On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem |
| |
Authors: | Roland Donninger Birgit Sch?rkhuber Peter C Aichelburg |
| |
Institution: | 1. Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL, 60637, USA 2. Faculty of Mathematics and Geoinformation, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8, 1040, Wien, Austria 3. Fakult?t f??r Physik, Gravitational Physics, Universit?t Wien, Boltzmanngasse 5, 1090, Wien, Austria
|
| |
Abstract: | 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
\frac12{\frac{1}{2}}. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of
unstable modes. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|