On nonlinear cross-diffusion systems: an optimal transport approach

Kim, Inwon; Mészáros, Alpár Richárd

Abstract

We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, we find a stable initial configuration which allows the densities to be segregated. This leads to the evolution of a stable interface between the two densities, and to a stronger convergence result to the continuum limit. In particular derivation of a standard weak solution to the system is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow.

Citation

Kim, I., & Mészáros, A. R. (2018). On nonlinear cross-diffusion systems: an optimal transport approach. Calculus of Variations and Partial Differential Equations, 57(3), Article 79. https://doi.org/10.1007/s00526-018-1351-9

• Journal Article Type: Article
• Acceptance Date: Apr 8, 2018
• Online Publication Date: Apr 28, 2018
• Publication Date: Jun 30, 2018
• Deposit Date: Oct 1, 2019
• Publicly Available Date: Feb 28, 2020
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 57
• Issue: 3
• Article Number: 79
• DOI: https://doi.org/10.1007/s00526-018-1351-9
• Public URL: https://durham-repository.worktribe.com/output/1290209
• Related Public URLs: https://arxiv.org/abs/1705.02457