Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

TL;DR

This paper investigates the instability of potential finite-time blow-up solutions to the incompressible Euler equations, demonstrating that such solutions, if they exist, are linearly unstable even in the axisymmetric case.

Contribution

It extends previous instability results to axisymmetric solutions and introduces a new blow-up criterion based solely on the toroidal vorticity component.

Findings

Blow-up solutions, if they exist, are linearly unstable near the blow-up time.

Established a blow-up criterion involving only the toroidal vorticity component.

Analyzed the instability of blow-up profiles.

Abstract

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated.