Series In Applied And Computational Mathematics: Hyperboloidal Foliation Method, The (ePub)

The “Hyperboloidal Foliation Method” introduced in this monograph is based on a (3 + 1) foliation of Minkowski spacetime by hyperboloidal hypersurfaces. This method allows the authors to establish global-in-time existence results for systems of nonlinear wave equations posed on a curved spacetime. It also allows to encompass the wave equation and the Klein-Gordon equation in a unified framework and, consequently, to establish a well-posedness theory for a broad class of systems of nonlinear wave-Klein-Gordon equations. This book requires certain natural (null) conditions on nonlinear interactions, which are much less restrictive that the ones assumed in the existing literature. This theory applies to systems arising in mathematical physics involving a massive scalar field, such as the Dirac-Klein-Gordon systems. Contents:IntroductionThe Hyperboloidal Foliation and the Bootstrap StrategyDecompositions and Estimates for the CommutatorsThe Null Structure in the Semi-Hyperboloidal FrameSobolev and Hardy Inequalities on HyperboloidsRevisiting Scalar Wave EquationsFundamental L∞ and L2 EstimatesSecond-Order Dervatives of the Wave ComponentsNull Forms and Decay in TimeL2 Estimate on the Interaction TermsThe Local Well-Posedness TheoryBibliographyReadership: Graduate students and researchers in analysis and differential equations. This book could serve as part of graduate course on nonlinear wave equations.