Hypocoercivity of linear degenerately dissipative kinetic equations

In this paper we develop a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator are considered when the spatial domain takes the whole space or torus and when there is a confining force or not. The key part of the developed approach is to construct some equivalent temporal energy functionals for obtaining time rates of the solution trending towards equilibrium in some Hilbert spaces. The result in the case of the linear Boltzmann equation with confining forces is new. The proof mainly makes use of the macro-micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic-parabolic system. At the end, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.