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Existence of permanent and breaking waves for a shallow water equation: a geometric approach. (English) Zbl 0944.35062

Summary: The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set.
Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient conditions for the initial profile to develop into a global classical solution or into a breaking wave. With respect to the geometric properties of the diffeomorphism group, we prove that the metric spray is smooth infering that, locally, two points can be joined by a unique geodesic. The qualitative analysis of the shallow water equation is used to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.

MSC:

35Q35 PDEs in connection with fluid mechanics
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] V. ARNOLD, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. · Zbl 0148.45301
[2] V. ARNOLD and B. KHESIN, Topological Methods in Hydrodynamics, Springer Verlag, New York, 1998. · Zbl 0902.76001
[3] B. BENJAMIN and J. L. BONA and J. J. MAHONY, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London, 272 (1972), 47-78. · Zbl 0229.35013
[4] R. CAMASSA and D. HOLM, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. · Zbl 0972.35521
[5] R. CAMASSA and D. HOLM and J. HYMAN, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. · Zbl 0808.76011
[6] M. CANTOR, Perfect fluid flows over Rn with asymptotic conditions, J. Funct. Anal., 18 (1975), 73-84. · Zbl 0306.58007
[7] A. CONSTANTIN, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235. · Zbl 0889.35022
[8] A. CONSTANTIN and J. ESCHER, Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. · Zbl 0934.35153
[9] A. CONSTANTIN and J. ESCHER, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328. · Zbl 0918.35005
[10] A. CONSTANTIN and J. ESCHER, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. · Zbl 0923.76025
[11] A. CONSTANTIN and J. ESCHER, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. · Zbl 0954.35136
[12] A. CONSTANTIN and H. P. MCKAN, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. · Zbl 0940.35177
[13] J. DIEUDONNÉ, Foundations of Modern Analysis, Academic Press, New York, 1969. · Zbl 0176.00502
[14] R. K. DODD and J. C. EILBECK and J. D. GIBBON and H. C. MORRIS, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984. · Zbl 0496.35001
[15] P. G. DRAZIN and R. S. JOHNSON, Solitons: an Introduction, Cambridge University Press, Cambridge - New York, 1989. · Zbl 0661.35001
[16] D. EBIN and J. E. MARSDEN, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. · Zbl 0211.57401
[17] L. EVANS and R. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Adv. Math., Boca Raton, Florida, 1992. · Zbl 0804.28001
[18] A. S. FOKAS and B. FUCHSSTEINER, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. · Zbl 1194.37114
[19] B. FUCHSSTEINER, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 296-343. · Zbl 0900.35345
[20] T. KATO, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Springer Lecture Notes in Mathematics, 448 (1975), 25-70. · Zbl 0315.35077
[21] C. KENIG and G. PONCE and L. VEGA, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. · Zbl 0808.35128
[22] D. J. KORTEWEG and G. de VRIES, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443. · JFM 26.0881.02
[23] S. KOURANBAEVA, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. · Zbl 0958.37060
[24] S. LANG, Differential and Riemannian Manifolds, Springer Verlag, New York, 1995. · Zbl 0824.58003
[25] H. P. MCKEAN, Integrable systems and algebraic curves, Global Analysis, Springer Lecture Notes in Mathematics, 755 (1979), 83-200. · Zbl 0449.35080
[26] J. MILNOR, Morse Theory, Ann. Math. Studies 53, Princeton University Press, 1963. · Zbl 0108.10401
[27] G. MISIOLEK, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. · Zbl 0901.58022
[28] P. OLVER, Applications of Lie Groups to Differential Equations, Springer Verlag, New-York, 1993. · Zbl 0785.58003
[29] E. M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1990. · Zbl 0232.42007
[30] G. B. WHITHAM, Linear and Nonlinear Waves, J. Wiley & Sons, New York, 1980.
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