Sharp well-posedness results of the Benjamin-Ono equation in $H^s(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution

We prove that the Benjamin–Ono equation on the torus is globally in time well-posed in the Sobolev space $H^s(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$andill-posedfor$s \le - 1/2$\mathrm{.}HencethecriticalSobolevexponent$s_c=-1/2$ of the Benjamin–Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^s(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$\mathrm{.}NovelconservationlawsandanonlinearFouriertransformon$H^{s}($\mathbb{T}$,$\mathbb{R}$)$with$s > - 1/2$ are key ingredients into the proofs of these results.