A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal

In Dias (C. R. Acad. Sci. Paris Sér. A. 285 (1977)) we have deduced, from Leslie's model (Arch. Rat. Mech. Anal. 28 (1968)), a weak formulation for the bidimensional coupled evolution equations of an incompressible nematic liquid crystal submitted to an homogeneous magnetic field. In this paper we prove some results about the existence, regularity and uniqueness of their solutions. This study extends the special case developed in Dias (J. Mécanique15 (1976)), where we assumed that the director field depends on time only.