Global Well-posedness and Long-time Behavior of the General Ericksen--Leslie System in 2D under a Magnetic Field

Abstract: In this paper, we investigate the global well-posedness and long-time behavior of the two-dimensional general Ericksen--Leslie system for a nematic liquid crystal in a constant magnetic field. The PDE system consists of Navier--Stokes equations and the harmonic heat flow equation for the orientations of liquid crystal molecules. For incompressible nematic liquid crystal fluids with either isotropic or anisotropic properties in torus $\mathbb{T}^2$, we derive the global well-posedness of strong solutions through higher-order energy estimates combined with compactness methods and acquire the long-time behavior of the solutions by using the \L ojasiewicz--Simon inequality after obtaining the boundedness of the nematic liquid crystal molecules' angle.