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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> for both the compressible system and its differential system with respect to time under uniformly in <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [<xref ref-type="bibr" rid="b6">6</xref>]. Moreover, we also obtain the convergence rates associated with <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with well-prepared initial data.