  We consider here the simplified Ericksen-Leslie system on the whole
three-dimensional space. 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 independing) case and the non-stationary (time depending) 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
