Finite-time blowup for a Navier-Stokes model equation for the self-amplification of strain

TL;DR

This paper introduces a simplified model of the Navier-Stokes strain equation, demonstrating finite-time blowup due to strain self-amplification, challenging assumptions about regularity criteria for fluid turbulence.

Contribution

It proves finite-time blowup for a Navier-Stokes strain model and links strain self-amplification to turbulence energy cascade mechanisms.

Findings

Finite-time blowup demonstrated in the model.

Strain self-amplification identified as a key turbulence mechanism.

Conditional blowup results for full Navier-Stokes based on model insights.

Abstract

In this paper, we consider a model equation for the Navier--Stokes strain equation. This model equation has the same identity for enstrophy growth and a number of the same regularity criteria as the full Navier-Stokes strain equation, and is also an evolution equation on the same constraint space. We prove finite-time blowup for this model equation, which shows that the identity for enstrophy growth and the strain constraint space are not sufficient on their own to guarantee global regularity for Navier-Stokes. The mechanism for the finite-time blowup of this model equation is the self-amplification of strain, which is consistent with recent research suggesting that strain self-amplification, not vortex stretching, is the main mechanism behind the turbulent energy cascade. Because the strain self-amplification model equation is obtained by dropping certain terms from the full Navier-Stokes strain equation, we will also prove a conditional blowup result for the full Navier-Stokes equation involving a perturbative condition on the terms neglected in the model equation.