A Note on the analytic solutions of the Camassa–Holm equation

Maria Carmela Lombardo; Marco Sammartino; Vincenzo Sciacca

doi:10.1016/j.crma.2005.10.006

Partial Differential Equations

A Note on the analytic solutions of the Camassa–Holm equation
[Solutions analytiques de l'équation de Camassa–Holm]

Présenté par : Yvonne Choquet-Bruhat

Maria Carmela Lombardo ¹ ; Marco Sammartino ¹ ; Vincenzo Sciacca ¹

¹ Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, Italy

Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 659-664.

Résumé
Abstract

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 $u_{0} (x)$ est bornée dans $L^{1}$ , appartient à $H^{s} (R)$ $s > 3 / 2$ et satisfait la condition $u_{0} - u_{0 x x} ⩾ 0$ , la solution résulte analytique globalement dans le temp.

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 $H^{s} (R)$ with $s > 3 / 2$ , $‖ u_{0} ‖_{L^{1}} < \infty$ and $u_{0} - u_{0 x x}$ does not change sign, we prove that the solution stays analytic globally in time.

Reçu le : 2005-08-11
Accepté le : 2005-10-11
Publié le : 2005-11-08

DOI : 10.1016/j.crma.2005.10.006

Affiliations des auteurs :

Maria Carmela Lombardo ¹ ; Marco Sammartino ¹ ; Vincenzo Sciacca ¹

¹ Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, Italy

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     title = {A {Note} on the analytic solutions of the {Camassa{\textendash}Holm} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {659--664},
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     doi = {10.1016/j.crma.2005.10.006},
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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/

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Cité par Sources :

^⁎ Work supported by the PRIN grant “Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media”.

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