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Global Existence of Strong and Weak Solutions to 2D Compressible Navier–Stokes System in Bounded Domains with Large Data and Vacuum

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Abstract

The barotropic compressible Navier–Stokes system subject to the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain is considered. For initial density allowed to vanish, the global existence of strong and weak solutions is established when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than one and a third. It should be mentioned that this result is obtained without any restrictions on the size of initial value. To get over the difficulties brought by boundary, on the one hand, Riemann mapping theorem and the pull-back Green’s function method are applied to get a pointwise representation of the effective viscous flux. On the other hand, since the orthogonality is preserved under conformal mapping due to its preservation on the angle, the slip boundary conditions are used to reduce the integral representation to the desired commutator form whose singularities can be cancelled out by using the estimates on the spatial gradient of the velocity.

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Acknowledgements

The authors would like to thank Prof. Guocai Cai for his valuable discussions. The research is partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, NSFC Grant Nos. 11688101 and 12071200, and Double-Thousand Plan of Jiangxi Province (No. jxsq2019101008).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, Beijing, People’s Republic of China

    Xinyu Fan, Jiaxu Li & Jing Li

  2. Department of Mathematics & Institute of Mathematics and Interdisciplinary Sciences, Nanchang University, Nanchang, 330031, Jiangxi Province, People’s Republic of China

    Jing Li

  3. Institute of Applied Mathematics, AMSS, & Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, Beijing, People’s Republic of China

    Jing Li

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  1. Xinyu Fan
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Correspondence to Jiaxu Li.

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The authors declare that they have no conflict of interest. The research grants from funding agencies have been stated in the text of the manuscript in a separate section before the reference list. The authors claim that none of the material in the paper has been published or is under consideration for publication elsewhere. Informed consent has been obtained from all authors in this paper. The authors declare that their manuscript complies to the Ethical Rules applicable for “Archive for Rational Mechanics and Analysis”.

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Communicated by P.-L. Lions.

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Fan, X., Li, J. & Li, J. Global Existence of Strong and Weak Solutions to 2D Compressible Navier–Stokes System in Bounded Domains with Large Data and Vacuum. Arch Rational Mech Anal 245, 239–278 (2022). https://doi.org/10.1007/s00205-022-01790-4

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  • DOI: https://doi.org/10.1007/s00205-022-01790-4

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