Transport distances for PDEs: the coupling method

Abstract

We informally reviewa fewPDEs 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.