Physical Space Proof of Bilinear Estimates and Applications to Nonlinear Dispersive Equations

TL;DR

This paper presents a simplified proof for the local well-posedness of certain nonlinear dispersive equations using bilinear estimates derived from a novel div-curl lemma, enhancing understanding of their solutions.

Contribution

It introduces a new div-curl lemma to establish bilinear estimates, simplifying the proof of well-posedness for modified KdV and Benjamin-Ono equations.

Findings

Simplified proof of local well-posedness in low regularity spaces

New bilinear estimate based on div-curl lemma

Applicable to modified KdV and Benjamin-Ono equations

Abstract

We give a simpler proof for the local well-posedness of the modified Korteweg-de Vries equations and modified Benjamin-Ono equation in $H^{\frac{1}{4}}(\mathbb{R})$ and $H^{\frac{1}{2}}(\mathbb{R})$, respectively. The proof is based on the Strichartz estimate, dyadic decomposition and a bilinear estimate given by a new type of div-curl lemma.