Liouville Theorems for a Stationary and Non-stationary Coupled System of Liquid Crystal Flows in Local Morrey Spaces

Abstract

We consider here the simplified Ericksen–Leslie system on the whole space $$\mathbb {R}^{3}$$ . This system deals with the incompressible Navier–Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both, the stationary (time independent) case and the non-stationary (time dependent) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi-product, our theorems also improve some well-known results on Liouville-type theorems for the particular case of classical Navier–Stokes equations.