Local well-posedness of the Benjamin-Ono equation with spatially quasiperiodic data

TL;DR

This paper proves local well-posedness for the Benjamin-Ono equation with quasiperiodic initial data, developing new Sobolev space properties and energy methods specific to quasiperiodic functions, and discusses challenges for global existence.

Contribution

It establishes local well-posedness in quasiperiodic Sobolev spaces and develops fundamental properties of these spaces for the Benjamin-Ono equation.

Findings

Local well-posedness in quasiperiodic Sobolev spaces

Development of Sobolev space properties for quasiperiodic functions

Conservation laws do not imply global control of Sobolev norms

Abstract

We consider the Benjamin-Ono equation in the spatially quasiperiodic setting. We establish local well-posedness of the initial value problem with initial data in quasiperiodic Sobolev spaces. This requires developing some of the fundamental properties of Sobolev spaces and the energy method for quasiperiodic functions. We discuss prospects for global existence. We demonstrate that while conservation laws still hold, these quantities no longer control the associated Sobolev norms, thereby preventing the establishment of global results by usual arguments.