  The interplay between the structural relaxation and the rheological response
of a binary LJ glass former is studied via MD simulations. In the quiescent
state, the model is well known for its sluggish dynamics and a two step
relaxation of correlation functions at low temperatures. An ideal glass
transition temperature of $T_c = 0.435$ has been identified in the previous
studies via the analysis of the system's dynamics in the frame work of the mode
coupling theory of the glass transition [W. Kob and H.C. Andersen, PRE 51, 4626
(1995)]. Here, we test wether a signature of this ideal glass transition can
also be found under shear. Indeed, the following distinction in the structural
relaxation is found: In the supercooled state, the structural relaxation is
dominated by the shear at relatively high shear rates, $\dot{\gamma}$, whereas
at sufficiently low $\dot{\gamma}$ the (shear-independent) equilibrium
relaxation is recovered. In contrast to this, the structural relaxation of a
\emph{glass} is always driven by shear. This distinct behavior of the
correlation functions is also reflected in the rheological response. In the
supercooled state, the shear viscosity, $\eta$, decreases with increasing shear
rate (shear thinning) at high shear rates, but then converges toward a constant
as the $\dot{\gamma}$ is decreased below a (temperature-dependent) threshold
value. Below $T_c$, on the other hand, the shear viscosity grows as $\eta
\propto 1/\dot{\gamma}$ suggesting a divergence at $\dot{\gamma} =0$. Thus,
within the accessible observation time window, a transition toward a
non-ergodic state seems to occur in the driven glass as the driving force
approaches zero.
