Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor

In this paper, we consider the regularity criteria for the 3D incompressible Navier-Stokes equations involving the middle eigenvalue (\begin{document}$ \lambda_2 $\end{document}) of the strain tensor. It is proved that if \begin{document}$ \lambda^+_2 $\end{document} belongs to Multiplier space or Besov space, then the weak solution remains smooth on \begin{document}$ [0, T] $\end{document}, where \begin{document}$ \lambda^{+}_2 = \max\{\lambda_2, 0\} $\end{document}. These regularity conditions allows us to improve the result obtained by Miller [7].