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

Kim, Inwon and Mészáros, Alpár Richárd (2018) 'On nonlinear cross-diffusion systems : an optimal transport approach.', Calculus of variations and partial differential equations., 57 (3). p. 79.


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.

Item Type:Article
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Publisher 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:
Date accepted:08 April 2018
Date deposited:28 February 2020
Date of first online publication:28 April 2018
Date first made open access:28 February 2020

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