Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D

In this article, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on 2, on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many “singular” times, at which the energy concentration occurs and the director field losses its second order derivatives.