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

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 alpha e(z)delta(x=y=0) is the one dimensional Hausdorff measure of an infinite, vertical line and alpha is an element of 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 SE arrow 0 originally due to Gallagher and Gallay.