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A CONTINUOUS INTERPOLATION BETWEEN CONSERVATIVE AND DISSIPATIVE SOLUTIONS FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM

Published online by Cambridge University Press:  07 January 2015

KATRIN GRUNERT
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected]
HELGE HOLDEN
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected] Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo, Norway; [email protected]
XAVIER RAYNAUD
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected] SINTEF ICT, Applied Mathematics, P.O. Box 124, NO-0314 Oslo, Norway; [email protected]

Abstract

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We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Ambrosio, L., Fusco, N. and Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (Clarendon Press, New York, 2000).Google Scholar
Aratyn, H., Gomes, J. F. and Zimerman, A. H., ‘On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa–Holm equation’, inSymmetry, Integrability and Geometry: Methods and Applications, 2 (2006), Paper 070, 12 pages.Google Scholar
Beals, R., Sattinger, D. and Szmigielski, J., ‘Peakon–antipeakon interaction’, J. Nonlinear Math. Phys. 8 (2001), 2327.Google Scholar
Bressan, A. and Constantin, A., ‘Global conservative solutions of the Camassa–Holm equation’, Arch. Ration. Mech. Anal. 183 (2007), 215239.CrossRefGoogle Scholar
Bressan, A. and Constantin, A., ‘Global dissipative solutions of the Camassa–Holm equation’, Anal. Appl. 5 (2007), 127.Google Scholar
Bressan, A., Holden, H. and Raynaud, X., ‘Lipschitz metric for the Hunter–Saxton equation’, J. Math. Pures Appl. 94 (2010), 6892.Google Scholar
Camassa, R. and Holm, D. D., ‘An integrable shallow water equation with peaked solitons’, Phys. Rev. Lett. 71(11) (1993), 16611664.Google Scholar
Chen, M., Liu, S.-Q. and Zhang, Y., ‘A two-component generalization of the Camassa–Holm equation and its solutions’, Lett. Math. Phys. 75 (2006), 115.Google Scholar
Chen, R. M. and Liu, Y., Wave breaking and global existence for a generalized two-component Camassa–Holm system. Inter. Math Research Notices, Article ID rnq118, 36 pages, 2010.Google Scholar
Constantin, A. and Escher, J., ‘Wave breaking for nonlinear nonlocal shallow water equations’, Acta Math. 181 (1998), 229243.Google Scholar
Constantin, A. and Ivanov, R. I., ‘On an integrable two-component Camassa–Holm shallow water system’, Phys. Lett. A 372 (2008), 71297132.Google Scholar
Escher, J., Lechtenfeld, O. and Yin, Z., ‘Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation’, Discrete Contin. Dyn. Syst. 19(3) (2007), 493513.CrossRefGoogle Scholar
Fu, Y. and Qu, C., ‘Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons’, J. Math. Phys. 50 012906 (2009).Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global conservative solutions of the Camassa–Holm equation for initial data with nonvanishing asymptotics’, Discrete Cont. Dyn. Syst. A 32 (2012), 42094277.Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global solutions for the two-component Camassa–Holm system’, Comm. Partial Differential Equations 37 (2012), 22452271.Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Lipschitz metric for the Camassa–Holm equation on the line’, Discrete Contin. Dyn. Syst. 33 (2013), 28092827.Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Periodic conservative solutions for the two-component Camassa–Holm system’, inSpectral Analysis, Differential Equations and Mathematical Physics (eds. Holden, H., Simon, B. and Teschl, G.) (American Mathematical Society, 2013), 165182. A Festschrift for Fritz Gesztesy on the Occasion of his 60th Birthday.Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global dissipative solutions of the two-component Camassa–Holm system for initial data with nonvanishing asymptotics’, Nonlinear Anal. Real World Appl. 17 (2014), 203244.Google Scholar
Guan, C., Karlsen, K. H. and Yin, Z., ‘Well-posedness and blow-up phenomenal for a modified two-component Camassa–Holm equation’, inNonlinear Partial Differential Equations and Hyperbolic Wave Phenomena (eds. Holden, H. and Karlsen, K. H.) Contemporary Mathematics, 526 (American Mathematical Society, 2010), 199220.Google Scholar
Guan, C. and Yin, Z., ‘Global existence and blow-up phenomena for an integrable two-component Camassa–Holm water system’, J. Differential Equations 248 (2010), 20032014.CrossRefGoogle Scholar
Guan, C. and Yin, Z., ‘Global weak solutions for a modified two-component Camassa–Holm equation’, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 623641.Google Scholar
Guan, C. and Yin, Z., ‘Global weak solutions for a two-component Camassa–Holm shallow water system’, J. Funct. Anal. 260 (2011), 11321154.Google Scholar
Gui, G. and Liu, Y., ‘On the Cauchy problem for the two-component Camassa–Holm system’, Math. Z. 268 (2011), 4566.CrossRefGoogle Scholar
Gui, G. and Liu, Y., ‘On the global existence and wave breaking criteria for the two-component Camassa–Holm system’, J. Funct. Anal. 258 (2010), 42514278.Google Scholar
Guo, Z. and Zhou, Y., ‘On solutions to a two-component generalized Camassa–Holm equation’, Stud. Appl. Math. 124 (2010), 307322.Google Scholar
Holden, H. and Raynaud, X., ‘Global conservative multipeakon solutions of the Camassa–Holm equation’, J. Hyperbolic Differ. Equ. 4 (2007), 3964.Google Scholar
Holden, H. and Raynaud, X., ‘Global conservative solutions of the Camassa–Holm equation — a Lagrangian point of view’, Comm. Partial Differential Equations 32 (2007), 15111549.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Global dissipative multipeakon solutions for the Camassa–Holm equation’, Comm. Partial Differential Equations 33 (2008), 20402063.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Dissipative solutions of the Camassa–Holm equation’, Discrete Contin. Dyn. Syst. 24 (2009), 10471112.Google Scholar
Holm, D. D., Náraigh, L. Ó. and Tronci, C., ‘Singular solutions of a modified two-component Camassa–Holm equation’, Phys. Rev. E 79 016601 (2009).CrossRefGoogle ScholarPubMed
Hu, Q. and Yin, Z., ‘Well-posedness and blow-up phenomena for a periodic two-component Camassa–Holm equation’, Proc. Roy. Soc. Edinburgh 141A (2011), 93107.Google Scholar
Ivanov, R. I., ‘Extended Camassa–Holm hierarchy and conserved quantities’, Z. Natforsch. 61A (2006), 133138.Google Scholar
Kuz’min, P. A., ‘Two-component generalizations of the Camassa–Holm equation’, Math. Notes 81 (2007), 130134.Google Scholar
Tian, L., Wang, Y. and Zhou, J., ‘Global conservative and dissipative solutions of a coupled Camassa–Holm equations’, J. Math. Phys. 52 063702 (2011).Google Scholar
Olver, P. J. and Rosenau, P., ‘Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support’, Phys. Rev. B 53(2) (1996), 19001906.Google Scholar
Wahlén, E., ‘The interaction of peakons and antipeakons’, Dyn. Contin. Discrete Impuls. Syst. A 13 (2006), 465472.Google Scholar
Wang, Y., Huang, J. and Chen, L., ‘Global conservative solutions of the two-component Camassa–Holm shallow water system’, Int. J. Nonlinear Sci. 9 (2009), 379384.Google Scholar