论文标题
对于二维二维非线性schrödinger方程的凯奇问题的尖锐适应性
Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity
论文作者
论文摘要
本文与$ \ Mathbb {r} \ times \ times \ Mathbb {r}^2 $中的二次非线性schrödinger方程有关的库奇问题与非线性$η| u | u |^2 $ where $ | u |^2 $中的$ | u |如果$ s <-1/4 $，则sobolev space $ h^{s}（\ mathbb {r}^2）$中的不适性会导致。我们将通过在初始数据上假设一些角度规律性，以$ -1/2 <s <-1/4 $在$ h^s（\ mathbb {r}^2）$中证明良好的态度。关键工具是经过修改的傅立叶限制规范和在增厚的超曲面上的卷积估计。
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.
