No abstract

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the toruswith a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively L 1 or weighted L 1 norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem. Contents

We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve similar results established by [26,5,6,14,10,11,1] to general force case, and improve the nonquantitative rate of convergence in [18] to quantitative explicit rate.

We prove the well-posedness for the non-cutoff Boltzmann equation with soft potentials when the initial datum is close to the global Maxwellian and has only polynomial decay at the large velocities in L 2 space. As a result, we get the propagation of the exponential moments and the sharp rates of the convergence to the global Maxwellian which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in L 2 space. Contents 1. Introduction 1 2. Analysis of the collision operator 8 3. Global well-posedness, propagation of moments and sharp convergence rate 26 4. Appendix: Toolbox and Proof of Theorem 1.2 43 References 70