On the regularity of very weak solutions for an elliptic coupled system of liquid crystal flows

TL;DR

This paper investigates the regularity of very weak solutions to an elliptic coupled system modeling liquid crystal flows, establishing conditions that enhance regularity and deriving a new criterion for Navier-Stokes regularity.

Contribution

It introduces a framework for very weak solutions in Morrey spaces and provides new regularity conditions, also applying findings to Navier-Stokes equations.

Findings

Derived sufficient conditions for regularity of very weak solutions

Established a new regularity criterion for steady Navier-Stokes equations

Extended regularity results within the Morrey space framework

Abstract

We consider here an elliptic coupled system describing the dynamics of liquid crystals flows. This system is posed on the whole n-dimensional space. We introduce first the notion of very weak solutions for this system. Then, within the fairly general framework of the Morrey spaces, we derive some sufficient conditions on the very weak solutions which improve their regularity. As a bi-product, we also prove a new regularity criterium for the time-independing Navier-Stokes equations.