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On nonlinear cross-diffusion systems: an optimal transport approach

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, where the densities are guaranteed to be segregated, a stable interface appears between the two densities, and a stronger convergence result, 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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https://hal.inria.fr/hal-01575286
Contributor : Alpár Richárd Mészáros <>
Submitted on : Friday, August 18, 2017 - 4:26:21 PM
Last modification on : Wednesday, August 7, 2019 - 12:14:40 PM

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  • HAL Id : hal-01575286, version 1
  • ARXIV : 1705.02457

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Inwon Kim, Alpár R. Mészáros. On nonlinear cross-diffusion systems: an optimal transport approach. 2017. ⟨hal-01575286⟩

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