Sharp Well-posedness Results for the Generalized Benjamin-ono Equation with High Nonlinearity

We establish the local well-posedness of the generalized BenjaminOno equation ∂tu+H∂ xu±u ∂xu = 0 in Hs(R), s > 1/2−1/k for k ≥ 12 and without smallness assumption on the initial data. The condition s > 1/2−1/k is known to be sharp since the solution map u0 7→ u is not of class Ck+1 on Hs(R) for s < 1/2 − 1/k. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill-posedness in Hs(R), s < 1/3.