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

Following the breakthrough work of Guillen and Silvestre \cite{GS24}, that shows that the Fisher information is monotonically decreasing for solutions to the homogeneous Landau equation, we study, for the same equation, the monotonicity properties of other physically relevant functionals. In the case of Maxwellian molecules, we show that the relative $L^2$ norm with respect to the equilibrium decays exponentially fast in time and is monotonically decreasing after some time. Moreover, still for the Maxwellian case, we provide a novel and short quantitative proof of time monotonicity of the entropic Wasserstein metric. For soft potentials, we show that the Wasserstein metric is contractive, conditional to $L^1(0,T,L^p({\mathbb{R}}^3))$ bound for the solution. This result provides an alternative proof of the Fournier and Fournier-Guerin uniqueness theorem in \cite{fournier2009well_posedness_soft_potentials} \cite{fournier2010uniqueness_Coulomb}