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

TL;DR

This paper investigates the roles of entropy inequality and energy dissipation in the well-posedness of the inertial Qian-Sheng model for nematic liquid crystals, introducing a new condition to ensure energy dissipation and extending solution existence results.

Contribution

It introduces a novel Condition (H) that guarantees energy dissipation and broadens the coefficient range for global solution existence in the inertial Qian-Sheng model.

Findings

Entropy inequality alone is insufficient for energy dissipation.

Condition (H) ensures energy dissipation and local solution existence.

At least one of the entropy inequality or rac{mu_2= mu_2} is needed for global solutions.

Abstract

For the inertial Qian-Sheng model of nematic liquid crystals in the $Q$-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 $\tilde{\mu}_2= \mu_2$, which significantly enlarge the range of the coefficients in previous works.