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Large-time behavior of the spherically symmetric compressible Navier–Stokes equations with degenerate viscosity coefficients

  • Guangyi Hong
  • Huanyao Wen
  • Changjiang ZhuEmail author
Article
  • 28 Downloads

Abstract

This paper is concerned with the vacuum free boundary problem for the compressible spherically symmetric Navier–Stokes equations with an external force and degenerate viscosities in \(\mathbb {R}^{n}(n\ge 2)\). When the initial data are a small perturbation of the stationary profile and the viscosity coefficients are proportional to \( \rho ^{\theta } \) with \(\theta \in {\left\{ \begin{array}{ll} (0,2(\gamma -1))\cap (0,\frac{\gamma }{2}]&{}n=2\\ (0,\frac{\gamma }{2}]&{}n\ge 3 \end{array}\right. }\), a result on the global existence as well as sharper time decay rates of the weak solution is obtained which improves the one in Wei et al. (SIAM J Math Anal 40:869–904, 2008). The proof is based on some weighted energy estimates, and in our analysis, no smallness constraint is prescribed upon the derivatives of the initial data. It is also worth pointing out that our result covers the interesting case of the Saint-Venant shallow water model (i.e., \(\gamma =2\) and \(\theta =1\)).

Keywords

Compressible Navier–Stokes equation Vacuum free boundary Density-dependent viscosity coefficients Large-time behavior Weighted energy estimates 

Mathematics Subject Classification

35Q30 35D30 35B40 76N10 

Notes

Acknowledgements

Wen was supported by the National Natural Science Foundation of China #11722104, 11671150, and by GDUPS (2016). Zhu was supported by the National Natural Science Foundation of China #11771150, 11831003, 11331005.

References

  1. 1.
    Amosov, A.A., Zlotnik, A.A.: Generalized solutions “in the large” of the equations of the one-dimensional motion of a viscous heat-conducting gas. Soviet Math. Dokl. 38, 1–5 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Amosov, A.A., Zlotnik, A.A.: Solvability “in the large” of a system of equations for the one-dimensional motion of an inhomogeneous viscous heat-conducting gas. Mat. Zametki 52, 3–16 (1992). 155MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model. Commun. Math. Phys. 238(1–2), 211–223 (2003)CrossRefGoogle Scholar
  4. 4.
    Bresch, D., Desjardins, B., Lin, C.K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun. Partial Differ. Equ. 28(3–4), 1009–1037 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, G.Q., Kratka, M.: Global solutions to the Navier–Stokes equations for compressible heat-conducting flow with symmetry and free boundary. Commun. Partial Differ. Equ. 27, 907–943 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ducomet, B., Zlotnik, A.: Viscous compressible barotropic symmetric flows with free boundary under general mass force. I. Uniform-in-time bounds and stabilization. Math. Methods Appl. Sci. 28, 827–863 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fang, D.Y., Zhang, T.: Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient. Arch. Ration. Mech. Anal. 182, 223–253 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fang, D.Y., Zhang, T.: Compressible Navier–Stokes equations with vacuum state in the case of general pressure law. Math. Methods Appl. Sci. 29, 1081–1106 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, Z.H., Jiu, Q.S., Xin, Z.P.: Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J. Math. Anal. 39, 1402–1427 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, Z.H., Zhu, C.J.: Global weak solutions and asymptotic behavior to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum. J. Differ. Equ. 248, 2768–2799 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hoff, D., Serre, D.: The failure of continuous dependence on initial data for the Navier–Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jiang, S., Xin, Z.P., Zhang, P.: Global weak solutions to 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity. Methods Appl. Anal. 12, 239–251 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jiang, S., Zhang, P.: On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys. 215, 559–581 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy inequality, About its history and some related results. Vydavatelský Servis, Plzeň (2007)Google Scholar
  16. 16.
    Li, H.L., Li, J., Xin, Z.P.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier–Stokes equations. Commun. Math. Phys. 281, 401–444 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics, vol. 2 Compressible Models. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  18. 18.
    Liu, T.P., Xin, Z.P., Yang, T.: Vacuum states for compressible flow. Discrete Contin. Dyn. Syst. 4, 1–32 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Luo, T., Xin, Z.P., Yang, T.: Interface behavior of compressible Navier–Stokes equations with vacuum. SIAM J. Math. Anal. 31, 1175–1191 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Luo, T., Xin, Z.P., Zeng, H.H.: On nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem. Adv. Math. 291, 90–182 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Luo, T., Xin, Z.P., Zeng, H.H.: Nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities. Commun. Math. Phys. 347, 657–702 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Matušu̇-Nečasová, S., Okada, M., Makino, T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. III. Jpn. J. Ind. Appl. Math. 14, 199–213 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mellet, A., Vasseur, A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 39, 1344–1365 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Okada, M., Makino, T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. Jpn. J. Ind. Appl. Math. 10, 219–235 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Okada, M., Matušu̇-Nečasová, S., Makino, T.: Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann. Univ. Ferrara Sez. VII (NS) 48, 1–20 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ou, Y.B., Zeng, H.H.: Global strong solutions to the vacuum free boundary problem for compressible Navier–Stokes equations with degenerate viscosity and gravity force. J. Differ. Equ. 259, 6803–6829 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Qin, X.L., Yao, Z.A., Zhao, H.X.: One dimensional compressible Navier–Stokes equations with density-dependent viscosity and free boundaries. Commun. Pure Appl. Anal. 7, 373–381 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Vong, S.W., Yang, T., Zhu, C.J.: Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum (II). J. Differ. Equ. 192, 475–501 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wei, M.J., Zhang, T., Fang, D.Y.: Global behavior of spherically symmetric Navier–Stokes equations with degenerate viscosity coefficients. SIAM J. Math. Anal. 40, 869–904 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Yang, T., Zhu, C.J.: Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zeng, H.: Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier–Stokes equations. Nonlinearity 28, 331–345 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhang, T., Fang, D.Y.: Global behavior of spherically symmetric Navier–Stokes equations with density-dependent viscosity. J. Differ. Equ. 236, 293–341 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zlotnik, A.A., Dyukome, B.: The stabilization rate and stability of viscous compressible barotropic symmetric flows with a free boundary for a general mass force. Sb. Math. 196, 1745–1799 (2005)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsSouth China University of TechnologyGuangzhouChina

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