  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.
