Remark on Well-posedness of Quadratic Schr\"odinger equation with nonlinearity $u\bar u$ in $H^{-1/4}(\R)$

TL;DR

This paper presents an alternative approach to establishing local well-posedness for the quadratic Schrödinger equation with a specific nonlinearity in the critical Sobolev space, using a novel resolution space inspired by previous work on KdV.

Contribution

It introduces a new method employing an $l^1$-analogue of $X^{s,b}$ spaces with a weaker low-frequency component, providing an alternative proof for the well-posedness result.

Findings

Successfully established local well-posedness in $H^{-1/4}$

Developed a new resolution space based on $l^1$-analogue of $X^{s,b}$

Extended techniques from KdV to Schrödinger equations

Abstract

In this remark, we give another approach to the local well-posedness of quadratic Schr\"odinger equation with nonlinearity $u\bar u$ in $H^{-1/4}$, which was already proved by Kishimoto \cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ space with low frequency part in a weaker space $L^{\infty}_{t}L^2_x$. Such type spaces was developed by Guo. \cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.