Regularity properties of the Zakharov system on the half line

TL;DR

This paper investigates the regularity, smoothing effects, and well-posedness of the Zakharov system on the half line, extending known results to rough initial data and analyzing the behavior of solutions over time.

Contribution

It establishes local and global well-posedness, smoothing effects, and Sobolev norm growth for the Zakharov system on the half line with rough initial data, including boundary considerations.

Findings

Smoothing effect matches that of periodic and real line cases.

Global existence and uniqueness of energy solutions are proved.

Higher Sobolev norms grow at most polynomially in time.

Abstract

In this paper we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. Smoothing means that the nonlinear part of the solution on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic Zakharov and the Zakharov on the real line. Uniqueness is proved in the class of smooth solutions. When the boundary value of the Schr\"odinger part of the solution is zero, uniqueness can be extended to the full range of local solutions. Under the same assumptions on the initial data we also prove global-in-time existence and uniqueness of energy solutions. For more regular data we prove that all higher Sobolev norms grow at most polynomially-in-time.