Global well-posedness of the two dimensional Beris-Edwards system with general Laudau-de Gennes free energy

In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter Q-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial L-infinity-norm of the Q-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the L-infinity-norm of the Q-tensor along the time evolution. (C) 2019 Elsevier Inc. All rights reserved.