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2004 The Cauchy problem for a modified Camassa-Holm equation with analytic initial data
Jennifer M. Gorsky
Differential Integral Equations 17(11-12): 1233-1254 (2004).

Abstract

We show that the periodic Cauchy problem for a modified Camassa-Holm equation with analytic initial data is analytic in the space variable $x$ for time near zero. By differentiating the equation and the initial condition with respect to $x$ we obtain a sequence of initial-value problems of KdV-type equations. These, written in the form of integral equations, define a mapping on a Banach space whose elements are sequences of functions equipped with a norm expressing the Cauchy estimates in terms of the KdV norms of the components introduced in the works of Bourgain, Kenig, Ponce, Vega, and others. By proving appropriate bilinear estimates we show that this mapping is a contraction, and therefore we obtain a solution whose derivatives in the space variable satisfy the Cauchy estimates.

Citation

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Jennifer M. Gorsky. "The Cauchy problem for a modified Camassa-Holm equation with analytic initial data." Differential Integral Equations 17 (11-12) 1233 - 1254, 2004.

Information

Published: 2004
First available in Project Euclid: 21 December 2012

zbMATH: 1150.35542
MathSciNet: MR2100024

Subjects:
Primary: 35Q53
Secondary: 35B10 , 35B65 , 76D05

Rights: Copyright © 2004 Khayyam Publishing, Inc.

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Vol.17 • No. 11-12 • 2004
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