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.
Author: | Lisa BeckGND, Daniel Matthes, Martina Zizza |
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Frontdoor URL | https://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): | ![]() |