R. Donninger, B. Schörkhuber, P. Aichelburg:

"On stable self-similar blow up for equivariant wave maps: the linearized problem";

in: "ASC Report 31/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

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 f0 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 f0. This is an indispensable prerequisite for the

study of nonlinear stability of the self-similar blow up which is conducted in the companion paper

[11]. In particular, we prove that f0 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.

http://www.asc.tuwien.ac.at/preprint/2010/asc31x2010.pdf

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