Skip to main content
SpringerLink
Log in

Asymptotic stability of periodic solution for compressible viscous van der Waals fluids

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper is concerned with the asymptotic stability of the periodic solution to a one-dimensional model system for the compressible viscous van der Waals fluid in Eulerian coordinates. If the initial density and initial momentum are suitably close to the average density and average momentum, then the solution is proved to tend toward a stationary solution as t→∞.

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. Dafermos, C. Conservation laws with dissipation. Lakshmikantham V. Nonlinear Phenomena in Mathematical Sciences. Academic Press, New York, 1982

    Google Scholar 

  2. Fan, T. A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids. Jour. Diff. Equs., 103: 179–204 (1993)

    Article  MATH  Google Scholar 

  3. Hattori, H. The Riemann problem and the existence of weak solutions to a system of mixed-type in dynamic phase transition. Journal Diff. Equs., 146: 287–319 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Huang, F.M., Li, J., Shi, X.D. Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun. Math. Sci., 8: 639–654 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Huang, F.M., Matsumura, A., Shi, X.D. Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Commun. Math. Phys., 239: 261–285 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Huang, F.M., Shi, X.D., Wang, Y. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 3: 409–425 (2000)

    MathSciNet  Google Scholar 

  7. Hoff, D., Khodja, M. Stability of coexisting phases for compressible van der Waals fluids. SIAM J. Appl. Math., 53: 1–14 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hsieh, D.Y., Wang, X.P. Phase transition in van der Waals fluids. SIAM J. Appl. Math., 57: 871–892 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Matsumura, A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Commun. Methods and Applications of Analysis, 8(4): 645–666 (2001)

    MATH  MathSciNet  Google Scholar 

  10. Mei, M., Liu, L., Wong, Y. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) existence and uniform boundedness. Discrete Cont. Dyn. Syst., B7: 825–837 (2007)

    MathSciNet  Google Scholar 

  11. Mei, M., Liu, L., Wong, Y. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) convergence. Discrete Cont. Dyn. Syst., B7: 839–857 (2007)

    MathSciNet  Google Scholar 

  12. Nishibata, S., Kawashima, S., Zhu, P.C. Asymptiotic stbility of stationary solution to the compressible Navier-Stokes equations in the half space. Communications in Mathematical Physics, 240: 483–500 (2003)

    MATH  MathSciNet  Google Scholar 

  13. Shi, X.D. On the stability of rarefaction wave solutions for viscous p-system with boundary effect. Acta Mathematicae Applicatae Sinica, 19: 1–12 (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Chemical Technology, Beijing, 100029, China

    Jin-yang Huang & Xiao-ding Shi

  2. Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

    Xiao-ping Wang

  3. School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China

    Bing Zhang

Authors
  1. Jin-yang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-ding Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-ping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-ding Shi.

Additional information

Supported by the National Natural Science Foundation of China (No. 10971215).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Jy., Shi, Xd., Wang, Xp. et al. Asymptotic stability of periodic solution for compressible viscous van der Waals fluids. Acta Math. Appl. Sin. Engl. Ser. 30, 1113–1120 (2014). https://doi.org/10.1007/s10255-014-0430-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-014-0430-8

Keywords

2000 MR Subject Classification

Search

Navigation