Sharp local well-posedness of the two-dimensional periodic nonlinear Schr\"odinger equation with a quadratic nonlinearity $|u|^2$

TL;DR

This paper proves sharp local well-posedness for the 2D quadratic nonlinear Schrödinger equation on the torus in $L^2$, resolving a 30-year-old open problem by establishing new bilinear and trilinear estimates.

Contribution

It introduces a novel bilinear estimate that handles non-resonant and nearly resonant interactions, leading to the first sharp local well-posedness result in $L^2$ for this equation.

Findings

Established local well-posedness in $L^2(\

Derived a tri-linear $L^3$-Strichartz estimate without derivative loss.

Resolved a long-standing open problem since Bourgain (1993).

Abstract

We study the nonlinear Schr\"odinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version of the $L^3$-Strichartz estimate without any derivative loss.