  We present a large-scale molecular dynamics study of nematic-paranematic
interfaces under shear. We use a model of soft repulsive ellipsoidal particles
with well-known equilibrium properties, and consider interfaces which are
oriented normal to the direction of the shear gradient (common stress case).
The director at the interface is oriented parallel to the interface (planar). A
fixed average shear rate is imposed with Lees-Edwards boundary conditions, and
the heat is dissipated with a profile-unbiased thermostat. First we study the
properties of the interface at one particular shear rate in detail. The local
interfacial profiles and the capillary wave fluctuations of the interfaces are
calculated and compared with those of the corresponding equilibrium interface.
Under shear, the interfacial width broadens and the capillary wave amplitudes
at large wavelengths increase. The strain is distributed inhomogeneously in the
system (shear banding), the local shear rate in the nematic region being
distinctly higher than in the paranematic region. Surprisingly, we also observe
(symmetry breaking) flow in the {\em vorticity} direction, with opposite
direction in the nematic and the paranematic state. Finally, we investigate the
stability of the interface for other shear rates and construct a nonequilibrium
phase diagram.
