Conjugate and cut points in ideal fluid motion
Conjugate and cut points in ideal fluid motion
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DOI：
10.1007/s40316-021-00176-4
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发表时间：
2021-05
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通讯作者：
Theodore D. Drivas;G. Misiołek;Bin Shi;Tsuyoshi Yoneda
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作者：
Theodore D. Drivas;G. Misiołek;Bin Shi;Tsuyoshi Yoneda
如果存在一个单参数族测地线（流体流动）将它们连接到无穷小阶，则沿着流动的两种流体配置是共轭的。从几何角度来看，它们可以被视为（无限维）体积保持微分同胚组的结果，该组具有足够强的正曲率，可以将附近的流动“拉”在一起。从物理上讲，它们表明粒子位置的构型空间中的一种（瞬态）稳定性：以相同构型开始的一系列流最初偏离，随后在稍后的某个时刻彼此重新收敛（共振）。在这里，我们首先在任意长宽比的矩形平面圆环上建立无限族柯尔莫哥洛夫流（欧拉方程的一类平稳解）中共轭点的存在性。用于识别体积保持微分同胚组中的共轭点的通用标准促进了分析。接下来，我们证明环面、圆盘和通道上沿阿诺德稳定状态不存在共轭点。最后，我们讨论切点，它们与指数图的非注入性的关系（不可能在给定时刻从粒子配置确定流动），并表明最接近恒等式的切点是时间周期拉格朗日流体流动的共轭点或中点。
Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.