Scattering for Defocusing generalized Benjamin-Ono Equation in the Energy Space

TL;DR

This paper proves that solutions to the defocusing generalized Benjamin-Ono equation in the energy space scatter, using a monotonicity formula and extending local theory to the subcritical regime.

Contribution

It establishes the first large data scattering result in the energy space for the defocusing generalized Benjamin-Ono equation, introducing a monotonicity formula and extending local theory.

Findings

Proved scattering in the energy space $H^{1/2}( )$ for the defocusing gBO.

Extended critical local theory to the subcritical regime.

Established global well-posedness in the energy space.

Abstract

We prove the scattering for the defocusing generalized Benjamin-Ono equation in the energy space $H^{\frac{1}{2}}(\mathbb{R})$. We first establish the monotonicity formula that describes the unidirectional propagation. More precisely, it says that the center of energy moves faster than the center of mass. This type of monotonicity was first observed by Tao (DCDS 2007) in the defocusing gKdV equations. We use the monotonicity in the setting of compactness-contradiction argument to prove the large data scattering in the energy space $H^{\frac{1}{2}}(\mathbb{R})$. On the way, we extend critical local theory of Vento (IMRN 2010) to the subcritical regime. Indeed, we obtain subcritical local theory and global well-posedness in the energy space.