非等熵气体恒星的稳定性
Stability of non-isentropic gaseous stars
- Fudan University（复旦大学）
- Xiangtan University（湘潭大学）
- Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非等熵Euler-Poisson系统静态球对称平衡态的线性稳定性，通过Hamilton系统与Morse指标刻画径向不稳定性，并给出两种熵规定下的稳定性判据及不稳定情形的指数增长结果。
AI中文摘要：
我们研究了非等熵Euler-Poisson系统的紧支撑静态球对称平衡态的线性稳定性。在Schwarzschild稳定情形下，线性化方程被实现为适应物理真空的加权空间上的可分离Hamilton系统。我们证明了针对非径向扰动的稳定性，并表明径向不稳定子空间的代数维数是仅受质量约束的密度二次型的Morse指标。因此，对于径向扰动，无穷多个线性化熵约束归结为一个质量约束和显式的熵重构。我们将此判据应用于两种熵规定。对固定的熵-密度关系不施加小性条件：在所陈述的有效压力和分支假设下，简单的质量最大值不是稳定性转变点。相比之下，对于作为包围质量函数固定的微小熵分布，转折点原理成立。在Schwarzschild不稳定情形下，我们从其二次型构造自伴速度算子，证明严格负的谱下界和尖锐的指数增长，并重构原始一阶线性化系统的能量解。
英文摘要：
We study the linear stability of compactly supported static spherical equilibria of the non-isentropic Euler--Poisson system. In the Schwarzschild-stable case, the linearized equations are realized as a separable Hamiltonian system on weighted spaces adapted to the physical vacuum. We prove stability against non-radial perturbations and show that the algebraic dimension of the radial unstable subspace is the Morse index of a density quadratic form subject only to the mass constraint. Thus the infinitely many linearized entropy constraints reduce, for radial perturbations, to one mass constraint and an explicit entropy reconstruction. We apply this criterion to two entropy prescriptions. No smallness condition is imposed on a fixed entropy--density relation: under the stated effective-pressure and branch hypotheses, a simple mass maximum is not a stability transition. By contrast, for a small entropy distribution fixed as a function of enclosed mass, the turning-point principle holds. In the Schwarzschild-unstable case, we construct the self-adjoint velocity operator from its quadratic form, prove a strictly negative spectral bottom and sharp exponential growth, and reconstruct energy solutions of the original first-order linearized system.
