Global Well-posedness and Long-time Behavior of the Two-dimensional General Ericksen--Leslie System in the Isotropic Case under a Magnetic Field

Abstract

This paper establishes the global well-posedness and long-time dynamics of the general Ericksen--Leslie system for isotropic nematic liquid crystals under a constant magnetic field. On the two-dimensional torus $\mathbb{T}^2$, a liquid crystal molecule coincides with itself under rotations by integer multiples of $\pi$, which results in special boundary conditions. We prove the existence of global-in-time strong solutions by developing novel high-order energy estimates and employing compactness techniques. A key challenge lies in controlling the orientation of the liquid crystal molecules. After achieving a uniform bound for the molecular orientation angle in $\mathbb{S}^1$, we further characterize the long-time behavior of the solutions. This is accomplished by applying the Lojasiewicz--Simon inequality, which reveals the convergence of the solutions as time approaches infinity.