The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials
The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials
复制标题
适用于整个截止软势范围的 Vlasov-Poisson-Boltzmann 系统
DOI：
10.1016/j.jfa.2016.09.017
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发表时间：
2017
影响因子：
1.7
通讯作者：
Mao Huijiang
中科院分区：
文献类型：
--
作者：
Xiao Qinghua;Xiong Linjie;Mao Huijiang
稀电子的动力学可以通过基本的单一物种弗拉索夫-泊松-玻尔兹曼系统进行建模，该系统描述了电子通过自洽静电场中的碰撞而相互作用。对于截止分子间相互作用，尽管最近在构建麦克斯韦附近柯西问题的全局光滑解方面取得了一些进展，但非常软势情况下的问题仍然没有得到解决。通过引入一种新的时间-速度加权能量方法，并基于对解本身及其关于空间和速度变量的一些导数的一些新的最优时间衰减估计，本手稿表明，所有截止软势的单种 Vlasov-Poisson-Boltzmann 系统的柯西问题确实存在一般初始扰动的唯一全局平滑解，对于一般初始扰动，无需满足[13]中针对适度软截止的情况强加的中性条件势，但假设在某些加权 Sobolev 空间中很小。我们的方法也适用于截止硬势的情况，因此为微扰框架中截止分子间相互作用的整个范围的麦克斯韦附近的单一物种 Vlasov-Poisson-Boltzmann 系统提供了令人满意的全局适定性理论。
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.