球面上聚集-扩散能量平衡态的稳定性
Stability of equilibria of an aggregation-diffusion energy on sphere
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中文总结 AI 辅助
该研究针对球面上的聚集-扩散能量，将平衡态存在性结果从二维推广到任意维，给出了平衡态的分岔结构，并通过显式判据完成了所有平衡态的稳定性分类，识别出两类关键分岔。
中文摘要 AI 辅助
我们研究了球面上的聚集-扩散能量，并探究其平衡态的稳定性。该能量由一个m>1的多孔介质型非线性熵$\frac{1}{m-1}\rho(x)^m\rm{d}S(x)$，以及一个由二次相互作用势建模的相互作用能组成。该能量推广了用于模拟聚合物取向的、具有偶极势的Onsager自由能。本研究补充了作者此前发表于《Nonlinearity》第39卷（2026年）、编号055011的工作，此前的工作研究了该能量泛函的基态。在本文中，我们将此前m>2时平衡态存在性的结果从d=2维推广到任意d≥2维。这使得我们能够针对所有m>1的情况，给出任意d≥2维下所有平衡态关于相互作用强度的分岔结构。此外，我们通过推导一个可显式验证的稳定性判据，对该能量所有平衡态的稳定性进行了完整分类。特别地，对于m>2的情况，我们识别出了严格支撑平衡态的鞍结分岔，以及均匀分布失去稳定性时的亚临界叉形分岔。
英文摘要
We consider an aggregation-diffusion energy on the sphere and investigate the stability of its equilibria. The energy consists of a porous-medium type nonlinear entropy $\frac{1}{m-1}\int ρ(x)^m\mathrm{d}S(x)$ with $m>1$, together with an interaction energy modeled by a quadratic interaction potential. The energy generalizes the Onsager free energy with dipolar potential, which models polymer orientation. Our study complements the authors' previous work [Nonlinearity {\bf 39} (2026), 055011], where the ground states of the energy functional were investigated. In the current paper we extend the previous results on existence of equilibria for $m>2$ from $d=2$ to arbitrary dimension $d \geq 2$. This allows us to present the bifurcation structure (with respect to the interaction strength) of all equilibria in any dimension $d \geq 2$, for all $m>1$. Furthermore, we provide a complete classification of the stability of all equilibria of the energy, by deriving a criterion for stability that can be checked explicitly. In particular, for $m>2$ we identify a saddle-node bifurcation of strictly supported equilibria, and a subcritical pitchfork bifurcation where the uniform distribution loses stability.
