Preprint A397/2005
Local and global well-posedness for the Ostrovsky equation
Aniura Milanés | 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.