On the flow map of the Benjamin-Ono equation on the torus

Patrick Gerard, Thomas Kappeler, Peter Topalov

Published 2019-09-16Version 1

We prove that for any $t\in\mathbb{R}$, the flow map $\mathcal{S}^t$ of the Benjamin-Ono equation on the torus continuously extends to the Sobolev space $H^{-s}_{r,0}$ for any $0<s<1/2$, but does not do so to $H^{-s}_{r,0}$ for $s>1/2$. Furthermore, we show that $\mathcal{S}^t$ is sequentially weakly continuous on $H^{-s}_{r,0}$ for any $0\le s<1/2$. Note that $- 1/2$ is the critical Sobolev exponent of the Benjamin-Ono equation.