SHARP ENTROPY DISSIPATION BOUNDS AND EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR

SHARP ENTROPY DISSIPATION BOUNDS AND EXPLICIT RATE OF TREND TO EQUILIBRIUM FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION G. TOSCANI AND C. VILLANI Abstract. We derive a new lower bound for the entropy dissipa- tion associated with the spatially homogeneous Boltzmann equa- tion. 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 equa- tions. Several variants are discussed. Contents 1. Introduction 2 2. Preliminaries : Fokker-Planck and Landau equations 11 3. Symmetries for Boltzmann and Fokker-Planck equations 18 4. Integral representation of a lower bound for D 23 5. Main result 28 6. Extension to other kernels 35 7. The Kac model 37 8. Remarks about Fisher information and entropy dissipation 42 References 45 1

• kernels associated


• ?? ?


• another approach


• spatially homogeneous


• dv dv? d?


• boltzmann equa- tion


• constant kr


• kinetic equations


• boltzmann equation


• boltzmann's collision