Systems of interacting particles, gradient flow approach in Wasserstein space.

Summary Since the seminal paper by Jordan, Kinderlehrer and Otto in 1998, it is well known that a large class of parabolic equations can be seen as gradient flows in Wasserstein space. The aim of this thesis is to extend this theory to some equations and systems which do not have exactly a gradient flow structure. The interactions studied are of different natures. The first chapter deals with systems with non local interactions in the drift. We then study cross-diffusion systems that apply to congestion models for several populations. Another model studied is the one where the coupling is in the reaction term like prey-predator systems with diffusion or tumor growth models. Finally, we will study new types of systems where the interaction is given by a multi-margin transport problem. A large part of these problems is illustrated by numerical simulations.