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arXiv 2102.07807math.APmath-phmath.MP
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集中涡环的全球时间演化

Global time evolution of concentrated vortex rings

Paolo Buttà, Guido Cavallaro, Carlo Marchioro

备注：
23 pages, LaTex; small changes in abstract and introduction, typos corrected, added an appendix with the proof of Lemma 4.1, and, with respect to version 2, Lemma 3.2 corrected (which implies very slight changes in other parts of the manuscript). This article is an extension of the results arXiv:1904.04785 and therefore draws heavily from arXiv:1904.04785
发表信息：
Z. Angew. Math. Phys. 73 (2022), Article 70
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摘要：

我们研究的时间演化的不可压缩流体的轴对称无旋流，假设初始数据，使初始涡是非常集中的$N$小不相交环的厚度$\varepsilon$和涡量质量的顺序为$|\log\varepsilon|^{ -1}$。当$\varepsilon \to 0$，我们表明，每个涡环的运动收敛到一个简单的平移与恒定的速度（取决于单个环）沿对称轴。我们得到一个尖锐的本地化的涡度支持在径向方向上的时间$t$，而我们只在轴向方向上的浓度属性。这是针对任意（但固定）时间间隔获得的。这项研究是以前的论文arXiv：1904.04785 [math-ph]的完成，其中涡度支持的急剧本地化沿径向和轴向方向，但收敛$\varepsilon\to 0$只工作了很短的时间。

英文摘要：

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity mass of the order of $|\log\varepsilon|^{ -1}$. When $\varepsilon \to 0$ we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time $t$ in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper arXiv:1904.04785 [math-ph], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for $\varepsilon\to 0$ worked only for short times.

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