March  2006, 16(1): 253-277. doi: 10.3934/dcds.2006.16.253

Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum

1. 

Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

2. 

Liu Bie Ju Centre of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

3. 

Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, China

Received  May 2005 Revised  February 2006 Published  June 2006

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 Boltzmann 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.
Citation: Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253
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