Abstract:

In this paper, we consider the regularity criterion of weak solution to Navier-Stokes equations in $\mathbb{R}^3$. It is proved that a Leray-Hopf weak solution $u$ becomes a regular solution if the pressure $\pi\in L^p(0, T; \dot{F}_{q, \frac{10q}{5q+6}}^0(\mathbb{R}^3)) $ with $ \frac{2}{p}+\frac{3}{q}<\frac{7}{4}, \frac{12}{5}<q \leq \infty$. Meanwhile the authors also prove that if the gradient of the pressure $\nabla\pi\in L^p(0, T;\dot{F}_{q, \frac{8q}{12-3q}}^0(\mathbb{R}^3))$ with $\frac{2}{p}+\frac{3}{q}=\frac{11}{4}, \frac{12}{11}<q<4 $, then the weak solution u can be smoothly extended beyond t=T

Key words: avier-Stokes equations, Blow up criterion, Leray-Hopf weak solution, Triebel-Lizorkin spaces, Pressure