From non-local to classical SKT systems: triangular case with bounded coefficients

TL;DR

This paper rigorously derives classical cross-diffusion SKT systems as limits of non-local systems with bounded coefficients, focusing on triangular cases using a new compactness result for the Kolmogorov equation.

Contribution

It provides the first rigorous derivation of classical SKT systems from non-local models in the triangular case with bounded coefficients.

Findings

Established a new compactness result for the Kolmogorov equation.

Proved convergence of non-local systems to classical SKT systems in the triangular case.

Extended the existence results to the asymptotic regime of convolution kernels.

Abstract

This paper solves partially a question suggested by Fontbona and M\'el\'eard in a paper published in 2015. The issue is to obtain rigorously cross-diffusion systems \`a la Shigesada-Kawasaki-Teramoto as the limit of relaxed systems in which the cross-diffusion and reaction coefficients are non-local. We depart from the existence result established by Fontbona M\'el\'eard for a general class of non-local systems and study the corresponding asymptotic as the convolution kernels tend to Dirac masses, but only in the case of (strictly) triangular systems, with bounded coefficients. Our approach is based on a new result of compactness for the Kolmogorov equation, which is reminiscent of the celebrated duality lemma of Michel Pierre.