On a cross-diffusion segregation problem arising from a model of interacting particles

Abstract

We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.