On Singular Vortex Patches, I: Well-posedness Issues
On Singular Vortex Patches, I: Well-posedness Issues
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DOI：
10.1090/memo/1400
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发表时间：
2019-03
影响因子：
1.9
通讯作者：
T. Elgindi;In-Jee Jeong
中科院分区：
文献类型：
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作者：
T. Elgindi;In-Jee Jeong
这项工作的目的是讨论奇异涡斑的适定性理论。我们的主要结果有两种类型：适定性和不适定性。在适定性方面，我们表明，只要 m ≥ 3. m\geq 3，具有从原点发出的角点的全局 m m 重对称涡斑在自然正则类中是全局适定的。在这种情况下，涉及的所有角度都求解一个封闭的 ODE 系统，该系统决定了角点的全局时间动态，并且仅取决于角点的初始位置和大小。在此过程中，我们获得了一类在原点具有奇异边界的对称斑块的全局适定性结果，其中包括对数螺线。在不适定性方面，我们表明，涡斑中任何其他类型的角奇点都不能随时间连续演化，除非所有涉及的角始终精确地具有角度 π 2 \frac {\pi }{2} 。即使在角点为 π 2 \frac {\pi }{2} 或角点仅局部 m m 重对称的涡斑的情况下，我们也证明它们通常是不适定的。我们预计，在这些不适定的情况下，涡斑实际上会以自相似的方式立即尖点，并且我们推导出一些渐近模型，这些模型可能有助于更精确地描述动力学。在 2020 年关于奇异涡旋补片的配套工作中，我们讨论了带角点的对称涡旋补片的长期行为，并使用它们在 R 2 \mathbb {R}^2 上构建具有有趣动态行为的补片，例如无限时间内的尖点和螺旋形成。
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. m\geq 3. In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π 2 \frac {\pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 \frac {\pi }{2} or with corners which are only locally m m -fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on R 2 \mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time.