Bruna, Maria, Burger, Martin, Ranetbauer, Helene and Wolfram, Marie-Therese (2017) Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures. Journal of Nonlinear Science, 27 (2). pp. 687-719. doi:10.1007/s00332-016-9348-z ISSN 0938-8974.

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Abstract

In this paper, we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived in Bruna and Chapman (J Chem Phys 137:204116-1–204116-16, 2012a) from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally, we illustrate the behavior of the model with various numerical simulations.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differential equations, Parabolic -- Numerical solutions, Asymptotic expansions
Journal or Publication Title:	Journal of Nonlinear Science
Publisher:	Springer
ISSN:	0938-8974
Official Date:	April 2017
Dates:	Date Event April 2017 Published 10 November 2016 Available 25 October 2016 Accepted
Volume:	27
Number:	2
Page Range:	pp. 687-719
DOI:	10.1007/s00332-016-9348-z
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	17 November 2016
Date of first compliant Open Access:	17 November 2016
Funder:	Deutsche Forschungsgemeinschaft (DFG), Österreichische Akademie der Wissenschaften [Austrian Academy of Sciences]
Grant number:	EXC1003 CiM (DFG), New Frontiers Group NST-001 (Austrian Academy of Sciences)

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