Asymptotic Behaviours for the Landau-Lifshitz-Bloch Equation

The Landau-Lifshitz-Bloch (LLB) equation is an interpolation between Bloch equation valid for high temperatures and Landau-Lifshitz equation valid for low temperatures. Conversely in this paper, we discuss the behaviours of the solutions of (LLB) equation both as the temperature goes to infinity or 0. Surprisingly in the first case, the behaviour depends also on the scaling of the damping parameter δ and the volume exchange parameter a. Three cases are considered and accordingly we get either a linear stationary equation, Bloch equation or Stokes equation. As for the small temperature behaviour, δ and a being independent of the temperature, we show that the limit of (LLB) equation is Landau-Lifshitz-Gilbert equation.


Introduction
A macroscopic description of the dynamics of magnetization m of ferromagnets at low temperature as well as at elevated temperature is given by the Landau-Lifshitz-Bloch (LLB) equation coupled with the magnetostatic equation satisfied by the magnetic field H. (LLB) equation interpolates between the Landau-Lifshitz (LL) equation see [1,3,6] arising at temperatures θ below the Curie point θ c and the Bloch equation when the temperatures exceed θ c see [4]. (LLB) model involves the longitudinal variation of the magnetization so the saturation constraint |m| = 1 is not conserved as in case of (LL) equation. The (LLB) model first introduced in [4] has been discussed from the physical point of view in many recent papers see [10,11] for example. The growing interest for this model is sparked by the many applications as the magnetic write head and the recording medium.
The model equations. We denote by | |, · and × respectively the Euclidean norm, the scalar and the cross products in R 3 and we consider an open bounded domain D ⊂ R 3 which is simply connected and regular with boundary Γ. We denote by ν the unit outward normal to Γ and for T > 0 fixed, we set D T = (0, T ) × D and Γ T = (0, T ) × Γ.
where ω := ω(m) = m |m| as long as m ̸ = 0, H is the effective magnetic field and F is the applied magnetic field. Omitting the contribution of the anisotropic and the internal exchange fields, the effective magnetic field H is given, see [4] by H = a ∆m + H.
The parameters g, a > 0 are respectively the gyromagnetic and the volume exchange coefficients, δ > 0 is the damping parameter and α tr and α l are the transverse and longitudinal damping parameters given by means of the dimensionless temperature τ = θ/θ c , see [4] by Using the relation one can rewrite the magnetization equation (1) in the form and in view of (4) we see that for τ ≥ 1 this equation simplifies into where we set κ(τ ) = 2 3 δ τ.
Asymptotic behaviours. Our aim in this work is to discuss the behaviour of the system (1)-(2) first as the dimensionless temperature τ → +∞ and then as τ → 0. In section 4 we discuss the behavior as τ → 0 assuming δ and a (independent of τ ) equal to 1 for simplicity. We prove that at the limit, the magnetization m satisfies the classical Landau-Lifshitz-Gilbert (LLG) equation, see Theorem 2.7.
Structure of the paper. We present our results in the next section, starting by giving some notations to precise the functional framework and a reminder of the existence result for problem (1)-(2) available in [7]. We end section 2 by some properties related to the magnetostatic equations which will be useful later. In section 3, we prove Theorems 2.2, 2.4 and 2.5. First we provide some uniform estimates on the solutions (m τ , H τ ) of problem (1)-(2) allowing to pass to the limit as τ → ∞ in each case. Section 4 is devoted to the proof of Theorem 2.7 following globally the same steps as before in order to perform the limit as τ → 0. But regarding to the difficulty related to the possible canceling of the magnetization m, this case requires more technicalities.

Statement of the Main Results
Before stating our main results, let us precise some notations and the hypotheses under consideration then we will recall the existence result of solutions to problem (1)- (2).
Notations. Let L p (D), W s,p (D) and H s (D) be the usual Lebesgue and Sobolev spaces for scalar functions and let L p (D), W s,p (D) and H s (D) be the associated vectorial functional spaces, all equipped with the usual norms and we denote by ∥ · ∥ either the L 2 (D) or L 2 (D) norm. We define the Hilbert space M = {m ∈ H 1 (D), div m = 0, m · ν = 0 on Γ} equipped with the usual norm of H 1 (D), see [2] for example and the Hilbert spaces with the norm ∥∇ϕ∥ (which is equivalent to the H 1 -norm thanks to Poincaré-Wirtinger inequality) and the vectorial functional spaces associated are denoted by L 2 ♯ (D) and H 1 ♯ (D). For a general Banach space V we denote the norm by ∥ · ∥ V and the dual space by The Bochner spaces associated with V are denoted by L p (0, T ; V ) and we define as usual the spaces H s (0, T ; V ), W s,p (0, T ; V ) and C([0, T ]; V ).
To end the notations, we point out that in the sequel C > 0 is a generic constant which depends only on the domain D and not of the physical parameters appearing in the equations.
Hypotheses. In the sequel we make use of the following assumptions on the data In section 4, m 0 is also assumed to satisfy the saturation condition |m 0 | = 1 a.e. in D. (9) Solutions to problem (1)- (2). We consider the (LLB) system (1)-(2) and we recall the following existence result, see [7].
Main theorems. We shall prove that at high or low temperatures, the solutions (m τ , H τ ) of problem (1)-(2) provided by Proposition 2.1 behave according to the different cases already described as follows Theorem 2.2. Assume hypotheses (8) to be satisfied and a is independent of τ . As τ → +∞, if κ(τ ) → +∞ then there exists a subsequence still labeled (m τ , H τ ) converging to a limit (m, H) such that and (m, H) satisfies the linear stationary problem  (8) to be satisfied and a is independent of τ . As τ → +∞, if κ(τ ) → 0, then there exists a subsequence still labeled (m τ , H τ ) converging to a limit (m, H) ∈ L ∞ (0, T ; H 1 (D)×H 1 ♯ (D)) such that (m, H) satisfies Bloch equation coupled to the magnetostatic equation As τ → +∞, if κ(τ ) → +∞ then there exists a subsequence (m τ , H τ ) converging to (m, 0) with m ∈ L ∞ (0, T ; M) and there exists π ∈ L 2 (0, T ; L 2 (D)) (which is unique up to a constant) such that (m, π) satisfies Stokes equations with Navier's slip boundary conditions

Remark 2.6. This result is very surprising. To our knowledge, this is the first time that the Stokes equation is used to describe the dynamics of the magnetization. In our opinion, this is explained by the fact that the applied magnetic field is very small.
Theorem 2.7. Assume hypotheses (8) and (9) to be satisfied and a = δ = 1. Then there exists a subsequence

and (m, H) is a global weak solution of the Landau-Lifshitz-Gilbert (LLG) equation coupled to the magnetostatic equations
The proofs of Theorems 2.2, 2.4 and 2.5 will be done in section 3 and the proof of Theorem 2.7 in section 4.
To end this section, let us recall some useful results on the magnetostatic equations.
The magnetostatic equations.
then by Poincaré-Wirtinger inequality we get the bound where C > 0 depends only on the domain D, which leads to the estimate In particular we have where div m ∈ C([0, T ]; L 2 (D)) and m · ν ∈ C([0, T ]; H 1/2 (Γ)), then applying elliptic regularity results we where C > 0 depends only on the domain D.
with |

High Temperature Limits
We consider the (LLB) system (6)-(2) when τ ≥ 1. In this section we will prove the asymptotic behaviour results given in Theorems 2.2, 2.4 and 2.5. To begin, let us give some energy estimates satisfied by the solutions (m τ , H τ ) of the problem provided by Proposition 2.1.
where C > 0 depends only on the domain D, E 0 = a∥∇m 0 ∥ 2 + ∥H 0 ∥ 2 , H 0 being the demagnetizing field associated to m 0 and the source term Proof. For simplicity we drop the index τ in (m τ , H τ ).
We multiply the magnetization equation by m then by H ∈ L 2 (0, T ; L 2 (D)) and integrate by parts to get and − Hence (27) leads straightforwardly to estimate (23) using relations (16) and (17). Now using relations (28), (21) and (22) and setting we get which leads to estimate (24) by using (23). To prove (25), we rewrite (30) as and use Gronwall lemma which leads to 3.1. The stationary limit. We will prove Theorem 2.2, so uniform bounds of m τ and H τ are needed. First the results of Proposition 3.1 allow to deduce that (m τ , H τ ) satisfy the estimates below.

Corollary 3.2. We have the estimates
where Next we will prove the following L ∞ (0, T ; H 1 (D)) uniform estimates.
Proof. We need to apply Poincaré-Wirtinger inequality for m τ so we shall estimate its mean value on the domain D. To this purpose we introduce the notation ⟨f ⟩ = |D| −1 ∫ D f (x) dx for a scalar or a vectorial function f , where |D| is the Lebesgue measure of D. We multiply the magnetostatic equation by x i for i = 1, 2, 3 and integrate by parts to obtain the relation for all t ∈ [0, T ]. Therefore the inequality |⟨x F (t)⟩| ≤ C ∥F (t)∥ implies that where throughout this demonstration C > 0 denotes different constants depending only on the domain D. By using Poincaré-Wirtinger inequality ∥m τ − ⟨m τ ⟩∥ ≤ C∥∇m τ ∥, we deduce that Hence (33) leads to the first estimate of the lemma and the second one follows by using (20).
Now we introduce the weak formulation of problem (6) and the weak formulation of the magnetostatic equation writes as ∫ The results of Corollary 3.4 allow to pass to the limit as τ → ∞ in (41) and (42), since κ(τ ) → ∞ we get so ∆m ∈ L ∞ (0, T ; H 1 (D)) which ends the proof of Theorem 2.2.
The uniform estimate of H τ in L ∞ (0, T ; H 1 (D)) is derived by the same bound of m τ , using inequality (20). Now we look for a bound of the time derivative of m τ . Lemma 3.6. ∂ t m τ is uniformly bounded with respect to τ in L 2 (0, T ; (H 2 (D)) ′ ).
Passing to the limit. As previously we will pass to the limit in the weak formulation of (6)-(2) as τ → +∞, assuming that a is independent of τ and κ(τ ) → 0. By using Corollary 3.5 and Lemma 3.6, we deduce using Aubin's compactness lemma see [8,9] and the compact embedding H 1 (D) ⊂ L p (D) for all 1 ≤ p < 6, the following convergence results.
Proof. It remains to prove the convergences given in (47). By Lemma 2.8, we see that H τ → H strongly in L 2 (0, T ; L 2 (D)) but since H τ is bounded in L 2 (0, T ; L 6 (D)), the strong convergence of H τ is true in L 2 (0, T ; L p (D)) with 1 ≤ p < 6. Next the sequence m τ × H τ is uniformly bounded in L 2 (0, T ; L 2 (D)) and the strong convergence of m τ and H τ implies that m τ × H τ → m × H strongly in L 1 (0, T ; L 2 (D)) which leads to the desired result. Similarly the sequence m τ × ∇m τ is bounded in L 2 (0, T ; L 3/2 (D)) then by the weak-strong convergence principle we deduce the stated convergence.
These results enable us to end the proof of Theorem 2.4. First using Lemma 2.8, we infer that the magnetostatic equation is satisfied. Let Φ ∈ (D([0, T [×D)) 3 , we write the weak formulation of the magnetization equation and we pass to the limit by using the convergence results given in Corollary 3.7 to get for all Φ ∈ (D([0, T [×D)) 3 . From there, it is easy to conclude that (m, H) satisfies the system of equations (12). This ends proof of Theorem 2.4.

The Stokes limit.
Now we shall discuss the behavior of the problem (6) where C > 0 depends only on the domain D and where C T > 0 depends only on the domain D and T involving the inequality ) . (54) Hence from estimates (50) and (54), we infer that ). (55) Passing to the limit. We use the previous bounds to deduce that Proof. It remains only to prove the last convergence result and for this one, we use the inequality ∥m τ × H τ ∥ L 2 (0,T ;L 3/2 (D)) ≤ ∥m τ ∥ L ∞ (0,T ;L 6 (D)) ∥H τ ∥ L 2 (0,T ;L 2 (D)) .

Small Temperature Limit
This section deals with the asymptotic behaviour of (LLB) system (1)-(2) when τ → 0, δ and a being independent of τ so without loss of generality we take them equal to 1. As we expect that the limit equation of the magnetization is (LLG) equation, we assume that the initial data satisfies the saturation constraint |m 0 | = 1 a.e. in D. From now on, we see that κ(τ ) = α l = 2 3 τ and we set γ(τ ) := α tr = 1 − 1 2 κ(τ ). We recall that for 0 < τ < 1, the existence result of Proposition 2.1 holds true, but regarding to the indetermination contained in the equation (due to the fact that m can cancel so ω(m) is not defined), the magnetization equation is rewritten in the following form, see [7] where H τ = ∆m τ + H τ . To be complete, we give again the magnetostatic equation satisfied by H τ We aim to prove the results stated in Theorem 2.7 so the first step is to establish uniform bounds of the solutions with respect to the small parameter τ .

Uniform estimates
We introduce the following notations. For t > 0 we set D t = (0, t) × D and we define the function χ by χ(m) = 1 if m ̸ = 0 and χ(m) = 0 elsewhere. Our first estimates are given below where E 0 = ∥∇m 0 ∥ 2 + ∥H 0 ∥ 2 and H 0 = ∇φ 0 is the demagnetizing field associated to the magnetization m 0 and the source term F (0).
Proof. We drop the index τ for simplicity. We multiply the magnetization equation (63) by m to obtain or equivalently Therefore integrating over D t leads to and we get estimate (65) using (16) and (17). Now we multiply the magnetization equation (63) by −H ∈ L 2 (0, T ; L 2 (D)) to get the equality which is equivalent to where ω = ω(m) = m |m| if m ̸ = 0 and we set ω(0) = u, u being any unit vector. Integrating over D t and using the inequality (ω · H) 2 then from the equality and inequality (22) we deduce where C > 0 depends only on D. Setting E(t) = ∥∇m(t)∥ 2 + ∥H(t)∥ 2 , we see that and by using Gronwall inequality, we get estimate (66).
According to the previous results and using the same notations, we infer that Proof. Estimate (80) derives from (66) since by Proposition 4.2, χ(m τ ) = 1 a.e. in D T .
In order to pass to the limit in the problem when τ → 0, a uniform bound on the time derivative of m τ is needed. To begin, since for τ > 0 small enough, m τ ̸ = 0 a.e. in D T , we can rewrite equation (63) of m τ in its first form (1) that is where ω τ = ω(m τ ) and we used the property |m 0 | 2 = 1 which implies that |m τ (0)| 2 = 1. Below we will prove the following result.  (8) and (9), ∂ t m τ is uniformly bounded in L 3/2 (D T ) with respect to the small parameter τ .

The LLG limit.
By using the previous bounds and Aubin's compactness lemma we deduce the following convergence results Proof. The strong convergence of H τ is a consequence of Lemma 2.8 and we deduce that H = ∇φ satisfies the magnetostatic equation (64) while property (87) is a direct consequence of the strong convergence of m τ and (77).
As a first step towards (LLG) equation, let us rewrite the magnetization equation (81) in a new form. We observe that if we take the cross product of (81) by m τ we get and inserting this expression in (81) we obtain the new formulation with γ(τ ) = 1 − κ(τ ) 2 → 1 as τ → 0.