Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations

TL;DR

This paper introduces a numerical framework using scaled vorticity norms to analyze the regularity and potential singularities of solutions in 3D Navier-Stokes and Euler equations.

Contribution

It proposes new dimensionless norms and criteria for testing regularity and singularity formation in these fluid dynamics equations.

Findings

Numerical experiments suggest criteria for Navier-Stokes regularity.

Scaled norms help identify potential singular behavior.

Method applicable to both Navier-Stokes and Euler equations.

Abstract

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity $\Omega_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$ where $\varpi_{0}$ is a constant frequency and $\alpha_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_{m}$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on $\int_{0}^{t}D_{m}\,d\tau$. The behaviour of the $D_{m}$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.