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 |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-913651 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/91365 |
| Parent Title (English): | SIAM Journal on Mathematical Analysis |
| Publisher: | Society for Industrial & Applied Mathematics (SIAM) |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 2023 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2021/12/14 |
| Volume: | 55 |
| Issue: | 3 |
| First Page: | 1766 |
| Last Page: | 1809 |
| DOI: | https://doi.org/10.1137/21M1466980 |
| 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 |
