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 ε for both the compressible system and its differential system with respect to time under uniformly in ε small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as ε → 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 L² norm with well-prepared initial data.