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On nonlinear cross-diffusion systems: an optimal transport approach

Kim, Inwon; Mészáros, Alpár Richárd

On nonlinear cross-diffusion systems: an optimal transport approach Thumbnail


Authors

Inwon Kim



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
Related Public URLs https://arxiv.org/abs/1705.02457

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Copyright Statement
This is a post-peer-review, pre-copyedit version of an article published in [Calculus of variations and partial differential equations. The final authenticated version is available online at: https://doi.org/10.1007/s00526-018-1351-9





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