Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations
Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations
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DOI：
10.1088/1361-6544/abe1d1
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发表时间：
2019-02
期刊：
影响因子：
1.7
通讯作者：
B. Khesin;Cheng Yang
中科院分区：
文献类型：
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作者：
B. Khesin;Cheng Yang
双法线（或涡丝）方程提供了 3D 不可压缩欧拉方程的局部感应近似。我们提出了在有限时间内崩溃的高维二元正态方程的显式解。这种现象的局部性质表明，5 维及更高维的欧拉方程的附近涡流斑点解中可能出现奇点。此外，Hasimoto 变换将联正方程应用于 NLS 和正压流体方程。我们表明，在更高的维度中，这种变换的存在意味着斜平均曲率流中威尔莫尔能量的守恒，并提出了基于球体乘积的涡流膜的反例。这些（反）例子意味着没有直接推广到更高维度的一维桥本变换。我们推导了它的替代方程，即膜的平均曲率和扭转形式的演化方程，从而推广了正压流体和 Da Rios 方程。
The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests a possibility of the singularity appearance in nearby vortex blob solutions of the Euler equation in 5D and higher. Furthermore, the Hasimoto transform takes the binormal equation to the NLS and barotropic fluid equations. We show that in higher dimensions the existence of such a transform would imply the conservation of the Willmore energy in skew-mean-curvature flows and present counterexamples for vortex membranes based on products of spheres. These (counter)examples imply that there is no straightforward generalization to higher dimensions of the 1D Hasimoto transform. We derive its replacement, the evolution equations for the mean curvature and torsion form for membranes, thus generalizing the barotropic fluid and Da Rios equations.