Blowing-up solutions of the axisymmetric Euler equations for an incompressible fluid

TL;DR

This paper investigates the potential for finite-time singularities in axisymmetric solutions of the incompressible Euler equations, revisiting Leray's question with a detailed analysis of self-similar solutions.

Contribution

It provides a new analysis of self-similar solutions in axisymmetric geometries, considering logarithmic time dependence, to better understand solution regularity and singularity formation.

Findings

Analysis suggests conditions for blow-up in axisymmetric Euler flows.

Reexamination of Leray's self-similar solutions in light of logarithmic time dependence.

Insights into the potential for finite-time singularities in inviscid fluids.

Abstract

A 1934 paper by Leray posed the question of the regularity of solutions of the dynamical equations for incompressible inviscid fluids with smooth initial data. Since there has been many attempts to answer this question. Leray examined the possibility of self-similar solutions becoming singular in finite time at a definite space-time location. We reexamine this question in the light of a thorough analysis of the equations for the self-similar solution in axisymmetric geometries with a dependence on a logarithm of time besides the one due to the transformation to self-similar variables.