Bounded Solutions for the Degasperis-Procesi Equation
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 439-453.

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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.
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Coclite, Giuseppe Maria; Karlsen, Kenneth H. Bounded Solutions for the Degasperis-Procesi Equation. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 439-453. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a7/

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

[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. | DOI | MR | Zbl

[3] G. M. Coclite - H. Holden - K. H. Karlsen, Wellposedness for a parabolic-elliptic system, Discrete Contin. Dynam. Systems, 13 (2005), 659-682. | DOI | MR | Zbl

[4] G. M. Coclite - K. H. Karlsen, On the wellposedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[8] A. Degasperis - D. D. Holm - A. N. I. Khon, A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 170-183. | DOI | MR

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

[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. | DOI | MR | Zbl

[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. | DOI | MR

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[14] J. Escher - Z. Yin, On the initial boundary value problems for the Degasperis-Procesi equation, Phys. Lett. A, 368 (2007), 69-76. | DOI | MR | Zbl

[15] B. Fuchssteiner - A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. | DOI | MR | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

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

[21] Y. Liu - Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. | DOI | MR | Zbl

[22] H. Lundmark - J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems, 19 (2003), 1241-1245. | DOI | MR | Zbl

[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

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

[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

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

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

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

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

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