Finite speed of propagation in porous media by mass transportation methods

Abstract

In this note we make use of mass transportation techniques to give a simple proof of the finite speed of propagation of the solution to the one-dimensional porous medium equation. The result follows by showing that the difference of support of any two solutions corresponding to different compactly supported initial data is a bounded in time function of a suitable Monge-Kantorovich related metric. To cite this article:

Scalar conservation laws seen as gradient flows: known results and new perspectives

“…The examples considered in [22,28] showed that the pseudo-inverse form is a very powerful tool to investigate contraction properties of a semigroup in one-dimensional Wasserstein spaces. In the special case of a gradient flow (1) on P 2 , in which v = ∂F[ρ], one can formally see that the pseudo-inverse form of (1) becomes…”

“…Let us consider the pseudo-inverse equation for (22). Let µ be a gradient flow solution to (22) and let…”

“…The natural question arises on whether F is an entropy solution for (25) provided µ is a Wasserstein gradient flow for (22) and viceversa. This task has been addressed in [8].…”

“…Roughly speaking, estimating the p-Wasserstein distance between two solutions ρ 1 and ρ 2 to (7) is equivalent to computing the L p distance between the two corresponding pseudo-inverse variables X ρ1 and X ρ2 upon a direct use of the PDE (12). The most striking example of application of the pseudo-inverse form for an evolutionary PDE is provided in [22,28] for the Porous Medium Equation (PME) (see also [44])…”

“…is a rigorous solution to (22) in the 2-Wasserstein gradient flow sense. We omit the details for simplicity.…”