The Benjamin-Ono equation with quasi-periodic data

TL;DR

This paper proves local well-posedness of the Benjamin-Ono equation with a broader class of quasi-periodic initial data, using advanced harmonic analysis techniques and gauge transforms.

Contribution

It extends local well-posedness results to more general quasi-periodic functions and introduces new a-priori estimates leveraging recent decoupling methods.

Findings

Unique solutions depend continuously on initial data.

Broader class of quasi-periodic functions covered.

New local well-posedness results in anisotropic Sobolev spaces.

Abstract

We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic functions than previous theorems. Central to the argument is an a-priori estimate, the proof of which utilizes Strichartz estimates for quasi-periodic functions obtained recently via decoupling, and a quasi-periodic extension of Tao's gauge transform. As a byproduct of our method, we also establish new local wellposedness results in certain anisotropic Sobolev spaces.