Differential and Integral Equations

The Cauchy problem for a modified Camassa-Holm equation with analytic initial data

Jennifer M. Gorsky

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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.

Article information

Source
Differential Integral Equations Volume 17, Number 11-12 (2004), 1233-1254.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060243

Mathematical Reviews number (MathSciNet)
MR2100024

Zentralblatt MATH identifier
1150.35542

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B10: Periodic solutions 35B65: Smoothness and regularity of solutions 76D05: Navier-Stokes equations [See also 35Q30]

Citation

Gorsky, Jennifer M. The Cauchy problem for a modified Camassa-Holm equation with analytic initial data. Differential Integral Equations 17 (2004), no. 11-12, 1233--1254. https://projecteuclid.org/euclid.die/1356060243.


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