---
abstract: 'We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-disperion limit, and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by Lax and Levermore to analyze the corresponding limit for the Korteweg-de Vries equation.'
address: |
    Department of Mathematics\
    University of Michigan\
    East Hall\
    530 Church St.\
    Ann Arbor, MI 48109
author:
- 'Peter D. Miller'
- Zhengjie Xu
title: 'On the Zero-Dispersion Limit of the Benjamin-Ono Cauchy Problem for Positive Initial Data'
---

=1

Introduction {#sec:intro}
============

The Benjamin-Ono (BO) equation $$\label{BOE} \frac{\partial {u_\epsilon}}{\partial t}+2{u_\epsilon}\frac{\partial{u_\epsilon}}{\partial x}+
\epsilon \mathcal{H}\left[\frac{\partial^2{u_\epsilon}}{\partial x^2}\right]=0,
\quad x\in\mathbb{R},\quad t>0,$$ where $\epsilon>0$ is a constant and $\mathcal{H}$ is the Hilbert transform operator defined by the Cauchy principal value integral $$\label{BOE operator H}
\mathcal{H}[f](x):=\frac{1}{\pi}\dashint_\mathbb{R}\frac{f(y)}{y-x}\,dy$$ is a model for weakly nonlinear dispersive waves on the interface between two ideal immiscible fluids, one of which may be considered to be infinitely deep. Applications include the modeling of internal waves in deep water [@Benjamin; @1967; @Davis; @1967; @Ono; @1975; @Choi; @1996], and also the modeling of atmospheric waves like the dramatic “morning glory” phenomenon of northeastern Australia [@Porter; @2002]. The relevant Cauchy problem 