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Hysteresis for ferromagnetism: asymptotics of some 2-scale Landau-Lifshitz model

We study a 2-scale version of the Landau-Lifshitz system of ferromagnetism, introduced by Starynkevitch to modelize hysteresis: the response of the magnetization is fast compared to a slowly varying applied magnetic fi eld. Taking the exchange term into account, in space dimension 3, we prove that, under some natural stability assumption on the equilibria of the system, the strong solutions follow the dynamics of these equilibria. We also give explicit examples of relevant equilibria and exterior magnetic fields, when the ferromagnetic medium occupies some ellipsoidal domain.


Introduction
Hysteresis is a widely studied, yet not completely understood phenomenon. It has played a role from the very beginning of the works on magnetism. Lord Rayleigh [9] proposed a model for ferromagnetic hysteresis in 1887, while the most achieved micromagnetism theory goes back to Landau and Lifshitz, in 1935 (see [7]).
In [12], Visintin gives many historical references, underlines the links between several forms of hysteresis (in particular, from plasticity, and from ferromagnetism), and how it is related to phase transitions. He performs a mathematical study of the so-called hysteresis operators, including the most famous one, due to Preisach.
Recently in [3], Carbou, Effendiev and Fabrie have proved the existence of strong solutions to a model of ferromagnetic hysteresis due to Effendiev.
In this paper, we rather investigate properties of a two-scale model introduced by Starynkevitch in [11]. This model describes the dynamics obtained when some exterior magnetic field is applied to the ferromagnetic material under consideration, while the response of the magnetization occurs on a much shorter time scale (say, denoted by ε > 0). Mathematically, such models, associated to ordinary differential equations, had been studied in the nonstandard analysis framework, leading to "canard cycles" (see [6]). Considering a Landau-Lifshitz model in 0 space dimension (thus, an ODE), Starynkevitch studies the possible equilibria of the system, and the asymptotic behavior of the solutions (as the above mentioned parameter ε goes to zero) when the exterior magnetic field slowly varies.
Our aim is to extend Starynkevitch's approach to the Landau-Lifshitz model in space dimension three, taking exchange term into account. This means, giving the asymptotic description of solutions to the slow-fast corresponding system of partial differential equations. Here, we prove such a result away from the bifurcation points of hysteresis loops. More precisely, assuming that the system (described by its magnetization) possesses at each time t some stable equilibrium m eq (t), and is submitted to some slowly varying exterior magnetic field, we show that the magnetization follows the dynamics of m eq . We also give explicit examples (for ellipsoidal domains) of relevant equilibria and exterior magnetic fields.

Statement of the results
The initial and boundary value problem associated to the 2 scale Landau-Lifshitz equation considered reads: The unknown is the magnetization m ε , function of the time variable t 0 and of the space variable x ∈ Ω, with values in the sphere S 2 ⊂ R 3 . The domain Ω occupied by the ferromagnet is a subset of R 3 . Furthermore, h ε T = h T (t, m ε (t)), where the total magnetic field h T is defined by (2.2) h T (t, m) = ∆m + h d (m) + h ext (t).
Here, the first term ∆m is the "exchange term", which tends to impose a constant magnetization (domains where magnetization is constant are called "Weiss domains"), and ∆m denotes the extension of ∆m by 0 out of Ω. The second term, yielding spatial variations of the magnetization, is the "demagnetizing field" h d (m), which results from a quasi-stationary approximation of Maxwell's equations; it is defined (at least, for m ∈ L 2 (Ω, R 3 ), as an element of L 2 (R 3 , R 3 )) by curl h d (m) = 0 and div (h d (m) + m) = 0 in R 3 .
Classical properties of the mapping m → h d (m) are recalled in Section 3.1. The third term, h ext , denotes some given exterior field, which is assumed to depend on time (and possibly on space). The positive constant α is some damping coefficient, which appears in the model when passing from a microscopic to a macroscopic description. The small parameter ε > 0 expresses the fact that, while the exterior field h ext depends on t, and has time variations at scale 1, the magnetization m ε essentially depends on t/ε, and thus has variations at the much more rapid scale ε. Throughout this paper, for any s ∈ N, we denote by H s (Ω) the usual Sobolev space of functions with values in some vector space R N , whereas H s (Ω, S 2 ) is the Sobolev space of functions with values in the sphere S 2 (which is not a vector space), Finally, for s 2, H s N (Ω) denotes the subspace of functions in H s (Ω) with homogeneous von Neumann boundary condition, . All these spaces (even if not vector spaces) inherit the (metric) topology given by the usual norm on H s (Ω).
To prove Theorem 2.1, we first show that m ε converges to m eq (0) within an initial layer of size t ε = Cε ln(1/ε). This is achieved via classical energy estimates (in H 2 ), carefully controlling the dependence upon ε -more technically speaking, the quasilinear and elliptic degenerate system of PDE's in (2.1) is first converted into a perturbation of some linear, strongly elliptic system, yielding the usual smooting properties, and a Galerkine approximation is used. In a second step, we prove that m ε converges towards m eq on the whole time interval [t ε , T ]. This amount to proving of long-time existence and return to equilibrium result for small initial data. Toward this end, we use again energy estimates, together with the stability assumption (2.5). Figure 1 illustrates this corresponding asymptotic behaviour. The above assumptions on the equilibrium m eq are discussed in Section 3.2 below. In particular, Assumption (ii) in Theorem 2.1 may be understood as a choice of 'prepared' data m 0 allowing to deal with the initial layer (0, cε ln(1/ε)). The dissipation property (2.5) expresses, for all t 0 ∈ [0, T ], the stability of the linearization around m eq (t 0 ) of (2.1), with ε = 1 and with h ext replaced with h ext (t 0 ), independent of time. This is a strong assumption, which ensures global existence of the solutions to the corresponding Landau-Lifshitz equation, for initial data close to m eq (t 0 ): Let Ω be an open and bounded subset of R 3 , with smooth boundary. Consider an exterior magnetic field h ext ∈ C ∞ (R 3 ) (independent of time) bounded with bounded derivatives. Assume that there exists m eq ∈ H 2 N (Ω, S 2 ) (independent of time) satisfying the equilibrium condition as well as the stability condition there exist C lin > 0 and η > 0 such that, for all δ ∈ H ∞ (Ω) with |m eq + δ| ≡ 1 and ∂ ν δ | ∂Ω = ∂ ν ∆δ | ∂Ω = 0, δ(t) H 2 (Ω) η implies: for the linearized operator L(0, m eq ) given by (4.31) (with m eq (0) and h ext (0) replaced with m eq and h ext , respectively). Then, there exists η 0 > 0 such that, for all m 0 ∈ H 2 N (Ω, S 2 ) satisfying the solution n to the initial and boundary value problem , and n(t) converges in H 2 (Ω), as t goes to ∞, towards m eq .
In the case of m eq (0) constant over Ω, Proposition 2.2 expresses that in Theorem 2.1, assumptions (i) and (iii) imply assumption (ii), so that we get: Let Ω be an open and bounded subset of R 3 , with smooth boundary. Let T > 0, and h ext ∈ C 1 ([0, T ], C ∞ (R 3 )), bounded with bounded derivatives. Assume that there exist m eq ∈ C 1 ([0, T ], H 2 N (Ω, S 2 )) satisfying assumptions (i) and (iii) from Theorem 2.1. Assume furthermore that m eq (0) is constant over Ω.
In Lemma 3.5 below, we give examples (in ellipsoidal domains) of equilibria m eq satisfying the assumptions of Corollary 2.3.

Some functional analysis
In this section, we recall some functional analysis results useful in the sequel. The first of them deals with the continuity properties of the demagnetizating field operator h d , immediately deduced from the Fourier representation Furthermore, for all v in L 2 (Ω) we have In addition to the usual Sobolev embeddings, we recall the following estimate, which results from the coercivity of the operator A = 1 − ∆, with domain D(A) = {m ∈ H 2 (Ω) | ∂ ν m | ∂Ω = 0} (see for example [5]) In the sequel, we will need the following definition.
The family of operators (P k ) k∈N satisfies useful properties: Lemma 3.4. The following properties are true.
with (ψ i ) i∈N the L 2 -orthonormal basis of the eigenvectors of ∆ associated to the eigenvalues (λ i ) i∈N . Then, using the vanishing Neumann boundary conditions, Point (ii) follows from the regularity properties of the family (ψ i ) i∈N .
provides a norm equivalent to the usual one on H s (Ω).

About equilibria
Global solutions and equilibria. In [1,Th. 4.3], in the case of ellipsoidal domains Ω ⊂ R 3 and under a smallness assumption (on h ext L ∞ and ∆m 0 L 2 ), Alouges and Beauchard construct global smooth solutions to (2.1). Furthermore, these solutions satisfy so that ∇∆m belongs to L 2 ((0, ∞) × Ω). This is a part of our assumptions on the equilibrium m eq , when requiring the existence of the global solution n 0 . Saying that m eq (t 0 ) is an equilibrium for (2.3) means and requiring H 2 convergence of n 0 (t) towards m eq (0) as t goes to ∞ implies that m eq (0) is an equilibrium for (2.3) with t 0 = 0.
Energy minimization. It is worth noting that energy decay occurs along the evolution of n 0 (t), so that one may hope at least H 1 convergence of n 0 (t) towards some local minimum of the energy, as t goes to ∞. To the Landau-Lifshitz system (2.1) is associated the energy and when m is solution to (2.1), we have Since the exterior magnetic field does not depend on time during the evolution of n 0 , we get In the case of ellipsoidal domains, special configurations are available. See [8], and references therein: there exists a real 3 × 3 definite positive diagonalizable matrix D giving the demagnetizing field resulting from any magnetization constant constant over Ω: Hence, if u ∈ S 2 is an eigenvector of D associated to the eigenvalue d > 0, and if the exterior magnetic field is h ext = λu for some λ > 0 (or h ext (x) = λχ(x)u to get a spatially localized field), then the system possesses two explicit equilibria m + eq and m − eq : One easily computes the energy associated to perturbations of these equilibria: for all δ ∈ H 2 N (Ω, R 3 ) such that |m ± eq + δ| = 1 a.e., The first two terms are non-negative, so that for λ large enough (λ > d), m + eq is a global minimum of E; but for λ small, it may fail to be even a local minimum. Concerning m − eq , for all λ > 0, if d is the largest eigenvalue of D, and δ is constant in space, then the difference of energies above is less than − λ 2 Ω |δ| 2 , thus negative, whereas for δ with large variations, the gradient term dominates, and the energy difference becomes positive. Hence, m − eq is always a saddle point for E.
The dissipation property (2.5). We have the following lemma, the proof of which is postponed to Section 6.1: For λ > 0 large enough, the equilibrium m + eq from (3.2) satisfies the dissipation property (2.5) (for some constant C lin depending on λ).

Proof of Theorem 2.1
First, consider the solution n 0 to the Cauchy problem and define n ε by ∀t 0, n ε (t) = n 0 (t/ε).
, and we know that Next, as in [4], we observe that for smooth functions m with constant modulus (w.r.t. x), one has m · ∆m = −2|∇m| 2 , so that smooth solutions to (2.1) equivalently satisfy issued from m 0 with constant modulus, equal to one, are shown to keep the same modulus for all time, (due to uniqueness of the solution a = |m| 2 to ε∂ t a = α∆a+2α|∇u| 2 (a−1), ∂ ν a | ∂Ω = 0, a |t=0 = 1). We thus solve (4.2) in the Banach space C([0, T ], H 2 N (Ω)), and deduce from this conservation that the solution actually belongs to the space ). It is worth noting that (2.1) is an initial and boundary value problem for some quasilinear and parabolic degenerate operator, which is seen in (4.2) as a perturbation of a linear and strongly parabolic one.
Standard energy estimates ensure local-in-time existence and uniqueness of solutions continuous in time, with values in H 2 (Ω)) (with an existence time depending on ε): see for example [1] or [4]. By the usual continuation argument, we simply need to bound the H 2 norm of m ε to ensure existence up to time T . Actually, we shall prove convergence (as ε goes to zero) at the same time, via energy estimates.
We first show that, after some time t ε of the form t ε = Cε ln(1/ε), m ε and n ε are close: sup [0,tε] m ε − n ε H 2 (Ω) goes to zero with ε; thus, for ε small enough, m ε (t ε ) is as close (in H 2 (Ω)) to m eq (0) as desired. We then use the stability property of m eq (t) to show that m ε (t) stays close to it, for t ∈ [t ε , T ].

First step: the initial layer
For k ∈ N , let P k be the L 2 (Ω)-orthogonal projection onto V k , the vector space spanned by the first k eigenfunctions of A = 1 − ∆, with domain D(A) = {m ∈ H 2 (Ω) | ∂ ν m | ∂Ω = 0}, as in Definition 3.3. Define a Galerkine approximation of (4.2) by: The projection n ε k = P k n ε also satisfies since for u ∈ D(A), P k ∆u = ∆P k u, according to Lemma 3.4. Now, perform energy estimates (in L 2 ) for ϕ ε k = m ε k − n ε k , solution to

L 2 estimates
Take the scalar product (in L 2 (Ω)) of ϕ ε k with the first equation in (4.5) to get ε 2 .
Estimating I 3 . As for I 1 , cancellations allow to write .

H 2 estimates
Next, take the scalar product in L 2 (Ω) of ∆ 2 ϕ ε k with the first equation in (4.5) to get , , .
Estimating II 1 . Split , The first term is written .
Then, using in particular the Sobolev inequality from Lemma 3.2 we get: for all η > 0, for some C η > 0.

(4.21)
Estimating II 3 . Now, with , Concerning II 3,1 , first write m ε k = n ε k + ϕ ε k , then integrate once by parts, so that II 3,1 takes the form of a L 2 scalar product between ∇∆ϕ ε k and a sum of terms ∇(abc), where a, b, c may be n ε k (or h d (n ε k )) or ϕ ε k (or h d (ϕ ε k )), and at least one of them is n ε k (or h d (n ε k )). Estimating each of a, b, c and their gradients in L 6 , one gets: for all η > 0, there is C η > 0 such that Then, split II 3,2 , The second term is estimated as II 3,1 . The first one is split into a sum involving n ε k ∧ n ε k ∧ (h ext (t) − h ext (0)) , and products of h ext (t) with two terms, one of them being ϕ ε k , and the other, ϕ ε k or n ε k . This leads to: for all η > 0, there is C η > 0 (also depending on h ext ) such that (4.23) Finally, summing up (4.22) and (4.23), we have: for all η > 0, there is C η > 0 (depending on n 0 L ∞ ((0,∞),H 2 (Ω)) ) such that
In the case of m − eq , we have which, together with (6.6), leads to Lemma 3.6.
6.2 Proof of the commutator lemma 4.1