An improved bilinear estimate for Benjamin-Ono type equations

TL;DR

This paper establishes an improved bilinear estimate in Fourier spaces for Benjamin-Ono type equations, leading to enhanced local and global well-posedness results for the associated Cauchy problem.

Contribution

It introduces a new bilinear estimate in Fourier restriction spaces that extends well-posedness results for a range of lpha, including global solutions.

Findings

Local well-posedness for s > -3/4(lpha - 1)

Global well-posedness for s q 0

Extension of results to a broader class of lpha values

Abstract

A bilinear estimate in Fourier restriction norm spaces with applications to the Cauchy problem associated to u_t - |D|^{\alpha}u_x + uu_x =0 is proved, for 1< \alpha <2. As a consequence, local well-posedness in H^s(\R) \cap \dot{H}^{-\omega}(\R) follows for s >-{3/4}(\alpha-1) and \omega=1/\alpha-1/2. This extends to global well-posedness for all s \geq 0.