标题： 关于广义相对论中自由边界硬相模型稳态的稳定性分析 显示英文标题

标题： On stability analysis for steady states of the free boundary hard phase model in general relativity

摘要： 硬相模型描述了一种相对论性的等熵流体，其声速等于光速。 在广义相对论的框架下，流体的运动与描述底层时空结构的爱因斯坦方程耦合。 该带有自由边界的模型存在一个具有球对称性的$1$参数族的稳定状态。 在本文中，对于球对称内的扰动，我们研究了该族的稳定性和不稳定性。 我们证明了围绕大中心密度稳定状态的线性化算子存在增长模，而对于小中心密度的稳定状态，这样的增长模不存在。 基于线性分析，我们进一步展示了大中心密度稳定状态的动力学非线性不稳定性。 证明依赖于线性化算子的谱分析以及全非线性自由边界问题的先验估计。 显示英文摘要

摘要： The hard phase model describes a relativistic barotropic fluid with sound speed equal to the speed of light. In the framework of general relativity, the motion of the fluid is coupled to the Einstein equations which describe the structure of the underlying spacetime. This model with free boundary admits a $1$-parameter family of steady states with spherical symmetry. In this work, for perturbations within spherical symmetry, we study the stability and instability of this family. We prove that the linearized operator around steady states with large central densities admits a growing mode, while such growing modes do not exist for steady states with small central densities. Based on the linear analysis, we further demonstrate a dynamical nonlinear instability for steady states with large central densities. The proof relies on a spectral analysis of the linearized operator and an a priori estimate on the full nonlinear free boundary problem.