On global well-posedness and decay of 3D Ericksen-Leslie system

Xiufang Zhao; Ning Duan; Xiufang Zhao; Ning Duan

doi:10.3934/math.2021730

AIMS Mathematics

2021, Volume 6, Issue 11: 12660-12679. doi: 10.3934/math.2021730

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Research article

On global well-posedness and decay of 3D Ericksen-Leslie system

Xiufang Zhao ^{1
,
,},
Ning Duan ²

1.
School of Science, Qiqihar University, Qiqihar 161006, China
2.
College of Sciences, Northeastern University, Shenyang 110819, China

Received: 11 June 2021 Accepted: 31 August 2021 Published: 03 September 2021
MSC : 35Q35, 76D03, 35B40

In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon > 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s < \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.
- Ericksen-Leslie system,
- strong solutions,
- global well-posedness,
- decay
Citation: Xiufang Zhao, Ning Duan. On global well-posedness and decay of 3D Ericksen-Leslie system[J]. AIMS Mathematics, 2021, 6(11): 12660-12679. doi: 10.3934/math.2021730

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Abstract

In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon > 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s < \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.

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