Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

We deal with the conditional regularity of the weak solutions to the Navier–Stokes equations. We discuss a famous criterion by Vasseur in terms of div(u/|u|) and extend this criterion to bounded domains with Navier and Navier‐type boundary conditions. Inspired by the equality u·∇|u|λ=−λ|u|λ+1div(u/|u|),λ≥1, we further prove an optimal regularity criterion in terms of u·∇|u|λ both for the whole three‐dimensional space and bounded domains with Navier's, Navier‐type, and Dirichlet boundary conditions. It specially means for λ=2 that the control of the energy flow in the critical norms LtpLxr provides the regularity of solutions. This criterion is proved by two different techniques.