The Zakharov System on the Upper Half-Plane

TL;DR

The paper addresses the two-dimensional Zakharov system on the upper half-plane with non-homogeneous boundary data, establishing local well-posedness at low regularity by combining the restricted norm method with a Fourier–Laplace extension. It introduces boundary-adapted spaces $\mathcal{H}^s_S$, $\mathcal{H}^s_W$ and linear solution operators $S_0^t$, $W_0^t$, deriving sharp $X^{s,b}$-type and Kato smoothing estimates to control the nonlinear interactions. A key contribution is showing that the nonlinear part gains additional regularity, yielding a smoothing effect in $u$ and $n$ relative to the initial/boundary data. This is the first result demonstrating low-regularity local well-posedness for the 2d Zakharov system on a half-plane, expanding the IBVP theory for dispersive PDEs on domains with boundary.

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.

Theorems & Definitions (46)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof