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

Abstract

We are interested in the stability of a class of totally geodesic wave maps, as recently studied in [1, 7, 8]The relevant equations of motion are a system of coupled semilinear wave and Klein-Gordon equations in$\mathbb R ^{1+n}$ whose nonlinearities have critical time decay 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.