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

TL;DR

This paper establishes new low regularity local well-posedness results for generalized Benjamin-Ono equations on the circle, using advanced Fourier methods to handle derivative loss and extending previous results to lower regularity spaces.

Contribution

It introduces a novel approach combining short-time Fourier transform restriction and modified energies to prove well-posedness without gauge transforms at lower regularities.

Findings

Local existence and a priori estimates in H^s for s > 1/2

Local well-posedness in H^s for s ≥ 3/4

Global existence for quartic nonlinearity with small initial data

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 $H^{1}(\mathbb{T},\mathbb{R})$ via gauge transforms. We show local existence and a priori estimates in $H^{s}(\mathbb{T},\mathbb{R})$, $s>1/2$, and local well-posedness in $H^{s}(\mathbb{T},\mathbb{R})$, $s\geq3/4$ without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.