  This work is concerned with the solvability of a Navier-Stokes/$Q$-tensor
coupled system modeling the nematic liquid crystal flow on a bounded domain in
three dimensional Euclidian space with strong anchoring boundary condition for
the order parameter. We prove the existence of local in time strong solution to
the system with the anisotropic elastic energy. The proof is based on mainly
two ingredients: first, we show that the Euler-Lagrange operator corresponding
to the Landau-de Gennes free energy with general elastic coefficients fulfills
the strong Legendre condition. This result together with a higher order energy
estimate leads to the well-posedness of the linearized system, and then a local
in time solution of the original system which is regular in temporal variable
follows via a fixed point argument. Secondly, the hydrodynamic part of the
coupled system can be reformulated into a quasi-stationary Stokes type equation
to which the regularity theory of the generalized Stokes system, and then a
bootstrap argument can be applied to enhance the spatial regularity of the
local in time solution.
