Contractivity of Wasserstein distance and exponential decay for the Landau equation with Maxwellian molecules

We consider the Landau equation with Maxwell molecules and show two results: exponential decay of the relative $L^2$ norm and contractivity of the $2$-Wasserstein distance of two arbitrary solutions. The proof of the decay of the relative $L^2$ uses a careful analysis of the operator and weighted Poincar\'e inequalities. Using the framework recently introduced by Guillen and Silvestre in \cite{GS24}, we provide a new, short, intuitive and quantitative proof that the Landau equation is contractive in the 2-Wasserstein metric. To achieve this, we quantify the convexity of the $2$-Wasserstein distance.