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2007 Remarks on the Ostrovsky equation
Ibtissame Zaiter
Differential Integral Equations 20(7): 815-840 (2007).

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.

Citation

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Ibtissame Zaiter. "Remarks on the Ostrovsky equation." Differential Integral Equations 20 (7) 815 - 840, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35439
MathSciNet: MR2333658

Subjects:
Primary: 35Q53
Secondary: 76B15

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 7 • 2007
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