论文标题
3D Navier-Stokes方程中Oseen Vortex的独特性标准
Uniqueness Criteria for the Oseen Vortex in the 3d Navier-Stokes Equations
论文作者
论文摘要
在本文中，我们考虑了3D Navier-Stokes方程的解决方案的独特性，其初始涡度由$ω_0=αe_zΔ__{x = y = 0} $，其中$δ_{x = y = 0} $是一维的Hausdorff量度，$ iS $ iS $ $ $ $ naby $ natimential hausdorff量。该初始数据对应于理想化的无限涡流丝。一种平滑的温和溶液由自相似的Oseen Vortex柱给出，该圆柱与热的演化相吻合。 Germain，Harrop-Griffiths和第一作者的先前工作意味着，该解决方案在一类轻度解决方案中是独一无二的，这些解决方案在合适的自相似加权空间中融合到Oseen Vortex。在本文中，Oseen Vortex的唯一性类别被扩展到包含任何在足够强的意义上收敛到初始数据的解决方案。这提供了进一步的证据，以支持Oseen Vortex是唯一可以识别为涡旋细丝的唯一可能的温和解决方案。证明是$ t \ searrow 0 $中2D紧凑性/刚度参数的3D变化，最初是由于Gallagher和Gallay造成的。
In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by $ω_0 = αe_z δ_{x = y = 0}$, where $δ_{x=y= 0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $α\in \mathbb R$ 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.
