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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
[3] Beals R., Sattinger D., Szmigielski J. (2000) Multipeakons and the classical moment problem. Adv. Math. 154, 229–257 · Zbl 0968.35008
[4] Beals R., Sattinger D., Szmigielski J. (2001) Peakon-antipeakon interaction. J. Nonlinear Math. Phys. 8, 23–27 · Zbl 0977.35106
[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 · Zbl 0936.35153
[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
[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
[11] Constantin A., McKean H.P. (1999) A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 · Zbl 0940.35177
[12] Constantin A., Molinet L. (2000) Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 · Zbl 1002.35101
[13] Constantin A., Strauss W. (2000) Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 · Zbl 1049.35149
[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
[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
[20] McKean H.P (2003) Fredholm determinants and the Camassa–Holm hierarchy. Comm. Pure Appl. Math. 56, 638–680 · Zbl 1047.37047
[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
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