论文标题
$ l^2（\ mathbb r^2）$的全球良好和散射方程
Global well-posedness and scattering for the Dysthe equation in $L^2(\mathbb R^2)$
论文作者
论文摘要
本文着重于dysthe方程，该方程是调制（Schrödinger）制度和无限深度情况下水波系统的高阶近似值。我们首先回顾了dysthe和相关方程的推导。然后，我们研究初步问题。我们证明了一个小的数据全球供应良好和散射导致关键空间$ l^2（\ mathbb r^2）$。鉴于流程图不能在$ l^2（\ Mathbb r^2）$以下连续$ C^3 $连续的事实。我们的分析取决于在傅立叶限制规范方法的背景下的线性和双线性strichartz估计。此外，由于我们处于临界水平，因此我们需要在原子空间$ u^2_s $及其双$ v^2_s $的框架的框架中工作。我们还证明，初始值问题在$ h^s（\ mathbb r^2）$，$ s> 0 $中本地供应良好。我们的结果扩展到了Dysthe方程的有限深度版本。
This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extend to the finite depth version of the Dysthe equation.
