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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2005-08-11
Accepted: 2005-10-11
Published online: 2005-11-08

DOI: 10.1016/j.crma.2005.10.006

Author's affiliations:

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

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