The global well-posedness for the Q-tensor model of nematic liquid crystals in the half-space

TL;DR

This paper proves the first global-in-time well-posedness result for the Q-tensor model of nematic liquid crystals in the half-space using advanced functional analysis techniques.

Contribution

It establishes the global existence and uniqueness of solutions in the half-space for the Q-tensor model, employing maximal regularity and semigroup theory.

Findings

First global-in-time solution in half-space

Uses maximal Lp-Lq regularity and R-solvability

Employs Banach fixed point argument

Abstract

In this paper, we consider the Q-tensor model of nematic liquid crystals, which couples the Navier-Stokes equations with a parabolic-type equation describing the evolution of the directions of the anisotropic molecules, in the half-space. The aim of this paper is to prove the global well-posedness for the Q-tensor model in the $L_p$-$L_q$ framework. Our proof is based on the Banach fixed point argument. To control the higher-order terms of the solutions, we prove the weighted estimates of the solutions for the linearized problem by the maximal $L_p$-$L_q$ regularity. On the other hand, the estimates for the lower-order terms are obtained by the analytic semigroup theory. Here, the maximal $L_p$-$L_q$ regularity and the generation of an analytic semigroup are provided by the R-solvability for the resolvent problem arising from the Q-tensor model. It seems to be the first result to discuss the unique existence of a global-in-time solution for the Q-tensor model in the half-space.