Communications in Mathematical Sciences

Volume 18 (2020)

Number 4

Global stability of large solutions to the 3-D compressible flow of liquid crystals

Pages: 887 – 908

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n4.a1

Authors

Yuhui Chen (School of Aeronautics and Astronautics, Sun Yat-Sen University, Guangzhou, China)

Jingchi Huang (School of Mathematics, Sun Yat-Sen University, Guangzhou, China)

Haiyan Xu (School of Applied Mathematics, Guangdong University of Technology, Guangzhou, China)

Zheng-An Yao (School of Mathematics, Sun Yat-Sen University, Guangzhou, China)

Abstract

The current paper is devoted to the investigation of the global-in-time stability of large solutions to the compressible liquid crystal equations in the whole space. Suppose that the density is bounded from above uniformly in time in the Höder space $C^\alpha$ with $\alpha$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results: (1) Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation. (2) Such kind of the solution is stable, which means any perturbed solution will remain close to the reference solution if initially they are close to each other. This implies that the set of the smooth and bounded solutions is open.

Keywords

compressible liquid crystal, large solutions, stability

2010 Mathematics Subject Classification

35A01, 35B45, 35Q35, 76A05, 76D03

Full Text (PDF format)

Yuhui Chen is partially supported by the China Postdoctoral Science Foundation under Grant 2019M663198 and NNSF of China under Grant 11571380, 11801586 and Guangzhou Science and Technology Program under Grant 201607010144. Jingchi Huang is supported by NNSF of China under Grant 11701585 and Science and Technology Planning Project of Guangdong under Grant 2017A030310047. Zheng-an Yao is partially supported by NNSF of China under Grant 11971496.

Received 8 December 2018

Accepted 6 December 2019

Published 28 July 2020