Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

TL;DR

This paper demonstrates that the flow map of the Benjamin-Ono equation on the real line is weakly continuous in L2(R), using local smoothing estimates, highlighting a key difference from the periodic case.

Contribution

It establishes weak continuity of the flow map in L2(R) for the Benjamin-Ono equation, extending understanding of its well-posedness in borderline spaces.

Findings

Weak continuity in L2(R) established

Contrast with non-continuity in L2(T)

Supports similar results for nonlinear Schrödinger equations

Abstract

In this paper we show that the floow map of the Benjamin-Ono equation on the line is weakly continuous in L2(R), using "local smoothing" estimates. L2(R) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet [27] has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L2(T). Our results are in line with previous work on the cubic nonlinear Schrodinger equation, where Goubet and Molinet [11] showed weak continuity in L2(R) and Molinet [28] showed lack of weak continuity in L2(T).