Abstract

In the framework of the Global Regularity Problem for the incompressible Navier–Stokes (NS) equations in the whole space \({{\mathbb{R}}^{3}}\), Li and Sinai in [13] proved that there are smooth complex solutions that become singular (“blow-up”). We discuss the possible extension of the Li-Sinai approach to real solutions and report results obtained by computer simulations on the behavior of a particular solution related to the complex blow-up. Although a blow-up is excluded by axial symmetry, the solution is a good model of a “tornado-like” behavior, with a sharp increase of speed and vorticity concentrated in an annular region around the symmetry axis. We conclude with a discussion on the search of possible candidates for a real blow-up in the framework of the Li-Sinai approach.

中文翻译：

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具有“龙卷风式”行为的3D不可压缩Navier-Stokes方程的反对称解

摘要

在整个空间\（{{\ mathbb {R}} ^ {3}} \）中不可压缩的Navier–Stokes（NS）方程的全局正则性问题的框架中，[13]中的Li和Sinai证明了是变得奇异的平滑复杂解决方案（“爆炸”）。我们讨论了将Li-Sinai方法扩展到实际解决方案的可能性，并报告通过计算机模拟获得的与复杂爆炸相关的特定解决方案的行为的结果。尽管轴向对称排除了爆炸，但该解决方案是“龙卷风”行为的良好模型，其速度和涡度急剧增加，集中在围绕对称轴的环形区域中。最后，我们讨论了在Li-Sinai方法框架内寻找真正爆发的可能候选人的讨论。