Advertisement

Annals of PDE

, 5:7 | Cite as

Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier–Stokes Equations with Vacuum

  • Jing LiEmail author
  • Zhouping Xin
Manuscript
  • 48 Downloads

Abstract

We are concerned with the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier–Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates (in large time) for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably small. Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions (which may be of possibly large oscillations) in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions–Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number \(\gamma >1\) provided that the initial total energy is small.

Keywords

Compressible Navier–Stokes equations Global-wellposedness Large-time behavior Cauchy problem Vacuum 

Notes

References

  1. 1.
    Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bergh, J., Lofstrom, J.: Interpolation Spaces, An Introduction. Springer, Berlin (1976)CrossRefGoogle Scholar
  3. 3.
    Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83, 243–275 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cho, Y., Kim, H.: On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities. Manuscr. Math. 120, 91–129 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Choe, H.J., Kim, H.: Strong solutions of the Navier–Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, New York (2004)zbMATHGoogle Scholar
  7. 7.
    Feireisl, E., Novotny, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)CrossRefGoogle Scholar
  9. 9.
    Hoff, D.: Global existence for 1D, compressible, isentropic Navier–Stokes equations with large initial data. Trans. Am. Math. Soc. 303(1), 169–181 (1987)zbMATHGoogle Scholar
  10. 10.
    Hoff, D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hoff, D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rational Mech. Anal. 132, 1–14 (1995)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hoff, D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55(11), 1365–1407 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hoff, D.: Compressible flow in a half-space with Navier boundary conditions. J. Math. Fluid Mech. 7(3), 315–338 (2005)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang, X.D., Li, J., Xin, Z.P.: Blowup criterion for viscous barotropic flows with vacuum states. Commun. Math. Phys. 301(1), 23–35 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Huang, X.D., Li, J., Xin, Z.P.: Serrin type criterion for the three-dimensional compressible flows. SIAM J. Math. Anal. 43(4), 1872–1886 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huang, X.D., Li, J., Xin, Z.P.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kato, T.: Remarks on the Euler and Navier–Stokes equations in \(R^2\). In: Proceedings of Symposia in Pure Mathematics, vol. 45, pp. 1–7. American Mathematical Society, Providence (1986)Google Scholar
  18. 18.
    Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh. 41, 282–291 (1977)MathSciNetGoogle Scholar
  19. 19.
    Li, J., Liang, Z.: On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum. J. Math. Pures Appl. (9) 102(4), 640–671 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, J., Xin, Z.: Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows. J. Differ. Equ. 221(2), 275–308 (2006)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press, New York (1996)zbMATHGoogle Scholar
  22. 22.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Oxford University Press, New York (1998)zbMATHGoogle Scholar
  23. 23.
    Luo, Z.: Global existence of classical solutions to two-dimensional Navier–Stokes equations with Cauchy data containing vacuum. Math. Methods Appl. Sci. 37(9), 1333–1352 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nash, J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Rozanova, O.: Blow up of smooth solutions to the compressible Navier–Stokes equations with the data highly decreasing at infinity. J. Differ. Equ. 245, 1762–1774 (2008)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Salvi, R., Straskraba, I.: Global existence for viscous compressible fluids and their behavior as \(t\rightarrow \infty \). J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 40, 17–51 (1993)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Serre, D.:: Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Serre, D.: Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math. 303, 703–706 (1986)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Serrin, J.: On the uniqueness of compressible fluid motion. Arch. Rational. Mech. Anal. 3, 271–288 (1959)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Xin, Z.P.: Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Xin, Z.P., Yan, W.: On blowup of classical solutions to the compressible Navier–Stokes equations. Commun. Math. Phys. 321(2), 529–541 (2013)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Zlotnik, A.A.: Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differ. Equ. 36, 701–716 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.The Institute of Mathematical SciencesThe Chinese University of Hong KongShatinHong Kong

Personalised recommendations