Regularity criteria for the 33D MHD equations involving one current density and the gradient of one velocity component

In this paper, we consider the Cauchy problem for the 33D MHD equations, and provide a regularity criterion involving ∇u3∇u3 and j3j3, where u3u3 and j3j3 are the third component of the velocity and the current density, respectively. As by-products, we show that assumptions on two vorticity and one current density component are enough to ensure the smoothness of the solution, and improve some previous results (Skalak, 2014; Zhou and Pokorný, 2009, 2010) for the Navier–Stokes equations.