黎曼流形上的对数速度对齐
Logarithmic Velocity Alignment on Riemannian Manifolds
AI总结：
该研究提出黎曼流形上的对数速度对齐模型，推导动能耗散估计并证明相互作用加权的渐近速度对齐，在双曲空间中得到无条件的该对齐结果。
AI中文摘要：
我们提出了一种用于黎曼流形上集体运动的对数速度对齐模型。该相互作用律由对数位移向量的协变时间导数定义，因此提供了成对速度差耦合的内在几何类似物。在双侧非正截面曲率界以及相互作用核与对数项的曲率诱导增长之间的相容性条件下，假设对数相互作用在全局上保持良定，且传输的速度偏差具有一致控制的时间变化，我们推导了动能耗散估计并证明了相互作用加权的渐近速度对齐。在双曲空间中，两个假设均可直接从几何和能量估计中验证，从而在所述全局解类内得到了无条件的相互作用加权对齐结果。
英文摘要：
We introduce a logarithmic velocity alignment model for collective motion on Riemannian manifolds. The interaction law is defined by the covariant time derivative of logarithmic displacement vectors and therefore provides an intrinsic geometric analogue of pairwise velocity-difference coupling. Under two-sided nonpositive sectional-curvature bounds and a compatibility condition between the interaction kernel and the curvature-induced growth of logarithmic terms, we derive a kinetic-energy dissipation estimate and prove interaction-weighted asymptotic velocity alignment, assuming that the logarithmic interactions remain globally well-defined and that transported velocity discrepancies have uniformly controlled time variation. In hyperbolic space, both assumptions are verified directly from the geometry and the energy estimates, yielding an unconditional interaction-weighted alignment result within the stated class of global solutions.
