Deterministic particle approximation of aggregation-diffusion equations on unbounded domains
Abstract
We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle approximations with piecewise constant densities introduced in~Di Francesco-Rosini starting from general bounded initial densities converge strongly in L1 to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded domains and for arbitrary nonnegative bounded initial densities, thus extending the results in Gosse-Toscani, Matthes-Osberger, Mathes-Soellner (in which a no-vacuum condition is required) and giving an alternative approach to Carrillo-Craig-Patacchini in the one-dimensional case, including also subquadratic and superquadratic diffusions.