Hausdorff measure of Vorticity Nodal Sets for the 3D Hyperviscous Navier Stokes Equations with General forces

TL;DR

This paper investigates the Hausdorff measure of vorticity nodal sets in a modified 3D Navier-Stokes system with an added l-Laplacian, linking geometric properties to turbulence characteristics and extending classical estimates.

Contribution

It introduces upper bounds on vorticity level set measures for a modified Navier-Stokes system, connecting these bounds to turbulence theory and the system's parameters.

Findings

Hausdorff measure bounds depend on the l-Laplacian parameter

Estimates relate to Kolmogorov length-scale and degrees of freedom

Results support the modified system as a turbulence model

Abstract

In this paper, we modified the three dimensional Navier-Stokes equations by adding a l-Laplacian. We provide upper bounds on the two-dimensional Hausdorff measure the level sets of the vorticity of solutions. We express them in terms of the Kolmogorov length-scale and the Landau-Lifschitz estimates of the number of degrees of freedom in turbulent flow. We also, under certain hypothesis recover the two-dimensional Hausdorff measure estimates for the usual 3D Navier-Stokes equations with potential force. Moreover, we show that the estimates depend on l, this result suggests that the modified Navier Stokes system is successful model of turbulence and the size of the nodal set leads the way for developing the turbulence theory.