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 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): |  CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand) |
