能量-Casimir泛函的约化和集中紧性原理
Reduction and a concentration-compactness principle for energy-Casimir functionals
摘要：
能量-Casimir泛函是数学物理中构造各种保守系统定态和分析其非线性稳定性的有用工具。最近，Y. Guo和作者利用它们构造了恒星动力学中的Vlasov-Poisson系统的稳定定态，其中能量-Casimir泛函作用于相空间上的数密度函数。在本文中，我们构建自然，减少功能的作用在空间上的质量密度和研究的紧性和存在性极小在这方面。这就把Y.Guo和作者开发的技术放到了一个更一般的框架中。我们恢复了P. L. Lions的集中紧性原理，并将我们的稳定性分析与G. Wolansky的稳定性分析联系起来。
英文摘要：
Energy-Casimir functionals are a useful tool for the construction of steady states and the analysis of their nonlinear stability properties for a variety of conservative systems in mathematical physics. Recently, Y. Guo and the author employed them to construct stable steady states for the Vlasov-Poisson system in stellar dynamics, where the energy-Casimir functionals act on number density functions on phase space. In the present paper we construct natural, reduced functionals which act on mass densities on space and study compactness properties and the existence of minimizers in this context. This puts the techniques developed by Y.Guo and the author into a more general framework. We recover the concentration-compactness principle due to P.L.Lions in a more specific setting and connect our stability analysis with the one of G.Wolansky.
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