Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering

We show that the Euclidean Wasserstein distance between two compactly supported solutions of the one–dimensional porous medium equation having the same center of mass decays to zero for large times. As a consequence, we detect an improved L 1 L^1 –rate of convergence of solutions of the one–dimensional porous medium equation towards well–centered self–similar Barenblatt profiles, as time goes to infinity.