Some regularity criteria for the 3D generalized Navier–Stokes equations

We show some regularity criteria (Prodi–Serrin type regularity) to weak solutions of the 3D generalized Navier–Stokes equations in viewpoint of the velocity vector u or the vorticity vector ω:=∇×u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega :=\nabla \times u$$\end{document} in Lorentz space. Moreover, we briefly mention some results for coupled equations with Navier–Stokes equation (see Remark 1.5 and 1.8).