Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems

TL;DR

This paper proves exponential convergence to equilibrium for nonlinear reaction-diffusion systems from chemical networks, using the entropy method, and provides explicit rates and conditions for systems with boundary equilibria.

Contribution

It establishes convergence results for renormalised solutions to complex balanced reaction-diffusion systems, including systems with boundary equilibria, with explicit rates and novel analytical techniques.

Findings

Exponential convergence to equilibrium is shown for systems without boundary equilibria.

A sufficient condition for convergence in systems with boundary equilibria is derived.

An explicit proof method is provided for a reversible enzyme reaction model.

Abstract

The convergence to equilibrium for renormalised solutions to nonlinear reaction-diffusion systems is studied. The considered reaction-diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.