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<p>We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie hyperbolic system an 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 (J Funct Anal, 288:110858, 2025). As a byproduct, we obtain the sharp (same as the linear solution) <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^\infty _x\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>x</mi> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>-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.</p>

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Global Solution of 2D Hyperbolic Liquid Crystal System for Small Initial Data

  • 期刊论文

  • 发表日期：

    2026年3月14日

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  • Xuecheng Wang

摘要

We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie hyperbolic system an 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 (J Funct Anal, 288:110858, 2025). As a byproduct, we obtain the sharp (same as the linear solution) \(L^\infty _x\) L x ∞ -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.