Regularities of the solutions to the Fokker–Planck–Boltzmann equation

In this paper, we consider the regularities of the solutions to the Fokker–Planck–Boltzmann equation. In particular, we prove that for hard sphere case, the strong solution constructed by Li and Matsumura [H. Li, A. Matsumura, Behavior of the Fokker–Planck–Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal. (2008), in press] near Maxwellian becomes immediately smooth with respect to all variables as long as t>0.