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

2022/03/29 by Liu, Ruoyuan, Oh, Tadahiro
#35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.15389

Abstract

We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity |u|², posed on the two-dimensional torus \mathbbT². While the relevant L³-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in L²(\mathbbT²), 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 L³-Strichartz estimate without any derivative loss.

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