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Global well-posedness of the two dimensional Beris-Edwards system with general Laudau-de Gennes free energy. (English) Zbl 1428.35378
Summary: 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 $$L^\infty$$-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 $$L^\infty$$-norm of the $$Q$$-tensor along the time evolution.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 76A15 Liquid crystals 35D30 Weak solutions to PDEs
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