Resonant decomposition and the $I$-method for the two-dimensional Zakharov system

TL;DR

This paper develops a novel approach combining resonant decomposition and the $I$-method to establish global well-posedness for the 2D Zakharov system on tori and improves existing results on $\

Contribution

It introduces a new method that extends the $I$-method with resonant decomposition to the 2D Zakharov system, achieving results for weaker initial data.

Findings

Proves global well-posedness for small $L^2$ initial data on 2D tori.

Extends the $I$-method with resonant decomposition to the Zakharov system.

Improves the known results for the initial value problem on $\

Abstract

The initial value problem of the Zakharov system on two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some 'resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).