Abstract
We consider the classical Cauchy problem for the three dimensional Navier–Stokes equation with the initial vorticity ω 0 concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good uniform estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution. We impose no restriction on the size of the initial data.
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Feng, H., Šverák, V. On the Cauchy Problem for Axi-Symmetric Vortex Rings. Arch Rational Mech Anal 215, 89–123 (2015). https://doi.org/10.1007/s00205-014-0775-4
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DOI: https://doi.org/10.1007/s00205-014-0775-4