In this paper, we review some recent results on the Boltzmann equation near the equilibrium states in the whole space $\mathbb R^n$. The emphasize is put on the well-posedness of the solution in some Sobolev space without time derivatives and its uniform stability and optimal decay rates, and also on the existence and asymptotical stability of the time-periodic solution. Most of results obtained here are proved by combining the energy estimates and the spectral analysis.
"The Boltzmann Equation Near Equilibrium States in $\mathbb R^n$." Methods Appl. Anal. 14 (3) 227 - 250, September 2007.