Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D

In this paper, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on $\mathbb T^2$, based on a new approximate system which is different from the classical Ginzburg-Landau approximation. Local energy inequalities are employed to recover the estimates on the second order spacial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions conserve the total energy and while the kinetic and potential energies transfer to the heat energy precisely. Furthermore, it is also established that these weak solutions have at most finite many singular times at which the energy concentration occurs, and as a result, the temperature must increase suddenly at each singular time on some part of $\mathbb T^2$.