  In this paper we prove the existence of global in time weak solutions for an
evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid
crystal (LC) flows in three dimensions of space. In our model, the
incompressible Navier-Stokes system for the macroscopic velocity $\vu$ is
coupled to a nonlinear convective parabolic equation describing the evolution
of the Q-tensor $\QQ$, namely a tensor-valued variable representing the
normalized second order moments of the probability distribution function of the
LC molecules. The effects of the (absolute) temperature $\vt$ are prescribed in
the form of an energy balance identity complemented with a global entropy
production inequality. Compared to previous contributions, we can consider here
the physically realistic singular configuration potential $f$ introduced by
Ball and Majumdar. This potential gives rise to severe mathematical
difficulties since it introduces, in the Q-tensor equation, a term which is at
the same time singular in $\QQ$ and degenerate in $\vt$. To treat it a careful
analysis of the properties of $f$, particularly of its blow-up rate, is carried
out.
