On the geometric regularity conditions for the 3D Navier–Stokes equations

Abstract

We prove geometrically improved version of Prodi–Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier–Stokes equations and denote {f}+=max{f,0}, QT=R3×(0,T). If {(v×ω∣ω∣)⋅Λβv∣Λβv∣}+∈Lx,tγ,α(QT) with 3/γ+2/α≤1 for some γ>3 and 1≤β≤2, then the local smooth solution v of the Navier–Stokes equations on (0,T) can be continued to (0,T+δ) for some δ>0. We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier–Stokes equations in a space–time domain containing z0=(x0,t0), let Qz0,r=Bx0,r×(t0−r2,t0) be a parabolic cylinder in the domain. We show that if either {(v×ω∣ω∣)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤1, or {(v∣v∣×ω)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤2, (γ≥2, α≥2), then z0 is a regular point for v. This improves previous local regularity criteria for the suitable weak solutions.