Vanishing viscosity limit for axisymmetric vortex rings

• 成果类型: Article
• 署名作者: Gallay, Thierry; Sverak, Vladimir
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Institut Universitaire de France; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01261-5
• 发表日期: 2024
• 页码: 275-348
• 关键词: small cross-section; steady vortex; navier-stokes; STABILITY; EXISTENCE; motion; filament; desingularization; vorticity; EQUATIONS

摘要

For the incompressible Navier-Stokes equations in R3 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 root 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 lin(x-a(t),t), where omega lin(& sdot;,t)=exp(nu t Delta)omega 0 is the solution of the heat equation with initial data omega 0, and a(center dot)(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.