The existence of classical solution for the Fokker-Planck-Boltzmann equation

We will study the non-stationary state homogeneous Fokker-Planck-Boltzmann equation ∂f/∂t - νΔ_ξ f = Q(f, f)(ξ, t) in this article. By means of semi-group theory and compactness argument, we prove the global existence of the classical solutions with initial data in L₂¹ (R³) in the case of hard potentials with angular cut-off. Furthermore we have mass, momentum conservation and energy linear relation identity.