Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schr\"{o}dinger equation with angular regularity

Abstract

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger 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.