Exponential convergence to equilibrium for coupled systems of nonlinear degenerate drift diffusion equations

TL;DR

This paper proves that coupled nonlinear drift-diffusion systems with a gradient flow structure exhibit exponential convergence to equilibrium, even with small coupling perturbations that break the contractivity of the decoupled system.

Contribution

It demonstrates that exponential convergence persists under small coupling perturbations, extending the understanding of long-time behavior in coupled nonlinear diffusion systems.

Findings

Exponential convergence to equilibrium for small coupling parameter

Persistence of global attraction despite loss of contractivity

Quantitative rate of convergence depending on coupling strength

Abstract

We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter $\varepsilon\ge0$. The nonlinearities and potentials are chosen such that in the decoupled system for $\varepsilon=0$, the evolution is metrically contractive, with a global rate $\Lambda>0$. The coupling is a singular perturbation in the sense that for any $\varepsilon>0$, contractivity of the system is lost. Our main result is that for all sufficiently small $\varepsilon>0$, the global attraction to a unique steady state persists, with an exponential rate $\Lambda_\varepsilon=\Lambda-K\varepsilon$. The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.