Vanishing viscosity limit for axisymmetric vortex rings

Abstract

For the incompressible Navier-Stokes equations in \(\mathbb{R}^{3}\) with low viscosity \(\nu >0\), we consider the Cauchy problem with initial vorticity \(\omega _{0}\) that represents an infinitely thin vortex filament of arbitrary given strength \(\Gamma \) supported on a circle. The vorticity field \(\omega (x,t)\) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness \(\sqrt{\nu t}\) that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that \(\omega (x,t)\) is well-approximated on a large time interval by \(\omega _{\mathrm {lin}}(x-a(t),t)\), where \(\omega _{\mathrm {lin}}(\cdot ,t)=\exp (\nu t\Delta )\omega _{0}\) is the solution of the heat equation with initial data \(\omega _{0}\), and \(\dot{a}(t)\) is the instantaneous velocity given by Kelvin’s formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold’s geometric stability methods developed in the inviscid case \(\nu =0\) to the slightly viscous situation. It turns out that although the geometric structures behind Arnold’s approach are no longer preserved by the equation for \(\nu > 0\), the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold’s theory and interact favorably with the viscous terms.