On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

  • Yachun Li
  • Ronghua PanEmail author
  • Shengguo Zhu


In this paper, we consider the three-dimensional isentropic Navier–Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and a vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by a vacuum. Moreover, for certain classes of initial data with a local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.



The authors sincerely appreciate the efforts and highly constructive suggestions of the referees; the reports helped to improve the quality of the presentation of this paper. The research of Y. Li and S. Zhu was supported in part by National Natural Science Foundation of China under Grants 11231006 and 11571232. Y. Li was also supported by National Natural Science Foundation of China under Grant 11831011. S. Zhu was also supported by China Scholarship Council and the Royal Society–Newton International Fellowships NF170015. The research of R. Pan was partially supported by The National Science Foundation under Grants DMS-1516415 and DMS-1813603, and by The National Natural Science Foundation of China under the Grant 11628103.

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The authors declare that they have no conflict of interest.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences, MOE-LSC, and SHL-MACShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.School of MathematicsGeorgia TechAtlantaUSA
  3. 3.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  4. 4.Mathematical InstituteUniversity of OxfordOxfordUK

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