The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

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  • Lille 1 University


We prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some (-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough.

Our arguments rely on the study of a dissipative quasilinear Schrödinger equation obtained via the stereographic projection and techniques introduced by Koch and Tataru.


Original languageEnglish
Article number2522
Pages (from-to)2522-2563
Number of pages42
Issue number7
Publication statusPublished - 18 Jun 2019


  • Complex Ginzburg-Landau equation, Discontinuous initial data, Dissipative Schrödinger equation, Ferromagnetic spin chain, Heat-flow for harmonic maps, Landau-Lifshitz-Gilbert equation, Self-similar solutions, Stability