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Lipschitz Metrics for a Class of Nonlinear Wave Equations

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Abstract

The nonlinear wave equation \({u_{tt}-c(u)(c(u)u_x)_x=0}\) determines a flow of conservative solutions taking values in the space \({H^1(\mathbb{R})}\). However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of \({H^1(\mathbb{R})}\). For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.

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References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Second edition. Lecture Notes in Mathematics, ETH Zürich. Birkhäuser, Basel, 2008

  2. Bolley F., Brenier Y., Loeper G.: Contractive metrics for scalar conservation laws. J. Hyperbolic Diff. Equ. 2, 91–107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenier Y.: \({\mathbf{L}^2}\) formulation of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 193, 1–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenier, Y.: Hilbertian approaches to some non-linear conservation laws. In: Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, pp. 19–35, Contemporary Math. 526, AMS, Providence, 2010

  5. Bressan, A.: A locally contractive metric for systems of conservation laws, Ann. Scuola Normale Sup. Pisa, Serie IV, XXII, 109–135, 1995

  6. Bressan, A.: Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford, 2000

  7. Bressan A., Chen G.: Generic regularity of conservative solutions to a nonlinear wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire. 34, 335–354 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bressan A., Chen G., Zhang Q.: Unique conservative solutions to a variational wave equation. Arch. Rational Mech. Anal. 217, 1069–1101 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bressan A., Colombo R.M.: The semigroup generated by \({2 \times 2}\) conservation laws. Arch. Ration. Mech. Anal. 113, 1–75 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Bressan A., Constantin A.: Global conservative solutions to the Camassa-Holm equation. Arch. Rational Mech. Anal. 183, 215–239 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Bressan, A., Crasta, G., Piccoli, B.: Well posedness of the Cauchy problem for \({n \times n}\) systems of conservation laws. Am. Math. Soc. Memoir. 694, 2000

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Bressan A., Huang T.: Representation of dissipative solutions to a nonlinear variational wave equation. Commun. Math. Sci. 14, 31–53 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bressan A., Huang T., Yu F.: Structurally stable singularities for a nonlinear wave equation. Bull. Inst. Math. Acad. Sinica 10, 449–478 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Bressan A., Zheng Y.: Conservative solutions to a nonlinear variational wave equation. Commun. Math. Phys. 266, 471–497 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Crandall M.G.: The semigroup approach to first order quasilinear equations in several space variables. Israel J. Math. 12, 108–132 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Glassey R.T., Hunter J.K., Zheng Y.: Singularities in a nonlinear variational wave equation. J. Differ. Equ. 129, 49–78 (1996)

    Article  ADS  MATH  Google Scholar 

  20. Grunert K., Holden H., Raynaud X.: Lipschitz metric for the periodic Camassa-Holm equation. J. Differ. Equ. 250, 1460–1492 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Kruzhkov S.: First-order quasilinear equations with several space variables. Math. USSR Sb. 10, 217–273 (1970)

    Article  MATH  Google Scholar 

  24. Smoller, J.: Shock Waves and Reaction-Diffusion Equations, Second edition. Springer-Verlag, New York, 1994

  25. Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence, 2003

  26. Zhang P., Zheng Y.: Weak solutions to a nonlinear variational wave equation. Arch. Rational Mech. Anal. 166, 303–319 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Alberto Bressan

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Geng Chen

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  1. Alberto Bressan
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  2. Geng Chen
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Correspondence to Geng Chen.

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Communicated by Tai-Ping Liu

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Bressan, A., Chen, G. Lipschitz Metrics for a Class of Nonlinear Wave Equations. Arch Rational Mech Anal 226, 1303–1343 (2017). https://doi.org/10.1007/s00205-017-1155-7

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  • DOI: https://doi.org/10.1007/s00205-017-1155-7

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