Periodic conservative solutions of the Camassa-holm equation

Annales de l'Institut Fourier

Volumn 58, Issue 3, 2008, Pages 945-988

  Holden, Helge ^a,b   Raynaud, Xavier ^a  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b UNIVERSITY OF OSLO   (Norway)

Author keywords

Camassa Holm equation; Periodic solution

Indexed keywords

EID: 47649124291     PISSN: 03730956     EISSN: None     Source Type: Journal    
DOI: 10.5802/aif.2375     Document Type: Article

References (33)

1. R. ABRAHAM, J. E. MARSDEN & T. RATIU, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988, 2nd ed.
2. L. AMBROSIO, N. FUSCO & D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
3. V. I. ARNOLD & B. A. KHESIN, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998.
4. R. BEALS, D. H. SATTINGER & J. SZMIGIELSKI, "Multi-peakons and a theorem of Stieltjes", Inverse Problems 15 (1999), p. L1-L4.
5. A. BRESSAN & A. CONSTANTIN, "Global conservative solutions of the Camassa-Holm equation", Arch. Ration. Mech. Anal. 183 (2007), p. 215-239.
6. A. BRESSAN & M. FONTE, "An optimal transportation metric for solutions of the Camassa-Holm equation", Methods Appl. Anal. 12 (2005), p. 191-220.
7. R. CAMASSA & D. D. HOLM, "An integrable shallow water equation with peaked solitons", Phys. Rev. Lett. 71 (1993), p. 1661-1664.
8. R. CAMASSA, D. D. HOLM & J. HYMAN, "A new integrable shallow water equation", Adv. Appl. Mech. 31 (1994), p. 1-33.
9. G. M. COCLITE, H. HOLDEN & K. H. KARLSEN, "Global weak solutions to a generalized hyperelastic-rod wave equation", SIAM J. Math. Anal. 37 (2005), p. 1044-1069.
10. _, "Well-posedness for a parabolic-elliptic system", Discrete Cont. Dynam. Systems 13 (2005), p. 659-682.
11. A. CONSTANTIN, "On the Cauchy problem for the periodic Camassa-Holm equation", J. Differential Equations 141 (1997), p. 218-235.
12. _, "On the inverse spectral problem for the Camassa-Holm equation", J. Funct. Anal. 155 (1998), p. 352-363.
13. _, "Existence of permanent and breaking waves for a shallow water equation: a geometric approach", Ann. Inst. Fourier, Grenoble 50 (2000), p. 321-362.
14. A. CONSTANTIN & J. ESCHER, "Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation", Comm. Pure Appl. Math. 51 (1998), p. 475-504.
15. _, "On the blow-up rate and the blow-up set of breaking waves for a shallow water equation", Math. Z. 233 (2000), p. 75-91.
16. A. CONSTANTIN & B. KOLEV, "On the geometric approach to the motion of inertial mechanical systems", J. Phys. A 35 (2002), p. R51-R79.
17. _, "Geodesic flow on the diffeomorphism group of the circle", Comment. Math. Helv. 78 (2003), p. 787-804.
18. _, "Integrability of invariant metrics on the diffeomorphism group of the circle", J. Nonlinear Sci. 16 (2006), p. 109-122.
19. A. CONSTANTIN & H. MCKEAN, "A shallow water equation on the circle", Comm. Pure Appl. Math. 52 (1999), p. 949-982.
20. A. CONSTANTIN & L. MOLINET, "Global weak solutions for a shallow water equation", Comm. Math. Phys. 211 (2000), p. 45-61.
21. A. CONSTANTIN & W. A. STRAUSS, "Stability of peakons", Comm. Pure Appl. Math. 53 (2000), p. 603-610.
22. G. B. FOLLAND, Real Analysis, John Wiley & Sons Inc., New York, 1999, 2nd ed.
23. M. FONTE, "Conservative solution of the Camassa-Holm equation on the real line", 2005, 3, arXiv.math.AP/0511549.
24. H. HOLDEN, K. H. KARLSEN & N. H. RISEBRO, "Convergent difference schemes for the Hunter-Saxton equation", Math. Comp. 76 (2007), p. 699-744.
25. H. HOLDEN & X. RAYNAUD, "Convergence of a finite difference scheme for the Camassa-Holm equation", SIAM J. Numer. Anal. 44 (2006), p. 1655-1680.
26. _, "A convergent numerical scheme for the Camassa-Holm equation based on multipeakons", Discrete Contin. Dyn. Syst. 14 (2006), p. 505-523.
27. _, "Global conservative multipeakon solutions of the Camassa-Holm equation", J. Hyperbolic Differ. Equ. 4 (2007), p. 39-64.
28. _, "Global conservative solutions of the Camassa-Holm equation - a Lagrangian point of view", Comm. Partial Differential Equations 32 (2007), p. 1511-1549.
29. _, "Global conservative solutions of the generalized hyperelastic-rod wave equation", J. Differential Equations 233 (2007), p. 448-484.
30. R. S. JOHNSON, "Camassa-Holm, Korteweg-de Vries and related models for water waves", J. Fluid Mech. 455 (2002), p. 63-82.
31. G. MISIOLEK, "A shallow water equation as a geodesic flow on the Bott-Virasoro group", J. Geom. Phys. 24 (1998), p. 203-208.
32. _, "Classical solutions of the periodic Camassa-Holm equation.", Geom. Funct. Anal. 12 (2002), p. 1080-1104.
33. K. YOSIDA, Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the sixth (1980) edition.