Turbulence properties and global regularity of a modified Navier-Stokes equation

TL;DR

This paper introduces a modified Navier-Stokes equation with infinite conserved quantities, analyzes its turbulence properties through numerical simulations, and proves global well-posedness for smooth initial data.

Contribution

The paper presents a new modified Navier-Stokes equation with conserved quantities and demonstrates its turbulence characteristics and global regularity results.

Findings

Dissipative structures are curled vortex sheets instead of vortex tubes.

Scaling of structure functions aligns with She-Lévéque phenomenology.

Global well-posedness established for smooth initial conditions.

Abstract

We introduce a modification of the Navier-Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions. The dissipative structures arising in this new equation are curled vortex sheets contrary to vortex tubes arising in Navier-Stokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the She-L\'ev\^eque approach. Finally, for this equation we demonstrate global well-posedness for sufficiently smooth initial conditions in the periodic case and in $\mathbb R^3$. The key feature is the availability of an additional estimate which shows that the $L^4$-norm of the velocity field remains finite.