Sharp well-posedness for the $k$-dispersion generalized Benjamin-Ono equations: Short and long time results

TL;DR

This paper proves well-posedness and scattering results for the $k$-dispersion generalized Benjamin-Ono equations, covering various regimes and data regularities, and introduces new smoothing estimates.

Contribution

It establishes sharp local and global well-posedness results for $k$-DGBO equations, including the critical and subcritical cases, and extends existing results for the $k=3$ case.

Findings

Well-posedness in critical and subcritical Sobolev spaces

Scattering theory for small data

Global results for rough initial data

Abstract

We consider the $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations. For nonlinearities with power $k \geq 4$, we establish local and global well-posedness results for the associated initial value problem (IVP) in both the critical and subcritical regimes, addressing sharp regularity in homogeneous and inhomogeneous Sobolev spaces. Additionally, our method enables the formulation of a scattering criterion and a scattering theory for small data. We also investigate the case $k = 3$ via frequency-restricted estimates, obtaining local well-posedness results for the IVP associated with the $3$-DGBO equation and generalizing the existing results in the literature for the whole subcritical range. For higher dispersion, these local results can be extended globally even for rough data, particularly for initial data in Sobolev spaces with negative indices. As a byproduct, we derive new nonlinear smoothing estimates.