Scattering and stability properties of nonlinear wave equations on asymptotically flat spacetimes

This thesis is the amalgamation of projects focused on the nonlinear behaviour of hyperbolic partial differential equations (PDEs) in asymptotically flat spacetimes. Each project focuses on the behaviour of solutions nearby special ones and are therefore part of a perturbative understanding of these PDEs. Furthermore, all projects are connected by the search for a geometric understanding of the equations and the use of this geometry for the analysis. The first chapter is concerned with the global behaviour of effective field theories (EFT). These are approximate equations that are supposed to model the interaction of light and very heavy particles in the limiting case, where the degrees of freedom connected to the heavy particles are inactive. From a mathematical point of view, these present difficulties, as the equations are higher than second order in time and admit physically non-acceptable run away solutions. The main result is that for a certain coupled system of Klein-Gordon equations in the high mass limit, global solutions as well as the scattering matrix is well approximated by solutions of an EFT. The second chapter is motivated by the weak null condition of Lindblad-Rodnianski, which was introduced to classify systems regarding the stability of their trivial solution. This chapter presents a new heuristic approach to the study of systems of coupled semilinear wave equations in Minkowski space. The non rigorous analysis suggests a condition related to stability for systems of wave equations. The main result of this chapter is the validation of this heuristic for a number of system. In particular we exhibit a system satisfying the weak null condition, but failing our classification which forms singularities in finite time for arbitrary small data. The third and fourth chapters contain results for semilinear wave equations in Minkowski space admitting soliton solutions. In the former, we study the geometric scattering problem in a neighbourhood of timelike infinity around a single soliton and construct solutions admitting a prescribed polynomially decaying radiation field. This study is motivated by the understanding of multi soliton solutions, that is, solutions describing solitons moving away from one another. Indeed, using the technical tools developed in chapter 3, we construct approximate and exact multisoliton solutions for a number of semilinear wave equations in chapter 4. In chapter five we study the scattering problem for a large class of quasilinear system of equations in a neighbourhood of spacelike infinity. This work is based on a collaboration with Leonhard Kehrberger. This study is motivated by the understanding of gravitational radiation for the Einstein equations. We present robust estimates to propagate regularity to different regions of Minkowski space. We use these to classify perturbations that allow for the same estimates to be applied. The perturbations in particular contain the wave equation on Schwarzchild background and therefore allow for the summability of fixed angular mode analysis performed by Kehrberger.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Kadar, Istvan

Advisor dc:contributor.advisor

• Warnick, Claude

• Partial Differential Equations

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ca3eee55-6b59-407c-9c6f-479b09b64aba/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.112247

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/374033