Uniqueness criteria for the Oseen vortex in the 3d Navier-Stokes equations

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.

Keywords:

• Navier-Stokes equations
• vortex filament
• self-similar solutions
• singular initial data

Notes

1 By which we mean a large part of the initial data cannot be approximated by Schwartz class functions.

Additional information

Funding

J.B. was supported by NSF CAREER grant DMS-1552826. W.G. was partially supported by NSF-DMS grant 1840314.