Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous

We derive a new lower bound for the entropy dissipation associated with the spatially homogeneous Boltzmann equation. This bound is expressed in terms of the relative entropy with respect to the equilibrium, and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy, with an explicit rate. Our result gives a considerable refinement of the analogous estimate by Carlen and Carvalho (9,10), under very little additional assumptions. Our proof takes advantage of the structure of Boltzmann's collision operator with respect to the tensor product, and its links with Fokker-Planck and Landau equations. Several variants are discussed.