Balance of the vorticity direction and the vorticity magnitude in 3D fractional Navier-Stokes equations

TL;DR

This paper introduces a new regularity criterion for 3D fractional Navier-Stokes equations based on the balance between vorticity direction and magnitude, bridging Euler and Navier-Stokes models.

Contribution

It presents a hybrid geometric-analytic regularity criterion linking vorticity direction and magnitude for fractional Navier-Stokes solutions.

Findings

Provides a new criterion for flow regularity in fractional Navier-Stokes.

Bridges understanding between Euler and Navier-Stokes equations.

Offers insights into singularity formation in fluid flows.

Abstract

Fractional Navier-Stokes equations -- featuring a fractional Laplacian -- provide a `bridge' between the Euler equations (zero diffusion) and the Navier-Stokes equations (full diffusion). The problem of whether an initially smooth flow can spontaneously develop a singularity is a fundamental problem in mathematical physics, open for the full range of models -- from Euler to Navier-Stokes. The purpose of this work is to present a hybrid, geometric-analytic regularity criterion for solutions to the 3D fractional Navier-Stokes equations expressed as a balance -- in the average sense -- between the vorticity direction and the vorticity magnitude, key geometric and analytic descriptors of the flow, respectively.