Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schr\"odinger equations

TL;DR

This paper improves bilinear estimates for quadratic derivative nonlinear Schrödinger equations in 2D and 3D, leading to better well-posedness results in Sobolev spaces by applying a nonlinear Loomis-Whitney inequality.

Contribution

It introduces an improved bilinear estimate using nonlinear Loomis-Whitney inequality, enhancing well-posedness results for the system in lower regularity Sobolev spaces.

Findings

Well-posedness in H^s for s ≥ 1/2 in 2D

Well-posedness in H^s for s > 1/2 in 3D

Enhanced bilinear estimates for the system

Abstract

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.