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

Abstract

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $s\ge 0$. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\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^\infty(\T) $) cannot be of class $C^{1+\alpha }$, $\alpha >0$, from $ {\dot H}^s(\T) $ into $ {\dot H}^s(\T) $ as soon as $s<0$.

47 pages