BOLTZMANN EQUATION WITH EXTERNAL FORCE AND VLASOV-POISSON-BOLTZMANN SYSTEM IN INFINITE VACUUM

Abstract. In this paper, we study the Cauchy problem for the Boltzmann equation with an external force and the Vlasov-Poisson-Boltzmann system in infinite vacuum. The global existence of solutions is first proved for the Boltz-mann equation with an external force which is integrable with respect to time in some sense under the smallness assumption on initial data in weighted norms. For the Vlasov-Poisson-Boltzmann system, the smallness assumption on initial data leads to the decay of the potential field which in turn gives the global existence of solutions by the result on the case with external forces and an iteration argument. The results obtained here generalize those previous works on these topics and they hold for a class of general cross sections including the hard-sphere model. 1. Introduction. For a rarefied gas in the whole space R3x, let f(t, x, v) be the distribution function for particles at time t ≥ 0 with location x = (x1, x2, x3) ∈ R3x and velocity v = (v1, v2, v3) ∈ R3v. In the presence of an external force, the time evolution of f is governed by the Boltzmann equation as a fundamental equatio