Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity $|u{|}^2$

Abstract

We study the nonlinear Schrödinger 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.

arXiv comment16 pages. To appear in Math. Res. Lett