Title: The Zakharov System on the Upper Half-Plane Show Chinese title

Title: Zakharov系统在上半平面上

Abstract: In this paper we study the Zakharov system on the upper half--plane $U=\{(x ,y)\in \R^2: y>0\}$ with non-homogenous boundary conditions. In particular we obtain low regularity local well--posedness using the restricted norm method of Bourgain and the Fourier--Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data. To our knowledge this is the first paper which establishes low regularity results for the 2d initial-boundary value Zakharov system. Show Chinese abstract

Abstract: 本文研究了上半平面$U=\{(x ,y)\in \R^2: y>0\}$上具有非齐次边界条件的 Zakharov 系统。特别地，我们利用 Bourgain 的受限范数方法和求解初边值问题的 Fourier-Laplace 方法，得到了低正则性局部适定性结果。此外，我们证明了解的非线性部分属于比初始数据更光滑的空间。据我们所知，这是第一篇建立二维初边值 Zakharov 系统低正则性结果的论文。