AI 中文总结
本文研究可压缩Euler-Riesz方程的旋转Riesz星解，在质量次临界情形证明其存在性与非线性稳定性，在质量超临界情形证明其存在性与不稳定性，揭示旋转对解的稳定或不稳定效应取决于Riesz相互作用的奇异性。
AI 中文摘要
可压缩Euler-Riesz方程广泛用于模拟恒星动力学、等离子体物理、数学生物学等多种物理现象。本文研究吸引型可压缩Euler-Riesz方程的旋转定态，即旋转Riesz星，在质量次临界和质量超临界两种情形下分别建立存在性及非线性稳定性或不稳定性结果。在质量次临界情形下，我们在角动量分布满足合适的次齐次性假设时证明旋转Riesz星的存在性，随后通过适配轴对称情形的集中紧性论证建立其非线性稳定性；旋转会带来新的紧性困难，最显著的是极小化序列可能沿半径发散至无穷的环保持紧性。在质量超临界情形下，我们在多方情形下，当角动量分布满足合适的超齐次性假设时证明存在性，将理论扩展到小角速度区域之外；该证明需要对保质量标度进行细致分析，其比非旋转情形更复杂。最后，通过分析自由能沿这些标度的凹性，我们证明所得质量超临界旋转Riesz星的不稳定性。结果表明，旋转可根据Riesz相互作用的奇异性产生稳定或不稳定效应。
英文摘要
The compressible Euler--Riesz equations arise in the modelling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. We study rotating steady states of the attractive compressible Euler--Riesz equations, which we call rotating Riesz stars, and establish existence and nonlinear stability or instability results in both the mass-subcritical and mass-supercritical regimes. In the mass-subcritical regime, we prove the existence of rotating Riesz stars under suitable subhomogeneity assumptions on the angular momentum profile. We then establish their nonlinear stability through a concentration compactness argument adapted to the axisymmetric setting. Rotation creates new compactness difficulties, most notably the possibility that minimising sequences are tight along rings whose radii diverge to infinity. In the mass-supercritical regime, we prove existence in the polytropic setting under suitable superhomogeneity assumptions on the angular momentum profile, thereby extending the theory beyond the small angular velocity regime. The proof requires a careful analysis of mass-preserving scalings, which are more delicate than in the non-rotating case. Finally, by analysing the concavity of the free-energy along these scalings, we establish the instability of the resulting mass-supercritical rotating Riesz stars. Our results show that rotation can have either a stabilising or a destabilising effect, depending on the singularity of the Riesz interaction.
Comments53 pages
