Skip to main content
SpringerLink
Log in

Nonlinear Asymptotic Stability of Traveling Waves of System for Gas Dynamics in Thermal Nonequilibrium

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

For the one-dimensional gas dynamics in thermal nonequilibrium which is a \(4\times 4\) nonlinear hyperbolic system with relaxation, global existence for Cauchy problem is established, and the asymptotic stability of the traveling wave solution with shock profile is proved under the condition that the shock strength and the initial disturbance are small and the integral zero of the initial disturbance.

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. Caflisch, R.E., Liu, T.-P.: Stability of shock waves for the Broadwell equations. Commun. Math. Phys. 114, 103–130 (1988)

    Article  MathSciNet  Google Scholar 

  2. Degond, P., Génieys, S., Jüngel, A.: A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. J. Math. Pures Appl. 76, 991–1015 (1997)

    Article  MathSciNet  Google Scholar 

  3. Fan, L., Matsumura, A.: Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas. J. Differ. Equ. 258, 1129–1157 (2015)

    Article  MathSciNet  Google Scholar 

  4. Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73, 256–274 (1951)

    Article  MathSciNet  Google Scholar 

  5. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Ration. Mech. Anal. 95, 325–344 (1986)

    Article  MathSciNet  Google Scholar 

  6. Huang, F., Matsumura, A.: Stability of a composite wave of two viscous shock waves for the full compressible Navier–Stokes equation. Commun. Math. Phys. 289, 841–861 (2009)

    Article  MathSciNet  Google Scholar 

  7. Huang, F., Wang, Y., Yang, T.: Vanishing viscosity limit of the compressible Navier–Stokes equations for solutions to a Riemann problem. Arch. Ration. Mech. Anal. 203, 379–413 (2012)

    Article  MathSciNet  Google Scholar 

  8. Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995)

    Article  MathSciNet  Google Scholar 

  9. Kawashima, S., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101, 97–127 (1985)

    Article  MathSciNet  Google Scholar 

  10. Liu, H.: Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws. J. Differ. Equ. 192, 285–307 (2003)

    Article  MathSciNet  Google Scholar 

  11. Liu, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987)

    Article  MathSciNet  Google Scholar 

  12. Liu, T.-P., Zeng, Y.: Shock waves in conservation laws with physical viscosity. Mem. Am. Math. Soc. 234, vi+168 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Liu, T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Commun. Pure Appl. Math. 50, 1113–1182 (1997)

    Article  MathSciNet  Google Scholar 

  14. Liu, T.-P., Zeng, Y.: Time-asymptotic behavior of wave propagation around a viscous shock profile. Commun. Math. Phys. 290, 23–82 (2009)

    Article  MathSciNet  Google Scholar 

  15. Luo, T., Xin, Z.: Nonlinear stability of shock fronts for a relaxation system in several space dimensions. J. Differ. Equ. 139, 365–408 (1997)

    Article  MathSciNet  Google Scholar 

  16. Luo, T., Yang, T.: Global structure and asymptotic behavior of weak solutions to flood wave equations. J. Differ. Equ. 207, 117–160 (2004)

    Article  MathSciNet  Google Scholar 

  17. Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)

    Article  MathSciNet  Google Scholar 

  18. Smoller, J.: Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, New York (1994)

    Book  Google Scholar 

  19. Szepessy, A., Xin, Z.: Nonlinear stability of viscous shock waves. Arch. Ration. Mech. Anal. 122, 53–103 (1993)

    Article  MathSciNet  Google Scholar 

  20. Vincenti, W., Kruger Jr., C.: Introduction to Physical Gas Dynamics. Krieger, Malabar (1986)

    Google Scholar 

  21. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  22. Xu, W.-Q.: Relaxation limit for piecewise smooth solutions to systems of conservation laws. J. Differ. Equ. 162, 140–173 (2000)

    Article  MathSciNet  Google Scholar 

  23. Yong, W.-A., Zumbrun, K.: Existence of relaxation shock profiles for hyperbolic conservation laws. SIAM J. Appl. Math. 60, 1565–1575 (2000)

    Article  MathSciNet  Google Scholar 

  24. Yu, S.: Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Ration. Mech. Anal. 146, 275–370 (1999)

    Article  MathSciNet  Google Scholar 

  25. Zeng, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150, 225–279 (1999)

    Article  MathSciNet  Google Scholar 

  26. Zeng, Y.: Thermal non-equilibrium flows in three space dimensions. Arch. Ration. Mech. Anal. 219, 27–87 (2016)

    Article  MathSciNet  Google Scholar 

  27. Zeng, Y.: Gas flows with several thermal nonequilibrium modes. Arch. Ration. Mech. Anal. 196, 191–225 (2010)

    Article  MathSciNet  Google Scholar 

  28. Zeng, H.: Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws. J. Differ. Equ. 246, 2081–2102 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, No. 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

    Tao Luo & Yan-Lin Wang

Authors
  1. Tao Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan-Lin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan-Lin Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Luo was supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 11306117).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, T., Wang, YL. Nonlinear Asymptotic Stability of Traveling Waves of System for Gas Dynamics in Thermal Nonequilibrium. J Dyn Diff Equat 32, 941–963 (2020). https://doi.org/10.1007/s10884-019-09749-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-019-09749-y

Keywords

Search

Navigation