Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy
arXiv:1810.09961 · doi:10.1016/j.jde.2019.07.010
Abstract
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 coupled with an evolution equation for the order parameter -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 -norm of the -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 -norm of the -tensor along the time evolution.