Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions

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