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In this paper, we rigorously justify the connection between Qian-Sheng'sinertial $Q$-tensor model and the full Ericksen-Leslie model for the liquidcrystal flow. By using the Hilbert expansion method, we prove that, when theelastic coefficients tend to zero(also called the uniaxial limit), the solutionto the Qian-Sheng's inertial model will converge to the solution to the fullinertial Ericksen-Leslie system.

Rigorous justification of the uniaxial limit from Qian-Sheng's inertial $Q$-tensor theory to the Ericksen-Leslie theory