Regularity criterion in terms of the oscillation of pressure for the 3D Navier–Stokes equations.

We obtain a new regularity criterion in terms of the oscillation of time derivative of the pressure for the 3D Navier–Stokes equations in a domain D ⊂ R 3 . The key observation for the proof is that the head pressure defined by Q ¯ (x , t) = 1 2 | v (x , t) | 2 + p ¯ (x , t) , where p ¯ (x , t) = p (x , t) - ∫ t 0 t sup y ∈ Ω (∂ s p (y , s) - | ω (y , s) | 2) d s with p(x, t) the pressure and ω the vorticity satisfies parabolic maximum principle in Ω × (t 0 , T) with Ω ⋐ D . [ABSTRACT FROM AUTHOR]