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.