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Transport distances for PDEs: the coupling method. (English) Zbl 1459.35037

Summary: We informally review a few PDEs for which some transport cost 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 means that we double the variables and solve, globally in time, a well-chosen PDE posed on the Euclidean square of the physical space. 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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35Q20 Boltzmann equations
35Q49 Transport equations
35Q84 Fokker-Planck equations
35K55 Nonlinear parabolic equations
35R11 Fractional partial differential equations
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