On the blow-up problem for the axisymmetric 3D Euler equations

TL;DR

This paper investigates finite-time blow-up phenomena in axisymmetric 3D Euler equations with swirl, identifying conditions under which solutions become singular in finite time, and relates the problem to the Constantin-Lax-Majda equations.

Contribution

It provides new criteria for blow-up in axisymmetric Euler equations and links the problem to explicit integrable models under specific assumptions.

Findings

Solutions blow up in finite time under certain pressure conditions.

Reduction to Constantin-Lax-Majda equations when second radial derivative vanishes.

Explicit integration possible in special cases.

Abstract

In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations we show that the solution blows-up in finite time. If we further assume that the second radial derivative vanishes on the axis, then system reduces to the form of Constantin-Lax-Majda equations, and can be integrated explicitly.