  In this paper, we consider the Beris-Edwards system for incompressible
nematic liquid crystal flows. The system under investigation consists of the
Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an
evolution equation for the order parameter $Q$-tensor. One important feature of
the system is that its elastic free energy takes a general form and in
particular, it contains a cubic term that possibly makes it unbounded from
below. In the two dimensional periodic setting, we prove that if the initial
$L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a
unique global weak solution. The proof is based on the construction of a
specific approximating system that preserves the $L^\infty$-norm of the
$Q$-tensor along the time evolution.
