Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure

TL;DR

This paper proves that solutions to the one-dimensional fractional porous medium equation converge exponentially fast to steady states using a novel functional inequality based on displacement convexity.

Contribution

It introduces a new approach leveraging displacement convexity of the Riesz potential to establish exponential convergence in the fractional porous medium context.

Findings

Solutions converge exponentially to steady states.

A new functional inequality links entropy, dissipation, and transport distance.

The method applies to the fractional porous medium equation in one dimension.

Abstract

We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and V\'azquez, where the pressure is obtained as a Riesz potential associated to the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states.