  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.
