Asymptotic stability of stationary solutions to the Euler-Poisson equationsarising in plasma physics
Asymptotic stability of stationary solutions to the Euler-Poisson equationsarising in plasma physics
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DOI：
10.3934/krm.2011.4.569
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发表时间：
2011-04
影响因子：
1
通讯作者：
Masahiro Suzuki
中科院分区：
文献类型：
--
作者：
Masahiro Suzuki
本论文的主要关注点是分析在等离子体接触的材料表面周围形成的鞘层。在这里，对于鞘层的形成，玻姆准则要求正离子的速度应该比某个物理常数快。正离子在等离子体中的行为由欧拉-泊松方程控制。在数学上，鞘被视为特殊的稳态解。首先，我们证明了玻姆准则给出了一个充分条件，通过使用相平面分析的定态解的存在。然后证明了在加权Sobolev空间中，只要初始扰动足够小，定态解是时间渐近稳定的。此外，我们还得到了时间整体解向稳态解的收敛速度与初始扰动的衰减速率的关系。这些定理是用加权能量法证明的。
The main concern of the present paper is to analyze a sheath formed around a surface of a material with which plasma contacts. Here, for a formation of the sheath, the Bohm criterion requires the velocity of positive ions should be faster than a certain physical constant. The behavior of positive ions in plasma is governed by the Euler-Poisson equations. Mathematically, the sheath is regarded as a special stationary solution. We first show that the Bohm criterion gives a sufficient condition for an existence of the stationary solution by using the phase plane analysis. Then it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in the weighted Sobolev space. Moreover we obtain the convergence rate of the time global solution towards the stationary solution subject to the decay rate of the initial perturbation. These theorems are proved by a weighted energy method.