Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

Abstract

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

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

“…In the one-dimensional case N = 1, we established in [17] the existence and asymptotics of the family {m c,α } c>0 of self-similar solutions of (LLG α ) for any fixed α ∈ [0, 1], extending the results in Gutiérrez, Rivas and Vega [18] in the setting of the Schrödinger map equation and related binormal flow equation. The motivation for the results presented in this paper first originated from the desire to study further properties of the self-similar solutions found in [17], and in particular their stability. In the case α = 0, the stability of the self-similar solutions of the Schrödinger map has been considered in the series of papers by Banica and Vega [5,6,7], but no stability result is known for these solutions in the presence of damping, i.e.…”

“…there exists a unique solution m ∈ X(R N × R + ; S 2 ) of (LLG α ) with initial condition m 0 such that 3 We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results. 4 See footnote in Section 3.3 for the definition of the Morrey space M 2,2 (R N ).…”

“…As we will see in Section 2, the proof of Theorem 1.1 relies on the use of the stereographic projection to reduce Theorem 1.1 to establish a well-posedness result for the associated dissipative (quasilinear) Schrödinger equation (see Theorem 2.1). In order to be able to apply Theorem 1.1 to the study of both the initial value problem related to the LLG equation with a jump initial condition, and the stability of the self-similar solutions found in [17], we will need a more quantitative version of this result. A more refined version of Theorem 1.1 will be stated in Theorem 2.9 in Subsection 2.2.…”

“…As mentioned in the introduction, in [17] we proved the existence of the uniparametric smooth family of self-similar solutions {m c,α } c>0 of (LLG α ) for all α ∈ [0, 1] with initial condition of the type (3.2) given by…”

“…More precisely, using the invariance under rotations of (LLG α ), we can prove that, if the angle between A + and A − is small enough, then the solution given by Corollary 3.1 with initial condition m 0 A ± coincides (modulo a rotation) with m c,α , for some c. We have the following: In order to prove Theorem 3.3, we need some preliminary estimates for m c,α in terms of c and α. To obtain them, we use some properties of the profile profile f c,α = (f 1,c,α , f 2,c,α , f 3,c,α ) constructed in [17] using the Serret-Frenet equations with initial conditions…”