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(In-)stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation. (English) Zbl 1293.35059

Summary: We use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyse both the harmonic map heat flow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

MSC:

35B44 Blow-up in context of PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35Q56 Ginzburg-Landau equations
35B35 Stability in context of PDEs
78A25 Electromagnetic theory (general)
82D45 Statistical mechanics of ferroelectrics
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