The Cauchy problem for the Ostrovsky equation with positive dispersion

TL;DR

This paper investigates the initial value problem for the Ostrovsky equation with positive dispersion, establishing local well-posedness in low regularity Sobolev spaces using bilinear and Strichartz estimates.

Contribution

It proves local well-posedness in $H^{-rac{3}{4}}$ and improves bilinear estimates using Strichartz estimates instead of calculus inequalities.

Findings

Proves local well-posedness in $H^{-rac{3}{4}}$

Reestablishes bilinear estimates via Strichartz estimates

Extends well-posedness to $H^{s}$ for $s > -rac{3}{4}$

Abstract

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -\gamma u=0, \end{eqnarray*} with positive $\beta$ and $\gamma $. This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in $H^{-\frac{3}{4}}(\R)$. Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy-Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in $H^{s}(\R)$ for $ s>-\frac{3}{4}$, with help of a fixed point argument.