On the global well-posedness and scattering of the 3D Klein-Gordon-Zakharov system

TL;DR

This paper proves the global existence and scattering of solutions for the 3D Klein-Gordon-Zakharov system with small initial data, using novel analytical techniques to handle the lack of scaling symmetry.

Contribution

It introduces a modified Alinhac's ghost weight method and a new normal-form estimate to establish global well-posedness without compactness assumptions.

Findings

Energy remains uniformly bounded globally

Established scattering for small initial data

Removed the need for compactness assumptions

Abstract

In this paper we are interested in the global well-posedness of the 3D Klein-Gordon-Zakharov equations with small initial data. We show the uniform boundedness of the energy for the global solution without any compactness assumptions on the initial data. The main novelty of our proof is to apply a modified Alinhac's ghost weight method together with a newly developed normal-form type estimate to remedy the lack of the space-time scaling vector field; moreover, we give a clear description of the smallness conditions on the initial data.