ON THE TREND TO EQUILIBRIUM FOR SOME DISSIPATIVE SYSTEMS WITH SLOWLY INCREASING

ON THE TREND TO EQUILIBRIUM FOR SOME DISSIPATIVE SYSTEMS WITH SLOWLY INCREASING A PRIORI BOUNDS G.TOSCANI AND C. VILLANI Abstract. We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails : in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uni- form in time estimates by the use of precise logarithmic Sobolev- type inequalities, and the assumption that the initial datum de- cays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover sit- uations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials). Contents 1. Introduction 1 2. The Fokker-Planck equation with weak drift 7 3. The Landau equation for mollified soft potentials 11 4. The Boltzmann equation for mollified soft potentials 21 References 26 1. Introduction We consider in this work the problem of trend to equilibrium for collisional kinetic equations of the form (1) ∂f∂t = Q(f) where the unknown f(t, v) ≥ 0 (t ≥ 0, v ? RN) is a probability den- sity on RNv , and Q is a collision operator which is mass-preserving and dissipative, in the sense that solutions of (1) make a certain entropy functional decrease with time.

• entropy functional


• ?h˙ ≥


• inter- action operators


• functional decrease


• ?? ?


• relative entropy


• collisional kinetic


• weak