Our aim is to prove Liouville type theorems for the three dimensional steady-state Navier-Stokes equations provided the velocity field belongs to some Lorentz spaces. The corresponding statement contains several known results as a particular case.

In this paper, we prove the uniqueness of weak solutions to the two dimensional full Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. This question remains unknown even in the case when the Leslie stress is vanishing. The main technique used in the proof is Littlewood-Paley analysis performed in a very delicate way. Different from the earlier result in [28], we introduce a new metric and explore the algebraic structure of the molecular field.

We present new interior regularity criteria for suitable weak solutions of the magneto-hydrodynamics (MHD) equations in terms of the velocity field. The result means that the velocity field plays a more important role than the magnetic field in the local regularity theory of the MHD equations. This also gives a positive answer to the problem proposed by Kang and Lee in [J. Differential Equations, 247 (2009), pp. 2310--2330].