Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component. (March 2015) - Zhang, Zujin
Abstract: In this paper, we consider the Cauchy problem for the 3 D MHD equations, and provide a regularity criterion involving ∇ u 3 and j 3, where u 3 and j 3 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.