Advertisement

Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1091–1137 | Cite as

Lipschitz Metric for the Novikov Equation

  • Hong Cai
  • Geng Chen
  • Robin Ming Chen
  • Yannan Shen
Article

Abstract

We consider the Lipschitz continuous dependence of solutions for the Novikov equation with respect to the initial data. In particular, we construct a Finsler type optimal transport metric which renders the solution map Lipschitz continuous on bounded sets of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), although it is not Lipschitz continuous under the natural Sobolev metric from an energy law due to the finite time gradient blowup. By an application of Thom’s transversality theorem, we also prove that when the initial data is in an open dense subset of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), the solution is piecewise smooth. This generic regularity result helps us extend the Lipschitz continuous metric to the general weak solutions. Our method of constructing the metric can be used to treat other kinds of quasi-linear equations, provided a good knowledge about the energy concentration.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bressan, A., Chen, G.: Generic regularity of conservative solutions to a nonlinear wave equation, Ann. I. H. Poincaré–AN 34(2), 335–354 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bressan, A., Chen, G.: Lipschitz metric for a class of nonlinear wave equations, Arch. Rat. Mech. Anal. 226(3), 1303–1343, 2017Google Scholar
  3. 3.
    Bressan A., Chen G., Zhang Q.: Uniqueness of conservative solutions to the Camassa–Holm equation via characteristics. Discret. Contin. Dynam. Syst., 35, 25–42 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bressan A., Constantin A.: Global conservative solutions to the Camassa–Holm equation. Arch. Rat. Mech. Anal., 183, 215–239 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan A., Constantin A.: Global solutions to the Hunter–Saxton equations. SIAM J. Math. Anal., 37, 996–1026 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bressan A., Fonte M.: An optimal transportation metric for solutions of the Camassa–Holm equation. Methods Appl. Anal., 12, 191–220 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bressan A., Holden H., Raynaud X.: Lipschitz metric for the Hunter–Saxton equation. J. Math. Pures Appl., 94, 68–92 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, G., Chen, R.M., Liu, Y.: Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation, to appear in Indiana U. Math. J. Google Scholar
  10. 10.
    Chen G., Shen Y.: Existence and regularity of solutions in nonlinear wave equations. Discret. Contin. Dynam. Syst., 35, 3327–3342 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen R.M., Guo F., Liu Y., Qu C.Z.: Analysis on the blow-up of solutions to a class of integrable peakon equations. J. Funct. Anal., 270, 2343–2374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47–66, 1981/1982Google Scholar
  13. 13.
    Holden H., Raynaud X.: Global conservative solutions of the Camassa–Holm equation–Lagrangian point of view. Commun. Partial Differ. Equ., 32, 1511–1549 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grunert K., Holden H., Raynaud X.: Lipschitz metric for the periodic Camassa–Holm equation. J. Differ. Equ., 250, 1460–1492 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grunert K., Holden H., Raynaud X.: Lipschitz metric for the Camassa–Holm equation on the line. Discret. Contin. Dyn. Syst., 33, 2809–2827 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hone A.N.W., Lundmark H., Szmigielski J.: Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa–Holm type equation. Dyn. PDEs, 6, 253–289 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hone A.N.W., Wang J.: Integrable peakon equations with cubic nonlinearity. J. Phys. A, 37, 372002 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jiang, Z., Ni, L.: Blow-up phenomenon for the integrable Novikov equation, preprint Google Scholar
  19. 19.
    Kardell M.: Peakon–antipeakon solutions of the Novikov equation, ThesisGoogle Scholar
  20. 20.
    Li, M.J., Zhang, Q.T.: Generic regularity of conservative solutions to Camassa–Holm type equations. 2015. hal-01202927.Google Scholar
  21. 21.
    Novikov V.: Generalizations of the Camassa–Holm type equation. J. Phys. A, 42, 342002 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Villani C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Hong Cai
    • 1
  • Geng Chen
    • 2
  • Robin Ming Chen
    • 3
  • Yannan Shen
    • 4
  1. 1.Department of MathematicsQingdao University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA
  3. 3.Department of MathematicsUniversity of PittsburghPittsburghUSA
  4. 4.Department of MathematicsCalifornia State UniversityNorthridgeUSA

Personalised recommendations