AI 中文总结
研究三维不可压缩欧拉方程解析涡旋片去奇异化定理，从伯克霍夫 - 罗特系统解析解出发，构造特定涡度，证明其在\(\varepsilon \to 0\)时收敛到规定涡旋片，得出解析三维涡旋片运动是精确欧拉流极限且寿命有下限的结论。
AI 中文摘要
我们证明了三维不可压缩欧拉方程解析涡旋片的去奇异化定理。从相应的伯克霍夫 - 罗特系统的解析解出发，对于每个足够小的厚度参数\(\varepsilon>0\)，我们构造一个精确的欧拉涡度，它支撑在围绕涡旋片宽度为\(O(\varepsilon)\)的管状邻域上，并定义在一个当\(\varepsilon \to 0\)时不会收缩到\(0\)的时间区间上。我们表明，当\(\varepsilon \to 0\)时，这些涡度在分布意义上收敛到规定的涡旋片。特别地，我们得出解析三维涡旋片运动作为精确欧拉流的极限出现，其寿命有与\(\varepsilon\)无关的下限。证明依赖于对由几乎平行曲面的时间相关叶状结构定义的涡度以及与这些曲面相切的无散向量场的研究。
英文摘要
We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to 0 as $\varepsilon \to 0$. We show that, as $\varepsilon \to 0$, these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of $ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.
Comments49 pages, 1 figure
