On the Cauchy problem for axi-symmetric vortex rings

Abstract

We consider the classical Cauchy problem for the 3d Navier-Stokes equation with the initial vorticity ${\omega }_0$ 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 a-priori 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.

36 pages

Cited by in corpus (12)

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• On the global well-posedness of 3-D axi-symmetric Navier-Stokes system with small swirl component
• Vanishing viscosity limits for axisymmetric flows with boundary
• Filamentation near Hill's vortex
• Long time evolution of concentrated vortex rings with large radius
• Long-time asymptotics of $3$-D axisymmetric Navier-Stokes equations in critical spaces
• Head-On Collision of a Pair of Coaxial Circular Vortex Filament
• Uniqueness of axisymmetric viscous flows originating from positive linear combinations of circular vortex filaments
• Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions