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Global solution of 2D hyperbolic liquid crystal system for small initial data

Xuecheng Wang

TL;DR

The work addresses global regularity for the $2$-D simplified Ericksen-Leslie hyperbolic liquid crystal system, where the director obeys a wave-map type equation coupled to an incompressible velocity field. The authors identify a hidden null structure in the velocity equation that cancels troublesome pressure–nonlinearity interactions, enabling a novel normal-form analysis that decouples the heat and wave dynamics. They establish global existence for small data and scattering, with sharp decay rates for both the heat and wave components and a decay of the heat energy at rate $t^{-1/2+\delta}$, thereby matching linear decay. The results advance understanding of low-dimensional hyperbolic liquid crystal models and introduce techniques that may apply to other 2D hyperbolic systems with similar null structures.

Abstract

We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao. As byproducts, we obtain the sharp (same as the linear solution) decay estimates for both the heat part and the wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity. This paper's main contribution is the discovery of a novel null structure within the velocity equation's wave-type quadratic self-interaction. This structure compensates the insufficient decay rate in 2D, which previously hindered the establishment of global regularity for small data.

Global solution of 2D hyperbolic liquid crystal system for small initial data

TL;DR

The work addresses global regularity for the -D simplified Ericksen-Leslie hyperbolic liquid crystal system, where the director obeys a wave-map type equation coupled to an incompressible velocity field. The authors identify a hidden null structure in the velocity equation that cancels troublesome pressure–nonlinearity interactions, enabling a novel normal-form analysis that decouples the heat and wave dynamics. They establish global existence for small data and scattering, with sharp decay rates for both the heat and wave components and a decay of the heat energy at rate , thereby matching linear decay. The results advance understanding of low-dimensional hyperbolic liquid crystal models and introduce techniques that may apply to other 2D hyperbolic systems with similar null structures.

Abstract

We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao. As byproducts, we obtain the sharp (same as the linear solution) decay estimates for both the heat part and the wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity. This paper's main contribution is the discovery of a novel null structure within the velocity equation's wave-type quadratic self-interaction. This structure compensates the insufficient decay rate in 2D, which previously hindered the establishment of global regularity for small data.
Paper Structure (16 sections, 10 theorems, 149 equations)

This paper contains 16 sections, 10 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Main difficulty in $2D$ and previous results
  3. Main result of this paper
  4. Hidden null structure
  5. Ideas of proof: exploiting the benefit of the null structure
  6. Methods of the proof
  7. Main bootstrap assumption
  8. Outline of this paper
  9. Preliminary
  10. Notation
  11. Energy estimate of the velocity part and the wave part
  12. Energy estimate of the velocity field
  13. $Z$-norm estimate of the velocity part and the wave part
  14. The estimate of $Z_u(t)$
  15. The estimate of $Z_\phi(t)$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $N_0=10^{5}$. There exists absolute small constant $\epsilon_0$ such that if the initial data $(u_0, \phi_0, \phi_1)$ of the system mainsysnew satisfies the following smallness condition where $U_0:=\phi_1+ i |\nabla| \phi_0.$ Then the system mainsysnew has unique global solution. The nonlinear solution $\phi(t)$ scatters to a linear solution $\phi_\infty(t)$. Moreover, the following sharp de

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more