Smooth solutions to the BGK equation and the ES-BGK equation with infinite energy

The BGK model and the ES-BGK model of the Boltzmann equation are of great importance in the kinetic theory of rarefied gases. For the Cauchy problems with general initial data, smooth solutions were obtained only in bounded domains for both models. In this paper, we establish the existence and uniqueness of smooth solutions in the whole space. More important, some smooth solutions can permit infiniteness of energy.