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 absolute temperature theta, the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood 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 d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg-Landau type, with a suitable penalization term to relax the constraint vertical bar d vertical bar = 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 d. The proposed model is shown to be compatible with 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.