  This paper studies the derivation of the quadratic porous medium equation and
a class of cross-diffusion systems from nonlocal interactions. We prove
convergence of solutions of a nonlocal interaction equation, resp. system, to
solutions of the quadratic porous medium equation, resp. cross-diffusion
system, in the limit of a localising interaction kernel. The analysis is
carried out at the level of the (nonlocal) partial differential equations and
we use gradient flow techniques to derive bounds on energy, second order
moments, and logarithmic entropy. The dissipation of the latter yields
sufficient regularity to obtain compactness results and pass to the limit in
the localised convolutions. The strategy we propose relies on a discretisation
scheme, which can be slightly modified in order to extend our result to PDEs
without gradient flow structure. In particular, it does not require convexity
of the associated energies. Our analysis allows to treat the case of limiting
weak solutions of the non-viscous porous medium equation at relevant low
regularity, assuming the initial value to have finite energy and entropy.
