Skip to main content
SpringerLink
Log in

Uniqueness of conservative solutions for nonlinear wave equations via characteristics

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

For some classes of one-dimensional nonlinear wave equations, solutions are Hölder continuous and the ODEs for characteristics admit multiple solutions. Introducing an additional conservation equation and a suitable set of transformed variables, one obtains a new ODE whose right hand side is either Lipschitz continuous or has directionally bounded variation. In this way, a unique characteristic can be singled out through each initial point. This approach yields the uniqueness of conservative solutions to various equations, including the Camassa-Holm and the variational wave equation u tt − c(u)(c(u)u x ) x = 0, for general initial data in H 1(R).

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. A. Bressan. Unique solutions for a class of discontinuous differential equations. Proc. Amer.Math. Soc., 104 (1988), 772–778.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bressan and G. Chen. Lipschitzmetrics for a class of nonlinearwave equations, submitted.

  3. A. Bressan, G. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discr. Cont. Dyn. Syst., 35 (2015), 25–42.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bressan, G. Chen and Q. Zhang. Unique conservative solutions to a variational wave equation. Arch. Rational Mech. Anal., to appear.

  5. A. Bressan and A. Constantin. Global solutions to the Hunter-Saxton equations. SIAM J. Math. Anal., 37 (2005), 996–1026.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bressan and A. Constantin. Global conservative solutionsto the Camassa-Holm equation. Arch. Rational Mech. Anal., 183 (2007), 215–239.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Bressan and A. Constantin. Global dissipative solutions to the Camassa-Holm equation. Analysis and Applications, 5 (2007), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bressan and M. Fonte. An optimal transportation metric for solutions of the Camassa-Holm equation. Methods Appl. Anal., 12 (2005), 191–220.

    MathSciNet  MATH  Google Scholar 

  9. A. Bressan, H. Holden and X. Raynaud. Lipschitz metric for the Hunter-Saxton equation. J. Math. Pures Appl., 94 (2010), 68–92.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Bressan and T. Huang. Representation of dissipative solutions to a nonlinear variational wave equation. Comm. Math. Sciences, 14 (2016), 31–53.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bressan, P. Zhang and Y. Zheng. Asymptotic variational wave equations. Arch. Rational Mech. Anal., 183 (2007), 163–185.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Bressan and Y. Zheng. Conservative solutions to a nonlinear variational wave equation. Comm. Math. Phys., 266 (2006), 471–497.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Camassa and D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71 (1993), 1661–1664.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Dafermos. Generalized characteristics and the Hunter-Saxton equation. J. Hyperbolic Diff. Equat., 8 (2011) 159–168.

    Article  MathSciNet  MATH  Google Scholar 

  15. R.T. Glassey, J.K. Hunter and Y. Zheng. Singularities in a nonlinear variational wave equation. J. Differential Equations, 129 (1996), 49–78.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Grunert, H. Holden and X. Raynaud. Lipschitz metric for the Camassa-Holm equation on the line. Discrete Contin. Dyn. Syst., 33 (2013), 2809–2827.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Hardt, D. Kinderlehrer and F. Lin. Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys., 105 (1986), 547–570.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Holden and X. Raynaud. Global semigroup of conservative solutions of the nonlinear variational wave equation. Arch. Rational Mech. Anal., 201 (2011), 871–964.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.K. Hunter and R.A. Saxton. Dynamics of director fields. SIAM J. Appl. Math., 51 (1991), 1498–1521.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.K. Hunter and Y. Zheng. On a completely integrable nonlinear hyperbolic variational equation. Physica D., 79 (1994), 361–386.

    Article  MathSciNet  MATH  Google Scholar 

  21. Z. Xin and P. Zhang. On the uniqueness and large time behavior of the weak solutions to a shallowwater equation.Comm. Part. Diff. Equat., 27 (2002), 1815–1844.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Zhang and Y. Zheng. Weak solutions to a nonlinear variational wave equation with general data. Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 207–226.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, Pa., 16802, USA

    Alberto Bressan

Authors
  1. Alberto Bressan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Bressan.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bressan, A. Uniqueness of conservative solutions for nonlinear wave equations via characteristics. Bull Braz Math Soc, New Series 47, 157–169 (2016). https://doi.org/10.1007/s00574-016-0129-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-016-0129-y

Keywords

Mathematical subject classification

Search

Navigation