Skip to main content
SpringerLink
Log in

Global Large Solutions and Incompressible Limit for the Compressible Navier–Stokes Equations

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

The present paper is dedicated to the global large solutions and incompressible limit for the compressible Navier–Stokes system in \(\mathbb {R}^d\) with \(d\ge 2\). Motivated by the \(L^2\) work of Danchin and Mucha (Adv Math 320:904–925, 2017) in critical Besov spaces, we extend the solution space into an \(L^p\) framework. The result implies the existence of global large solutions initially from large highly oscillating velocity fields.

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. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011)

    Book  Google Scholar 

  2. Chemin, J.Y., Gallagher, I., Paicu, M.: Global regularity for some classes of large solutions to the Navier–Stokes equations. Ann. Math. 173, 983–1012 (2011)

    Article  MathSciNet  Google Scholar 

  3. Charve, F., Danchin, R.: A global existence result for the compressible Navier–Stokes equations in the critical \(L^p\) framework. Arch. Ration. Mech. Anal. 198, 233–271 (2010)

    Article  MathSciNet  Google Scholar 

  4. Chen, Q., Miao, C., Zhang, Z.: Global well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocity. Comm. Pure Appl. Math. 63, 1173–1224 (2010)

    Article  MathSciNet  Google Scholar 

  5. Chen, Q., Miao, C., Zhang, Z.: On the ill-posedness of the compressible Navier–Stokes equations. Rev. Mat. Iberoam. 31, 1375–1402 (2015)

    Article  MathSciNet  Google Scholar 

  6. Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  7. Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26, 1183–1233 (2001)

    Article  MathSciNet  Google Scholar 

  8. Danchin, R.: Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Commun. Partial Differ. Equ. 32, 1373–1397 (2007)

    Article  MathSciNet  Google Scholar 

  9. Danchin, R.: A Lagrangian approach forthe compressible Navier–Stokes equations. Ann. Inst. Fourier (Grrenoble) 64, 753–791 (2014)

    Article  Google Scholar 

  10. Danchin, R., He, L.: The incompressible limit in \( L^p\) type critical spaces. Math. Ann. 366, 1365–1402 (2016)

    Article  MathSciNet  Google Scholar 

  11. Danchin, R., Mucha, P.B.: From compressible to incompressible inhomogeneous flows in the case of large data. arXiv:1710.08819

  12. Danchin, R., Mucha, P.: Compressible Navier–Stokes system: large solutions and incompressible limit. Adv. Math. 320, 904–925 (2017)

    Article  MathSciNet  Google Scholar 

  13. Danchin, R., Xu, J.: Optimal time-decay estimates for the compressible Navier–Stokes equations in the critical \(L^p\) framework. Arch. Ration. Mech. Anal. 224, 53–90 (2017)

    Article  MathSciNet  Google Scholar 

  14. Dejardins, B., Grenier, E.: Low Mach number limlit of compressible flows in the whole space. Proc. R. Soc. Lond. 455, 2271–2279 (1999)

    Article  ADS  Google Scholar 

  15. Fang, D., Zhang, T., Zi, R.: Decay estimates for isentropic compressible Navier–Stokes equations in bounded domain. J. Math. Anal. Appl. 386, 939–947 (2012)

    Article  MathSciNet  Google Scholar 

  16. Fang, D., Zhang, T., Zi, R.: Global solutions to the isentropic compressible Navier–Stokes equations with a class of large initial data. arXiv:1608.06447

  17. Feireisl, E., Novotný, A., Petzeltová, H.: On the global existence of globally defined weak solutions to the Navier–Stokes equations of isentropic compressible fluids. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  18. Feireisl, E., Gwiazda, P., Świerczewska-Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Partial Differ. Equ. 55, 55–141 (2016)

    Article  MathSciNet  Google Scholar 

  19. Fujita, H., Kato, T.: On the Navier–Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)

    Article  MathSciNet  Google Scholar 

  20. Haspot, B.: Existence of global strong solutions in critical spaces for barotropic viscous fluids. Arch. Ration. Mech. Anal. 202, 427–460 (2011)

    Article  MathSciNet  Google Scholar 

  21. He, L., Huang, J., Wang, C.: Global stability of large solutions to the 3D compressible Navier–Stokes equations. arXiv:1710.10778

  22. Hoff, D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120, 215–254 (1995)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  24. Hoff, D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37, 1742–1760 (2006)

    Article  MathSciNet  Google Scholar 

  25. Huang, J., Paicu, M., Zhang, P.: Global solutions to 2-D inhomogeneous Navier–Stokes system with general velocity. J. Math. Pures Appl. 100, 806–831 (2013)

    Article  MathSciNet  Google Scholar 

  26. Huang, X., Li, J., Xin, Z.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equaitons. Commun. Pure Appl. Math. 65, 549–585 (2012)

    Article  Google Scholar 

  27. Lions, P.L.: Mathematical Topics in Fluid Mechanics. Compressible Models, vol. 2. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  28. Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)

    Article  MathSciNet  Google Scholar 

  29. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci. 55, 337–342 (1979)

    Article  MathSciNet  Google Scholar 

  30. Ponce, G., Racke, R., Sideris, T.C., Titi, E.S.: Global stability of large solutions to the 3D Navier–Stokes equations. Commun. Math. Phys. 159, 329–341 (1994)

    Article  ADS  Google Scholar 

  31. Nash, J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bulletin de la Soc. Math. de France 90, 487–497 (1962)

    Article  MathSciNet  Google Scholar 

  32. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202, no. 950 (2009)

    Article  MathSciNet  Google Scholar 

  33. Xu, H., Li, Y., Chen, F.: Global solution to the incompressible inhomogeneous Navier–Stokes equations with some large initial data. J. Math. Fluid Mech. 19, 315–328 (2016)

    Article  MathSciNet  Google Scholar 

  34. Zhai, X., Li, Y.: Global large solutions to the three dimensional compressible Navier–Stokes equations. (Submitted)

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China under the Grants 11601533 and 11571240.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Zhi-Min Chen & Xiaoping Zhai

Authors
  1. Zhi-Min Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoping Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoping Zhai.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by E. Feireisl.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, ZM., Zhai, X. Global Large Solutions and Incompressible Limit for the Compressible Navier–Stokes Equations. J. Math. Fluid Mech. 21, 26 (2019). https://doi.org/10.1007/s00021-019-0428-3

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-019-0428-3

Keywords

Mathematics Subject Classification

Search

Navigation