Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems

TL;DR

This paper proves that renormalized solutions to complex reaction-cross-diffusion systems in bounded domains converge exponentially to equilibrium, extending models to multiple species with a formal gradient-flow structure.

Contribution

It establishes exponential convergence to equilibrium for renormalized solutions of reaction-cross-diffusion systems with complex balanced reactions and non-symmetric diffusion matrices.

Findings

Renormalized solutions conserve mass and satisfy entropy inequalities.

All solutions converge exponentially to equilibrium under certain conditions.

The convergence rate is explicit up to a finite dimensional inequality.

Abstract

The convergence to equilibrium of renormalized solutions to reaction-cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, it can be proved under the assumption of no boundary equilibria that {\it all} renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.