The zero dispersion limit for the Benjamin--Ono equation on the line

TL;DR

This paper characterizes the zero dispersion limit of the Benjamin--Ono equation on the line for various initial data, revealing properties like maximum principle and local smoothing, and analyzing special cases including characteristic functions.

Contribution

It provides an explicit description of the zero dispersion limit for the Benjamin--Ono equation and explores its properties and special cases, including the absence of semigroup behavior.

Findings

Established maximum principle and local smoothing for the limit

Derived explicit formulas for the zero dispersion limit

Proved the lack of semigroup property in a specific case

Abstract

We identify the zero dispersion limit of a solution of the Benjamin--Ono equation on the line corresponding to every initial datum in $L^2(\R)\cap L^\infty(\R )$. We infer a maximum principle and a local smoothing property for this limit. The proof is based on an explicit formula for the Benjamin--Ono equation and on the combination of calculations in the special case of rational initial data, with approximation arguments. We also investigate the special case of an initial datum equal to the characteristic function of a finite interval, and prove the lack of semigroup property for this zero dispersion limit.