Wave breaking for a periodic shallow water equation. (English) Zbl 1042.35060

Summary: The focus of this paper is on the blow-up of a recently derived one-dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime. Some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case.


35Q35 PDEs in connection with fluid mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI


[1] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II: The KdV equation, Geom. Funct. Anal., 3, 209-262 (1993) · Zbl 0787.35098
[2] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[3] Camassa, R.; Holm, D.; Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011
[4] Constantin, A., The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141, 218-235 (1997) · Zbl 0889.35022
[5] Constantin, A.; Escher, J., Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[6] Constantin, A.; Escher, J., On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233, 75-91 (2000) · Zbl 0954.35136
[7] Constantin, A.; Escher, J., On the structure of a family of quasi linear equations arising in shallow water theory, Math. Ann., 312, 403-416 (1998) · Zbl 0923.76028
[8] Constantin, A.; McKean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 949-982 (1999) · Zbl 0940.35177
[9] Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[10] Dai, H.-H., Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127, 193-207 (1998) · Zbl 0910.73036
[11] Dai, H.-H.; Huo, Y., Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, Proc. Roy. Soc. London Ser. A, 456, 331-363 (2000) · Zbl 1004.74046
[12] Fuchssteiner, B.; Fokas, A., Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4, 47-66 (1981/1982) · Zbl 1194.37114
[13] Himonas, A.; Misiolek, G., Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161, 479-495 (2000) · Zbl 0945.35073
[14] Himonas, A.; Misiolek, G., The Cauchy problem for an integrable shallow water equation, Differential Integral Equations, 14, 821-831 (2001) · Zbl 1009.35075
[15] Kato, T., Quasi Linear Equations of Evolution, with Applications to Partial Differential Equations. Quasi Linear Equations of Evolution, with Applications to Partial Differential Equations, Lecture Notes in Mathematics, vol. 448 (1975), Springer-Verlag: Springer-Verlag Berlin, pp. 25-70 · Zbl 0315.35077
[16] Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119
[17] Misiolek, G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, 203-208 (1998) · Zbl 0901.58022
[18] Rodriguez-Blanco, G., On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46, 309-327 (2001) · Zbl 0980.35150
[19] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley New York · Zbl 0373.76001
[20] Xin, Z.; Zhang, P., On the weak solution to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092
[21] Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27, 1815-1844 (2002) · Zbl 1034.35115
[22] Y. Zhou, On a shallow water equation, Thesis, The Chinese University of Hong Kong, 2001; Y. Zhou, On a shallow water equation, Thesis, The Chinese University of Hong Kong, 2001
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.