Low regularity well-posedness for generalized Benjamin–Ono equations on the circle

Abstract

New low regularity well-posedness results for the generalized Benjamin–Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet–Ribaud established local well-posedness in [Formula: see text] via gauge transforms. We show local existence and a priori estimates in [Formula: see text], [Formula: see text], and local well-posedness in [Formula: see text], [Formula: see text] without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.