Long time behavior of solutions for a damped Benjamin-Ono equation

TL;DR

This paper studies the long-term behavior of solutions to a damped Benjamin-Ono equation on the torus, establishing global well-posedness, describing asymptotic limits, and proving boundedness of higher Sobolev norms using a nonlinear Fourier transform.

Contribution

It introduces a damping mechanism on Fourier modes and analyzes the resulting long-term dynamics, including weak and strong limit points, using the Birkhoff map as a nonlinear Fourier transform.

Findings

Global well-posedness in $L^2_{r,0}( ext{T})$

Weak limit points are also strong limit points

Boundedness of higher-order Sobolev norms

Abstract

We consider the Benjamin-Ono equation on the torus with an additional damping term on the smallest Fourier modes (cos and sin). We first prove global well-posedness of this equation in $L^2_{r,0}(\mathbb{T})$. Then, we describe the weak limit points of the trajectories in $L^2_{r,0}(\mathbb{T})$ when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin-Ono equation, that we use as an adapted nonlinear Fourier transform.