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 where is the one dimensional Hausdorff measure of an infinite, vertical line and 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 originally due to Gallagher and Gallay.