Existence of permanent and breaking waves for a shallow water equation: a geometric approach
Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 321-362.

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 to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.

On obtient des théorèmes d’existence de solutions globales en temps et des résultats sur la formation de singularités pour une équation qui modélise le phénomène des ondes de surface en eau peu profonde. La solution peut exploser uniquement sous la forme d’un déferlement. En utilisant le fait que cette équation d’ondes décrit le flot géodésique du groupe des difféomorphismes de la droite vérifiant certaines conditions asymptotiques à l’infini, muni d’une structure de variété riemannienne, on donne des conditions suffisantes sur la donnée initiale pour que la solution soit globale en temps ou bien qui impliquent un déferlement au bout d’un temps fini. Ces résultats se traduisent en terme de propriétés des géodésiques du groupe des difféomorphismes.

@article{AIF_2000__50_2_321_0,
     author = {Constantin, Adrian},
     title = {Existence of permanent and breaking waves for a shallow water equation: a geometric approach},
     journal = {Annales de l'Institut Fourier},
     pages = {321--362},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1757},
     mrnumber = {2002d:37125},
     zbl = {0944.35062},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1757/}
}
TY  - JOUR
AU  - Constantin, Adrian
TI  - Existence of permanent and breaking waves for a shallow water equation: a geometric approach
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 321
EP  - 362
VL  - 50
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1757/
DO  - 10.5802/aif.1757
LA  - en
ID  - AIF_2000__50_2_321_0
ER  - 
%0 Journal Article
%A Constantin, Adrian
%T Existence of permanent and breaking waves for a shallow water equation: a geometric approach
%J Annales de l'Institut Fourier
%D 2000
%P 321-362
%V 50
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1757/
%R 10.5802/aif.1757
%G en
%F AIF_2000__50_2_321_0
Constantin, Adrian. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 321-362. doi : 10.5802/aif.1757. http://www.numdam.org/articles/10.5802/aif.1757/

[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. | Numdam | MR | Zbl

[2] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer Verlag, New York, 1998. | MR | Zbl

[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. | MR | Zbl

[4] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. | MR | Zbl

[5] R. Camassa and D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. | Zbl

[6] M. Cantor, Perfect fluid flows over Rn with asymptotic conditions, J. Funct. Anal., 18 (1975), 73-84. | MR | Zbl

[7] A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. | MR | Zbl

[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. | MR | Zbl

[12] A. Constantin and H. P. Mckan, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. | MR | Zbl

[13] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969. | MR | Zbl

[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

[15] P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge - New York, 1989. | MR | Zbl

[16] D. Ebin and J. E. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. | MR | Zbl

[17] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Adv. Math., Boca Raton, Florida, 1992. | MR | Zbl

[18] A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66.

[19] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 296-343. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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

[23] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. | MR | Zbl

[24] S. Lang, Differential and Riemannian Manifolds, Springer Verlag, New York, 1995. | MR | Zbl

[25] H. P. Mckean, Integrable systems and algebraic curves, Global Analysis, Springer Lecture Notes in Mathematics, 755 (1979), 83-200. | MR | Zbl

[26] J. Milnor, Morse Theory, Ann. Math. Studies 53, Princeton University Press, 1963. | Zbl

[27] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. | MR | Zbl

[28] P. Olver, Applications of Lie Groups to Differential Equations, Springer Verlag, New-York, 1993. | MR | Zbl

[29] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1990.

[30] G. B. Whitham, Linear and Nonlinear Waves, J. Wiley & Sons, New York, 1980.

Cited by Sources: