Dissipative solutions for the modified two–component Camassa–Holm system

详细信息    查看全文

• 作者：Yujuan Wang (1)
  Yongduan Song (1)
  
• 关键词：35L05 ; 35L65 ; 35L40 ; The modified two ; component Camassa–Holm system ; Global solutions ; Dissipative solutions ; Lagrangian variables
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2014
• 出版时间：June 2014
• 年：2014
• 卷：21
• 期：3
• 页码：339-360
• 全文大小：
• 参考文献：1. Holm D., Naraigh L., Tronci C.: Singular solution of a modified two-component Camassa–Holm equation. Phys. Rev. E 79, 1-3 (2009) CrossRef
  2. Chen M., Liu S.-Q., Zhang Y.: A 2-component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75, 1-5 (2006) CrossRef
  3. Constantin A., Ivanov R.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372, 7129-132 (2008) CrossRef
  4. Holm D., Marsden J., Ratiu T.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math 137, 1-1 (1998) CrossRef
  5. Constantin A.: The Hamiltonian structure of the Camassa–Holm equation. Expo. Math 15, 53-5 (1997)
  6. Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett 71, 1661-664 (1993) CrossRef
  7. Constantin A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457, 953-70 (2001) CrossRef
  8. Dullin H., Gottwald G., Holm D.: An integral shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501-504 (2001) CrossRef
  9. Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal 192, 165-86 (2009) CrossRef
  10. Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229-43 (1998) CrossRef
  11. Constantin A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321-62 (2000) CrossRef
  12. Bressan A., Constantin A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal 183, 215-39 (2007) CrossRef
  13. Holden H., Raynaud X.: Global conservative solutions of the Camassa–Holm equation-a Lagrangian point of view. Comm. P. D. E 32, 11511-549 (2007)
  14. Holden H., Raynaud X.: Periodic conservative solutions of the Camassa–Holm equation. Ann. Inst. Fourier (Grenoble) 58, 945-88 (2008) CrossRef
  15. Holden H., Raynaud X.: Global conservative multipeakon solutions of the Camassa–Holm equation. J. Hyperbolic Differ. Equ. 4, 39-4 (2007) CrossRef
  16. Bressan A., Constantin A.: Global dissipative solutions of the Camassa–Holm equation. Appl. Anal. 5, 1-7 (2007) CrossRef
  17. Holden H., Raynaud X.: Dissipative solutions for the Camassa–Holm equation. Discrete Contin. Dyn. Syst. 24, 1047-112 (2009) CrossRef
  18. Holden H., Raynaud X.: Global dissipative multipeakon solutions of the Camassa–Holm equation. Communications in P.D.E 33, 2040-063 (2008) CrossRef
  19. Lv G., Wang M.: Some remarks for a modified periodic Camassa–Holm system. Discrete Contin. Dyn. Syst. 30, 1161-180 (2011) CrossRef
  20. Guan C., Karlsen K.H., Yin Z.: Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation, Nonlinear partial differential equations and hyperbolic wave phenomena. Contemp. Math. 526, 199-20 (2010) CrossRef
  21. Tan W., Yin Z.: Global dissipative solutions of a modified two-component Camassa–Holm shallow water system. J. Math. Phys. 52:033507 (2011)
  22. Tan W., Yin Z.: Global conservative solutions of a modified two-component Camassa–Holm shallow water system. J. Differ. Equ. 251, 3558-582 (2011) CrossRef
  23. Tan W., Yin Z.: Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation. J. Funct. Anal. 261, 1204-226 (2011) CrossRef
  24. He B.: New peakon, solitary wave and periodic wave solutions for the modified Camassa–Holm equation. Nonlinear Anal. 71, 6011-018 (2009) CrossRef
  25. Wen Z., Liu Z.: Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation. Nonlinear Anal. R.W.A 12, 1698-707 (2011) CrossRef
  26. Guo Z., Zhu M.: Wave breaking for a modified two-component Camassa–Holm system. J. Differ. Equ. 252, 2759-770 (2012) CrossRef
  27. Jin, L., Guo, Z.: A note on a modified twocomponent CamassaHolm system Nonlinear Anal R W A 13887892 (2012)
  28. Guo Z., Zhu M.: Blow-up criteria of solutions to a modified two-component Camassa–Holm system. Nonlinear Anal. R. W. A. 12, 3531-540 (2011) CrossRef
  29. Tian, L., Wang, Y.: Global conservative and dissipative solutions of a coupled Camassa–Holm equations. J. Math. Phys. 52:063702 (2011)
  30. Wang Y., Huang J.: Global conservative solutions of the two-component Camassa–Holm shallow water system. Int. J. Nonlinear Sci. 9, 379-84 (2010)
  31. Guan C., Yin Z.: Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Differ. Equ 248, 2003-014 (2010) CrossRef
  32. Shen, Z., Wang, Y., Karimi, H., Song, Y.: On the multipeakon dissipative behavior of the modified coupled Camassa–Holm model for shallow water system. Mathematical Problems in Engineering http://dx.doi.org/10.1155/2013/107450 (2013)
  
• 作者单位：Yujuan Wang (1)
  Yongduan Song (1)
  
  1. School of automation, Institute of smart system and renewable energy, Chongqing University, Chongqing, 400044, People’s Republic of China
  
• ISSN：1420-9004

文摘

Camassa–Holm model is capable of characterizing the dynamic behavior of shallow water wave, thus has been extensively studied. This paper is concerned with shallow water wave behavior after wave breaking. To better reflect the whole process, the modified two-component Camassa–Holm system is considered. The continuation of solutions of such system after wave braking is investigated. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which global dissipative solutions are obtained. The results obtained herein are deemed useful in understanding the dynamic behavior of shallow water wave during and after wave breaking.