ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR HARD POTENTIALS

ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR HARD POTENTIALS PART II : H-THEOREM AND APPLICATIONS L. DESVILLETTES AND C. VILLANI Abstract. We find a lower bound for the entropy dissipation of the spatially homogeneous Landau equation with hard potentials in terms of the entropy itself. We deduce from this explicit estimates on the speed of convergence towards equilibrium for the solution of this equation. In the case of so-called overmaxwellian potentials, the convergence is exponential. We also compute a lower bound for the spectral gap of the associated linear operator in this setting. Contents 1. Introduction and main result 1 2. Entropy dissipation : first method 8 3. Entropy dissipation : second method 13 4. The trend towards equilibrium : overmaxwellian case 16 5. Improved results 18 6. The trend towards equilibrium : the case of true hard potentials 21 7. Poincare-type inequalities and applications 24 8. Entropy dissipation and regularity estimates 26 Appendix A. Definition of the entropy dissipation 27 Appendix B. Approximation of the entropy dissipation 29 References 30 1. Introduction and main result We recall the spatially homogeneous Landau equation (Cf. [8, 18]), (1) ∂f∂t (t, v) = Q(f, f)(t, v), v ? R N , t ≥ 0, 1

• landau equation


• course maxwellian molecules


• dissipation


• spatially homogeneous


• boltzmann equa- tion


• cross section


• maxwellian molecules


• entropy dissipation