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On stable self‐similar blowup for equivariant wave maps
Author(s) -
Donninger Roland
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20366
Subject(s) - equivariant map , mathematics , ode , minkowski space , eigenvalues and eigenvectors , stability (learning theory) , mathematical analysis , space (punctuation) , pure mathematics , geometry , physics , linguistics , philosophy , quantum mechanics , machine learning , computer science
We consider corotational wave maps from (3 + 1) Minkowski space into the 3‐sphere. This is an energy supercritical model that is known to exhibit finite‐time blowup via self‐similar solutions. The ground state self‐similar solution f 0 is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f 0 is stable under the assumption that f 0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f 0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f 0 . © 2011 Wiley Periodicals, Inc.