Differential and Integral Equations

A few remarks on the Camassa-Holm equation

Raphaël Danchin

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In the present paper, we use some standard a priori estimates for linear transport equations to prove the existence and uniqueness of solutions for the Camassa-Holm equation with minimal regularity assumptions on the initial data. We also derive some explosion criteria and a sharp estimate from below for the existence time. We finally address the question of global existence for certain initial data. This yields, among other things, a different proof for the existence and uniqueness of Constantin and Molinet's global weak solutions (see [9]).

Article information

Differential Integral Equations, Volume 14, Number 8 (2001), 953-988.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B40: Asymptotic behavior of solutions 35G25: Initial value problems for nonlinear higher-order equations 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]


Danchin, Raphaël. A few remarks on the Camassa-Holm equation. Differential Integral Equations 14 (2001), no. 8, 953--988. https://projecteuclid.org/euclid.die/1356123175

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