Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts. (October 2022) - Alasio, Luca; Bruna, Maria; Fagioli, Simone; Schulz, Simon
Abstract: 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 Ω in any spatial dimension and originates from a many-particle system. 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 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 δ → 1 .