In this paper we study the incompressible non-inertial Qian-Sheng model, which describes the hydrodynamics of nematic liquid crystals without inertial effect in the Q-tensor framework. Under some proper assumptions on the viscous coefficients, we prove the local well-posedness with large initial data and the global existence with small size of the initial data in the classical solutions regime.

For the inertial Qian–Sheng model of nematic liquid crystals in the [Formula: see text]-tensor framework, we illustrate the roles played by the entropy inequality and energy dissipation in the well-posedness of smooth solutions when we employ energy method. We first derive the coefficients requirements from the entropy inequality, and point out the entropy inequality is insufficient to guarantee energy dissipation. We then introduce a novel Condition (H) which ensures the energy dissipation. We prove that when both the entropy inequality and Condition (H) are obeyed, the local in time smooth solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Furthermore, to extend the solutions globally in time and obtain the decay of solutions, we require at least one of the two conditions: entropy inequality, or [Formula: see text], which significantly enlarge the range of the coefficients in previous works.

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 for both the compressible system and its differential system with respect to time under uniformly in small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as → 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 2norm with well-prepared initial data.

The Qian–Sheng model is a system describing the hydrodynamics of nematic liquid crystals in the Q-tensor framework. When the inertial effect is included, it is a hyperbolic-type system involving a second-order material derivative coupling with forced incompressible Navier–Stokes equations. If formally letting the inertial constant [Formula: see text] go to zero, the resulting system is the corresponding parabolic model. We provide the result on the rigorous justification of this limit in [Formula: see text] with small initial data, which validates mathematically the parabolic Qian–Sheng model. To achieve this, an initial layer is introduced to not only overcome the disparity of the initial conditions between the hyperbolic and parabolic models, but also make the convergence rate optimal. Moreover, a novel [Formula: see text]-dependent energy norm is carefully designed, which is non-negative only when [Formula: see text] is small enough, and handles the difficulty brought by the second-order material derivative.