Skip to main content
SpringerLink
Log in

Optimal decay rates of solutions to hyperbolic conservation laws with damping

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with the asymptotic behavior of solutions to the system of hyperbolic conservation laws with damping. In particular, a system includes compressible Euler equations with damping, \(M_1\)-model, etc. Under some smallness conditions on initial perturbations, we prove that the solutions to the Cauchy problem of the system globally exist and time-asymptotically converge to corresponding equilibrium state, and further give the optimal convergence rate. The approach adopted is the technical time-weighted energy method combined with the Green’s function method.

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. Berthon, C., Charrier, P., Dubroca, B.: An asymptotic preserving relaxation scheme for a moment model of radiative transfer. C. R. Math. Acad. Sci. Paris 344, 467–472 (2007)

    Article  MathSciNet  Google Scholar 

  2. Fang, D.Y., Xu, J.: Existence and asymptotic behavior of \(C^{1}\) solutions to the multi-dimensional compressible Euler equations with damping. Nonlinear Anal. 70, 244–261 (2009)

    Article  MathSciNet  Google Scholar 

  3. Geng, S.F., Wang, Z.: Convergence rates to asymptotic profile for solutions of quasilinear hyperbolic equations with nonlinear damping. Acta Math. Appl. Sin. Engl. Ser. 32, 55–66 (2016)

  4. Geng, S.F., Zhang, L.N.: Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping. Proc. R. Soc. Edinb. Sect. A 145, 959–978 (2015)

  5. Goudona, T., Lin, C.J.: Analysis of the \(M_{1}\) model: well-posedness and diffusion asymptotics. J. Math. Anal. Appl. 402, 579–593 (2013)

    Article  MathSciNet  Google Scholar 

  6. Hsiao, L., Liu, T.P.: Convergence to diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 259, 599–605 (1992)

    Article  MathSciNet  Google Scholar 

  7. Hsiao, L., Liu, T.P.: Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chin. Ann. Math. Ser. B, 14, 465–480 (1993)

  8. Hsiao, L., Luo, T.: Nonlinear diffusion phenomena of solutions for the system of compressible adiabatic flow through porous media. J. Differ. Equ. 125, 329–365 (1996)

    Article  Google Scholar 

  9. Huang, F.M., Marcati, P., Pan, R.H.: Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 176, 1–24 (2005)

    Article  MathSciNet  Google Scholar 

  10. Huang, F.M., Pan, R.H.: Convergence rate for compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 166, 359–376 (2003)

    Article  MathSciNet  Google Scholar 

  11. Huang, F.M., Pan, R.H.: Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum. J. Differ. Equ. 220, 207–233 (2006)

    Article  MathSciNet  Google Scholar 

  12. Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph.D. Thesis, Kyoto University (1983)

  13. Marcati, P., Nishihara, K.: The \(L^{p}\)-\(L^{q}\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. J. Differ. Equ. 191, 445–469 (2003)

    Article  Google Scholar 

  14. Matsumura, A.: On the asymptotic behavior of the solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12, 169–189 (1976)

    Article  MathSciNet  Google Scholar 

  15. Matsumura, A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first order dissipation. Publ. RIMS Kyoto Univ. 13, 349–379 (1977)

    Article  MathSciNet  Google Scholar 

  16. Mei, M.: Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping. J. Differ. Equ. 247, 1275–1296 (2009)

    Article  MathSciNet  Google Scholar 

  17. Mei, M.: Best asymptotic profile for hyperbolic p-system with damping. SIAM J. Math. Anal. 42, 1–23 (2010)

    Article  MathSciNet  Google Scholar 

  18. Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Pub. Math. D’Orsay (1978)

  19. Nishihara, K.: Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J. Differ. Equ. 131, 171–188 (1996)

    Article  MathSciNet  Google Scholar 

  20. Nishihara, K.: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Differ. Equ. 137, 384–395 (1997)

    Article  MathSciNet  Google Scholar 

  21. Nishihara, K.: Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media. Proc. R. Soc. Edinb. Sect. A, 133, 177–196 (2003)

  22. Nishihara, K., Wang, W.K., Yang, T.: \(L^{p}\) convergence rate to nonlinear diffusion waves for p-system with damping. J. Differ. Equ. 161, 191–218 (2000)

    Article  Google Scholar 

  23. Nishikawa, M.: Convergence rate to the traveling wave for viscous conservation laws. Funkcial. Ekvac. 41, 107–132 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Yang, T., Zhu, C.J.: Existence and non-existence of global smooth solutions for p-system with relaxation. J. Differ. Equ. 161, 321–336 (2000)

    Article  MathSciNet  Google Scholar 

  25. Zhao, H.J.: Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J. Differ. Equ. 174, 200–236 (2001)

    Article  MathSciNet  Google Scholar 

  26. Zhao, H.J.: Asymptotic behaviors of solutions of quasilinear hyperbolic equations with linear damping I2. J. Differ. Equ. 167, 467–494 (2000)

    Article  Google Scholar 

  27. Zhu, C.J., Jiang, M.N.: \(L^{p}\)-decay rates to nonlinear diffusion waves for p-system with nonlinear damping. Sci. China Ser. A 49, 721–739 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research was supported by the National Natural Science Foundation of China \(\#\)12171160, 11771150, 11831003 and Guangdong Basic and Applied Basic Research Foundation \(\#\)2020B1515310015.

Author information

Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Nangao Zhang & Changjiang Zhu

Authors
  1. Nangao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changjiang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changjiang Zhu.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, N., Zhu, C. Optimal decay rates of solutions to hyperbolic conservation laws with damping. Z. Angew. Math. Phys. 73, 34 (2022). https://doi.org/10.1007/s00033-021-01657-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01657-w

Keywords

Mathematics Subject Classification

Search

Navigation