Trend to Equilibrium for Systems with Small Cross-Diffusion
| Autor: | Markus Schmidtchen, Luca Alasio, Marie-Therese Wolfram, Helene Ranetbauer |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Numerical Analysis
Class (set theory) Cross diffusion Applied Mathematics 010102 general mathematics Type (model theory) 01 natural sciences 010101 applied mathematics Computational Mathematics Nonlinear system Mathematics - Analysis of PDEs Modeling and Simulation Convergence (routing) FOS: Mathematics Applied mathematics 0101 mathematics Special case Variety (universal algebra) Analysis Stationary state Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1906.08060 |
| Popis: | This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic 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 convergence result obtained for the general system and we derive sharper 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. |
| Databáze: | OpenAIRE |
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