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

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.