The hyperboloidal foliation method

TL;DR

The paper introduces the hyperboloidal foliation method, a geometric approach to analyze nonlinear wave equations on curved spacetimes, providing global existence, energy bounds, and decay rates, applicable to various physical systems.

Contribution

It develops a unified, Lorentz-invariant framework using hyperboloidal hypersurfaces for studying nonlinear wave and Klein-Gordon equations, extending the analysis to complex systems.

Findings

Establishes global-in-time existence results for nonlinear wave systems.

Derives uniform energy bounds and optimal decay rates.

Unifies wave and Klein-Gordon equations within a geometric framework.

Abstract

The Hyperboloidal Foliation Method presented in this monograph is based on a (3+1)-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It allows us to establish global-in-time existence results for systems of nonlinear wave equations posed on a curved spacetime and to derive uniform energy bounds and optimal rates of decay in time. We are also able to encompass the wave equation and the Klein-Gordon equation in a unified framework and to establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a large class of nonlinear interactions. The hyperboloidal foliation of Minkowski spacetime we rely upon in this book has the advantage of being geometric in nature and, especially, invariant under Lorentz transformations. As stated, our theory applies to many systems arising in mathematical physics and involving a massive scalar field, such as the Dirac-Klein-Gordon system. As it provides uniform energy bounds and optimal rates of decay in time, our method appears to be very robust and should extend to even more general systems.