Some interior regularity criteria for suitable weak solutions to the 3-D Navier-Stokes equations
\[ \frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=f,\quad \text{div}\,v=0, \quad x\in \Omega,\;t\in I, \] are studied in the paper. Here \(v=(v_1,v_2,v_3)\) is the velocity of a fluid, \(p\) is the pressure, \(\Omega\) is either a domain in \(\mathbb R^3\) or the 3-D torus \(T^3\), \(I\) is a finite time interval.
Let \(z=(x,t)\in \Omega\times I\) and \(Q_{z,r}=B_r\times(t-r^2,t)\), \(\alpha=\frac{3}{s}+\frac{2}{q}\), \(B_r=\{y\in\mathbb R^3:| y-x| <r\}\). It is proved that the suitable weak solutions \((v,p)\) of the problem are regular at \(z\) if one of the following conditions hold for a small constant \(\varepsilon\):
\[ \lim_{r\to 0}\sup r^{-\alpha+1}\| v-(v)_r\|_{L^{s,q}}\leq\varepsilon,\tag{i} \] where \((v)_r\) is the average on \(B_r\), \(1\leq\alpha\leq 2\), \(1\leq s,q\leq\infty\); \[ \lim_{r\to 0}\sup r^{-\alpha+2}\| \nabla v\|_{L^{s,q}}\leq\varepsilon,\quad 2\leq\alpha\leq 3,\quad 1\leq q\leq\infty;\tag{ii} \]
\[ \lim_{r\to 0}\sup r^{-\alpha+2}\| \text{curl}\, v\|_{L^{s,q}}\leq\varepsilon,\quad 2\leq\alpha\leq 3,\quad 1\leq q\leq\infty;\tag{iii} \]
\[ \lim_{r\to 0}\sup r^{-\alpha+3}\| \nabla\text{curl}\, v\|_{L^{s,q}}\leq\varepsilon,\quad 3\leq\alpha\leq 4,\quad 1\leq q,\quad 1\leq s.\tag{iv} \]
There are several corrections made to the authors’ paper [J. Differ. Equations 226, 594–618 (2006; Zbl 1159.35396)].