Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

Junling Sun; Xuefeng Han; Junling Sun; Xuefeng Han

doi:10.3934/math.2022863

AIMS Mathematics

2022, Volume 7, Issue 9: 15759-15794. doi: 10.3934/math.2022863

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Research article

Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

Junling Sun ,
Xuefeng Han ^,

School of Mathematics and Information Science, Henan University of Polytechnic, Jiaozuo, Henan 454003, China

Received: 05 April 2022 Revised: 12 June 2022 Accepted: 21 June 2022 Published: 27 June 2022
MSC : 35A01, 35Q31

This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \Omega $. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $ H^{s}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.
- nematic liquid crystal flow,
- global Sobolev solution,
- asymptotic expansion
Citation: Junling Sun, Xuefeng Han. Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions[J]. AIMS Mathematics, 2022, 7(9): 15759-15794. doi: 10.3934/math.2022863

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Abstract

This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \Omega $. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $ H^{s}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.

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