We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell system and investigate various features of the solutions. This extends a previous investigation \cite{AR1} of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter there are no physically meaningful solutions. Furthermore, we investigate if the inequality \sqrt{M}\leq \frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}, derived in \cite{An2}, is sharp within the class of solutions to the Einstein-Vlasov-Maxwell system. Here M is the ADM mass, Q the charge, and R the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell system. Finally, we consider one parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider.
A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system
在之前的一篇论文中，两位作者（G。R.和A. D. R.）的情况下进行。结果表明，对于小的初始数据，存在球对称Vlasov-Einstein系统的整体经典解.本文件继续这项调查，并允许也大的初始数据。它表明，如果解决方案的球对称弗拉索夫-爱因斯坦系统开发的奇点，然后第一个奇点出现在对称中心。这一结果增加了宇宙审查的猜测，如果一个取代尘埃作为物质模型，裸奇点确实由弗拉索夫方程描述的无碰撞气体形成。主要的工具是一个估计，它表明，解决方案是全球性的，如果所有的物质仍然远离对称中心。
Veiled singularities for the spherically symmetric massless Einstein-Vlasov system
本文继续研究在[RV]中引发的无碰撞物质坍缩中裸奇点的形成。证明了Einstein-Vlasov系统的某些类非光滑解的存在性。这些解是自相似的，因此不是渐近平坦的。为了得到与物理更相关的解，尝试以适当的方式截断这些解，使它们渐近平坦是有意义的。这项任务，这是原来是技术上具有挑战性的，将在本文中进行。[RV]A. D. Rendall和J.J.L. Velázquez，无质量Einstein-Vlasov系统的一类尘埃自相似解。Annales Henri Poincare 12，919-964，（2011）.
Future stability of expanding spatially homogeneous FLRW solutions of the spherically symmetric Einstein--massless Vlasov system with spatial topology $\mathbb{R}^3$
