On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain

TL;DR

This paper investigates the stability of blow-up solutions of the Navier-Stokes equations with linear strain, demonstrating their stability and observing secondary blow-up phenomena under certain conditions.

Contribution

It extends the stability theory of Burgers vortices to solutions with time-dependent strain, revealing new stability results and secondary blow-up behavior.

Findings

Blow-up solutions are stable under certain conditions.

Secondary blow-up occurs with weaker strain.

Stability theory applies to time-dependent strain scenarios.

Abstract

We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.