Global conservative solutions of the Camassa-Holm equation. (English) Zbl 1105.76013

Summary: This paper develops a new approach to the analysis of Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time.


76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
35Q51 Soliton equations
Full Text: DOI


[1] Aronszajn N. (1976) Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57, 147–190 · Zbl 0342.46034
[2] Beals R., Sattinger D., Szmigielski J. (1999) Multi-peakons and a theorem of Stieltjes. Inverse Problems 15, L1–L4 · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[3] Beals R., Sattinger D., Szmigielski J. (2000) Multipeakons and the classical moment problem. Adv. Math. 154, 229–257 · Zbl 0968.35008 · doi:10.1006/aima.1999.1883
[4] Beals R., Sattinger D., Szmigielski J. (2001) Peakon-antipeakon interaction. J. Nonlinear Math. Phys. 8, 23–27 · Zbl 0977.35106 · doi:10.2991/jnmp.2001.8.s.5
[5] Bressan A., Zhang, P., Zheng, Y. On asymptotic variational wave equations. Arch. Ration Mech. Anal., to appear · Zbl 1168.35026
[6] Camassa R., Holm D.D. (1993) An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 · doi:10.1103/PhysRevLett.71.1661
[7] Constantin A. (2000) Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann.Inst. Fourier (Grenoble) 50, 321–362 · Zbl 0944.35062
[8] Constantin A. (2001) On the scattering problem for the Camassa–Holm equation. Proc. Roy. Soc. Lond Ser. A math. phys. Eng. Sci. 457, 953–970 · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[9] Constantin A., Escher J. (1998) Global existence and blow-up for a shallow water equation. Ann. Sc Norm. Pisa Super. Sci. 26(5): 303–328 · Zbl 0918.35005
[10] Constantin A., Escher J. (1998) Wave breaking for nonlinear nonlocal shallow water equations. Acta. Math. 181, 229–243 · Zbl 0923.76025 · doi:10.1007/BF02392586
[11] Constantin A., McKean H.P. (1999) A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[12] Constantin A., Molinet L. (2000) Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 · Zbl 1002.35101 · doi:10.1007/s002200050801
[13] Constantin A., Strauss W. (2000) Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[14] Diéudonne J. (1960) Foundations of Modern Analysis. Academic Press, New York
[15] Evans L.C., Gariepy R.F. (1992) Measure Theory andFine Properties of Functions. CRC Press, Boca Raton, FL · Zbl 0804.28001
[16] Fokas A., Fuchssteiner B. (1981) Symplectic structures, their Bäcklund transformation and hereditary symmetries. PhysD 4, 47–66 · Zbl 1194.37114
[17] Holden H., Raynaud X. (2006) A convergent numerical scheme for the Camassa–Holm equation based on multipeakons. DiscreteContin. Dyn. Syst. 14, 505–523 · Zbl 1111.35061
[18] Lenells J. (2005) Conservation laws of the Camassa–Holm equation. J. Phys. A 38, 869–880 · Zbl 1076.35100 · doi:10.1088/0305-4470/38/4/007
[19] Johnson R.S. (2002) Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[20] McKean H.P (2003) Fredholm determinants and the Camassa–Holm hierarchy. Comm. Pure Appl. Math. 56, 638–680 · Zbl 1047.37047 · doi:10.1002/cpa.10069
[21] Wahlen, E. On the peakon-antipeakon interaction. Dyn. Contin. Discrete. Impuls. Syst, to appear
[22] Xin Z., Zhang, P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53, 1411–1433 (2000) · Zbl 1048.35092
[23] Xin Z., Zhang P. (2002) On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differential Equations 27, 1815–1844 · Zbl 1034.35115 · doi:10.1081/PDE-120016129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.