Eulerian Calculus for the Contraction in the Wasserstein Distance

We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories, and on an Eulerian formulation for the Wasserstein distance using smooth curves. Our approach avoids the existence result for optimal transport maps on Riemannian manifolds.