Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-18T23:05:44.033Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE LOW REGULARITY SOLUTIONS FOR A MODIFIED TWO-COMPONENT CAMASSA–HOLM SHALLOW WATER SYSTEM

Published online by Cambridge University Press:  10 March 2011

XINGXING LIU  and
ZHAOYANG YIN
XINGXING LIU
Affiliation:
Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China e-mail: liuxingxing123456@163.com, mcsyzy@mail.sysu.edu.cn
ZHAOYANG YIN
Affiliation:
Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China e-mail: liuxingxing123456@163.com, mcsyzy@mail.sysu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We first study the regularised version of a modified two-component Camassa–Holm shallow water system and obtain the energy estimates of the corresponding approximate solutions. Then, we present a sufficient condition which guarantees that these approximate solutions converge to a low regularity weak solution of the modified two-component Camassa–Holm shallow water system.

Keywords

35G2535L05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 611 - 621
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Bona, J. L. and Smith, R., The initial value problem for the Korteweg-de Vries equation, Phil. Trans. R. Soc. Lond. A 278 (1975), 555–601.Google Scholar
2.Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661–1664.CrossRefGoogle ScholarPubMed
3.Camassa, R., Holm, D. and Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1–33.CrossRefGoogle Scholar
4.Chen, M., Liu, S-Q. and Zhang, Y., A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys. 75 (2006), 1–15.CrossRefGoogle Scholar
5.Constantin, A., The Hamiltonian structure of the Camassa-Holm equation, Expo. Math. 15 (1997), 53–85.Google Scholar
6.Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A 457 (2001), 953–970.CrossRefGoogle Scholar
7.Constantin, A. and Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 1992 (2009), 165–186.CrossRefGoogle Scholar
8.Constantin, A. and Escher, J., Global existence and blow-up for a shallow water equation, Annali Sci. Norm. Sup. Pisa 26 (1998), 303–328.Google Scholar
9.Constantin, A. and Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), 229–243.CrossRefGoogle Scholar
10.Constantin, A. and Escher, J., Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007), 423–431.CrossRefGoogle Scholar
11.Constantin, A. and Ivanov, R., On an integrable two-component Camass-Holm shallow water system, Phys. Lett. A 372 (2008), 7129–7132.CrossRefGoogle Scholar
12.Constantin, A. and Strauss, W., Stability of peakons, Commun. Pure Appl. Math. 53 (2000), 603–610.3.0.CO;2-L>CrossRefGoogle Scholar
13.Dai, H., Model equations for nonlinear dispersive waves in compressible Mooney-Rivlin rod, Acta Mech. 127 (1998), 193–207.CrossRefGoogle Scholar
14.Danchin, R., A few remarks on the Camassa-Holm equation, Differ. Int. Equ. 14 (2001), 953–988.Google Scholar
15.Dullin, H. R., Gottwald, G. A. and Holm, D. D., An integral shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001), 4501–4504.CrossRefGoogle ScholarPubMed
16.Escher, J., Lechtenfeld, O. and Yin, Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Disc. Cont. Dyn. Syst. 19 (2007), 493–513.CrossRefGoogle Scholar
17.Escher, J. and Yin, Z., Initial boundary value problems of the Camassa-Holm equation, Commun. Part. Differ. Equ. 33 (2008), 377–395.CrossRefGoogle Scholar
18.Escher, J. and Yin, Z., Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal. 256 (2008), 479–508.CrossRefGoogle Scholar
19.Falqui, G., On a Camassa-Holm type equation with two dependent variables, J. Phys. A 39 (2006), 327–342.CrossRefGoogle Scholar
20.Guan, C. and Yin, Z., Global existence and blow-up phenomena for an integrable two component Camassa-Holm shallow water system, J. Differ. Equ. 248 (2010), 2003–2014.CrossRefGoogle Scholar
21.Guan, C., Karlsen, K. H. and Yin, Z., Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, in Proceedings of the 2008–2009 special year in nonlinear partial differential equations, vol. 526 (Holden, H. and Karlsen, K. H., Editors) (Contemporary Mathematics, American Mathematical Society, Providence, RI, 2010), 199–220.Google Scholar
22.Holm, D. and Ivanov, R., Two-component CH system: Inverse scattering, peakons and geometry, arXiv:1009.5374v1 [nlin.SI] (27 Sep. 2010).CrossRefGoogle Scholar
23.Holm, D., Naraigh, L. and Tronci, C., Singular solution of a modified two-component Camassa-Holm equation, Phys. Rev. E 79 (2009), 1–13.CrossRefGoogle ScholarPubMed
24.Ivanov, R., Water waves and integrability, Philos. Trans. R. Soc. Lond. A 365 (2007), 2267–2280.Google ScholarPubMed
25.Kato, T. and Ponce, G., Commutator estimates and Navier-Stokes equations, Commun. Pure Appl. Math. 41 (1988), 203–208.CrossRefGoogle Scholar
26.Li, Y. and Oliver, P., Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differ. Equ. 162 (2000), 27–63.CrossRefGoogle Scholar
27.Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires (Dunod, Gauthier-Villars, Paris, 1969).Google Scholar
28.Yin, Z., Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Disc. Cont. Dyn. Syst. 10 (2004), 393–411.CrossRefGoogle Scholar