Remarks on the Ostrovsky equation

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.