Diffusion-aggregation models, gradient flows, Wasserstein distance, stationary solutions.

Diffusion-aggregation models from population biology can be interpreted as gradient flows with respect to the Euclidean Wasserstein distance (cf. the works of Ambrosio-Gigli-Savaré or Carrillo-McCann-Villani). We first present an improved existence theory in some special cases, in 1d, where the interaction potential need not be globally convex. Then, we prove that models without diffusion enjoy asymptotically stable measure valued equilibria, with various large time behaviors depending on the properties of the interaction potential. Finally, we prove existence of stationary compactly supported solutions for the model with (small) quadratic diffusion. The results are the product of a work in collaboration with Martin Burger (University of Muenster).

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