Entropic Methods for the Study of the Long Time Behavior of Kinetic Equations

We show how the use of the concepts of entropy and entropy dissipation help to nd estimates of return to the equilibrium in the context of kinetic equations with a collision kernel. This approach enables to give explicit constants in those estimates. 1. INTRODUCTION We wish to present in this work the entropy dissipation method for the study of the long time behavior of kinetic equations with a collision kernel. Let us rst consider the Cauchy problem for a spatially homogeneous equation with a collision kernel Q acting only on the velocity variable v. Such an equation writes