Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions

Melcher, Christof

doi:10.1512/iumj.2012.61.4717

Cited by 41 publications

(61 citation statements)

References 31 publications

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“…The blow up criterion follows from the higher order energy estimates and uniqueness follows from stability estimates. The details are hence omitted here and one can find similar treatment in [23,31,32] for Landau-Lifshitz equation or general parabolic equations, or our recent paper for a similar model in [26]. This completes the proof of Theorem 5.1.…”

Section: Local Well-posednesssupporting

confidence: 61%

Partial regularity to the Landau-Lifshitz equation with spin accumulation

Wang

2020

Journal of Differential Equations

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In this paper, we consider a model for the spin-magnetization system that takes into account the diffusion process of the spin accumulation. This model consists of the Landau-Lifshitz equation describing the precession of the magnetization, coupled with a quasi-linear parabolic equation describing the diffusion of the spin accumulation. This paper establishes the global existence and uniqueness of weak solutions for large initial data in R 2 . Moreover, partial regularity is shown. In particular, the solution is regular on R 2 × (0, ∞) with the exception of at most finite singular points.where s = (s 1 , s 2 , s 3 ) ∈ R 3 is the spin accumulation, m = (m 1 , m 2 , m 3 ) ∈ S 2 is the precession of the magnetization, and J s is the spin current given by J s = m ⊗ J e − D 0 (x) [∇s − βm ⊗ (∇s · m)] ,

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Section: Local Well-posednesssupporting

confidence: 61%

Partial regularity to the Landau-Lifshitz equation with spin accumulation

Wang

2020

Journal of Differential Equations

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“…We consider the linearization of (6) around a steadystate, which is given by (22). Using the spectral decomposition of (22) into an infinite number of two-dimensional systems, we isolate the n first (2×2 blocks) modes, which are known to contain all unstable modes, whereas all other ones are naturally stable. On the finite-dimensional linear control system containing all unstable modes, we verify that, under a generic condition on the lengths of the solenoid, the Kalman condition holds true.…”

Section: The Main Resultsmentioning

confidence: 99%

“…Consider a steady-state, assumed to be M 0 without loss of generality, as in Section 2.2. Then, locally around M 0 , every solution m can be written as (21) in the frame (M 0 , M 1 , M 2 ), and the Landau-Lifshitz equation is then locally equivalent to (22).…”

Section: Proof Of the Resultsmentioning

confidence: 99%

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Control and stabilization of steady-states in a finite-length ferromagnetic nanowire

Privat

Trélat

2015

ESAIM: COCV

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We consider a finite-length ferromagnetic nanowire, in which the evolution of the magnetization vector is governed by the Landau-Lishitz equation. We first compute all steady-states of this equation, and prove that they share a quantization property in terms of a certain energy. We study their local stability properties. Then we address the problem of controlling and stabilizing steady-states by means of an external magnetic field induced by a solenoid rolling around the nanowire. We prove that, for a generic placement of the solenoid, any steady-state can be locally exponentially stabilized with a feedback control. Moreover we design this feedback control in an explicit way by considering a finite-dimensional linear control system resulting from a spectral analysis. Finally, we prove that we can steer approximately the system from any steady-state to any other one, provided that they have the same energy level.

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“…In contrast to elliptic regularity theory, there is no known result that guarantees regularity of solutions based on smoothness of the initial condition alone. For the LLG equation, such results are only known with additional assumptions to m 0 (see [CF01a,CF01b,Mel12,FT17]) which, in our case, need not be fulfilled. Overall, weak-strong uniqueness is a non-trivial observation.…”

Section: Main Theoremmentioning

confidence: 99%

Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

Fratta

Innerberger

Praetorius

2020

Nonlinear Analysis: Real World Applications

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We consider the time-dependent Landau-Lifshitz-Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii-Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weakstrong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.

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Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions

Cited by 41 publications

References 31 publications

Partial regularity to the Landau-Lifshitz equation with spin accumulation

Partial regularity to the Landau-Lifshitz equation with spin accumulation

Control and stabilization of steady-states in a finite-length ferromagnetic nanowire

Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

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