Low regularity solutions for a 2D quadratic non-linear Schrödinger equation

Abstract

We establish that the initial value problem for the quadratic non-linear Schrödinger equation $$i{u}_t-\Delta u=u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s({\mathbb{R}}^2)$ when $s>-1$. The critical exponent for this problem is $s_c=-1$ and previous work in \cite{c1} established local well-posedness for $s>-3/4$.