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On the Cauchy problem of the Camassa-Holm equation. (English) Zbl 1222.35169
Summary: The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato [in: Spectral Theor. Differ. Equat., Proc. Symp. Dundee 1974, Lect. Notes Math. 448, 25–70 (1975; Zbl 0315.35077)] and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in ${L}^{p}$ frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B03 Existence, uniqueness, and regularity theory (fluid mechanics) 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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