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

TL;DR

This work proves global well-posedness for the Beris–Edwards Q-tensor model of nematic liquid crystals in the half-space $\mathbb{R}^N_+$ within the maximal $L_p$-$L_q$ regularity framework. The authors decompose the problem into linear and nonlinear parts, establish $\mathcal{R}$-bounded resolvent solvability and analytic semigroup generation for the linearized operator, and derive weighted high-order estimates to control nonlinear terms. A fixed-point argument in a carefully designed function space yields a unique global-in-time solution for small initial data, with precise regularity for $\boldsymbol u$, $\boldsymbol Q$, and $\nabla \frak p$, including a bound $\| (1+t)\nabla \frak p\|_{L_p}$ by a multiple of the initial data and forcing. The results extend the global theory in the half-space, complementing existing bounded-domain and whole-space analyses, and provide a rigorous foundation for the time-evolution and boundary behavior of nematic flows in half-space geometries.

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.

Theorems & Definitions (17)

• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7: Weis W
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3