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For the incompressible Navier-Stokes equations in R3 with low viscosity ν > 0, we consider the Cauchy problem with initial vorticity ω0 that represents an infinitely thin vortex filament of arbitrary given strength \Gamma supported on a circle. The vorticity field ω(x, t) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness√νt 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 ω(x, t) is well-approximated on a large time interval by ω_lin (x − a(t), t), where ω_lin(·, t) = exp(νt\Delta)ω0 is the solution of the heat equation with initial data ω0, and ˙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 ν = 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 ν >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. 

We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with a stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.