Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

Abstract

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R}\times {\mathbb{R}}^2$ with the nonlinearity $\eta |u{|}^2$ where $\eta \in \mathbb{C}\setminus \{0\}$ and low regularity initial data. If $s<-1/4$, the ill-posedness result in the Sobolev space $H^s({\mathbb{R}}^2)$ is known. We will prove the well-posedness in $H^s({\mathbb{R}}^2)$ for $-1/2<s<-1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.

35 pages