On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

TL;DR

This paper studies an axisymmetric 3D Euler model, demonstrating conditions under which solutions develop finite time singularities or remain globally smooth, depending on initial data and boundary conditions.

Contribution

It proves finite time singularity formation and global regularity for an axisymmetric 3D Euler model with specific boundary conditions, extending previous results to unbounded domains.

Findings

Finite time singularity occurs for certain initial data.

Global smooth solutions exist for large initial data under specific boundary conditions.

Boundary conditions critically influence solution behavior.

Abstract

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler and Navier-Stokes equations is that the convection term is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D inviscid model can develop a finite time singularity starting from smooth initial data on a rectangular domain. A global well-posedness result was also proved for a class of smooth initial data under some smallness condition. The analysis in \cite{HSW09} does not apply to the case when the domain is axisymmetric and unbounded in the radial direction. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin boundary condition will develop a finite time singularity starting from smooth initial data in an axisymmetric domain. Moreover, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition.