Exponential Convergence to Equilibrium for Coupled Systems of Nonlinear Degenerate Drift Diffusion Equations

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 . The nonlinearities and potentials are chosen such that in the decoupled system for , the evolution is metrically contractive, with a global rate . The coupling is a singular perturbation in the sense that for any , contractivity of the system is lost. Our main result is that for all sufficiently small , the global attraction to a unique steady state persists, with an exponential rate for some . The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.