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Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

Laurent Lafleche, Alexis F. Vasseur, Misha Vishik

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.

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

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.

Paper Structure

This paper contains 7 sections, 9 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Set up of the method
  3. Blow-up criterion
  4. Proof of the Linear instability
  5. Control of the vorticity by beta
  6. Control of beta by the linear stability
  7. Proof of Corollary \ref{['cor']}

Key Result

Theorem 1

Assume $u$ is an axisymmetric solution of eq:Euler verifying the Hypothesis hyp:regularity with $s>5/2$, and with initial condition $u^\mathrm{in}$ verifying $r\,u^\mathrm{in} \in L^\infty(\Omega)\cap L^q(\Omega)$ with $q<\frac{6}{5}$. Let $T^*>0$ be the maximal time $T$ such that $u$ verifies Hypot

Theorems & Definitions (19)

  • Theorem 1: Instability of the Blow-up
  • Corollary 1.1
  • Proposition 1: Blow-up criterion
  • Remark 1.1
  • Proposition 2
  • Proposition 3
  • proof : Proof of Theorem \ref{['thm:bound_omega_lambda']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 9 more