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A note on the Cauchy problem for the two-component Novikov system

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Abstract

Considered herein is the initial value problem for the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map \(z_{0}\mapsto z(t)\) of this problem in the periodic case is not uniformly continuous in Besov spaces \(B^{s}_{p,r}({\mathbb {T}})\times B^{s}_{p,r}({\mathbb {T}}) \) with \(s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty \) and \(B^{5/2}_{2,1}({\mathbb {T}})\times B^{5/2}_{2,1}({\mathbb {T}})\) through the method of approximate solutions. Furthermore, it is in the non-periodic case that the non-uniform continuity of this solution map in Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\times B^{s}_{p,r}({\mathbb {R}})\) with \(s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty \) is discussed by constructing new subtle initial data. Finally, the Hölder continuity of the solution map in Besov spaces is proved.

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References

  1. H. Bahouri, J. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften 343, Springer-Verlag, Berlin Heidelberg, 2011.

  2. A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. A. Constantin, J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.

    Article  MathSciNet  Google Scholar 

  5. A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. M. Chen, Y. Liu, P. Zhang, The Hölder continuity of the solution map to the \(b-\)family equation, Math. Ann., 357 (2013), 1245-1289.

    Article  MathSciNet  Google Scholar 

  9. R. Chen, Z. Qiao, S. Zhou, Persistence properties and wave-breaking criteria for the Geng-Xue system, Math. Method Appl. Sci., 42 (2019), 6999-7110.

    Article  MathSciNet  Google Scholar 

  10. R. Danchin, A few reamarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988.

    MathSciNet  MATH  Google Scholar 

  11. R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.

    Article  MathSciNet  Google Scholar 

  12. R. Danchin, Fourier analysis method for PDEs , Lecture Notes, 14 November, 2005.

  13. B. Fuchssteiner, A. Fokas, Symplectic structures, their Bäklund transformation and hereditary symmetries, Physica D, 4 (1981/82), 47-66.

  14. X. Geng, B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856.

    Article  MathSciNet  Google Scholar 

  15. A. Himonas, C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22 (2009), 201-224.

    MathSciNet  MATH  Google Scholar 

  16. A. Himonas, C. Kenig, G. Misiołek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162.

    Article  MathSciNet  Google Scholar 

  17. A. Himonas, J. Holmes, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54 (2013), 061501.

    Article  MathSciNet  Google Scholar 

  18. A. Himonas, C. Holliman, The Cauchy problem of the Novikov equation, Nonlinearlity, 25 (2012), 449-479.

    Article  MathSciNet  Google Scholar 

  19. A. Hone, J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002.

    Article  MathSciNet  Google Scholar 

  20. J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499.

    Article  MathSciNet  Google Scholar 

  21. N. Li, Q. Liu, On bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A, 377 (2013), 257-261.

    Article  MathSciNet  Google Scholar 

  22. Y. Li, P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.

    Article  MathSciNet  Google Scholar 

  23. W. Luo, Z. Yin, Local well-posedness and blow-up criterion for a two-component Novikov system in the critical space, Nonlinear Anal., 122 (2015), 1-22.

    Article  MathSciNet  Google Scholar 

  24. J. Li, Y. Li, W. Zhu, Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces, J. Differential Equations, 269 (2020), 8686-8700.

    Article  MathSciNet  Google Scholar 

  25. J. Li, Z. Yin, Well-posedness and analytic solutions of the two-component Euler-Poincaré system, Monatsh Math., 183 (2017), 509-537.

    Article  MathSciNet  Google Scholar 

  26. Y. Mi, C. Mu, W. Tao, On the Cauchy problem for the two-component Novikov equation, Adv. Math. Phys., 2 (2013), 105-121.

    MathSciNet  MATH  Google Scholar 

  27. L. Ni, Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021.

    Article  MathSciNet  Google Scholar 

  28. V. Novikov, Generalizations of Camassa-Holm equation, J. Phys. A, 42 (2009), 342002.

    Article  MathSciNet  Google Scholar 

  29. Z. Popowicz, Doubled extended cubic peakon equation, Phys. Lett. A, 379 (2015), 1240-1245.

    Article  MathSciNet  Google Scholar 

  30. H. Tang, Y. Zhao, Z. Liu, A note on the solution map for the periodic Camassa-Holm equation, Appl. Anal., 93 (2014), 1745-1760.

    Article  MathSciNet  Google Scholar 

  31. H. Tang, Z. Liu, The Cauchy problem for a two-component Novikov equation in the critical Besov space, J. Math. Anal. Appl., 423 (2015), 120-135.

    Article  MathSciNet  Google Scholar 

  32. W. Yan, Y. Li, Y. Zhang, The Cauchy problem of the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318.

    Article  MathSciNet  Google Scholar 

  33. H. Wang, Y. Fu, Non-uniform dependence on initial data for the two-component Novikov system, J. Math. Phys., 58 (2017), 021502.

    Article  MathSciNet  Google Scholar 

  34. H. Wang, Y. Fu, A note on the Cauchy problem for the periodic two-component Novikov system , Appl. Anal., 99 (2020), 1042-1065.

    Article  MathSciNet  Google Scholar 

  35. S. Yu, X. Yin, The Cauchy problem for a generalized two-component short pulse system with high-order nonlinearities, J. Math. Anal. Appl., 475 (2019), 1427-1447.

    Article  MathSciNet  Google Scholar 

  36. S. Zhou, Y. Li, Persistence properties for the two-component Novikov equation in weighted  \(L^{p}\)  spaces, Appl. Anal., 98 (2019), 2105-2117.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work is supported by National Natural Science Foundation Grant-11471259 and the National Science Basic Research Program of Shaanxi (Program Nos. 2019JM-007, 2020JC-37).

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xi’an, 710069, People’s Republic of China

    Haiquan Wang & Lili Wu

  2. Department of Mathematics, Xi’an Jiaotong University City College, Xi’an, 710018, People’s Republic of China

    Gezi Chong

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  1. Haiquan Wang
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  2. Gezi Chong
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  3. Lili Wu
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Correspondence to Haiquan Wang.

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Wang, H., Chong, G. & Wu, L. A note on the Cauchy problem for the two-component Novikov system. J. Evol. Equ. 21, 1809–1843 (2021). https://doi.org/10.1007/s00028-020-00657-z

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