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Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals

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Abstract

The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu (SIAM J Math Anal 45(5):2678–2699, 2013) through relaxing the regularity requirement of the initial data in terms of the director field. Based on the global existence, we then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible flow of liquid crystals converges to the solution to the corresponding incompressible model in some function spaces. Moreover, the accurate converge rates are obtained.

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Authors and Affiliations

  1. College of Science, China Three Gorges University, Yichang, 443002, People’s Republic of China

    Qunyi Bie

  2. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Haibo Cui

  3. School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Qiru Wang & Zheng-An Yao

Authors
  1. Qunyi Bie
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  2. Haibo Cui
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  3. Qiru Wang
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  4. Zheng-An Yao
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Corresponding author

Correspondence to Qiru Wang.

Additional information

Research Supported by the NNSF of China (Nos. 11271379, 11271381, 11671406, 11601164, 11701325), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002) and the Scientific Research Funds of Huaqiao University (No. 15BS201).

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Bie, Q., Cui, H., Wang, Q. et al. Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals. Z. Angew. Math. Phys. 68, 113 (2017). https://doi.org/10.1007/s00033-017-0862-0

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  • DOI: https://doi.org/10.1007/s00033-017-0862-0

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