Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

Pages: 257 – 283

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a11

Authors

Xia Ye (School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China)

Mingxuan Zhu (School of Mathematical Sciences, Qufu Normal University, Qufu, China; and College of Mathematics & Physics, Wenzhou University, Wenzhou, China)

Abstract

This paper is concerned with the initial value problem of the three-dimensional nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. We prove the existence of global strong solution on $\mathbb{R}^3 \times (0,\infty)$ under the initial norm ${\lVert u_0 \rVert}_{\dot{H}^\alpha} + {\lVert \nabla d_0 \rVert}_{\dot{H}^\alpha} (1/2 \lt \alpha \leq 1)$ being suitably small. In addition, the algebraic decay rate estimates of the global strong solution are obtained.

Keywords

nonhomogeneous incompressible liquid crystal system, density-dependent viscosity, global existence, vacuum, decay

2010 Mathematics Subject Classification

35Q35, 76A15, 76D03

Full Text (PDF format)

This work was supported by the National Natural Science Foundation of China (Nos. 12061037 and 11971209), and by the Natural Science Foundation of Jiangxi Province (Grant 20212BAB201016).

Received 17 October 2022

Received revised 27 May 2023

Accepted 15 June 2023

Published 7 December 2023