Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data

TL;DR

This paper proves the global existence of small amplitude solutions for a quadratic quasi-linear coupled wave-Klein-Gordon system in two dimensions with mildly decaying initial data, using energy estimates and normal form techniques.

Contribution

It introduces a novel approach combining quasi-linear normal forms and semi-classical micro-local analysis to establish global solutions in low-dimensional coupled wave-Klein-Gordon systems.

Findings

Global existence of small solutions established

Energy estimates with Klainerman vector fields derived

New coupled system approach for decay estimates developed

Abstract

The aim of this paper is to study the global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small, smooth and mildly decaying at infinity. Some physical models strictly related to general relativity have shown the importance of studying such systems, but very few results are know at present in low space dimension. We study here a model two-dimensional system, in which the non-linearity writes in terms of 'null forms', and show the global existence of small solutions. Our goal is to prove some energy estimates on the solution when a certain number of Klainerman vector fields is acting on it, and some optimal uniform estimates. The former ones are obtained using systematically quasi-linear normal forms, in their para-differential version; the latter ones are recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system, by means of a semi-classical micro-local analysis of the problem. We expect the strategy developed here to be robust enough to enable us, in the future, to treat the case of the most general non-linearities.