In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number $ \epsilon $ for both the compressible system and its differential system with respect to time under uniformly in $ \epsilon $ small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as $ \epsilon \rightarrow 0 $, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [6]. Moreover, we also obtain the convergence rates associated with $ L^2 $-norm with well-prepared initial data.

中文翻译：

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可压缩惯性钱胜模型的马赫数下限：经典解的收敛性

本文研究了液晶流可压缩惯性钱胜模型的不可压缩极限。我们首先在统一的小初始数据下，根据时间对可压缩系统及其微分系统的马赫数ε导出统一能量估计。然后，基于这些统一的估计，我们将可压缩系统的极限传递为$ \ epsilon \ rightarrow 0 $，以便我们通过紧凑性参数建立不可压缩系统的全局经典解。我们强调，在初始数据较小的情况下，不可压缩惯性钱盛模型在全局时间上存在，与De Anna和Zarnescu的工作结果相比，我们对系数的假设范围大大扩大了[6]。此外，我们还使用准备好的初始数据来获得与$ L ^ 2 $ -norm相关的收敛速度。