On the well-posedness of the Degasperis-Procesi equation

Journal of Functional Analysis

Volumn 233, Issue 1, 2006, Pages 60-91

  Coclite, Giuseppe M ^a,b   Karlsen, Kenneth H ^a  

^a UNIVERSITY OF OSLO   (Norway)

^b UNIVERSITY OF BARI   (Italy)

Author keywords

Discontinuous solution; Entropy condition; Existence; Hyperbolic equation; Integrable equation; Shallow water equation; Uniqueness; Weak solution

Indexed keywords

EID: 32544454510     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jfa.2005.07.008     Document Type: Article

References (39)

1. R. Beals D.H. Sattinger J. Szmigielski Acoustic scattering and the extended Korteweg-de Vries hierarchy Adv. in Math. 140 2 1998 190-206
2. R. Beals D.H. Sattinger J. Szmigielski Multi-peakons and a theorem of Stieltjes Inverse Problems 15 1 1999 L1-L4
3. R. Camassa D.D. Holm An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 11 1993 1661-1664
4. G.M. Coclite, H. Holden, K.H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., to appear.
5. G.M. Coclite K.H. Karlsen H. Holden Wellposedness for a parabolic-elliptic system Discrete Contin. Dyn. Systems 13 3 2005 659-682
6. A. Constantin Existence of permanent and breaking waves for a shallow water equation: A geometric approach Ann. Inst. Fourier (Grenoble) 50 2 2000 321-362
7. A. Constantin On the scattering problem for the Camassa-Holm equation R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 2008 2001 953-970
8. A. Constantin J. Escher Global existence and blow-up for a shallow water equation Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 2 1998 303-328
9. A. Constantin J. Escher Global weak solutions for a shallow water equation Indiana Univ. Math. J. 47 4 1998 1527-1545
10. A. Constantin J. Escher Wave breaking for nonlinear nonlocal shallow water equations Acta Math. 181 2 1998 229-243
11. A. Constantin H.P. McKean A shallow water equation on the circle Comm. Pure Appl. Math. 52 8 1999 949-982
12. A. Constantin L. Molinet Global weak solutions for a shallow water equation Comm. Math. Phys. 211 1 2000 45-61
13. A. Constantin W.A. Strauss Stability of peakons Comm. Pure Appl. Math. 53 5 2000 603-610
14. H.-H. Dai Y. Huo Solitary shock waves and other travelling waves in a general compressible hyperelastic rod R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 1994 2000 331-363
15. R. Danchin A few remarks on the Camassa-Holm equation Differential Integral Equations 14 8 2001 953-988
16. R. Danchin A note on well-posedness for Camassa-Holm equation J. Differential Equations 192 2 2003 429-444
17. A. Degasperis D.D. Holm A.N.W. Hone Integrable and non-integrable equations with peakons Nonlinear Physics: Theory and Experiment, II (Gallipoli, 2002) 2003 World Scientific Publishing River Edge, NJ 37-43
18. A. Degasperis D.D. Holm A.N.I. Khon A new integrable equation with peakon solutions Teoret. Mat. Fiz. 133 2 2002 170-183
19. A. Degasperis M. Procesi Asymptotic integrability Symmetry and perturbation theory (Rome, 1998) 1999 World Scientific Publishing River Edge, NJ 23-37
20. B. Fuchssteiner A.S. Fokas Symplectic structures their Bäcklund transformations and hereditary symmetries, Phys. D 4 1 1981/1982 47-66
21. A.A. Himonas G. Misiołek The Cauchy problem for an integrable shallow-water equation Differential Integral Equations 14 7 2001 821-831
22. D.D. Holm M.F. Staley Wave structure and nonlinear balances in a family of evolutionary PDEs SIAM J. Appl. Dyn. Systems 2 3 2003 323-380 (electronic)
23. R.S. Johnson Camassa-Holm Korteweg-de Vries and related models for water waves J. Fluid Mech. 455 2002 63-82
24. C.E. Kenig G. Ponce L. Vega Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle Comm. Pure Appl. Math. 46 4 1993 527-620
25. S.N. Kružkov First order quasi-linear equations in several independent variables Math. USSR-Sb. 10 2 1970 217-243
26. Y.A. Li P.J. Olver Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation J. Differential Equations 162 1 2000 27-63
27. Y.G. Lu Convergence of solutions to nonlinear dispersive equations without convexity conditions Appl. Anal. 31 4 1989 239-246
28. H. Lundmark J. Szmigielski Multi-peakon solutions of the Degasperis-Procesi equation Inverse Problems 19 6 2003 1241-1245
29. -1,q est compacte pour tout q<2 J. Math. Pures Appl. (9) 60 3 1981 309-322
30. O.G. Mustafa A note on the Degasperis-Procesi equation J. Nonlinear Math. Phys. 12 1 2005 10-14
31. G. Rodríguez-Blanco On the Cauchy problem for the Camassa-Holm equation Nonlinear Anal. Ser. A: Theory Methods 46 3 2001 309-327
32. M.E. Schonbek Convergence of solutions to nonlinear dispersive equations Comm. Partial Differential Equations 7 8 1982 959-1000
33. L. Tartar Compensated compactness and applications to partial differential equations Nonlinear Analysis and Mechanics: Heriot-Watt Symposium vol. IV 1979 Pitman Boston, MA 136-212
34. Z. Xin P. Zhang On the weak solutions to a shallow water equation Comm. Pure Appl. Math. 53 11 2000 1411-1433
35. Z. Xin P. Zhang On the uniqueness and large time behavior of the weak solutions to a shallow water equation Comm. Partial Differential Equations 27 9-10 2002 1815-1844
36. Z. Yin Global existence for a new periodic integrable equation J. Math. Anal. Appl. 283 1 2003 129-139
37. Z. Yin On the Cauchy problem for an integrable equation with peakon solutions Illinois J. Math. 47 3 2003 649-666
38. Z. Yin Global solutions to a new integrable equation with peakons Indiana Univ. Math. J. 53 2004 1189-1209
39. Z. Yin Global weak solutions for a new periodic integrable equation with peakon solutions J. Funct. Anal. 212 1 2004 182-194