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Trend to equilibrium for systems with small cross-diffusion
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Alasio, Luca, Ranetbauer, Helene, Schmidtchen, Markus and Wolfram, Marie-Therese (2020) Trend to equilibrium for systems with small cross-diffusion. ESAIM: Mathematical Modelling and Numerical Analysis, 54 (5). 1661-1688 . doi:10.1051/m2an/2020008 ISSN 0764-583X.
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Official URL: http://dx.doi.org/10.1051/m2an/2020008
Abstract
This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which have been proposed to describe the dynamics of a variety of large systems of interacting particles. Under suitable assumptions, we prove existence of classical solutions and we show exponential convergence in time to the stationary state. Furthermore, we consider the special case of one mobile and one immobile species, for which the system reduces to a nonlinear equation of Fokker-Planck type. In this framework, we improve the equilibration result obtained for the general system and we derive L∞-bounds for the solutions in two spatial dimensions. We conclude by illustrating the behaviour of solutions with numerical experiments in one and two spatial dimensions.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | ESAIM: Mathematical Modelling and Numerical Analysis | ||||||
Publisher: | EDP Sciences | ||||||
ISSN: | 0764-583X | ||||||
Official Date: | 28 July 2020 | ||||||
Dates: |
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Volume: | 54 | ||||||
Number: | 5 | ||||||
Page Range: | 1661-1688 | ||||||
DOI: | 10.1051/m2an/2020008 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Open Access Version: |
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