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

  • 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 ε≥0. The nonlinearities and potentials are chosen such that in the decoupled system for ε=0, the evolution is metrically contractive, with a global rate Λ>0. The coupling is a singular perturbation in the sense that for any ε>0, contractivity of the system is lost. Our main result is that for all sufficiently small ε>0, the global attraction to a unique steady state persists, with an exponential rate Λε=Λ−Kε. The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.

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Author:Lisa BeckGND, Daniel Matthes, Martina Zizza
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/91365
Parent Title (English):SIAM Journal on Mathematical Analysis
Publisher:Society for Industrial & Applied Mathematics (SIAM)
Year of first Publication:2023
Publishing Institution:Universität Augsburg
Release Date:2021/12/14
First Page:1766
Last Page:1809
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehr- und Forschungseinheit Angewandte Analysis
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht