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Monatshefte für Mathematik

, Volume 187, Issue 4, pp 735–764 | Cite as

The local well-posedness in Besov spaces and non-uniform dependence on initial data for the interacting system of Camassa–Holm and Degasperis–Procesi equations

  • Shouming Zhou
Article
  • 156 Downloads

Abstract

This paper deals with the Cauchy problem for the interacting system of the Camassa–Holm and Degasperis–Procesi equations
$$\begin{aligned} m_t=-3m(2u_x+v_x)-m_x(2u+v), n_t=-2n(2u_x+v_x)-n_x(2u+v), \end{aligned}$$
where \(m=u-u_{xx}\) and \(n= v-v_{xx}\). By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces \(B^s_{p,r}\times B^s_{p,r}\) with \(1\le p,r \le +\infty \) and \(s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} \) is obtained, and the local well-posedness in critical Besov space \(B^{5/2}_{2,1}\times B^{5/2}_{2,1}\) is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space \(H^s({\mathbb {R}})\times H^s({\mathbb {R}})\) with \(s>5/2\), and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous.

Keywords

Degasperis–Procesi equation Camassa–Holm equation Well-posedness Non-uniform dependence 

Mathematics Subject Classification

35G25 35L05 35Q50 

Notes

Acknowledgements

This work is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJ1703043), Natural Science Foundation of Chongqing (Grant Nos. cstc2017jcyjAX0123 and cstc2016jcyjA0181) and National Natural Science Foundation of China (Grant Nos. 11771063 and 11601048).

References

  1. 1.
    Boutet de Monvel, A., Kostenko, A., Shepelsky, D., Teschl, G.: Long-time asymptotics for the Camassa–Holm equation. SIAM J. Math. Anal. 41, 1559–1588 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)CrossRefGoogle Scholar
  5. 5.
    Constantin, A.: On the inverse spectral problem for the Camassa–Holm equation. J. Funct. Anal. 155, 352–363 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Constantin, A.: Particle trajectories in extreme Stokes waves. IMA J. Appl. Math. 77, 293–307 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Constantin, A., Ivanov, R.: Dressing method for the Degasperis–Procesi equation. Stud. Appl. Math. 138, 205–226 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Constantin, A., Ivanov, R., Lenells, J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Constantin, A., Strauss, W.A.: Stability of the Camassa–Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192, 429–444 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Danchin, R.: Fourier Analysis Methods for PDEs, Lecture Notes (2003)Google Scholar
  20. 20.
    Degasperis, A., Holm, D., Hone, A.: A new integral equation with peakon solutions. Theor. Math. Phys. 133, 1461–1472 (2002)Google Scholar
  21. 21.
    Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37. World Scientific Publishing, River Edge, NJ (1999)Google Scholar
  22. 22.
    Dullin, H.R., Gottwald, G.A., Holm, D.D.: Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73–79 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Dullin, H.R., Gottwald, G.A., Holm, D.D.: On asymptotically equivalent shallow water wave equations. Physica D 190, 1–14 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Eckhardt, J.: The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. Arch. Ration. Mech. Anal. 224, 21–52 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Fokas, A., Fuchssteiner, B.S.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47–66 (1981)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Fu, Y., Qu, C.: Well-posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons. J. Math. Phys. 50, 012906 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fu, Y., Qu, C., Ma, Y.: Well-posedness and blow-up phenomena for the interacting system of the Camassa–Holm and Degasperis–Procesi equations. Discrete Contin. Dyn. Syst. 27, 1025–1035 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Himonas, A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Himonas, A., Mantzavinos, D.: The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation. Nonlinear Anal. TMA 95, 499–529 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Himonas, A., Kenig, C.: Non-uniform dependence on initial data for the CH equation on the line. Differ. Integral Equ. 22, 201–224 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ni, L.D., Zhou, Y.: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 250, 3002–3201 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Popowicz, Z.: A two-component generalization of the Degasperis–Procesi equation. J. Phys. A Math. Gen. 39, 13717–13726 (2006)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Popowicz, Z.: A Camassa–Holm equation interacted with the Degasperis–Procesi equation. Czechoslov. J. Phys. 56, 1263–1268 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Taylor, M.: Commutator estimates. Proc. Am. Math. Soc. 131(5), 1501–1507 (2003)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Yin, Z.: On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math. 47, 649–666 (2003)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zhou, S.: Continuity and analyticity for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs. Monatsh. Math. 182, 195–238 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.College of Mathematics ScienceChongqing Normal UniversityChongqingPeople’s Republic of China

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