On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Abstract.

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing $z_{0}=(x_{0}, t_{0})$, and let $Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0})$ be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$ with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$ with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2$, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.