Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts

Abstract

We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self-and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain Ω in any spatial dimension. The cross interactions between different species are scaled by a parameter δ < 1, with the δ = 0 case corresponding to no interactions across species. A smallness condition on δ ensures existence of solutions up to an arbitrary time T > 0 in a subspace of L 2 (0, T ; H 1 (Ω)). This is shown via a Schauder fixed point argument for a regularised system combined with a vanishing diffusivity approach. The behaviour of solutions for extreme values of δ is studied numerically.