Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows

Zhongying Liu; Yang Liu; Yiqi Jiang; Zhongying Liu; Yang Liu; Yiqi Jiang

doi:10.3934/math.2022695

AIMS Mathematics

2022, Volume 7, Issue 7: 12536-12565. doi: 10.3934/math.2022695

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Research article

Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows

1.
College of Mathematics, Changchun Normal University, Changchun, 130032, China
2.
School of Mathematics, Jilin University, Changchun, 130012, China

Received: 09 December 2021 Revised: 20 March 2022 Accepted: 24 March 2022 Published: 27 April 2022
MSC : 35Q35, 76D03

We are concerned with the initial value problem of non-isothermal incompressible nematic liquid crystal flows in $ \Bbb R^3 $. Through some time-weighted a priori estimates, we prove the global existence of a strong solution provided that $ \Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\nabla d_0\|_{L^2}^2\Big)\Big(\|\nabla u_0\|_{L^2}^2+\|\nabla^2d_0\|_{L^2}^2\Big) $ is reasonably small, which extends the corresponding Li's (Methods Appl. Anal. 2015 ^[4]) and Ding-Huang-Xia's (Filomat 2013 ^[2]) results to the whole space $ \Bbb R^3 $ and non-isothermal case. Furthermore, we also derive the algebraic decay estimates of the solution.
- inhomogeneous case,
- incompressible nematic liquid crystal flows,
- global strong solution,
- vacuum
Citation: Zhongying Liu, Yang Liu, Yiqi Jiang. Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows[J]. AIMS Mathematics, 2022, 7(7): 12536-12565. doi: 10.3934/math.2022695

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Abstract

We are concerned with the initial value problem of non-isothermal incompressible nematic liquid crystal flows in $ \Bbb R^3 $. Through some time-weighted a priori estimates, we prove the global existence of a strong solution provided that $ \Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\nabla d_0\|_{L^2}^2\Big)\Big(\|\nabla u_0\|_{L^2}^2+\|\nabla^2d_0\|_{L^2}^2\Big) $ is reasonably small, which extends the corresponding Li's (Methods Appl. Anal. 2015 ^[4]) and Ding-Huang-Xia's (Filomat 2013 ^[2]) results to the whole space $ \Bbb R^3 $ and non-isothermal case. Furthermore, we also derive the algebraic decay estimates of the solution.

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