发表机构
College of Mathematics, Jilin University; School of Mathematical Sciences, Fudan University; School of Mathematics and Computational Science, Xiangtan University（吉林大学数学学院; 复旦大学数学科学学院; 湘潭大学数学与计算科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究刚性旋转黏性恒星（NSP系统）的轴对称稳定性，证明黏性耗散可通过角动量再分布使恒星在无黏情形失稳前先失稳，并给出转折点原理。
AI 中文摘要
我们研究了由自由边界Navier--Stokes--Poisson（NSP）系统建模的刚性旋转黏性恒星的轴对称稳定性。线性化NSP生成元的不稳定指标（按Riesz代数重数计数）等于在固定质量和总角动量下增广能量的Morse指标。我们证明了具有有限负刚度指标的阻尼陀螺方程的一个抽象Kelvin--Tait--Chetaev定理，该定理不需要紧性假设，也不要求零处存在谱间隙；刚度核可以是无限维的。在NSP应用中，黏性耗散是退化的：其轴对称核由刚性轴向平移和刚性旋转生成，在应用抽象定理之前被投影掉，而守恒定律和Routh约化将由此产生的不稳定指标转移回完整的NSP生成元。对于具有固定总角动量的慢旋转分支，黏性不稳定性始于第一个非退化球对称质量极大值的延续处。相比之下，对于每一个足够小的非零角动量，相应的Euler--Poisson恒星在该极大值之后的某个区间内保持轴对称谱稳定。因此，耗散可以在相应的无黏恒星变得不稳定之前使旋转恒星失稳。这种稳定性阈值的分离源于角动量的黏性再分布，它消除了无黏情况下对物质分布的限制，同时保持其总值不变。
英文摘要
We study axisymmetric stability of rigidly rotating viscous stars modeled by the free-boundary Navier--Stokes--Poisson (NSP) system. The unstable index of the linearized NSP generator, counted with Riesz algebraic multiplicity, equals the Morse index of the augmented energy at fixed mass and total angular momentum. We prove an abstract Kelvin--Tait--Chetaev theorem for damped gyroscopic equations with finite negative stiffness index, requiring no compactness assumptions and no spectral gap at zero; the stiffness kernel may be infinite-dimensional. In the NSP application, the viscous dissipation is degenerate: its axisymmetric kernel, generated by rigid axial translation and rigid rotation, is projected out before applying the abstract theorem, while conservation laws and a Routh reduction transfer the resulting instability index back to the full NSP generator. For slowly rotating branches with fixed total angular momentum, viscous instability begins at the continuation of a first nondegenerate spherical mass maximum. By contrast, for every sufficiently small nonzero angular momentum, the corresponding Euler--Poisson star remains axisymmetrically spectrally stable on an interval beyond this maximum. Thus dissipation can destabilize a rotating star before the corresponding inviscid star becomes unstable. This separation of the stability thresholds results from viscous redistribution of angular momentum, which removes the inviscid constraints on its material distribution while preserving its total value.
