Preprint A397/2005
Local and global well-posedness for the Ostrovsky equation

Aniura MilanĂ©s | Linares, Felipe

**Keywords: **
nonlinear dispersive equations

We consider the initial value problem for
\begin{equation}
\p_t u-\beta \p^3_xu-\gamma \partial_x^{-1} u+uu_x=0, \quad x,t\in\R,
\end{equation}
where $u$ is a real valued function, $\beta$ and $\gamma$ are real numbers such that
$\beta\cdot \gamma \ne 0$ and $\partial_x^{-1}f=((i\xi)^{-1}\widehat{f}(\xi))^{\vee}$.
This equation differs from Korteweg-de Vries equation in a nonlocal term. Nevertheless,
we obtained local well posedness in $X_s=\{f\in H^s(\R):\partial_x^{-1}f\in
L^2(\R)\}$,$\; s>3/4$, using techniques developed in \cite{kpv2}. For the case
$\beta\cdot \gamma >0$, we also obtain a global result in $X_1$, using appropriate
conservation laws.