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

E. Compaan

doi:10.3934/cpaa.2019112

Article Contents

2019, Volume 18, Issue 5: 2473-2489. Doi: 10.3934/cpaa.2019112

This issue Previous Article Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian Next Article Large deviations for stochastic 3D Leray-

$ \alpha $

model with fractional dissipation

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

E. Compaan

Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA 02139, USA

Received: May 31, 2018

Revised: December 31, 2018

Published: March 2019

The first author is supported by NSF MSPRF #1704865

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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.

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Mathematics Subject Classification: Primary: 35Q55, 35Q40.

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