Global well‐posedness to the compressible inertial Qian–Sheng model of liquid crystals

Abstract

In this paper, we study the well‐posedness of the Cauchy problem to the compressible inertial Qian–Sheng model. Inspired by Jiang–Luo–Ma–Tang's work in [J. Hyperbolic. Differ. Equ 18:221‐256, 2021], we introduce a novel Condition (H) which ensures the energy dissipation, and then, we prove that when both the viscosity coefficients and Condition (H) are satisfied, the local‐in‐time solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Finally, the local solution can be extended globally under at least one of the two conditions: or .