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

TL;DR

This paper demonstrates that the periodic Benjamin-Ono equation is ill-posed in negative Sobolev spaces by proving the flow map's discontinuity in weak L^2 topology, completing the understanding of its well-posedness.

Contribution

It establishes the ill-posedness of the periodic Benjamin-Ono equation in H^s for s<0, confirming the limits of its well-posedness.

Findings

Flow map discontinuity in weak L^2 topology

Ill-posedness in H^s for s<0

Completes the well-posedness characterization

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.