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arXiv · 1903.11349

Monge-Kantorovich distance for PDEs: the coupling method

Nicolas Fournier·Benoît Perthame
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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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Nicolas Fournier, Benoît Perthame. 2025-08-26. Monge-Kantorovich distance for PDEs: the coupling method. https://doi.org/10.4171/emss%2F35

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