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Well-posedness and persistence property for a four-component Novikov system with peakon solutions

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Abstract

In this paper we mainly study the Cauchy problem of a four-component Novikov system. We first show the local well-posedness of the system in Besov spaces \(B^{s+1}_{p,r}\times B^{s+1}_{p,r}\times B^s_{p,r} \times B^s_{p,r}\) with \(p,r\in [1,\infty ],~s>\max \{\frac{1}{p},\frac{1}{2}\}\) by using the Littlewood–Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we prove the local well-posedness of the system in the critical Besov space \(B^{\frac{3}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1} \times B^{\frac{1}{2}}_{2,1}\times B^{\frac{1}{2}}_{2,1}\). Next, we establish two blow-up criteria for strong solutions to the system by using the structure of the system. Moreover, we investigate the persistence property for strong solutions to the system. Finally, we verify that the system possesses a special class of peakon solutions.

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References

  1. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  2. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

    Article  MATH  Google Scholar 

  6. Constantin, A.: The Hamiltonian structure of the Camassa–Holm equation. Expos. Math. 15(1), 53–85 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. Ser. A 457, 953–970 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. l’Inst. Fourier (Grenoble) 50, 321–362 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 26, 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Am. Math. Soc. Bull. New Ser. 44, 423–431 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173(2), 559–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Constantin, A., Ivanov, R.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372, 7129–7132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192, 429–444 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Escher, J., Lechtenfeld, O., Yin, Z.: Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. Ser. A 19, 493–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D 4(1), 47-66 (1981/1982)

  23. Guan, C., Yin, Z.: Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Differ. Equ. 248, 2003–2014 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guan, C., Karlsen, K.H., Yin, Z.: Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation. Contemp. Math. 526, 199–220 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guan, C., Yin, Z.: Global weak solutions for a two-component Camassa–Holm shallow water system. J. Funct. Anal. 260, 1132–1154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Guan, C., Yin, Z.: Global weak solutions for a modified two-component Camassa–Holm equation. Ann. l’Inst. Henri Poincaré (C) Non Linear Anal. 28, 623–641 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gui, G., Liu, Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal. 258, 4251–4278 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Holm, D.D., Naraigh, L., Tronci, C.: Singular solution of a modified two-component Camassa–Holm equation. Phys. Rev. E 79, 1–13 (2009)

    Article  Google Scholar 

  30. Hone, A.N.W., Wang, J.: Integrable peakon equations with cubic nonlinearity. J. Phys. A Math. Theor. 41, 372002 (2008). 10pp

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, Y., Yin, Z.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lai, S.: Global weak solutions to the Novikov equation. J. Funct. Anal. 265, 520–544 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Novikov, V.: Generalizations of the Camassa–Holm equation. J. Phys. A Math. Theor. 42, 342002 (2009). 14pp

    Article  MathSciNet  MATH  Google Scholar 

  34. Popowicz, Z.: Double extended cubic peakon equation. Phys. Lett. A 379, 1240–1245 (2015)

    Article  MathSciNet  Google Scholar 

  35. Rodríguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. Theory Methods Appl. 46, 309–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tan, W., Yin, Z.: Global conservative solutions of a modified two-component Camassa–Holm shallow water system. J. Differ. Equ. 251, 3558–3582 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tan, W., Yin, Z.: Global dissipative solutions of a modified two-component Camassa–Holm shallow water system. J. Math. Phys. 52, 033507 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)

    MathSciNet  MATH  Google Scholar 

  39. Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A Math. Theor. 44, 055202 (2011). 17pp

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, X., Yin, Z.: Well-posedness and global existence for the Novikov equation. Annali della Scuola Normale Superiore di Pisa Classe di Sci. Ser. V 11, 707–727 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Wu, X., Yin, Z.: A note on the Cauchy problem of the Novikov equation. Appl. Anal. 92, 1116–1137 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 253, 298–318 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. NoDEA 20, 1157–1169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yan, K., Yin, Z.: Well-posedness for a modified two-component Camassa–Holm system in critical spaces. Discrete Contin. Dyn. Syst. Ser. A 33, 1699–1712 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was partially supported by NNSFC (No. 11271382), RFDP (No. 20120171110014), the Macao Science and Technology Development Fund (No. 098/2013/A3) and the key project of Sun Yat-sen University. The authors thank the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Wei Luo & Zhaoyang Yin

  2. Faculty of Information Technology, Macau University of Science and Technology, Macau, China

    Zhaoyang Yin

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  1. Wei Luo
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  2. Zhaoyang Yin
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Correspondence to Wei Luo.

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Communicated by A. Constantin.

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Luo, W., Yin, Z. Well-posedness and persistence property for a four-component Novikov system with peakon solutions. Monatsh Math 180, 853–891 (2016). https://doi.org/10.1007/s00605-015-0809-5

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  • DOI: https://doi.org/10.1007/s00605-015-0809-5

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