Annales Henri Poincaré

, Volume 13, Issue 1, pp 103–144 | Cite as

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

  • Roland Donninger
  • Birgit Schörkhuber
  • Peter C. Aichelburg


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 \({\frac{1}{2}}\). The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.


Cauchy Problem Minkowski Space Unstable Mode Global Regularity Regular Singular Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Roland Donninger
    • 1
  • Birgit Schörkhuber
    • 2
  • Peter C. Aichelburg
    • 3
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Faculty of Mathematics and Geoinformation, Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  3. 3.Fakultät für Physik, Gravitational PhysicsUniversität WienWienAustria

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