no. 3
Periodic conservative solutions of the Camassa–Holm equation
[Solutions périodiques conservatives de l’équation de Camassa–Holm]
Holden, Helge1 ; Raynaud, Xavier2
1 Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway) University of Oslo Centre of Mathematics for Applications P.O. Box 1053, Blindern 0316 Oslo (Norway)
2 Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway)
Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 945-988.

Nous montrons que l’équation de Camassa–Holm périodique u t -u xxt +3uu x -2uxu xx -uu xxx =0 possède un semi-groupe continu de solutions globales pour des conditions initiales u| t=0 dans H per 1 . Le résultat est obtenu en utilisant un changement de variable où l’équation est réécrite en variables lagrangiennes. Pour décrire les solutions, il est nécessaire d’introduire la densité d’énergie donnée par la mesure de Radon positive μ qui satisfait μ ac =(u 2 +u x 2 )dx. L’énergie totale est préservée par la solution.

We show that the periodic Camassa–Holm equation u t -u xxt +3uu x -2u x u xx -uu xxx =0 possesses a global continuous semigroup of weak conservative solutions for initial data u| t=0 in H per 1 . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with μ ac =(u 2 +u x 2 )dx. The total energy is preserved by the solution.

DOI : 10.5802/aif.2375
Classification : 65M06, 65M12, 35B10, 35Q53
Keywords: Camassa–Holm equation, periodic solution
Mot clés : équation de Camassa–Holm, solutions périodiques

Holden, Helge 1 ; Raynaud, Xavier 2

1 Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway) University of Oslo Centre of Mathematics for Applications P.O. Box 1053, Blindern 0316 Oslo (Norway)
2 Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway) 
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Holden, Helge; Raynaud, Xavier. Periodic conservative solutions of the Camassa–Holm equation. Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 945-988. doi : 10.5802/aif.2375. https://aif.centre-mersenne.org/articles/10.5802/aif.2375/

[1] Abraham, R.; Marsden, J. E.; Ratiu, T. Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988 (2nd ed) | MR | Zbl

[2] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000 | MR | Zbl

[3] Arnold, V. I.; Khesin, B. A. Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998 | MR | Zbl

[4] Beals, R.; Sattinger, D. H.; Szmigielski, J. Multi-peakons and a theorem of Stieltjes, Inverse Problems, Volume 15 (1999), p. L1-L4 | DOI | MR | Zbl

[5] Bressan, A.; Constantin, A. Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal., Volume 183 (2007), pp. 215-239 | DOI | MR | Zbl

[6] Bressan, A.; Fonte, M. An optimal transportation metric for solutions of the Camassa–Holm equation, Methods Appl. Anal., Volume 12 (2005), pp. 191-220 | MR

[7] Camassa, R.; Holm, D. D. An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993), pp. 1661-1664 | DOI | MR | Zbl

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

[9] Coclite, G. M.; Holden, H.; Karlsen, K. H. Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., Volume 37 (2005), pp. 1044-1069 | DOI | MR | Zbl

[10] Coclite, G. M.; Holden, H.; Karlsen, K. H. Well-posedness for a parabolic-elliptic system, Discrete Cont. Dynam. Systems, Volume 13 (2005), pp. 659-682 | DOI | MR | Zbl

[11] Constantin, A. On the Cauchy problem for the periodic Camassa–Holm equation, J. Differential Equations, Volume 141 (1997), pp. 218-235 | DOI | MR | Zbl

[12] Constantin, A. On the inverse spectral problem for the Camassa–Holm equation, J. Funct. Anal., Volume 155 (1998), pp. 352-363 | DOI | MR | Zbl

[13] Constantin, A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier, Grenoble, Volume 50 (2000), pp. 321-362 | DOI | Numdam | MR | Zbl

[14] Constantin, A.; Escher, J. Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., Volume 51 (1998), pp. 475-504 | DOI | MR | Zbl

[15] Constantin, A.; Escher, J. On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., Volume 233 (2000), pp. 75-91 | DOI | MR | Zbl

[16] Constantin, A.; Kolev, B. On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, Volume 35 (2002), p. R51-R79 | DOI | MR | Zbl

[17] Constantin, A.; Kolev, B. Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., Volume 78 (2003), pp. 787-804 | DOI | MR | Zbl

[18] Constantin, A.; Kolev, B. Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci., Volume 16 (2006), pp. 109-122 | DOI | MR

[19] Constantin, A.; McKean, H. A shallow water equation on the circle, Comm. Pure Appl. Math., Volume 52 (1999), pp. 949-982 | DOI | MR | Zbl

[20] Constantin, A.; Molinet, L. Global weak solutions for a shallow water equation, Comm. Math. Phys., Volume 211 (2000), pp. 45-61 | DOI | MR | Zbl

[21] Constantin, A.; Strauss, W. A. Stability of peakons, Comm. Pure Appl. Math., Volume 53 (2000), pp. 603-610 | DOI | MR | Zbl

[22] Folland, G. B. Real Analysis, John Wiley & Sons Inc., New York, 1999 (2nd ed.) | MR | Zbl

[23] Fonte, M. Conservative solution of the Camassa–Holm equation on the real line, 2005 (3, arXiv:math.AP/0511549)

[24] Holden, H.; Karlsen, K. H.; Risebro, N. H. Convergent difference schemes for the Hunter–Saxton equation, Math. Comp., Volume 76 (2007), pp. 699-744 | DOI | MR | Zbl

[25] Holden, H.; Raynaud, X. Convergence of a finite difference scheme for the Camassa–Holm equation, SIAM J. Numer. Anal., Volume 44 (2006), pp. 1655-1680 | DOI | MR | Zbl

[26] Holden, H.; Raynaud, X. A convergent numerical scheme for the Camassa–Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., Volume 14 (2006), pp. 505-523 | MR | Zbl

[27] Holden, H.; Raynaud, X. Global conservative multipeakon solutions of the Camassa–Holm equation, J. Hyperbolic Differ. Equ., Volume 4 (2007), pp. 39-64 | DOI | MR | Zbl

[28] Holden, H.; Raynaud, X. Global conservative solutions of the Camassa–Holm equation — a Lagrangian point of view, Comm. Partial Differential Equations, Volume 32 (2007), pp. 1511-1549 | DOI | MR

[29] Holden, H.; Raynaud, X. Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, Volume 233 (2007), pp. 448-484 | DOI | MR | Zbl

[30] Johnson, R. S. Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech., Volume 455 (2002), pp. 63-82 | DOI | MR | Zbl

[31] Misiołek, G. A shallow water equation as a geodesic flow on the Bott–Virasoro group, J. Geom. Phys., Volume 24 (1998), pp. 203-208 | DOI | Zbl

[32] Misiołek, G. Classical solutions of the periodic Camassa–Holm equation., Geom. Funct. Anal., Volume 12 (2002), pp. 1080-1104 | DOI

[33] Yosida, K. Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995 Reprint of the sixth (1980) edition | MR

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