Low-regularity solutions of the periodic Camassa-Holm equation

Communications in Partial Differential Equations

Volumn 32, Issue 1, 2007, Pages 87-126

  De Lellis, Camillo ^a   Kappeler, Thomas ^a   Topalov, Peter ^a  

^a UNIVERSITY OF ZURICH   (Switzerland)

Author keywords

Camassa Holm equation; Low regularity solutions; Periodic peakons

Indexed keywords

EID: 33847159961     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300601091470     Document Type: Article

References (42)

1. Adams, R. (1975). Sobolev Spaces. Pure Appl. Math. Vol. 65. Academic Press.
2. Arnold, V. (1966). Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits. Ann. Inst. Fourier 16(1):319-361.
3. Arnold, V., Khesin, B. (1998). Topological Methods in Hydrodynamics. Springer.
4. Beals, R., Sattinger, D., Szmigielski, J. (1998). Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140:190-206.
5. Beals, R., Sattinger, D., Szmigielski, J. (2000). Multipeakons and the classical moment problem. Adv. Math. 154:229-257.
6. Bressan, A., Constantin, A. (2005). Global conservative solutions of the Camassa-Holm equation. http://www.math.ntnu.no/conservation/2005/016.html
7. Bressan, A., Fonte, M. (2005). An optimal transportation metric for solutions of the Camassa-Holm equation. Methods Appl. Anal. 12(2):191-219.
8. Camassa, R., Holm, D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1661-1664.
9. Camassa, R., Holm, D., Hyman, J. (1994). A new integrable shallow water equation. Advances in Applied Mechanics 31:1-33.
10. Coclite, G., Holden, H., Karlsen, K. H. (2005). Global weak solutions to a generalized hyperelastic-rod wave equation. SIAM J. Math. Anal. 37(4):1044-1069.
11. Constantin, A. (2000). Existence of permanent and breaking waves for a shallow water equation: a geometrical approach. Annales de l'Institut Fourier 50:321-362.
12. Constantin, A., Escher, J. (1998a). Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51:475-504.
13. Constantin, A., Escher, J. (1998b). Global existence and blow-up for a shallow water equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 26:303-328.
14. Constantin, A., Escher, J. (1998c). Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181:229-243.
15. Constantin, A., McKean, H. (1999). A shallow water equation on the circle. Comm. Pure Appl. Math. 52:949-982.
16. Constantin, A., Molinet, L. (2000). Global weak solutions for a shallow water equation. Comm. Math. Phys. 211:45-61.
17. Constantin, A., Strauss, W. (2000). Stability of peakons. Comm. Pure Appl. Math. 53:603-610.
18. Constantin, A., Kolev, B. (2002). On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35:51-79.
19. Constantin, A., Kolev, B. (2003). Geodesic flows on the diffeomorphism group of the circle. Comment. Math. Helv. 78:787-804.
20. Danchin, R. (2001). A few remarks on the Camassa-Holm equation. Differential and Integral Equations 14:953-988.
21. Danchin, R. (2003). A note on well-posedness for Camassa-Holm equation. J. Diff. Eq. 192:429-444.
22. Ebin, D., Marsden, J. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92:102-163.
23. Evans, L. C., Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press.
24. Fokas, A., Fuchssteiner, B. (1981). Symplectic structures, their Bäcklund transformation and heriditary symmetries. Physica D. Nonlinear Phenomena 4:47-66.
25. Himonas, A., Misiolek, G. (2000). Well-posedness of the Cauchy problem for a shallow water equation on the circle. J. Diff. Eq. 162:479-495.
26. Holm, D., Kouranbaeva, S., Marsden, J., Ratiu, T., Shkoller, S. (1999). A nonlinear analysis of the averaged Euler equations. Arnold Festschrift Proceedings. Communications of the Fields Institute.
27. Khesin, B., Misiolek, G. (2003). Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176:116-144.
28. Kouranbaeva, S. (1999). The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40:857-868.
29. Lang, S. (1972). Differential Manifolds. Addison-Wesley Series in Mathematics.
30. Li, Y., Olver, O. (2000). Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162:27-63.
31. McKean, H. (1998). Breakdown of shallow water equation. Mikio Sato: a great Japanese mathematician of the twentieth century. Asian J. Math. 2:867-874.
32. McKean, H. (2000). Shallow Water & Diffemorphism Group. Applied and Industrial Mathematics. Venice-2, 1998. Dordrecht: Kluwer Acad. Publ., pp. 135-143.
33. Michor, P., Ratiu, T. (1998). On the geometry of the Virasoro-Bott group. Journal of Lie Theory 8:293-309.
34. Misiolek, G. (1998). A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24:203-208.
35. Misiolek, G. (2002). Classical solutions of the periodic Camassa-Holm equation. GAFA 12:1080-1104.
36. Ovsienko, V., Khesin, B. (1987). Korteweg-de Vries superequations as an Euler equation. Functional Anal. Appl. 21:81-82.
37. Rodriguez-Blanco, G. (2001). On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. TMA 46:309-327.
38. Shilov, G., Gurevich, B. (1977). Integral, measure, and derivative: a unified approach. Revised English Edition. Dover.
39. 1 metric and mean hydrodynamics. J. Funct. Anal. 160:337-365.
40. Taylor, M. (1996). Partial Differential Equations III. Nonlinear Equations. Applied Mathematical Sciences 117. Springer-Verlag.
41. Taylor, M. (2000). Tools for PDE, pseudodifferential operators, paradifferential operators, and layer potentials. Amer. Math. Soc.
42. Xin, Z., Zhong, P. (2000). On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53:1411-1433.