发表机构
Stony Brook University; Max Planck Institute for Mathematics in the Sciences; University of Notre Dame（石溪大学; 马克斯·普朗克数理科学研究所; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过度量介于性给出流体动力学Hopf-Rinow强问题的等价准则，证明平坦环面上体积保持微分同胚群中法邻域的测地凸性及距离等价性，并建立平方测地距离的C^{1,1}延拓。
AI 中文摘要
流体动力学Hopf-Rinow问题询问理想流体的给定重构是否可以通过Euler流实现。在平坦的二维环面上，我们给出了该问题强版本（要求流体运动最小化所需的总动能）的一个等价准则，该准则以度量介于性（metric betweenness）表述。为此，我们证明了平坦n维环面（n≥2）的体积保持微分同胚群中，配备动能度量的足够小的法邻域是测地凸的；从而在H^s拓扑中局部解决了强问题。我们进一步建立了内在测地距离与外在L^2距离之间的局部双Lipschitz等价性。最后，我们证明了从单位元出发的平方测地距离允许从足够小的法邻域到环境L^2空间的C^{1,1}延拓。
英文摘要
The hydrodynamical Hopf-Rinow problem asks whether a given reconfiguration of an ideal fluid can be attained by an Euler flow. On the flat two-dimensional torus, we give an equivalent criterion for the strong version of this problem (which requires that the fluid motion minimizes the total kinetic energy required) in terms of metric betweenness. Toward this, we prove that sufficiently small normal neighborhoods in the volume-preserving diffeomorphism group of the flat $n$-dimensional torus, $n \geq 2$, equipped with the kinetic energy metric, are geodesically convex; resolving the strong problem locally in the $H^s$ topology. We further establish local bi-Lipschitz equivalence of the intrinsic geodesic distance and the extrinsic $L^2$ distance. Finally, we prove that the squared geodesic distance from the identity admits a $C^{1,1}$ extension from a sufficiently small normal neighborhood to the ambient $L^2$ space.
