In this talk I present my recent results on the regularity conditions for a solution to the 3D Navier-Stokes equations with powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. For the proof of the we exploit geometric properties of the vortex stretching term as well as the estimate using the Triebel-Lizorkin type of norms.

We describe a stochastic model for complex networks possessing three
qualitative features: power-law degree distributions, local clustering, and
slowly-growing diameter.
The model is mathematically natural, permits a wide variety
of explicit calculations, has the desired three qualitative features,
and fits the complete range of degree scaling exponents and clustering parameters.
Write-ups exist as a

short version
and as a
long version