Balance of the Vorticity Direction and the Vorticity Magnitude in 3D Fractional Navier–Stokes Equations

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.