Random data Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance

Abstract

This article is concerned with the Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance. By randomizing the initial data, we prove that the Cauchy problem is almost surely locally well-posed in $H^s(\mathbb{R}^d)$ for $d \ge 5$ and $\frac{d-4}{d-3}s_c \lt s \lt s_c$, where $s_c := \frac{d}{2} - 2$ is the scaling critical regularity.