Differential and Integral Equations

Remarks on the Ostrovsky equation

Ibtissame Zaiter

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Abstract

The main result of this paper concerns the limit of the solution of the Ostrovsky equation as the rotation parameter $\gamma $ goes to zero. We are interested also in the ill-posedness of the Cauchy problem associated with this equation. First, using a compactness method, we show that the initial-value problem of Ostrovsky equation is locally well-posed in $H^s(\mathbb R)$ for $s>3/4$. The compactness method is essentially used to prove that the solution of the Ostrovsky equation converges to that of the Korteweg-de Vries equation, as $\gamma $ tends to zero, locally in time, in $ H^s(\mathbb R)$ for $ s>3/4$. Thanks to some conservation laws and estimates, we will prove a persistence property of the solutions. Therefore, we show the convergence of the solutions in $ L^{\infty}_{loc}(\mathbb R, H^s(\mathbb R))$ for $ s \geq 3/4$. In the case of positive dispersion, we gain a strong convergence in $ C(\mathbb R, H^1(\mathbb R))$. The last section is devoted to studying the ill-posedness of the Cauchy problem associated with the Ostrovsky equation.

Article information

Source
Differential Integral Equations, Volume 20, Number 7 (2007), 815-840.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039411

Mathematical Reviews number (MathSciNet)
MR2333658

Zentralblatt MATH identifier
1212.35439

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

Citation

Zaiter, Ibtissame. Remarks on the Ostrovsky equation. Differential Integral Equations 20 (2007), no. 7, 815--840. https://projecteuclid.org/euclid.die/1356039411


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