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

TL;DR

This paper proves the global existence and describes the long-term behavior of solutions to a complex liquid crystal model under a magnetic field in two dimensions, using advanced energy estimates and convergence techniques.

Contribution

It introduces novel high-order energy estimates and applies the Lojasiewicz--Simon inequality to analyze the long-time dynamics of the Ericksen--Leslie system in 2D.

Findings

Existence of global strong solutions in 2D

Uniform bound for molecular orientation angle

Solutions converge to equilibrium as time approaches infinity

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.