On Singularity Formation of a 3D Model for Incompressible Navier-Stokes Equations

TL;DR

This paper rigorously analyzes a simplified 3D model related to the Navier-Stokes equations, demonstrating finite-time singularity formation for certain initial conditions and global regularity for small data.

Contribution

It proves finite-time singularity formation and global regularity results for a simplified 3D Navier-Stokes model with and without viscosity.

Findings

Finite-time singularity formation for certain initial data.

Global regularity for small initial data.

The model shares key properties with the full Navier-Stokes equations.

Abstract

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier-Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.