Spectral Properties and Stability of Self-Similar Wave Maps

TL;DR

This thesis analyzes the spectral properties and stability of self-similar solutions in wave maps from 3+1 Minkowski space to S^3, revealing their role in singularity formation and providing a detailed linear stability analysis.

Contribution

It offers a comprehensive spectral analysis of perturbation operators and establishes well-posedness of the linearized problem using advanced functional analytic methods.

Findings

Spectral analysis of perturbation operators completed.

Well-posedness of the linearized Cauchy problem proved.

Growth estimates inform stability of self-similar solutions.

Abstract

In this thesis the Cauchy problem and in particular the question of singularity formation for co--rotational wave maps from 3+1 Minkowski space to the three--sphere $S^3$ is studied. Numerics indicate that self--similar solutions of this model play a crucial role in dynamical time evolution. In particular, it is conjectured that a certain solution $f_0$ defines a universal blow up pattern in the sense that the future development of a large set of generic blow up initial data approaches $f_0$. Thus, singularity formation is closely related to stability properties of self--similar solutions. In this work, the problem of linear stability is studied by functional analytic methods. In particular, a complete spectral analysis of the perturbation operators is given and well--posedness of the linearized Cauchy problem is proved by means of semigroup theory and, alternatively, the functional calculus for self--adjoint operators. These results lead to growth estimates which provide information on the stability of self--similar wave maps. Finally, convergence properties of $f_n$ for large $n$ and the spectra of the corresponding perturbation operators are investigated. The thesis is intended to be self--contained as far as possible, i.e. all the mathematical requirements are carefully introduced, including proofs for many results which could be found elsewhere.