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

"On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem";

Annales Henri Poincare,1(2011).

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 (Donninger in Commun. Pure Appl. Math., 64(8), 2011). 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://dx.doi.org/10.1007/s00023-011-0125-0

Erstellt aus der Publikationsdatenbank der Technischen Universität Wien.