# American Institute of Mathematical Sciences

November  2018, 23(9): 3879-3899. doi: 10.3934/dcdsb.2018115

## On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

 1 NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China 2 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: W. Wang

Received  April 2017 Revised  December 2017 Published  April 2018

Fund Project: Y. Liu is supported by NSF of China under Grant 11601334. W. Wang is supported by NSF of China under Grant 11501502 and "the Fundamental Research Funds for the Central Universities" 2016QNA3004

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 and uniqueness of local in time strong solutions to the system with an 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.

Citation: Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
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