AI 中文总结
本文研究一般压力律下多维可压缩Euler-Riesz方程定态的非线性稳定性与不稳定性，结合多种方法证明不同区域的稳定性结论及弱解整体存在性，相关方法可推广至其他非线性偏微分方程。
AI 中文摘要
可压缩Euler-Riesz方程可用于建模多种物理现象，包括恒星动力学、等离子体物理和数学生物学。本文研究一般压力律下多维可压缩Euler-Riesz方程定态的非线性稳定性与不稳定性：在多方气体情形中，通过分析质量保持伸缩下自由能的凹性，证明吸引势下质量超临界区域定态的非线性不稳定性；在质量临界指数处，证明对任意定态，存在任意接近它但支撑集不断增长的解。对一般压力律，采用集中紧性方法证明能量极小元的存在性并建立定态的非线性稳定性；通过推导有限能量解的相对熵界量化有限时间稳定性，无需密度的一致逐点上下界；进一步利用二阶矩的凸性得到正能解的定量增长估计，从而证明稳定性结果的局部性；最后通过补偿紧性方法证明一般压力律下具球对称性的可压缩Euler-Riesz方程有限能量弱解的整体存在性，在弱解类中得到定态周围的无条件稳定性。本文发展的方法应有助于解决其他涉及类似困难的非线性偏微分方程问题。
英文摘要
The compressible Euler-Riesz equations arise in the modeling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. In this paper, we investigate the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations under general pressure laws. In the polytropic case, we establish the nonlinear instability of steady states in the mass-supercritical regime for attractive potentials; this is achieved by analyzing the concavity of the free energy along mass-preserving dilations. At the mass-critical exponent, we show that, for any steady state, there exist solutions that start arbitrarily close to it, but develop growing support. For general pressure laws, we employ a concentration-compactness approach to prove the existence of energy minimizers and establish the nonlinear stability of steady states. Moreover, we quantify the finite-time stability by deriving a relative entropy bound for finite-energy solutions, without requiring uniform pointwise upper and lower bounds on the density. We further exploit the convexity of the second moment to obtain quantitative growth estimates for solutions with positive energy, thereby proving the local nature of the stability result. Finally, we prove the global existence of finite-energy weak solutions to the compressible Euler-Riesz equations with spherical symmetry for general pressure laws via the compensated compactness method, thereby yielding unconditional stability around steady states within the class of weak solutions. The approach developed in this paper should be useful for solving other nonlinear partial differential equations involving similar difficulties.
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