L 1 STABILITY FOR THE VLASOV-POISSON-BOLTZMANN SYSTEM AROUND VACUUM RENJUN DUAN Department of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China renjun duan@yahoo.com.cn MEI ZHANG Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, P.R. China mzh2613@163.com CHANGJIANG ZHU Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, P.R. China cjzhu@mail.ccnu.edu.cn Based on the global existence theory of the Vlasov-Poisson-Boltzmann system around vacuum in the N -dimensional phase space, in this paper, we prove the uniform L 1 sta- bility of classical solutions for small initial data when N ≥ 4. In particular, we show that the stability can be established directly for the soft potentials, while for the hard potentials and hard sphere model it is obtained through the construction of some non- linear functionals. These functionals thus generalize those constructed by Ha for the case without force to capture the effect of the force term on the time evolution of solutions. In addition, the local-in-time L 1 stability is also obtained for the case of N = 3. Keywords : Vlasov-Poisson-Boltzmann system; stability; nonlinear functionals. AMS Subject Classification: 76P05, 82D05, 93D05 1. Introduction In this paper, we consider the Cauchy problem of the Vlasov-Poisson-Boltzmann system (VPB in short):      ∂ t f + v ·∇ x f + E f ·∇ v f = J (f,f ), E f = ∇ x φ f ,  x φ f = ρ f =  fdv, (1.1) with initial data f (0, x, v)= f 0 (x, v), (1.2) 1