论文标题
球形对称爱因斯坦-Vlasov系统的静态解决方案：粒子数 - 卡西米尔方法
Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach
论文作者
论文摘要
对爱因斯坦 - 维拉索夫系统的球形对称解的存在是众所周知的。但是，无论静态解决方案是否作为变异问题的最小化者出现，这都是一个空旷的问题。除了自身感兴趣之外，与非线性稳定性的联系赋予了这个话题的重要性。在\ cite {wol}中考虑了这个问题，但是如\ cite {ak}所指出的那样，纸\ cite {wol}包含严重的缺陷。在这项工作中，我们通过求解能量密度$ρ$作为固定点问题的Euler-Lagrange方程来构建静态解决方案。 Euler-lagrange方程源自\ cite {wol}中引入的粒子数字函数。然后，我们在相位空间上定义了一个密度函数$ f $，该函数诱导了能量密度$ρ$，我们表明它构成了Einstein-Vlasov系统的静态解决方案。因此，我们严格解决\ cite {wol}试图证明的作者的部分。
Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to non-linear stability that gives this topic its importance. This problem was considered in \cite{Wol}, but as has been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In this work we construct static solutions by solving the Euler-Lagrange equation for the energy density $ρ$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in \cite{Wol}. We then define a density function $f$ on phase space which induces the energy density $ρ$ and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what the author of \cite{Wol} attempted to prove.
