On stable self-similar blowup for equivariant wave maps

✍ Scribed by Roland Donninger

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 347 KB
• Volume: 64
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20366

✦ Synopsis

We consider corotational wave maps from .3 C 1/ Minkowski space into the 3-sphere. This is an energy supercritical model that is known to exhibit finitetime 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 .