Mathematics – Analysis of PDEs
Scientific paper
2010-06-08
Ann. Henri Poincare 13 (2012) no. 1, 103-144
Mathematics
Analysis of PDEs
32 pages, 2 figures, acknowledgments added
Scientific paper
10.1007/s00023-011-0125-0
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 a companion paper. 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.
Aichelburg Peter C.
Donninger Roland
Schoerkhuber Birgit
No associations
LandOfFree
On stable self-similar blow up for equivariant wave maps: The linearized problem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On stable self-similar blow up for equivariant wave maps: The linearized problem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On stable self-similar blow up for equivariant wave maps: The linearized problem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-40471