Skip to main content
SpringerLink
Log in

On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed by Cho et al. (2004), Cho and Kim (2006) and Choe and Kim (2003) is removed in a suitable sense. Then, the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and the compatibility condition, the strong solution is proved to be a classical one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon S, Douglis A, Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm Pure Appl Math, 1964, 17: 35–92

    Article  MathSciNet  Google Scholar 

  2. Cho Y, Choe H-J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluids. J Math Pures Appl (9), 2004, 83: 243–275

    Article  MathSciNet  Google Scholar 

  3. Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscripta Math, 2006, 120: 91–129

    Article  MathSciNet  Google Scholar 

  4. Choe H-J, Kim H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differential Equations, 2003, 190: 504–523

    Article  MathSciNet  Google Scholar 

  5. Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004

    MATH  Google Scholar 

  6. Feireisl E, Novotny A, Petzeltová H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3: 358–392

    Article  MathSciNet  Google Scholar 

  7. Germain P. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J Math Fluid Mech, 2011, 13: 137–146

    Article  MathSciNet  Google Scholar 

  8. Hoff D. Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J Differential Equations, 1995, 120: 215–254

    Article  MathSciNet  Google Scholar 

  9. Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch Ration Mech Anal, 1995, 132: 1–14

    Article  MathSciNet  Google Scholar 

  10. Hoff D. Compressible flow in a half-space with Navier boundary conditions. J Math Fluid Mech, 2005, 7: 315–338

    Article  MathSciNet  Google Scholar 

  11. Hoff D, Santos M-M. Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch Ration Mech Anal, 2008, 188: 509–543

    Article  MathSciNet  Google Scholar 

  12. Hoff D, Tsyganov E. Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow. J Differential Equations, 2008, 245: 3068–3094

    Article  MathSciNet  Google Scholar 

  13. Ladyženskaja O-A, Solonnikov V-A, Ural’ceva N-N. Linear and Quasilinear Equations of Parabolic Type. Providence: Amer Math Soc, 1968

    Google Scholar 

  14. Li J, Liang Z L. On local classical solutions to the Cauchy problem of the two dimensional barotropic compressible Navier-Stokes equations with vacuum. J Math Pures Appl (9), 2014, 102: 640–671

    Article  MathSciNet  Google Scholar 

  15. Li J, Xin Z P. Global existence of regular solutions with large oscillations and vacuum for compressible flows. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Cham: Springer, 2018, 2037–2083

    Chapter  Google Scholar 

  16. Lions P-L. Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques. C R Math Acad Sci Paris, 1993, 316: 1335–1340

    MathSciNet  MATH  Google Scholar 

  17. Lions P-L. Mathematical Topics in Fluid Mechanics. Compressible Models, vol. 2. New York: Oxford University Press, 1998

  18. Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67–104

    MathSciNet  MATH  Google Scholar 

  19. Nash J. Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull Soc Math France, 1962, 90: 487–497

    Article  MathSciNet  Google Scholar 

  20. Salvi R, Straškraba I. Global existence for viscous compressible fluids and their behavior as t → 8. J Fac Sci Univ Tokyo Sect IA Math, 1993, 40: 17–51

    MathSciNet  MATH  Google Scholar 

  21. Serrin J. On the uniqueness of compressible fluid motion. Arch Ration Mech Anal, 1959, 3: 271–288

    Article  MathSciNet  Google Scholar 

  22. Valli A. Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann Sc Norm Super Pisa Cl Sci (5), 1983, 10: 607–647

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11688101, 11731007 and 11671412) and Youth Innovation Promotion Association, Chinese Academy of Sciences.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Xiangdi Huang

Authors
  1. Xiangdi Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangdi Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, X. On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum. Sci. China Math. 64, 1771–1788 (2021). https://doi.org/10.1007/s11425-019-9755-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9755-3

Keywords

MSC(2010)

Search

Navigation