Global classical solution for a three-dimensional viscous liquid-gas two-fluid flow model with vacuum

Article
First Online:
Received:
Revised:

Abstract

This is a continuation of the paper (J. Math. Phys., 52(2011), 093102). We consider the Cauchy problem to the three-dimensional viscous liquid-gas two-fluid flow model. The global existence of classical solution is proved, where the initial vacuum is allowed.

Keywords

Viscous liquid-gas two-fluid flow model classical solution vacuum 

2000 MR Subject Classification

76T10 76N10 35L65 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Brennen, C.E. Fundamentals of Multiphase Flow. Cambridge University Press, New York, 2005CrossRefMATHGoogle Scholar
  2. [2]
    Bresch, D., Desjardins, B., Ghidaglia, J.-M., Grenier, E. Global weak solutions to a generic tow-fluid model. Arch. Ration. Mech. Anal., 196: 599–629 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    Bresch, D., Huang, X.D., Li, J. Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system. Commun. Math. Phys., 10.1007/s00220-011-1379-6 (2011)Google Scholar
  4. [4]
    Cho, Y., Choe, H.J., Kim, H. Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures. Appl., 83: 243–275 (2004)CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    Cho, Y., Kim, H. On classical solution of the compressible Navier-Stokes equations with nonnegative nitial densities. Manuscript Math., 120: 91–129 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    Cui, H.B., Wen, H.Y., Yin, H.Y. Global classical solutions of viscous liquid-gas two-phase flow model. Math. Methods Appl. Sci., 36: 567–583 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Evje, S., Flåtten, T., Friis, H.A. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal., 70: 3864–3886 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Evje, S., Karlsen, K.H. Global existence of weak solutions for a viscous two-phase model. J. Differential Equations, 245, 2660–2703 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Evje, S., Karlsen, K.H. Global weak solutions for a viscous liquid-gas model with singular pressure law. Commun. Pure Appl. Anal., 8: 1867–1894 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    Guo, Z.H., Yang, J., Yao, L. Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum. J. Math. Phys., 52: 093102 (2011)CrossRefMathSciNetGoogle Scholar
  11. [11]
    Hao, C.C, Li, H.L. Well-posedness for a multidimensional viscous gas-liquid two-phase flow model. SIAM J. Math. Anal., 44: 1304–1332 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    Hoff, D. Compressible flow in a half-space with Navier boundary conditions. J. Math. Fluid Mech., 7: 315–338 (2005)CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    Hoff, D. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differential Equations, 120: 215–254 (1995)CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    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. Comm. Pure Appl. Math., 65: 549–585 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    Ishii, M. Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris, 1975MATHGoogle Scholar
  16. [16]
    Prosperetti, A., Tryggvason G. (Editors): Computational Methods for Multiphase Flow. Cambridge University Press, New York, 2007Google Scholar
  17. [17]
    Simon, J. Nonhomogeneous viscous incompressible fluids: existence of vecocity, density and pressure. SIAM J. Math. Anal., 21: 1093–1117 (1990)CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Wen, H.Y., Yao, L., Zhu, C.J. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J. Math. Pures Appl., 97: 204–229 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    Yao, L., Zhu, C.J. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J. Differential Equations, 247: 2705–2739 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    Yao, L., Zhu, C.J. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math. Ann., 349: 903–928 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    Yao, L., Zhang, T., Zhu, C.J. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J. Math. Anal., 42: 1874–1897 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    Yao, L., Zhang, T., Zhu, C.J. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J. Differential Equations, 250: 3362–3378 (2011)CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    Zhang, T. Global solution of compressible Navier-Stokes equation with a density-dependent viscosity coefficient. J. Math. Phys., 52: 043510 (2011)CrossRefMathSciNetGoogle Scholar
  24. [24]
    Zlotnik, A.A. Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Diff. Equations, 36: 701–716 (2000)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematics and Center for Nonlinear StudiesNorthwest UniversityXi’anChina

Personalised recommendations