[Solutions analytiques de l'équation de Camassa–Holm]
On étude ici l'équation de Camassa–Holm dans les espaces des fonctions analytiques. On montre que, si les données initiales sont analytiques, il existe, localement dans le temp, une solution unique analytique.
En outre si la la donnée initiale analytique
In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to
Accepté le :
Publié le :
Maria Carmela Lombardo 1 ; Marco Sammartino 1 ; Vincenzo Sciacca 1
@article{CRMATH_2005__341_11_659_0, author = {Maria Carmela Lombardo and Marco Sammartino and Vincenzo Sciacca}, title = {A {Note} on the analytic solutions of the {Camassa{\textendash}Holm} equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {659--664}, publisher = {Elsevier}, volume = {341}, number = {11}, year = {2005}, doi = {10.1016/j.crma.2005.10.006}, language = {en}, }
TY - JOUR AU - Maria Carmela Lombardo AU - Marco Sammartino AU - Vincenzo Sciacca TI - A Note on the analytic solutions of the Camassa–Holm equation JO - Comptes Rendus. Mathématique PY - 2005 SP - 659 EP - 664 VL - 341 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2005.10.006 LA - en ID - CRMATH_2005__341_11_659_0 ER -
Maria Carmela Lombardo; Marco Sammartino; Vincenzo Sciacca. A Note on the analytic solutions of the Camassa–Holm equation. Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 659-664. doi : 10.1016/j.crma.2005.10.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.10.006/
[1] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993) no. 11, pp. 1661-1664
[2] On the Cauchy problem for the periodic Camassa–Holm equation, J. Differential Equations, Volume 141 (1997) no. 2, pp. 218-235
[3] On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., Volume 10 (2000), pp. 391-399
[4] Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume XXVI (1998), pp. 303-328
[5] Well-posedness, Global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., Volume 51 (1998), pp. 475-504
[6] Global weak solutions for a shallow water equation, Indiana Univ. Math. J., Volume 47 (1998) no. 4, pp. 1527-1545
[7] A few remarks on the Camassa–Holm equation, J. Differential Integral Equations, Volume 14 (2001), pp. 953-988
[8] A note on well-posedness for Camassa–Holm equation, J. Differential Equations, Volume 192 (2003), pp. 429-444
[9] Regularity of solutions and the convergence of the Galerkin method in the Ginzburg–Landau equation, Numer. Funct. Anal. Optim., Volume 14 (1993), pp. 299-321
[10] Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal., Volume 87 (1989), pp. 359-369
[11] Symplectic structures, their bäcklund transformations and hereditary symmetries, Physica D, Volume 4 (1981), pp. 47-66
[12] The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, Volume 14 (2001), pp. 821-831
[13] Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., Volume 327 (2003), pp. 575-584
[14] Analyticity of solutions for a generalized Euler equation, J. Differential Equations, Volume 133 (1997), pp. 321-339
[15] Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, Volume 162 (2000), pp. 27-63
[16] Classical solutions of the periodic Camassa–Holm equation, Geom. Funct. Anal., Volume 12 (2002), pp. 1080-1104
[17] On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal., Volume 46 (2001), pp. 309-327
[18] An abstract Cauchy–Kovalevskaya thm in a weighted Banach space, Comm. Pure Appl. Math., Volume XLVIII (1995), pp. 629-637
- Radius of analyticity for the Camassa–Holm equation on the line, Nonlinear Analysis, Volume 174 (2018), p. 1 | DOI:10.1016/j.na.2018.04.007
- PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION, Journal of Applied Analysis Computation, Volume 7 (2017) no. 1, p. 278 | DOI:10.11948/2017019
- Global analyticity for a generalized Camassa–Holm equation and decay of the radius of spatial analyticity, Journal of Differential Equations, Volume 263 (2017) no. 1, p. 732 | DOI:10.1016/j.jde.2017.02.052
- Wave breaking and persistent decay of solution to a shallow water wave equation, Discrete and Continuous Dynamical Systems - Series S, Volume 9 (2016) no. 6, p. 2149 | DOI:10.3934/dcdss.2016089
- Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations, Communications in Nonlinear Science and Numerical Simulation, Volume 19 (2014) no. 6, p. 1746 | DOI:10.1016/j.cnsns.2013.10.029
- Analytic Solutions and Singularity Formation for the Peakon b-Family Equations, Acta Applicandae Mathematicae (2012) | DOI:10.1007/s10440-012-9753-8
Cité par 6 documents. Sources : Crossref
⁎ Work supported by the PRIN grant “Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media”.
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier