Stabilization of Dispersion Generalized Benjamin Ono

TL;DR

This paper investigates the well-posedness and stabilization of the dispersion-generalized Benjamin-Ono equation on a periodic domain, introducing a novel dissipation-normalized Bourgain space to achieve smoothing and stabilization results.

Contribution

It develops a new dissipation-normalized Bourgain space to analyze smoothing effects and establish stabilization for the dispersion-generalized Benjamin-Ono equation.

Findings

Established $L^2$ well-posedness

Proved stabilization for small initial data

Developed a bilinear estimate for the nonlinearity

Abstract

In this article, we examine $L^2$ well-posedness and stabilization property of the dispersion-generalized Benjamin-Ono equation with periodic boundary conditions. The main ingredient of our proof is a development of dissipation-normalized Bourgain space, which gains smoothing properties simultaneously from dissipation and dispersion within the equation. We will establish a bilinear estimate for the derivative nonlinearity using this space and prove the linear observability inequality leading to small-data stabilization.