Velocity Direction Assumption as Sufficient Condition for Regularity of Weak Solutions to the Navier–Stokes Equations

ABSTRACT In this paper, we show that a simple geometrical assumption on the direction of the velocity leads to the regularity of weak solutions of the 3D Navier–Stokes equations on a smooth bounded domain. We prove two versions of this result: In the first one, we use a Lipschitz‐continuity assumption on the velocity direction in all sufficiently close space points. In the second version, only points whose distance is equal to a sufficiently small positive number dependent on the data are considered.