论文标题
二维周期性非线性schrödinger方程的尖锐局部良好性，具有二次非线性$ | u |^2 $
Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity $|u|^2$
论文作者
论文摘要
我们使用二维非线性$ | u |^2 $研究了非线性schrödinger方程（NLS），在二维圆环$ \ mathbb {t}^2 $上构成。虽然相关的$ l^3 $ -strichartz估计仅以衍生损失而闻名，但我们证明了$ l^2（\ Mathbb {t}^2）$中二次NLS的本地良好性，因此自Bourgain（Bourgain（1993）以来，我们就解决了三十年以来的开放问题。鉴于sobolev空间不足的不良性，该结果是锐利的。我们通过单独研究非谐振和几乎共鸣的病例来建立至关重要的双线性估计值。作为推论，我们获得了$ l^3 $ -strichartz估算的三线性版本，而不会造成任何衍生损失。
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.
