This monograph develops a semi global scattering theory for a broad class of quasilinear wave equations in a neighbourhood of spacelike infinity, including both past and future null infinity. Scattering data are prescribed on an ingoing null cone and at past null infinity.The authors establish weighted, optimal in decay energy estimates and prove the propagation of polyhomogeneous asymptotics from past to future null infinity. They further introduce an explicit algorithm for computing the coefficients in the resulting expansions and apply it to several linear and nonlinear models. A key consequence is the summability in the spherical harmonic index of fixed mode estimates previously obtained in the series The Case Against Smooth Null Infinity. The framework extends beyond finite energy solutions and applies directly to systems such as the Einstein vacuum equations in harmonic gauge. A novel ansatz accommodating the stronger than Schwarzschildean divergence o
f light cones enables the treatment of slowly decaying data, thereby enlarging the regime of known stability results for Minkowski space in harmonic gauge.This book is intended for researchers and graduate students in partial differential equations, mathematical relativity, and geometric analysis who seek a precise and versatile framework for understanding asymptotics near null and spacelike infinity.
Introduction and setup.- Discussion of a toy model problem.- Definitions, Preliminaries and Notation.- ODE Lemmata.- Energy estimates for the finite problem.- Scattering theory for perturbations of = 0.- Propagation of polyhomogeneity for = f.- Propagation of polyhomogeneity for perturbations of = 0 and applications.- Wave equations on Schwarzschild and the summing of the -modes.- The specificity of peeling to even spacetime dimensions and asymptotics for the scale-invariant wave equation.- The no incoming radiation condition on Cauchy data.- Scattering theory for general quasilinear perturbations.- Analysis of the Einstein vacuum equations in harmonic gauge.
Istvan Kadar is a mathematician working in the field of analysis, with a focus on geometric singular analysis. His research emphasises the use of geometric methods to study analytic problem. He has contributed to the development of techniques at the interface of analysis and geometry, with applications to problems arising in various different areas of mathematical physics. Outside mathematics, he is a keen explorer of curvature in the form of cycling.Lionor Kehrberger is a mathematician working in the field of partial differential equations, with a particular focus on problems arising in general relativity. Lionor has contributed to the both physically motivated and mathematically rigorous understanding of the asymptotic structure of gravitational radiation near infinity, in particular having written a series of papers titled "The Case Against Smooth Null Infinity". Just like Istvan, Lionor enjoys cycling, though Lionor prefers slightly rougher terrain.
