论文标题
双曲线空间上相互作用模型的平衡和能量最小化
Equilibria and energy minimizers for an interaction model on the hyperbolic space
论文作者
论文摘要
我们研究了双曲线空间$ \ bbh^\ dm $的内在模型。我们研究了包括牛顿排斥的相互作用电位的聚合方程的平衡（或等效地，相关相互作用能的临界点）。通过使用移动平面的方法，我们建立了在$ \ bbh^\ dm $的测地球上支持的径向对称性和平衡的单调性。我们发现了几种明确的平衡形式，并表明一种这样的平衡是一种全球能量最小化器。我们还考虑了更多的一般潜力，并利用了一种用于$ \ bbr^\ dm $的技术来确定紧凑型全球最小化器的存在。提出了数值模拟，表明这里研究的一些均衡是全球吸引子。我们调查中的关键工具是我们为此目的开发的$ \ bbh^\ dm $的同密家族。
We study an intrinsic model for collective behaviour on the hyperbolic space $\bbh^\dm$. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of $\bbh^\dm$. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a technique used for $\bbr^\dm$ to establish the existence of compactly supported global minimizers. Numerical simulations are presented, suggesting that some of the equilibria studied here are global attractors. The key tool in our investigations is a family of isometries of $\bbh^\dm$ that we have developed for this purpose.
