Some remarks on the regularity of weak solutions for the stationary Ericksen-Leslie and MHD systems

Abstract: We consider two elliptic coupled systems of relevance in the fluid dynamics. These systems are posed on the whole three-dimensional space and they consider the action of external forces. The first system deals with the simplified Ericksen-Leslie (SEL) system, which describes the dynamics of liquid crystal flows. The second system is the time-independent magneto-hydrodynamic (MHD) equations. For the (SEL) system, we obtain a new criterion to improve the regularity of weak solutions, provided that they belong to some homogeneous Morrey space. As a bi-product, we also obtain some new regularity criterion for the stationary Navier-Stokes equations and for a nonlinear harmonic map flow. This new regularity criterion also holds true for the (MHD) equations. Furthermore, for this last system we are able to use the Gevrey class to prove that all finite energy weak solutions are analytic functions, provided the external forces belong to some Gevrey class.