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

TL;DR

This paper introduces a novel physical space approach using bilinear estimates and a div-curl lemma to establish low-regularity well-posedness for nonlinear dispersive equations, bypassing Bourgain spaces.

Contribution

It proposes an alternative method based on physical space bilinear estimates and a div-curl lemma, providing a new tool for analyzing dispersive PDEs.

Findings

Reproduces best known local well-posedness results for 2d and 3d Zakharov systems.

Develops a new div-curl type lemma for bilinear estimates in physical space.

Demonstrates the effectiveness of the approach without relying on Bourgain spaces.

Abstract

The work by Kenig-Ponce-Vega [15] initiated the use of Bourgain spaces to study the low-regularity well-posedness of semilinear dispersive equations. Since then, the Bourgain space method has become the dominant, and almost the only method to deal with this problem. The goal of this series of papers is to propose an alternative approach for this problem that does not rely on Bourgain spaces. Our method is based on a bilinear estimate, which is proved in a physical space approach by a new div-curl type lemma introduced by the third author. Combining these ingredients with a Strichartz estimate of mixed spatial integrability, we will illustrate our method in the present paper by reproducing best known local well-posedness results for the 2d and 3d Zakharov system from Bejenaru-Herr-Holmer-Tataru [2] and Bejenaru-Herr [1].