The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space

TL;DR

This paper establishes maximal regularity results for the linearized Beris-Edwards model of liquid crystal flows in a half-space, enabling analysis of well-posedness for the nonlinear system using advanced operator theory.

Contribution

It proves the unique maximal $L^p$-$L^q$ regularity for the linearized Beris-Edwards model in the half-space, a key step for analyzing nonlinear well-posedness.

Findings

Maximal $L^p$-$L^q$ regularity for the linearized problem established.

Local well-posedness for the nonlinear model with small initial data proved.

Method relies on $ ext{R}$-boundedness and operator-valued Fourier multipliers.

Abstract

In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data.