On the low regularity phase space of the Benjamin-Ono equation

TL;DR

This paper establishes the global well-posedness of the Benjamin-Ono equation in a low regularity phase space with logarithmic weights, extending the understanding of its solution behavior at minimal regularity.

Contribution

It proves global well-posedness of the Benjamin-Ono equation in a new low regularity space with logarithmic weights, identifying a maximal phase space for solutions.

Findings

Global $C^0$-well-posedness in $H^{-1/2, ootlog}$

Maximal low regularity phase space identified

Extension of solution theory to minimal regularity

Abstract

In this paper we prove that the Benjamin-Ono equation is globally in time $C^0$-well-posed in the Hilbert space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ of periodic distributions in $H^{-1/2}(\mathbb{T},\mathbb{R})$ with $\sqrt{\log}$-weights. The space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ can thus be considered as a maximal low regularity phase space for the Benjamin-Ono equation corresponding to the scale $H^s(\mathbb{T},\mathbb{R})$, $s>-1/2$.