Soliton dynamics for Einstein field equations

Abstract

This thesis is devoted to the study of hyperbolic Partial Differential Equations arising from General Relativity Theory. Given the complexity of the Einstein equations, it is often a good choice to study a question of interest in the framework of a restricted class of solutions. One way to impose such restrictions is to consider solutions that satisfy a given symmetry condition. This work is concerned with the particular class of spacetimes that admit two space-like Killing vector fields. More precisely, we will focus on the Einstein vacuum model $R_{\mu \nu}(\tilde g)=0$, where $\tilde g$ is the metric tensor and $R_{\mu \nu}$ is the Ricci tensor, in the Belinski-Zakharov setting. This ansatz is compatible with the well-known Gowdy symmetry.

The main goal of this thesis is to describe rigorously the conditions for the global existence of small solutions, and their decay in the light cone, as well as the stability of a first set of solitonic solutions (gravisolitons), for the so-called \textit{reduced Einstein equation}, viewed as an identification of the Principal Chiral Field (PCF) model. In the following, the results of this thesis will be described. The manuscript is divided into four central chapters, which can be read independently of each other.

Chapter 2 firstly describes rigorously the theory of local and global existence of solutions with small initial data for the PCF model. Secondly, the study of the long-term dynamics of finite energy solutions is addressed, proposing suitable virial estimates for the model. Finally, an explicit soliton-type solution belonging to the family of solutions described in the global existence theory is proposed.

Chapter 3 addresses the Einstein field equations, where, under appropriate assumptions of regularity and small initial data, the local and global theory for the problem is obtained, as well as an adequate description of the energy and momentum in the case of cosmological type solutions in General Relativity. In addition, virial estimates are proposed for this class of solutions, which allows to account for the decay of solutions with finite energy.

In Chapter 4, we return to the PCF model, this time, interested in studying the orbital stability of the explicit solutions described in Chapter 2. Unlike the classical approach, we will combine asymptotic stability techniques and preservation of local energy to provide a near complete characterization of perturbations of regular soliton solutions of PCF model.

Regarding Chapter 5, a problem with a completely different focus will be addressed. This chapter proposes to study the blow-up rate for the modified Zakharov-Kuznetsov model. This problem is particularly interesting since there are still no concrete results on the existence of blow-up solutions for the model. However, numerical studies suggest that if there is a singular solution, the blow-up rate could be restricted to a certain range. The purpose of this study is to contribute to the state of the art on this challenging problem.

Finally, in Chapter 6, we present the conclusions of the different topics addressed and some open problems to be considered in the future.