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.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2020008
Mots-clés : Cross-diffusion systems, asymptotic behaviour
@article{M2AN_2020__54_5_1661_0,
author = {Alasio, Luca and Ranetbauer, Helene and Schmidtchen, Markus and Wolfram, Marie-Therese},
title = {Trend to equilibrium for systems with small cross-diffusion},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1661--1688},
publisher = {EDP-Sciences},
volume = {54},
number = {5},
year = {2020},
doi = {10.1051/m2an/2020008},
mrnumber = {4127953},
zbl = {1466.35027},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2020008/}
}
TY - JOUR AU - Alasio, Luca AU - Ranetbauer, Helene AU - Schmidtchen, Markus AU - Wolfram, Marie-Therese TI - Trend to equilibrium for systems with small cross-diffusion JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1661 EP - 1688 VL - 54 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2020008/ DO - 10.1051/m2an/2020008 LA - en ID - M2AN_2020__54_5_1661_0 ER -
%0 Journal Article %A Alasio, Luca %A Ranetbauer, Helene %A Schmidtchen, Markus %A Wolfram, Marie-Therese %T Trend to equilibrium for systems with small cross-diffusion %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1661-1688 %V 54 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2020008/ %R 10.1051/m2an/2020008 %G en %F M2AN_2020__54_5_1661_0
Alasio, Luca; Ranetbauer, Helene; Schmidtchen, Markus; Wolfram, Marie-Therese. Trend to equilibrium for systems with small cross-diffusion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1661-1688. doi : 10.1051/m2an/2020008. http://www.numdam.org/articles/10.1051/m2an/2020008/
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