Vortex stretching and anisotropic diffusion in the 3D Navier-Stokes equations

TL;DR

This paper discusses recent mathematical and numerical insights into how vortex stretching and anisotropic diffusion mechanisms may prevent finite-time singularities in the 3D Navier-Stokes equations by controlling vorticity growth.

Contribution

It presents a cohesive analysis of a geometric scenario where vortex stretching leads to small scales that trigger anisotropic diffusion, supporting the sub-criticality of the Navier-Stokes regularity problem.

Findings

Vortex stretching produces small scales that activate anisotropic diffusion.

Anisotropic diffusion can control vorticity, preventing singularities.

The scenario is supported by experimental, numerical, and mathematical evidence.

Abstract

The goal of this article is to present -- in a cohesive, and somewhat self-contained fashion -- several recent results revealing an experimentally, numerically, and mathematical analysis-supported \emph{geometric scenario} manifesting \emph{large data} logarithmic \emph{sub-criticality} of the 3D Navier-Stokes regularity problem. Shortly -- in this scenario -- the \emph{transversal small scales} produced by the mechanism of vortex stretching (coupled with the decay of the volume of the regions of intense vorticity) reach the threshold sufficient for the \emph{locally anisotropic diffusion} to engage and control the sup-norm of the vorticity, preventing the (possible) formation of finite time singularities.