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

Ruoyuan Liu; Tadahiro Oh

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arXiv · 2203.15389

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

Ruoyuan Liu·Tadahiro Oh

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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.

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Ruoyuan Liu, Tadahiro Oh. 2022-03-29. Sharp local well-posedness of the two-dimensional periodic nonlinear Schr\"odinger equation with a quadratic nonlinearity $|u|^2$. https://arxiv.org/abs/2203.15389

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