Entropy inequality and energy dissipation of inertial Qian–Sheng model for nematic liquid crystals

Abstract

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.

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

“…In this paper, we study the compressible inertial Qian–Sheng model which involving a hyperbolic‐type equation. Comparing to our previous work on the incompressible inertial Qian–Sheng model [8], there appears some new difficulties. We now make some comments on the key ingredients of the analysis for the above theorems.…”

“…Then, in [7], they study the low Mach number limits of the compressible inertial Qian-Sheng model in context of small classical solutions on whole space. More recently, [8] analyzed the relations between entropy inequality and energy dissipation and proved that the existence of large local-in-time classical solution could be derived under a novel condition; otherwise, they obtained the small local existence. Very recently, in [9], the authors proved the zero inertial limit of the inertial Qian-Sheng liquid crystal model with general initial data.…”

“…As inspired in [8], one can construct a novel Condition (H), which makes the energy dissipation. When Condition (H) holds, it is easy to see that dissipative functional $$$ F $$$ () is coercivity.…”