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

TL;DR

This paper establishes the existence and regularity of solutions for a complex nonlinear PDE system modeling multiple interacting species with porous medium behavior, using a fixed point approach under small cross-diffusion conditions.

Contribution

It introduces a novel existence proof for a multi-species porous medium system with nonlocal interactions, valid for small cross-diffusion parameters.

Findings

Existence of solutions for small cross-diffusion parameter elta<1.

Sobolev regularity of solutions in a bounded domain.

Numerical analysis of solutions for extreme elta values.

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 $\Omega$ in any spatial dimension. 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 combined with a vanishing diffusivity approach. The behaviour of solutions for extreme values of $\delta$ is studied numerically.