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

We prove the 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 models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain omega in any spatial dimension and originates from a many-particle system. The cross interactions between different species are scaled by a parameter delta < 1, with the delta=0 case corresponding to no interactions across species. A smallness condition on delta ensures existence of solutions up to an arbitrary time T > 0 in a subspace of L-2(0,T;H-1(omega)). This is shown via a Schauder fixed point argument for a regularised system followed by a vanishing diffusivity approach. The proof uses the lower semicontinuity of the Fisher information in combination with the div-curl Lemma. An ad hoc weak-strong uniqueness result ensures equivalence between weak formulations of the regularised problem; this is proved by studying a related dual problem. We provide numerical evidence showing blow-up of the Sobolev norm for delta -> 1. (C) 2022 The Author(s). Published by Elsevier Ltd.