Existence and blow-up of the solutions to the viscous quantum magnetohydrodynamic nematic liquid crystal model

Abstract

In this paper, we investigate the coupled viscous quantum magnetohydrodynamic equations and nematic liquid crystal equations which describe the motion of the nematic liquid crystals under the magnetic field and the quantum effects in the two-dimensional case. We prove the existence of the global finite energy weak solutions by use of a singular pressure close to vacuum. Then we obtain the local-in-time existence of the smooth solution. In the final, the blow-up of the smooth solutions is studied. The main techniques are Faedo-Galerkin method, compactness theory, Arzela-Ascoli theorem and construction of the functional differential inequality.