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On Well-Posedness and Decay of Strong Solutions for 3D Incompressible Smectic-A Liquid Crystal Flows

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Abstract

In this paper, we study a hydrodynamic system that models smectic-A liquid crystal flow in \({\mathbb {R}}^3\). This model consists of the Navier–Stokes equations for fluid velocity coupled with a fourth-order equation for the layer variable. The main purpose is to analyze the well-posedness and asymptotic behavior of strong solutions. We first prove the local well-posedness through the higher-order a prior estimates of the solution and Galerkin method. Then, we establish the existence of global strong solution provided that \( \Vert u_0 \Vert _{ \dot{H }^{\frac{1}{2} }} +\Vert \varphi _0\Vert _{\dot{ H} ^{\frac{3}{2} }} +\Vert \varphi _0\Vert _{\dot{ H} ^{\frac{7}{2} }} \) is sufficiently small. Finally, we show the temporary decay estimates for the higher-order derivatives of strong solution by using the negative Sobolev norm estimates.

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Acknowledgements

The authors would like to thank the referee for helpful comments. This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. N2005031).

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Authors and Affiliations

  1. College of Sciences, Northeastern University, Shenyang, 110004, China

    Xiaopeng Zhao

  2. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, China

    Yong Zhou

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Yong Zhou

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  1. Xiaopeng Zhao
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  2. Yong Zhou
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Correspondence to Yong Zhou.

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Communicated by Dejan Slepcev.

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Zhao, X., Zhou, Y. On Well-Posedness and Decay of Strong Solutions for 3D Incompressible Smectic-A Liquid Crystal Flows. J Nonlinear Sci 32, 7 (2022). https://doi.org/10.1007/s00332-021-09771-9

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