Effect of vorticity coherence on energy-enstrophy bounds for the 3D Navier-Stokes equations

TL;DR

This paper derives bounds on energy and enstrophy for the 3D Navier-Stokes equations based on vorticity coherence, revealing significant differences between critical and subcritical cases.

Contribution

It introduces a new analysis of energy-enstrophy bounds under vorticity coherence assumptions, especially highlighting the critical case with exponential enstrophy growth.

Findings

Critical case yields exponential maximal enstrophy.

Subcritical case results in algebraic enstrophy bounds.

Vorticity coherence significantly influences energy-enstrophy dynamics.

Abstract

Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is H\"older continuous with exponent $r=1/2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1/2$ (algebraic in Grashof).