Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions

TL;DR

This paper proves the global existence of small solutions for coupled wave and Klein-Gordon systems in three dimensions under weaker null conditions, broadening applicability to several physical models.

Contribution

It introduces a weaker null condition than previously used, enabling the analysis of more general coupled wave and Klein-Gordon systems.

Findings

Global existence of small solutions established

Applicable to Dirac-Klein-Gordon and related systems

Weaker null condition than Georgiev's is sufficient

Abstract

We consider the Cauchy problem for coupled systems of wave and Klein-Gordon equations with quadratic nonlinearity in three space dimensions. We show global existence of small amplitude solutions under certain condition including the null condition on self-interactions between wave equations. Our condition is much weaker than the strong null condition introduced by Georgiev for this kind of coupled system. Consequently our result is applicable to certain physical systems, such as the Dirac-Klein-Gordon equations, the Dirac-Proca equations, and the Klein-Gordon-Zakharov equations.