Hypocoercivity of linear kinetic equations via Harris's Theorem

Ikusi/ Ireki
Data
2019-02-27Laburpena
We study convergence to equilibrium of the linear relaxation
Boltzmann (also known as linear BGK) and the linear Boltzmann
equations either on the torus
$(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space
$(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with 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.