The Stability of Hyperbolic PDEs in String Theory, Particle Physics and Cosmology

Abstract

In this thesis we study hyperbolic PDEs arising from general relativity and the standard model of particle physics. In particular we prove the asymptotic stability of special solutions of these PDEs against small initial data perturbations. The study of stability elucidates our understanding of whether such PDEs can provide mathematically reasonable models for physical phenomena in our universe.

In the first chapter, we prove the stability of a system of quasilinear wave equations satisfying the weak null condition. In particular, we prove that the Kaluza-Klein spacetime, the cartesian product of Minkowski spacetime with a circle, viewed as a solution to the vacuum Einstein equations, is stable to circle-independent perturbations. In the second chapter, we show that the Milne spacetime is a stable solution to the Einstein-Klein-Gordon equations. We upgrade a technique that uses the continuity equation complementary to L^2 estimates to control massive matter fields. In contrast to earlier applications of this idea we require a correction to the energy density to obtain sufficiently strong pointwise bounds. In the third chapter, we use the hyperboloidal foliation method to study an interesting PDE system relevant in particular physics. In particular we establish the stability of the ground state of the U(1) standard model of electroweak interactions. In particular, we investigate here the Dirac equation and consider a new energy functional for this field defined with respect to the hyperboloidal foliation of Minkowski spacetime. We provide a novel decay result for the Dirac equation which is uniform in the mass coefficient, and thus allows for the Dirac mass coefficient to be arbitrarily small. In the final chapter, we bring together ideas developed in the first three chapters and prove the stability, with respect to the evolution determined by the vacuum Einstein equations, of the Cartesian product of high-dimensional Minkowski space with a compact Riemannian manifold admitting nonzero parallel spinors. Such a product includes the example of special holonomy compactifications, which play a central role in string theory.