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

Qingtong Wu

TL;DR

This paper analyzes the 2D general Ericksen–Leslie system for isotropic nematics under a constant magnetic field on the torus $\mathbb{T}^2$. It develops high-order energy estimates and a Galerkin framework to prove global well-posedness of strong solutions, and then leverages the Łojasiewicz–Simon inequality to establish convergence of the velocity and molecular orientation to a steady state, with precise decay rates. A key novel aspect is handling modulo-$\pi$ boundary conditions for the orientation angle, including a detailed steady-state analysis for periodic and modulo-$\pi$ cases via elliptic sine–Gordon problems. The results provide rigorous insight into Fréedericksz-type transitions under magnetic fields for the general Ericksen–Leslie system and yield quantitative long-time behavior that connects to equilibrium states determined by the elliptic problems. The work thus advances the mathematical theory of nematic flows under external fields, offering a robust framework for global dynamics and stability in 2D isotropic settings.

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 $π$, 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.

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 analyzes the 2D general Ericksen–Leslie system for isotropic nematics under a constant magnetic field on the torus . It develops high-order energy estimates and a Galerkin framework to prove global well-posedness of strong solutions, and then leverages the Łojasiewicz–Simon inequality to establish convergence of the velocity and molecular orientation to a steady state, with precise decay rates. A key novel aspect is handling modulo- boundary conditions for the orientation angle, including a detailed steady-state analysis for periodic and modulo- cases via elliptic sine–Gordon problems. The results provide rigorous insight into Fréedericksz-type transitions under magnetic fields for the general Ericksen–Leslie system and yield quantitative long-time behavior that connects to equilibrium states determined by the elliptic problems. The work thus advances the mathematical theory of nematic flows under external fields, offering a robust framework for global dynamics and stability in 2D isotropic settings.

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 , a liquid crystal molecule coincides with itself under rotations by integer multiples of , 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 , 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.

Paper Structure

This paper contains 45 sections, 27 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. The general Ericksen--Leslie system
  3. Brief review of previous results
  4. Notations
  5. Main results
  6. Basic energy law
  7. System for
  8. Proof.
  9. Proof.
  10. Proof.
  11. System for
  12. System for_v_d
  13. Proof.
  14. Steady-state problem
  15. Periodic boundary conditions
  16. ...and 30 more sections

Key Result

Theorem 1.1

For steady-state problem (iso_elliptic_thm_first), we have the following conclusions. (i) Assume $a_1=a_2=0$. If $\mathrm{H}^2>\lambda_2$, there exists at least one nonconstant solution to (iso_elliptic_thm_first). If $\mathrm{H}^2\leqslant\lambda_2$, all solutions to (iso_elliptic_thm_first) are co

Theorems & Definitions (35)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.1
  • Lemma 2.4
  • Proposition 2.1
  • ...and 25 more