On stable self-similar blow up for equivariant wave maps: The linearized problem

Autor(en)

Roland Donninger, Birgit Schörkhuber, Peter Christian Aichelburg

Abstrakt

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f(0) is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f(0). This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f(0) is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1/2. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

Organisation(en)

Gravitationsphysik

Journal

Annales Henri Poincare

Band

13

Seiten

103-144

Anzahl der Seiten

42

ISSN

1424-0637

DOI

https://doi.org/10.1007/s00023-011-0125-0

Publikationsdatum

2012

Peer-reviewed

Ja

ÖFOS 2012

103019 Mathematische Physik

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/9c9aed01-94c4-4e3a-ac48-8fba5723a1eb