Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX BOLTZMANN EQUATION WITH EXTERNAL FORCE AND VLASOV-POISSON-BOLTZMANN SYSTEM IN INFINITE VACUUM Renjun Duan Department of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China Tong Yang Liu Bie Ju Centre of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China Changjiang Zhu Laboratory of Nonlinear Analysis, Department of Mathematics Central China Normal University, Wuhan 430079, P.R. China 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 R 3 x , let f (t, x, v) be the distribution function for particles at time t ≥ 0 with location x =(x 1 ,x 2 ,x 3 ) ∈ R 3 x and velocity v =(v 1 ,v 2 ,v 3 ) ∈ R 3 v . In the presence of an external force, the time evolution of f is governed by the Boltzmann equation as a fundamental equation in statistical physics, ∂ t f + v ·∇ x f + E ·∇ v f = J (f,f ), (1.1) with initial data f (0, x, v)= f 0 (x, v). (1.2) Here E = E(t, x, v) is the external force. The collision operator J (f,f ) describing the binary elastic collision takes the form: J (f,f )= Q(f,f ) − fR(f ), (1.3) with Q(f,f )(t, x, v)=  D B(θ, |v − v 1 |)f (t, x, v ′ )f (t, x, v ′ 1 ) dεdθdv 1 , (1.4) 2000 Mathematics Subject Classification. 76P05, 82C40, 74G25. Key words and phrases. Boltzmann equation, Vlasov-Poisson-Boltzmann System, global exis- tence, classical solutions. 1