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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.'
address: 'Department of Mathematics, Massachusetts Institute of Technology'
author:
- 'E. Compaan'
title: 'A Note on Global Existence for the Zakharov System on ${\mathbb{T}}$'
---

Introduction & Statement of Results
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In this note, we consider the periodic Zakharov system $$\begin{cases}
iu_t + \Delta u = nu, \qquad x \in{\mathbb{T}}, \; t \in {\mathbb{R}}, \\
n_{tt}-\Delta n = \Delta |u|^2,\\
u(x,0) = u_0(x),\\
n(x,0) = n_0(x), \quad n_t(x,0) = n_1(x).
\end{cases}$$ This system was introduced in the 1970s as a model of Langmuir turbulence in ionized plasma [@Zak]. The function $u$, the Schrödinger part, represents the envelope of a oscillating electric field, while the wave part $n$ represents the deviation from the mean of the ion density. The purpose of this note is to derive a global well-posedness result that holds for 