On a non-isothermal model for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the {\it absolute temperature} $\teta$, the {\it velocity field} $\ub$, and the {\it director field} $\bd$, representing preferred orientation of molecules in a neighborhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier-Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field $\bd$, where the transport (viscosity) coefficients vary with temperature. The dynamics of $\bd$ is described by means of a parabolic equation of Ginzburg-Landau type, with a suitable penalization term to relax the constraint $|\bd | = 1$. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field $\bd$. The proposed model is shown compatible with \emph{First and Second laws} of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.