Spectral Properties and Linear Stability of Self-Similar Wave Maps

TL;DR

This paper analyzes the spectral properties and linear stability of self-similar wave maps from 3+1 Minkowski space to S^3, focusing on the stability of a family of solutions using operator theory.

Contribution

It provides a spectral analysis of perturbation operators and establishes stability results for self-similar wave maps, including the limiting solution as n approaches infinity.

Findings

Spectral properties of perturbation operators are characterized.

Well-posedness of the linear Cauchy problem is proved.

Growth estimates for solutions are derived.

Abstract

We study co--rotational wave maps from $(3+1)$--Minkowski space to the three--sphere $S^3$. It is known that there exists a countable family $\{f_n\}$ of self--similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well--posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from $f_n$ by letting $n \to \infty$.