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Monge-Kantorovich distance for PDEs: the coupling method

Abstract : We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.
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https://hal.sorbonne-universite.fr/hal-02080155
Contributor : Benoît Perthame <>
Submitted on : Tuesday, March 26, 2019 - 3:02:41 PM
Last modification on : Friday, April 10, 2020 - 5:27:02 PM
Document(s) archivé(s) le : Thursday, June 27, 2019 - 4:54:22 PM

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  • HAL Id : hal-02080155, version 1
  • ARXIV : 1903.11349

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Nicolas Fournier, Benoît Perthame. Monge-Kantorovich distance for PDEs: the coupling method. 2019. ⟨hal-02080155⟩

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