A note on global existence for the Zakharov system on $\mathbb{T}$

TL;DR

This paper proves global well-posedness of the one-dimensional periodic Zakharov system in low-regularity Fourier-Lebesgue spaces, combining the I-method and Bourgain's high-low decomposition, with probabilistic results in Sobolev spaces.

Contribution

It introduces a novel approach to establish global existence for the Zakharov system in low-regularity spaces, extending previous results to broader function classes.

Findings

Global well-posedness in Fourier-Lebesgue spaces.

Probabilistic global existence in L^2-based Sobolev spaces.

Sharpness of results in H^{1/2+} imes L^2.

Abstract

We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in $L^2$-based Sobolev spaces. We also obtain global well-posedness in $H^{\frac12+} \times L^2$, which is sharp (up to endpoints) in the class of $L^2$-based Sobolev spaces.