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

  • Zhi-Min Chen
  • Xiaoping ZhaiEmail author


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.


Compressible Navier–Stokes equations Incompressible limit Besov spaces Global well-posedness 

Mathematics Subject Classification

35Q35 76N10 



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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011)zbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, Q., Miao, C., Zhang, Z.: On the ill-posedness of the compressible Navier–Stokes equations. Rev. Mat. Iberoam. 31, 1375–1402 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26, 1183–1233 (2001)MathSciNetCrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Danchin, R.: A Lagrangian approach forthe compressible Navier–Stokes equations. Ann. Inst. Fourier (Grrenoble) 64, 753–791 (2014)CrossRefGoogle Scholar
  10. 10.
    Danchin, R., He, L.: The incompressible limit in \( L^p\) type critical spaces. Math. Ann. 366, 1365–1402 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Danchin, R., Mucha, P.B.: From compressible to incompressible inhomogeneous flows in the case of large data. arXiv:1710.08819
  12. 12.
    Danchin, R., Mucha, P.: Compressible Navier–Stokes system: large solutions and incompressible limit. Adv. Math. 320, 904–925 (2017)MathSciNetCrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dejardins, B., Grenier, E.: Low Mach number limlit of compressible flows in the whole space. Proc. R. Soc. Lond. 455, 2271–2279 (1999)ADSCrossRefGoogle Scholar
  15. 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)MathSciNetCrossRefGoogle Scholar
  16. 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. 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)ADSMathSciNetCrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fujita, H., Kato, T.: On the Navier–Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Haspot, B.: Existence of global strong solutions in critical spaces for barotropic viscous fluids. Arch. Ration. Mech. Anal. 202, 427–460 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    He, L., Huang, J., Wang, C.: Global stability of large solutions to the 3D compressible Navier–Stokes equations. arXiv:1710.10778
  22. 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)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Hoff, D.: Compressible flow in a half-space with Navier boundary condtions. J. Math. Fluid Mech. 7, 315–338 (2005)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Hoff, D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37, 1742–1760 (2006)MathSciNetCrossRefGoogle Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics. Compressible Models, vol. 2. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  28. 28.
    Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)MathSciNetCrossRefGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 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)ADSCrossRefGoogle Scholar
  31. 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)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202, no. 950 (2009)Google Scholar
  33. 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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhai, X., Li, Y.: Global large solutions to the three dimensional compressible Navier–Stokes equations. (Submitted) Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShenzhen UniversityShenzhenChina

Personalised recommendations