Well-posedness results for the generalized Benjamin-Ono equation with arbitrary large initial data

We prove new local well-posedness results for the generalized Benjamin-Ono equation (GBO) ∂tu+ℋ∂x2u+uk∂xu=0, k ≥ 2. By combining a gauge transformation with dispersive estimates, we establish the local well-posedness of GBO in Hs(ℝ) for s ≥ 1/2 if k ≥ 5, s > 1/2 if k=2,4, and s≥ 3/4 if k=3. Moreover, we prove that in all these cases, the flow map is locally Lipschitz on Hs(ℝ).