Scattering, Polyhomogeneity and Asymptotics for Quasilinear Wave Equations
From Past to Future Null Infinity
Series: Progress in Mathematical Physics;
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Product details:
- Publisher Springer Nature Switzerland
- Date of Publication 24 July 2026
- ISBN 9783032272553
- Binding Hardback
- No. of pages169 pages
- Size 235x155 mm
- Language English
- Illustrations XII, 169 p. 700
Categories
Long description:
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 â„“ 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 of 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.
Table of Contents:
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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 â–¡ηÏ• = 0.- Propagation of polyhomogeneity for â–¡ηÏ• = f.- Propagation of polyhomogeneity for perturbations of â–¡ηÏ• = 0 and applications.- Wave equations on Schwarzschild and the summing of the â„“-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.
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