Three-Dimensional Stability of Burgers Vortices: the Low Reynolds Number Case

TL;DR

The paper proves rigorous stability of Burgers vortices in the three-dimensional Navier-Stokes equations for small Reynolds numbers, showing that any small perturbation of a Burgers vortex converges to a (possibly shifted) Burgers vortex as $t \to \infty$ and providing an explicit formula for the change in the circulation number. Central to the analysis is a decomposition into a base Burgers vortex plus perturbations, reduction to a two-dimensional scalar equation, and a fixed-point argument that constructs non-axisymmetric Burgers vortices for all asymmetry values in $[0,1)$ at small $Re$, with precise decay. The stability result for three-dimensional perturbations uses an axial modulation of the base vortex and a transverse vorticity remainder, with the perturbations decaying exponentially to yield convergence to a modulated vortex. An appendix provides semigroup estimates for the linearized operator and a detailed treatment of the Biot-Savart law, enabling spectral analysis and $L^p$–$L^q$ bounds necessary for decay and regularity.

Abstract

In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change in the circulation number (which characterizes the limiting vortex completely.) We also give a rigorous proof of the existence and stability of non-axisymmetric Burgers vortices provided the Reynolds number is sufficiently small, depending on the asymmetry parameter.

Theorems & Definitions (24)

• Remark 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Proposition 2.1
• Proposition 2.2
• Remark 2.3