Spectral Properties and Stability of Self-Similar Wave Maps

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.

Aichelburg, Peter C.
University of Vienna

 Record created 2011-05-23, last modified 2018-03-13

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