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

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.