Global well-posedness in L 2 for the periodic Benjamin-Ono equation

Source abstract

We prove that the Benjamin-Ono equation is globally well-posed in $H^s({\Bbb T})$ for $s\ge 0$. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $H^s_0({\Bbb T})$, $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on ${H}_0^\infty({\Bbb T})$) cannot be of class $C^{1+\alpha}$, $\alpha>0$, from $H_0^s({\Bbb T})$ into $H_0^s({\Bbb T})$ as soon as $s< 0$.