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

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 \(\delta0\) 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.