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, u0 L1 < ∞ and u0 − u0xx does not change sign, we prove that the solution stays analytic globally in time.
LOMBARDO, M.C., SAMMARTINO, M., SCIACCA, V. (2005). A note on the analytic solutions of the Camassa-Holm equation. COMPTES RENDUS MATHÉMATIQUE, 341(11), 659-664 [10.1016/j.crma.2005.10.006].
A note on the analytic solutions of the Camassa-Holm equation
LOMBARDO, Maria Carmela;SAMMARTINO, Marco Maria Luigi;SCIACCA, Vincenzo
2005-01-01
Abstract
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, u0 L1 < ∞ and u0 − u0xx does not change sign, we prove that the solution stays analytic globally in time.
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