Global existence for a system without self-diffusion and different mobilities

TL;DR

This paper proves the global existence of weak solutions for a one-dimensional cross-diffusion system modeling two interacting populations with different mobilities, using entropy estimates and Young measures.

Contribution

It introduces a new framework for proving global existence of solutions without self-diffusion, applicable to systems with different mobilities.

Findings

Existence of global weak solutions for the system.

Any admissible approximation sequence converges to a weak solution.

The approach uses entropy estimates and the div--curl lemma.

Abstract

We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a linear pressure law and different mobilities. For arbitrary bounded non-negative initial data, we show that any good approximation scheme, yields existence of global weak solutions. More precisely, we introduce a notion of \textit{admissible approximation sequence} and show that any such sequence admits a subsequence converging to a weak solution of the system. The strategy relies on entropy estimates and the div--curl lemma, in the framework of Young measures.