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

TL;DR

This paper investigates regularity conditions for weak solutions of stationary fluid dynamics systems, including liquid crystals and MHD, establishing new criteria and analyticity results under specific functional space assumptions.

Contribution

It introduces new regularity criteria for weak solutions of the stationary Ericksen-Leslie and MHD systems, extending to Navier-Stokes and harmonic map flows, and proves analyticity of solutions in the MHD case.

Findings

New regularity criterion for weak solutions in Morrey spaces

Analyticity of finite energy weak solutions in MHD with Gevrey class forces

Extension of regularity results to harmonic map flow and Navier-Stokes

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.