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

Abstract

In this remark, we give another approach to the local well-posedness of quadratic Schrödinger 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 $L_t^{\infty }{L}_x^2$. Such type spaces was developed by Guo. \cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.

arXiv comment5 pages