Sharp ill-posedness result for the periodic Benjamin-Ono equation

Abstract

We prove the discontinuity for the weak $ L^2(\T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(\T) $ as soon as $s<0$ and thus completes exactly the well-posedness result obtained by the author.

This paper has been withdrawn by the author since there were errors in the calculus of the defect coefficient in Page 11. The corrected calculus gives actually zero which do not lead to a contradiction on the continuity of the flow-map of the Benjamin-Ono equation. The author warmly thank Professor Patrick Gérard for having pointing out this error to him