Global Existence and Scattering of the Klein-Gordon-Zakharov System in Two Space Dimensions

TL;DR

This paper proves the global existence, optimal decay, and scattering of solutions for the Klein-Gordon-Zakharov system in two spatial dimensions, overcoming challenges posed by slow decay and nonlinearities.

Contribution

It establishes the first pointwise decay and scattering results for the system in two dimensions without initial data compactness assumptions.

Findings

Proved global existence of small solutions in 2D

Established optimal pointwise decay rates

Demonstrated linear scattering of the Klein-Gordon component

Abstract

We are interested in the Klein-Gordon-Zakharov system in $\mathbb{R}^{1+2}$, which is an important model in plasma physics with extensive mathematical studies. The system can be regarded as semilinear coupled wave and Klein-Gordon equations with nonlinearities violating the null conditions. Without the compactness assumptions on the initial data, we aim to establish the existence of small global solutions, and in addition, we want to illustrate the optimal pointwise decay of the solutions. Furthermore, we show that the Klein-Gordon part of the system enjoys linear scattering while the wave part has uniformly bounded low-order energy. None of these goals is easy because of the slow pointwise decay nature of the linear wave and Klein-Gordon components in $\mathbb{R}^{1+2}$. We tackle the difficulties by carefully exploiting the properties of the wave and the Klein-Gordon components, and by relying on the ghost weight energy estimates to close higher-order energy estimates. This appears to be the first pointwise decay result and the first scattering result for the Klein-Gordon-Zakharov system in $\mathbb{R}^{1+2}$ without compactness assumptions.