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Global Well-Posedness and Exponential Stability of 3D Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum in Unbounded Domains

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Abstract

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional (3D) nonhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity and vacuum. After establishing some key a priori exponential decay-in-time rates of the strong solutions, we obtain both the global existence and exponential stability of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in some homogeneous Sobolev space which may be optimal compared with the case of homogeneous Navier-Stokes equations. Note that this result is proved without any smallness conditions on the initial density which contains vacuum and even has compact support.

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Acknowledgements

The authors would like to thank the referee for his/her careful reading and helpful suggestions on the manuscript. The research of J. Li is partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, National Natural Science Foundation of China Grant Nos. 11688101, 11525106, and 12071200, and Double-Thousand Plan of Jiangxi Province (No. jxsq2019101008). The research of B. Lü is partially supported by Natural Science Foundation of Jiangxi Province (No. 20202ACBL211002), Science and Technology Project of Jiangxi Provincial Education Department (No. GJJ160719), and National Natural Science Foundation of China (Grant Nos. 11601218 and 11971217 ).

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Authors and Affiliations

  1. Division of Mathematics, Department of Mathematical and Physical Sciences, National Natural Science Foundation of China, Beijing, 100085, People’s Republic of China

    Cheng He

  2. Department of Mathematics, Nanchang University, Nanchang, 330031, People’s Republic of China

    Jing Li

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

    Jing Li

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Jing Li

  5. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, People’s Republic of China

    Boqiang Lü

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  1. Cheng He
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Correspondence to Jing Li.

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He, C., Li, J. & Lü, B. Global Well-Posedness and Exponential Stability of 3D Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum in Unbounded Domains. Arch Rational Mech Anal 239, 1809–1835 (2021). https://doi.org/10.1007/s00205-020-01604-5

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