The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials

详细信息    查看全文

• 作者：Qinghua Xiao^a ; ^{pdexqh@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; Linjie Xiong^b ; ^{xlj@hnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Huijiang Zhao^c ; ^{hhjjzhao@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：One-species Vlasov&ndash ; Poisson&ndash ; Boltzmann system ; Cutoff soft potentials ; Global solutions near Maxwellians ; Time&ndash ; velocity weighted energy method
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：272
• 期：1
• 页码：166-226
• 全文大小：849 K

文摘

The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff intermolecular interactions, although there is some progress on the construction of global smooth solutions to its Cauchy problem near Maxwellians recently, the problem for the case of very soft potentials remains unsolved. By introducing a new time–velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables, it is shown in this manuscript that the Cauchy problem of the one-species Vlasov–Poisson–Boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in [13] for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted Sobolev spaces. Our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well-posedness theory to the one-species Vlasov–Poisson–Boltzmann system near Maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework.