Stability of some two dimensional wave maps: a wave--Klein-Gordon model

TL;DR

This paper proves the global stability of certain two-dimensional wave maps modeled by coupled wave and Klein-Gordon equations, using energy methods to establish decay and regularity results.

Contribution

It introduces a pure energy method to demonstrate global existence and stability for a critical coupled wave-Klein-Gordon system in two dimensions.

Findings

Established global existence for the system in 2D

Derived uniform energy bounds at lower regularity levels

Obtained pointwise decay estimates for solutions

Abstract

We are interested in the stability of a class of totally geodesic wave maps, as recently studied by Abbrescia and Chen, and later by Duan and Ma. The relevant equations of motion are a system of coupled semilinear wave and Klein-Gordon equations in $\mathbb{R}^{1+n}$ whose nonlinearities are critical when $n=2$. In this paper we use a pure energy method to show global existence when $n=2$. By carefully examining the structure of the nonlinear terms, we are able to obtain uniform energy bounds at lower orders. This allows us to prove pointwise decay estimates and also to reduce the required regularity.