  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.
