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Journal Articles EMS Surveys in Mathematical Sciences Year : 2020

Transport costs 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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Dates and versions

hal-02080155 , version 1 (26-03-2019)
hal-02080155 , version 2 (25-03-2021)

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Nicolas Fournier, Benoît Perthame. Transport costs for PDEs: the coupling method. EMS Surveys in Mathematical Sciences, 2020, 7 (1), pp.1-31. ⟨10.4171/EMSS/35⟩. ⟨hal-02080155v2⟩
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