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Local well-posedness and blow-up phenomenon for a generalization two-component Camassa–Holm system

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Abstract

In this paper, a new generalized two-component Camassa–Holm system is derived via the energy variational approach. This system has two parameters which depend on the energy functional. The initial value problem is investigated. The local well-posedness is obtained when the initial density is away from vacuum. Taking advantage of the method of characteristics and the conservation laws, we prove the blow-up criteria. According to the blow-up criteria, we can prove the finite time blow-up result under some suitable condition. Moreover, we give some exact expression of traveling solutions.

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References

  1. P. F. Byrd and M. D. Friedman. Handbook of elliptic integrals for engineers and scientists. Die Grundlehren der mathematischen Wissenschaften, Band 67. Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised.

  2. R. Camassa and D. D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71(11):1661–1664, 1993.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. A. Constantin and J. Escher. Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(2):303–328, 1998.

    MathSciNet  MATH  Google Scholar 

  7. A. Constantin and J. Escher. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math., 181(2):229–243, 1998.

    Article  MathSciNet  Google Scholar 

  8. A. Constantin and J. Escher. Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math., 51(5):475–504, 1998.

    Article  MathSciNet  Google Scholar 

  9. A. Constantin and R. I. Ivanov. On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A, 372(48):7129–7132, 2008.

    Article  MathSciNet  Google Scholar 

  10. A. Constantin and D. Lannes. The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal., 192(1):165–186, 2009.

    Article  MathSciNet  Google Scholar 

  11. A. Constantin and L. Molinet. Global weak solutions for a shallow water equation. Comm. Math. Phys., 211(1):45–61, 2000.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. R. Danchin. A few remarks on the Camassa–Holm equation. Differential Integral Equations, 14(8):953–988, 2001.

    MathSciNet  MATH  Google Scholar 

  14. R. Danchin. A note on well-posedness for Camassa–Holm equation. J. Differential Equations, 192(2):429–444, 2003.

    Article  MathSciNet  Google Scholar 

  15. J. Escher, O. Lechtenfeld, and Z. Yin. Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst., 19(3):493–513, 2007.

    Article  MathSciNet  Google Scholar 

  16. B. Fuchssteiner and A. S. Fokas. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D, 4(1):47–66, 1981.

    Article  MathSciNet  Google Scholar 

  17. K. Grunert, H. Holden, and X. Raynaud. Global solutions for the two-component Camassa–Holm system. Comm. Partial Differential Equations, 37(12):2245–2271, 2012.

    Article  MathSciNet  Google Scholar 

  18. C. Guan and Z. Yin. Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Differential Equations, 248(8):2003–2014, 2010.

    Article  MathSciNet  Google Scholar 

  19. C. Guan and Z. Yin. Global weak solutions for a two-component Camassa–Holm shallow water system. J. Funct. Anal., 260(4):1132–1154, 2011.

    Article  MathSciNet  Google Scholar 

  20. G. Gui and Y. Liu. On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal., 258(12):4251–4278, 2010.

    Article  MathSciNet  Google Scholar 

  21. G. Gui and Y. Liu. On the Cauchy problem for the two-component Camassa–Holm system. Math. Z., 268(1-2):45–66, 2011.

    Article  MathSciNet  Google Scholar 

  22. Z. Guo, X. Liu, L. Molinet, and Z. Yin. Ill-posedness of the Camassa–Holm and related equations in the critical space. J. Differential Equations, 266(2-3):1698–1707, 2019.

    Article  MathSciNet  Google Scholar 

  23. D. Ionescu-Kruse. Variational derivation of the Camassa–Holm shallow water equation. J. Nonlinear Math. Phys., 14(3):303–312, 2007.

    Article  MathSciNet  Google Scholar 

  24. D. Ionescu-Kruse. Variational derivation of the Camassa–Holm shallow water equation with non-zero vorticity. Discrete Contin. Dyn. Syst., 19(3):531–543, 2007.

    Article  MathSciNet  Google Scholar 

  25. D. Ionescu-Kruse. Variational derivation of two-component Camassa–Holm shallow water system. Appl. Anal., 92(6):1241–1253, 2013.

    Article  MathSciNet  Google Scholar 

  26. J. Li and Z. Qiao. Bifurcations and exact traveling wave solutions of the generalized two-component Camassa–Holm equation. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22(12):1250305, 13, 2012.

    Article  MathSciNet  Google Scholar 

  27. J. Li and Z. Yin. Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces. J. Differential Equations, 261(11):6125–6143, 2016.

    Article  MathSciNet  Google Scholar 

  28. W. Luo and Z. Yin. Gevrey regularity and analyticity for Camassa–Holm type systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18(3):1061–1079, 2018.

    MathSciNet  MATH  Google Scholar 

  29. G. Rodríguez-Blanco. On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal., 46(3):309–327, 2001.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Jingchi Huang & Wei Luo

  2. School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou, 510275, China

    Yuhui Chen

  3. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Fang Yu

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  1. Yuhui Chen
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  2. Jingchi Huang
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  3. Wei Luo
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  4. Fang Yu
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Correspondence to Wei Luo.

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Chen, Y., Huang, J., Luo, W. et al. Local well-posedness and blow-up phenomenon for a generalization two-component Camassa–Holm system. J. Evol. Equ. 19, 935–963 (2019). https://doi.org/10.1007/s00028-019-00503-x

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