Finite Time Blow-up of a 3D Model for Incompressible Euler Equations

TL;DR

This paper rigorously proves that a 3D model derived from the incompressible Euler equations develops a finite time singularity, shedding light on potential blow-up behavior in fluid dynamics models.

Contribution

The paper establishes the finite time blow-up for a specific 3D inviscid model, advancing understanding of singularity formation in fluid equations.

Findings

Finite time singularity occurs for the model with smooth initial data.

The model preserves energy and properties similar to Navier-Stokes equations.

Rigorous proof of blow-up supports numerical evidence of singularity formation.

Abstract

We investigate the role of convection on its large time behavior of 3D incompressible Euler equations. In \cite{HL09a}, we constructed a new 3D model by neglecting the convection term from the reformulated axisymmetric Navier-Stokes equations. This model preserves almost all the properties of the full Navier-Stokes equations, including an energy identity for smooth solutions. The numerical evidence presented in \cite{HL09a} seems to support that the 3D model may develop a finite time singularity. In this paper, we prove rigorously that the 3D inviscid model develops a finite time singularity for a family of smooth initial data whose energy is finite and conserved in time.