On stable self-similar blow up for equivariant wave maps: The linearized problem - Mathematics > Analysis of PDEsReportar como inadecuado




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Abstract: We consider co-rotational wave maps from 3+1 Minkowski space into thethree-sphere. This is an energy supercritical model which is known to exhibitfinite time blow up via self-similar solutions. The ground state self-similarsolution $f 0$ is known in closed form and based on numerics, it is supposed todescribe the generic blow up behavior of the system. In this paper we develop arigorous linear perturbation theory around $f 0$. This is an indispensableprerequisite for the study of nonlinear stability of the self-similar blow upwhich is conducted in a companion paper. In particular, we prove that $f 0$ islinearly stable if it is mode stable. Furthermore, concerning the modestability problem, we prove new results that exclude the existence of unstableeigenvalues with large imaginary parts and also, with real parts larger than1-2. The remaining compact region is well-studied numerically and all availableresults strongly suggest the nonexistence of unstable modes.



Autor: Roland Donninger, Birgit Schoerkhuber, Peter C. Aichelburg

Fuente: https://arxiv.org/







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