A note on asymptotics of linear dissipative kinetic equations in bounded domains

Abstract: We establish $L^{2$-exponential} decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in $L^2$ is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from $L^{2$-hypocoercivity} and relative entropy methods.