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

TL;DR

This paper proves the local existence of strong solutions for a coupled Navier-Stokes and Q-tensor system modeling nematic liquid crystal flow, using energy estimates and regularity theory in three dimensions.

Contribution

It establishes the well-posedness of the system with anisotropic elastic energy, addressing the solvability of a complex coupled PDE system for liquid crystals.

Findings

Existence of local strong solutions proven.

The Euler-Lagrange operator satisfies the strong Legendre condition.

Regularity of solutions enhanced via bootstrap argument.

Abstract

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.