AI 中文总结
该研究针对黎曼流形上带键合力的Cucker--Smale系统，结合平行运输变分公式与雅可比场估计克服几何障碍，在特定条件下证明渐近集群行为并通过数值模拟验证。
AI 中文摘要
我们研究在曲率一致有界的完备黎曼流形上带键合力的Cucker--Smale型系统。在一般流形上，运动智能体之间平行运输的时间变化会产生依赖曲率的项，因此标准的能量耗散论证无法直接得到渐近速度对齐。键合能量将所有两两距离限制在单射半径以下，为运输的速度差异提供了全局适定性和时间可积性。为克服剩余的几何障碍，我们将平行运输的变分公式与沿运动极小测地线的雅可比场的一致端点估计相结合，得到了将能量耗散转化为渐近对齐所需的一致正则性。在依赖能量的单射条件和动态相关距离范围的正通信界下，我们建立了渐近集群行为。数值模拟说明了该动力学在非常截面曲率情形下的表现。
英文摘要
We study a Cucker--Smale type system with bonding forces on complete Riemannian manifolds with uniformly bounded curvature. On general manifolds, the time variation of parallel transport between moving agents produces curvature-dependent terms, so the standard energy-dissipation argument does not directly yield asymptotic velocity alignment. The bonding energy confines all pairwise distances below the injectivity radius, providing global well-posedness and time integrability of the transported velocity discrepancies. To overcome the remaining geometric obstruction, we combine the variation formula for parallel transport with a uniform endpoint estimate for Jacobi fields along moving minimizing geodesics. This yields the uniform regularity needed to convert energy dissipation into asymptotic alignment. Under an energy-dependent injectivity condition and a positive communication bound on the dynamically relevant distance range, we establish asymptotic flocking. Numerical simulations illustrate the resulting dynamics in a nonconstant-sectional-curvature setting.
