Abstract

In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by ${\omega }_0=\alpha e_z{\delta }_{x=y=0},$ where ${\delta }_{x=y=0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $\alpha \in \mathbb{R}$ is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of mild solutions that converge to the Oseen vortex in suitable self-similar weighted spaces. In this paper, the uniqueness class of the Oseen vortex is expanded to include any solution that converges to the initial data in a sufficiently strong sense. This gives further evidence in support of the expectation that the Oseen vortex is the only possible mild solution that is identifiable as a vortex filament. The proof is a 3d variation of a 2d compactness/rigidity argument in $t\searrow 0$ originally due to Gallagher and Gallay.

摘要

在本文中，我们考虑了具有以下初始涡度的 3d Navier-Stokes 方程的解的唯一性 ${\omega }_0=\alpha {\mathrm{电子}}_z{\delta }_{X=是=0},$ 在哪里 ${\delta }_{X=是=0}$ 是无限垂直线的一维豪斯多夫测度，并且 $\alpha \in \mathbb{电阻}$是任意循环。此初始数据对应于理想化的无限涡旋灯丝。自相似的 Oseen 涡柱提供了一种平滑、温和的解决方案，它与放热相吻合。Germain、Harrop-Griffiths 和第一作者之前的工作表明，该解在一类温和解中是独一无二的，这些解在合适的自相似加权空间中会聚到 Oseen 涡流。在本文中，Oseen 涡旋的唯一性类被扩展为包括任何在足够强的意义上收敛到初始数据的解。这提供了进一步的证据来支持 Oseen 涡流是唯一可识别为涡流丝的可能温和解决方案的预期。证明是 2d 紧凑性/刚性参数的 3d 变体$吨\searrow 0$ 最初是由于加拉格尔和加莱。