, possible extension in the discussion section. The discussion also presents complementary topics about the theoretical framework of gradient flows in metric spaces, and about other models in evolutionary PDEs which are connected to optimal transport but are not gradient flows.