A Sharp Bilinear Estimate for the Bourgain-type Space with Application to the Benjamin Equation

TL;DR

This paper establishes a precise bilinear estimate within Bourgain-type spaces and applies it to determine the optimal local well-posedness or ill-posedness of the Benjamin equation's Cauchy problem.

Contribution

It provides the first sharp bilinear estimate for Bourgain-type spaces and uses it to analyze the well/ill-posedness of the Benjamin equation.

Findings

Established a sharp bilinear estimate for Bourgain-type spaces.

Determined the optimal local well-posedness threshold for the Benjamin equation.

Identified conditions leading to ill-posedness in the Cauchy problem.

Abstract

This note shows the existence of a sharp bilinear estimate for the Bourgain-type space and gives its application to the optimal local well/ill-posedness of the Cauchy problem for the Benjamin equation.