Skip to main content
SpringerLink
Log in

Large-Time Behavior of a Linear Boussinesq System for the Water Waves

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

Abstract

In this paper we are concerned with a linear Boussinesq system of Benjamin–Bona–Mahony type modelling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. We propose several dissipation mechanisms leading to systems for which one has both the global existence of solutions and a nonincreasing energy. Following the analysis developed in Rosier (J Math Study 49:195–204, 2016) we prove that all the trajectories are attracted by the origin provided that the unique continuation of weak solutions holds.

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. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear, dispersive media. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)

    Article  MATH  Google Scholar 

  2. Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boussinesq, J.: Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. 72, 755–759 (1871)

    MATH  Google Scholar 

  5. Capistrano-Filho, R.A., Pazoto, A.F., Rosier, L.: Control of a Boussinesq system of KdV–KdV type on a bounded interval. arXiv:1401.6833 (2017)

  6. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a retangular canal, and on a new type of long stacionary waves. Philos. Mag. 39, 422–423 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  7. Micu, S., Pazoto, A.F.: Stabilization of a Boussinesq system with localized damping. J. Anal. Math. (to appear)

  8. Micu, S., Pazoto, A.F.: Stabilization of a Boussinesq system with generalized damping. Syst. Control Lett. 105, 62–69 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Micu, S., Ortega, J., Rosier, L., Zhang, B.-Y.: Control and stabilization of a family of Boussinesq systems. Discret. Contin. Dyn. Syst. 24, 273–313 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pazoto, A.F., Rosier, L.: Stabilization of a Boussinesq system of KdV–KdV type. Syst. Control Lett. 57, 595–601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330 (1966)

    Article  Google Scholar 

  12. Rosier, L.: On the Benjamin–Bona–Mahony equation with a localized damping. J. Math. Study 49, 195–204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rosier, L., Zhang, B.-Y.: Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J. Differ. Equ. 254, 141–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referees for their helpful comments and suggestions. GJB was supported by Capes and Faperj (Brazil) and AFP was partially supported by CNPq (Brazil).

Author information

Authors and Affiliations

  1. Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, P.O. Box 68530, Rio de Janeiro, RJ, CEP 21945-970, Brazil

    George J. Bautista & Ademir F. Pazoto

Authors
  1. George J. Bautista
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ademir F. Pazoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ademir F. Pazoto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bautista, G.J., Pazoto, A.F. Large-Time Behavior of a Linear Boussinesq System for the Water Waves. J Dyn Diff Equat 31, 959–978 (2019). https://doi.org/10.1007/s10884-018-9689-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9689-4

Keywords

Search

Navigation