The dispersion generalized Benjamin-Ono equation

TL;DR

This paper proves well-posedness for a family of dispersion generalized Benjamin-Ono equations in negative Sobolev spaces using advanced analytical techniques, extending classical results to broader functional settings.

Contribution

It introduces a novel combination of pseudodifferential gauge transform, paradifferential normal form, and variable coefficient Strichartz analysis in Sobolev spaces.

Findings

Well-posedness established in negative Sobolev spaces.

Results match classical well-posedness at Benjamin-Ono and KdV endpoints.

Extended the functional setting from $L^2$ to Sobolev spaces.

Abstract

We consider the well-posedness of the family of dispersion generalized Benjamin-Ono equations. Earlier work of Herr-Ionescu-Kenig-Koch established well-posedness with data in $L^2$, by using a discretized gauge transform in the setting of Bourgain spaces. In this article, we remain in the simpler functional setting of Sobolev spaces, and instead combine a pseudodifferential gauge transform, a paradifferential normal form, and a variable coefficient Strichartz analysis to establish well-posedness in negative-exponent Sobolev spaces. Our result coincides with the classical well-posedness results obtained at the Benjamin-Ono and KdV endpoints.