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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Metadaten
Author:Lisa BeckGND, Daniel Matthes, Martina Zizza
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/91365
URL:https://arxiv.org/abs/2112.05810
Parent Title (English):arXiv
Type:Preprint
Language:English
Year of first Publication:2021
Publishing Institution:Universität Augsburg
Release Date:2021/12/14
Issue:arXiv:2112.05810
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
Latest Publications (not yet published in print):Aktuelle Publikationen (noch nicht gedruckt erschienen)
Licence (German):CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand)