Uniqueness of bounded solutions to aggregation equations by optimal transport methods

Abstract

We show how to extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diffusion competition. They contain several mathematical biology models proposed in macroscopic descriptions of swarming and chemotaxis for the evolution of mass densities of individuals or cells. Uniqueness is shown for bounded nonnegative mass-preserving weak solutions without diffusion. In diffusive cases, we use a coupling method [16], [33], and thus we need a stochastic representation of the solution to hold. In summary, our results show, modulo certain technical hypotheses, that nonnegative mass-preserving solutions remain unique as long as their L∞-norm is controlled in time.