On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation

Abstract

As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-∞). Our results hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity, which at the moment have only been established in certain particular cases. Among the most important steps in our proof are 1) quantitative variants of Boltzmann’s H-theorem, as proven in [52,60], based on symmetry features, hypercontractivity and information-theoretical tools; 2) a new, quantitative version of the instability of the hydrodynamic description for non-small Knudsen number; 3) some functional inequalities with geometrical content, in particular the Korn-type inequality which we established in [22]; and 4) the study of a system of coupled differential inequalities of second order, by a treatment inspired from [21]. We also briefly point out the particular role of conformal velocity fields, when they are allowed by the geometry of the problem.