Differential and Integral Equations

The Cauchy problem for an integrable shallow-water equation

A. Alexandrou Himonas and Gerard Misiołek

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We prove that the periodic initial value problem for a completely integrable shallow-water equation is not locally well-posed for initial data in the Sobolev space $H^s(\mathbb{T})$ whenever $s <3/2$. Since on the other hand this problem is locally well-posed in the sense of Hadamard for $s>3/2$ our result suggests that $s=3/2$ is the critical Sobolev index for well-posedness. We also show that the nonperiodic initial value problem is not locally well-posed in $H^s(\mathbb{R})$ for $s <3/2$.

Article information

Differential Integral Equations, Volume 14, Number 7 (2001), 821-831.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B35: Stability 35R25: Improperly posed problems 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]


Himonas, A. Alexandrou; Misiołek, Gerard. The Cauchy problem for an integrable shallow-water equation. Differential Integral Equations 14 (2001), no. 7, 821--831. https://projecteuclid.org/euclid.die/1356123193

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