Reduction and a Concentration-Compactness Principle for Energy-Casimir Functionals
Reduction and a Concentration-Compactness Principle for Energy-Casimir Functionals
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DOI：
10.1137/p0036141001389275
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发表时间：
2001-05
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影响因子：
--
通讯作者：
G. Rein
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文献类型：
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作者：
G. Rein
能量-Casimir泛函是数学物理中构造各种保守系统的定态及其非线性稳定性的有用工具。最近，郭勇和作者利用它们构造了恒星动力学中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，我们构造了作用于空间上质量密度的自然约化泛函，并在此背景下研究了紧性性质和极小子的存在性。这使郭英成和作者开发的技术进入了一个更一般的框架。我们恢复了P.-L.Lions[Ann]的集中紧致原理。安装H.庞加莱肛门。Non Lineaire，1(1984)，第109--145页]，并将我们的稳定性分析与G.Wolansky[Ann]的稳定性分析联系起来。安装H.庞加莱肛门。Non Lineaire，16(1999)，第15-48页]。
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 [Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), pp. 109--145] in a more specific setting and connect our stability analysis with that of G. Wolansky [Ann. Inst. H. Poincare Anal. Non Lineaire, 16 (1999), pp. 15--48].