Sharp Well-posedness for the Benjamin Equation

TL;DR

This paper establishes sharp well-posedness results for the Benjamin equation, extending local solutions to the critical regularity level and proving global well-posedness for a broad range of initial data.

Contribution

It proves the sharp threshold for well-posedness of the Benjamin equation, extending local results to the critical regularity and establishing global well-posedness.

Findings

Well-posedness extends to the critical regularity s = -3/4.

Global well-posedness holds for s in [-3/4, ∞).

Ill-posedness occurs for s < -3/4.

Abstract

Having the ill-posedness in the range $s<-3/4$ of the Cauchy problem for the Benjamin equation with an initial $H^{s}({\mathbb R})$ data, we prove that the already-established local well-posedness in the range $s>-3/4$ of this initial value problem is extendable to $s=-3/4$ but also that such a well-posed property is globally valid for $s\in [-3/4,\infty)$.