Coclite, Karlsen: Bounded Solutions for the Degasperis-Procesi Equation

Coclite, Giuseppe Maria and Karlsen, Kenneth H.:
Bounded Solutions for the Degasperis-Procesi Equation
Bollettino dell'Unione Matematica Italiana Serie 9 1 (2008), fasc. n.2, p. 439-453, (English)
pdf (426 Kb), djvu (147 Kb). | MR 2424303 | Zbl 1164.35071

Sunto

This paper deals with the well-posedness in $L^{1} \cap L^{\infty}$ of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.

Referenze Bibliografiche

[1] R. CAMASSA - D. D. HOLM, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. | fulltext (doi) | MR 1234453 | Zbl 0972.35521

[2] G. M. COCLITE - H. HOLDEN - K. H. KARLSEN, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. | fulltext (doi) | MR 2192287 | Zbl 1100.35106

[3] G. M. COCLITE - H. HOLDEN - K. H. KARLSEN, Wellposedness for a parabolic-elliptic system, Discrete Contin. Dynam. Systems, 13 (2005), 659-682. | fulltext (doi) | MR 2152336 | Zbl 1082.35056

[4] G. M. COCLITE - K. H. KARLSEN, On the wellposedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. | fulltext (doi) | MR 2204675 | Zbl 1090.35142

[5] G. M. COCLITE - K. H. KARLSEN, On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Differential Equations, 233 (2007), 142-160. | fulltext (doi) | MR 2298968 | Zbl 1133.35028

[6] G. M. COCLITE - K. H. KARLSEN, A Semigroup of Solutions for the Degasperis-Procesi Equation, WASCOM 2005-13th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ (2006), 128-133. | fulltext (doi) | MR 2172828 | Zbl 1345.35123

[7] A. DEGASPERIS - D. D. HOLM - A. N. W. HONE, Integrable and non-integrable equations with peakons, Nonlinear physics: theory and experiment, II (Gallipoli, 2002), World Sci. Publishing, River Edge, NJ (2003), 37-43. | fulltext (doi) | MR 2028761 | Zbl 1053.37039

[8] A. DEGASPERIS - D. D. HOLM - A. N. I. KHON, A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 170-183. | fulltext (doi) | MR 2001531

[9] A. DEGASPERIS - M. PROCESI, Asymptotic integrability, Symmetry and perturbation theory (Rome, 1998), World Sci. Publishing, River Edge, NJ (1999), 23-37. | MR 1844104 | Zbl 0963.35167

[10] G. M. COCLITE - K. H. KARLSEN - N. H. RISEBRO, Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 80-105. | fulltext (doi) | MR 2387906 | Zbl 1246.76114

[11] 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), 331-363. | fulltext (doi) | MR 1811323

[12] J. ESCHER - Y. LIU - Z. YIN, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. | fulltext (doi) | MR 2271927 | Zbl 1126.35053

[13] J. ESCHER - Y. LIU - Z. YIN, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. | fulltext (doi) | MR 2305931 | Zbl 1124.35041

[14] J. ESCHER - Z. YIN, On the initial boundary value problems for the Degasperis-Procesi equation, Phys. Lett. A, 368 (2007), 69-76. | fulltext (doi) | MR 2328777 | Zbl 1209.35114

[15] B. FUCHSSTEINER - A. S. FOKAS, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. | fulltext (doi) | MR 636470 | Zbl 1194.37114

[16] H. A. HOEL, A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation, Electron. J. Diff. Eqns., 2007 (2007), 1-22. | fulltext EuDML | MR 2328701 | Zbl 1133.35430

[17] D. D. HOLM - M. F. STALEY, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2003 (2), 323-380. | fulltext (doi) | MR 2031278 | Zbl 1088.76531

[18] S. N. KRUZÏKOV, First order quasi-linear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243.

[19] C. E. KENIG - C. E. PONCE - 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. | fulltext (doi) | MR 1211741 | Zbl 0808.35128

[20] R. S. JOHNSON, Camassa-Holm, Korteweg-de Vries and related models for water waves., J. Fluid Mech., 455 (2002), 63-82. | fulltext (doi) | MR 1894796 | Zbl 1037.76006

[21] Y. LIU - Z. YIN, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. | fulltext (doi) | MR 2249792 | Zbl 1131.35074

[22] H. LUNDMARK - J. SZMIGIELSKI, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems, 19 (2003), 1241-1245. | fulltext (doi) | MR 2036528 | Zbl 1041.35090

[23] F. MURAT, L'injection du cône positif de $H^{-1}$ dans $W^{-1, q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. | MR 633007

[24] O. G. MUSTAFA, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. | fulltext (doi) | MR 2122861 | Zbl 1067.35078

[25] L. TARTAR, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Pitman, Boston, Mass., IV (1979), 136-212. | MR 584398

[26] M. E. SCHONBEK, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. | fulltext (doi) | MR 668586 | Zbl 0496.35058

[27] Z. YIN, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. | fulltext (doi) | MR 1994177 | Zbl 1033.35121

[28] Z. YIN, On the Cauchy problem for an integrable equation with peakon solutions., Illinois J. Math., 47 (2003), 649-666. | MR 2007229 | Zbl 1061.35142

[29] Z. YIN, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209. | fulltext (doi) | MR 2095454 | Zbl 1062.35094

[30] Z. YIN, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. | fulltext (doi) | MR 2065241 | Zbl 1059.35149

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche