preprint

Inserted: 22 may 2026
Last Updated: 22 may 2026

Year: 2021

Abstract:

We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self- and cross-diffusion, transportconfinement 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.