Global existence of small amplitude solutions to nonlinear coupled wave-Klein-Gordon systems in four space-time dimension with hyperboloidal foliation method

TL;DR

This paper introduces a hyperboloidal foliation energy method that simplifies proving global existence of small amplitude solutions for coupled wave-Klein-Gordon systems in four dimensions.

Contribution

It develops a new energy method based on hyperboloidal foliation, enabling simpler proofs of global existence for coupled wave-Klein-Gordon systems.

Findings

Established global existence of solutions using the new method.

Simplified the analysis compared to classical approaches.

Extended results to general coupled systems.

Abstract

In this article one will develop a new type of energy method based on a foliation of spacetime into hyperboloidal hypersurfaces . As we will see, with this method, some classical results such as global existence and almost global existence of regular solutions to the quasi-linear wave equations and Klein-Gordon equations will be established in a much simpler and much more natural way. Most importantly, the global existence of regular solutions to a general type of coupled quasilinear wave-Klein-Gordon system will be established. All of this suggests that compared with the classical method, this hyperboloidal foliation of space-time may be a more natural way to regard the wave operator.