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Global well-posedness in L2 for the periodic Benjamin-Ono equation


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 $.