Classical solutions to the Fokker–Planck–BGK equation with infinite energy

We consider the BGK equation perturbed by Fokker–Planck operator, which is important in the kinetic theory of rarefied gases. This model equation, which we call the Fokker–Planck–BGK equation, has many physical features that the Fokker–Planck–Boltzmann equation possesses. By establishing a new moments lemma and L∞L∞ bounds for macroscopic quantities without the boundedness of energy, we get several global existence and uniqueness results to the Cauchy problem of the Fokker–Planck–BGK equation under various circumstances with infinite energy.