Calculus of Variations and Geometric Measure Theory
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I. Kim - A. R. Mészáros

On nonlinear cross-diffusion systems: an optimal transport approach

created by mészáros on 13 May 2017
modified on 10 Apr 2018


Accepted Paper

Inserted: 13 may 2017
Last Updated: 10 apr 2018

Journal: Calc. Var. Partial Differential Equations
Year: 2018


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.


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