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Smooth control of nanowires by means of a magnetic
field
Gilles Carbou, Stéphane Labbé, Emmanuel Trélat
To cite this version:
Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a
magnetic field. Communications on Pure and Applied Mathematics, Wiley, 2009, 8 (3), pp.871–879.
�10.3934/cpaa.2009.8.871�. �hal-00313779�
Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals
Volume X, Number 0X, XX 200X pp.X–XX
SMOOTH CONTROL OF NANOWIRES BY MEANS OF A MAGNETIC FIELD
Gilles Carbou
MAB, UMR 5466, CNRS, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 Talence cedex, France.
St´ephane Labb´e
Universit´e Joseph Fourier, Laboratoire Jean Kuntzmann, CNRS, UMR 5224, 51 rue des Math´ematiques, B.P. 53, 38041 Grenoble Cedex 9, France.
Emmanuel Tr´elat
Universit´e d’Orl´eans, Laboratoire MAPMO, CNRS, UMR 6628, F´ed´eration Denis Poisson, FR 2964,
Bat. Math., BP 6759, 45067 Orl´eans cedex 2, France.
Abstract. We address the problem of control of the magnetic moment in a ferromagnetic nanowire by means of a magnetic field. Based on theoretical results for the 1D Landau-Lifschitz equation, we show a robust controllability result.
1. Model and control result. The magnetic moment u of a ferromagnetic ma-terial is usually modeled as a unitary vector field, solution of the Landau-Lifschitz equation
∂u
∂t = −u ∧ He− u ∧ (u ∧ He), (1) where the effective field He is given by He= ∆u + hd(u) + Ha. The demagnetizing
field hd(u) is solution of the magnetostatic equations
div B = div (H + u) = 0 and curl H = 0,
where B is the magnetic induction. The applied field is denoted by Ha (see [3,
12, 17, 22] for more details on the modelization). Existence results have been established for the Landau-Lifschitz equation in [4, 5, 13, 21], numerical aspects have been investigated in [11,15,16], and asymptotic properties have been proved in [1,6,10,18,20]; control issues were addressed in [9].
Here we restrict ourselves to a one dimensional model, i.e., we consider a ferro-magnetic nanowire, submitted to an external ferro-magnetic field applied along the axis of the wire and which is our control. The model is then written as (see [20])
∂u
∂t = −u ∧ hδ(u) − u ∧ (u ∧ hδ(u)), (2)
2000 Mathematics Subject Classification. Primary: 35B37, 93D15; Secondary: 93C20. Key words and phrases. Landau-Lifschitz equation, control, stabilization.
The authors were partially supported by the ANR project SICOMAF (”SImulation et COntrˆole des MAt´eriaux Ferromagn´etiques”).
where hδ(u) = ∂
2
u
∂x2− u2e2− u3e3+ δe1. Here, (e1, e2, e3) is the canonical basis of IR
3
and the nanowire is the real axis IRe1. The magnetic field is written δ(t)e1, where
the function δ(·) is our control. Setting h(u) = uxx− u2e2− u3e3, this yields
ut= −u ∧ h(u) − u ∧ (u ∧ h(u)) − δ(u ∧ e1+ u ∧ (u ∧ e1)). (3)
When δ ≡ 0, stationary solutions do exist, of the form
M0(x) = th x 0 1 ch x (4)
and are called Bloch walls. Their stability properties were studied in [7]. When δ(·) ≡ δ is constant, the solution writes
uδ(t, x) = RδtM0(x + δt), (5) where Rθ= 1 0 0 0 cos θ − sin θ 0 sin θ cos θ
is the rotation of angle θ around the axis IRe1. It corresponds to a rotation plus
translation of the above wall along the nanowire.
Notice the invariance of (3) through translations x 7→ x − σ and rotations Rθ
around the axis e1. This generates a three-parameters family of particular solutions
defined by
uδ,θ,σ(t, x) = M
Λuδ(t, x) = Rδt+θM0(x + δt − σ) (6)
called travelling wall profiles.
Controlling these walls (position plus speed) might be relevant for coding and transporting some information. This is our aim here to derive a controllability re-sult, with an eye on possible applications such as rapid recording. In [9], control properties were proven with piecewise constant controls. However, practical appli-cations require the control to be smooth. Recall that the control here is an external magnetic field applied along the nanowire. The main result of [9] strongly uses the fact that the control is a piecewise constant function and our aim is here to extend this result to the case of smooth controls, hence closer to practical issues.
Theorem 1.1. There exist ε0> 0 and δ0> 0 such that, for all δ1, δ2∈ IR satisfying
|δi| ≤ δ0, i = 1, 2, for all σ1, σ2 ∈ IR, for every ε ∈ (0, ε0), there exist T > 0 and a
control function δ(·) ∈ C∞(IR, IR) such that, for every solution u of (3) associated
with the control δ(·) and satisfying
∃θ1∈ IR | ku(0, ·) − uδ1,θ1,σ1(0, ·)kH2 ≤ ε, (7)
there exists a real number θ2 such that
ku(T, ·) − uδ2,θ2,σ2
(T, ·)kH2≤ ε. (8)
Moreover, there exists real numbers θ′
2 and σ2′, with |θ′2− θ2| + |σ2′ − σ2| ≤ ε, such
that
ku(t, ·) − uδ2,θ2′,σ′2(t, ·)k
H2 −→
SMOOTH CONTROL OF NANOWIRES 3
In the proof of the main result, we shall choose control laws δ(·) so that δ(t) =
δ1 if t ≤ 0,
δ2+σ1−σt 2 if t ≥ T, (10)
where T > 0 is large, δ|[0,T ] is a smooth function such that t ˙δ remains small, and
the function δ is smooth overall IR.
Notice that this control shares robustness properties in H2 norm. The time T is
required to be large enough. It follows from this result that the family of travelling wall profiles (6) is approximately controllable in H2norm, locally in δ and globally
in σ, in time sufficiently large.
2. Proof of Theorem 1.1. Similarly as in [7,8,9], it is relevant to first reexpress the Landau-Lifschitz equation in adapted coordinates.
2.1. Preliminaries. The following formulas, easy to establish, will be useful next: • d dθRθ= 0 0 0 0 − sin θ − cos θ 0 cos θ − sin θ = Rθ+π 2 − e1e T 1 = Rπ 2Rθ− e1e T 1; • v ∧ e1= −Rπ 2v + v1e1; • Rθu ∧ Rθv = Rθ(u ∧ v); • a ∧ (b ∧ c) = b(a.c) − c(a.b); • Rθ(IRe1) = IRe1.
It is clear from Equation (2) that the solution u has a constant norm. Up to normalizing, assume this norm is equal to 1. Set v(t, x) = R−δ(t)t(u(t, x − δ(t)t));
then, v has a constant norm too, equal to 1. Using the above formulas, computations lead to
vt= −v ∧ h(v) − v ∧ (v ∧ h(v)) − δ(vx+ v1v − e1) − t ˙δ(vx− v3e2+ v2e3), (11)
where we recall that h(v) = vxx− v2e2− v3e3. Define
M1(x) = 1 ch x 0 −th x and M2= 0 1 0 .
In the frame (M0(x), M1(x), M2), the solution v : IR+× IR −→ S2⊂ IR3 writes in
the form v(t, x) =p1 − r1(t, x)2− r2(t, x)2M0(x) + r1(t, x)M1(x) + r2(t, x)M2. Note that: • M′ 0(x) = 1 ch xM1(x), M ′ 1(x) = − 1 ch xM0, M ′′ 0(x) = − sh x ch2xM1(x) − 1 ch2xM0; • e1= th x M0+ 1 ch xM1(x), e2= M2, e3= 1 ch xM0− th x M1(x); • h(M0) = − 2 ch2xM0; • M0∧ M1= M2, M0∧ M2= −M1, M1∧ M2= M0;
Then, easy but lengthy computations, not reported here, show that v is solution of (11) if and only if r =r1
r2
satisfies
where R(t, δ, ˙δ, x, r, rx, rxx) = − δℓ 00 ℓ r + G(r)rxx+ H1(x, r)rx+ H2(r)(rx, rx) + P (x, r) − δB(x, r) − t ˙δC(x, r), (13) and • A = L L −L L with L = ∂xx+ (1 − 2th2x)Id; • ℓ = ∂x+ th x Id;
• G(r) is the matrix defined by
G(r) = r1r2 p1 − krk2 r2 2 p1 − krk2 +p1 − krk 2− 1 − r 2 1 p1 − krk2 −p1 − krk 2+ 1 − r1r2 p1 − krk2 ;
• H1(x, r) is the matrix defined by
H1(x, r) = 2 p1 − krk2ch x r2p1 − krk2− r1r22 −r2+ r2r21 r2− r32 p1 − krk2r2+ r1r22 ;
• H2(r) is the quadratic form on IR2 defined by
H2(r)(X, X) = (1 − krk 2)XTX + (rTX)2 (1 − krk2)3/2 p1 − krk2r 1+ r2 p1 − krk2r 2− r1 ; • P (x, r) = P1(x, r) P2(x, r) , with P (x, r) =2r2(p1 − krk2− 1) 1 ch2x− 2r1r2 sh x ch2x− 2r1krk 2 1 ch2x − 2r21p1 − krk2 sh x ch2x+ r 3 1+ r2(1 −p1 − krk2) + r1r22, and P2(x, r) = − 2r1(p1 − krk2− 1) 1 ch2x+ 2r 2 1 sh x ch2x− 2r2krk 2 1 ch2x − 2r1r2p1 − krk2 sh x ch2x+ r2krk 2, • B(x, r) = (∂x+ th x)r + 1 ch x p1 − krk2− 1 + r2 1 r1r2 + th x (p1 − krk2− 1)r, • C(x, r) = ∂x+ th x0 −11 0 r +p1 − krk 2 ch x 1 −1 .
SMOOTH CONTROL OF NANOWIRES 5
It is clear that there holds
G(r) = O(krk2), H1(x, r) = O(krk), H2(r) = O(krk), P (x, r) = O(krk2), B(x, r) = O(krk + krxk), C(x, r) = O(krk + krxk),
uniformly with respect to the variable x ∈ IR. Then, we infer that there exists a constant C > 0 such that, if krk2
IR2 = krk2≤1
2 and |δ| ≤ 1, then, for all p, q ∈ IR 2,
for all x, t, ε ∈ IR,
kR(t, δ, ε, x, r,p, q)kIR2 ≤ C |δ|krkIR2+ |δ|kpk IR2+ t|ε| + t|ε|kpk IR2 + krk2IR2kqkIR2+ krkIR2kpkIR2+ krkIR2kpk2 IR2+ krk2 IR2 . (14)
From this a priori estimate, one might consider R(t, δ, ˙δ, x, r, rx, rxx) as a remainder
term in Equation (12). The proof uses stability properties established for the linear operator A, so as to establish. We next follow the same lines as in [9].
2.2. Change of coordinates. The operator L is a self-adjoint operator on L2(IR),
of domain H2(IR), and L = −ℓ∗ℓ with ℓ = ∂
x+ th x Id (one has ℓ∗= −∂x+ th x Id).
It follows that L is nonpositive, and that ker L = ker ℓ is the one dimensional subspace of L2(IR) generated by 1
ch x. In particular, the operator L, restricted to
the subspace E = (ker L)⊥, is negative.
Remark 1. On the subspace E: • the norms k(−L)1/2f k
L2
(IR) and kf kH1
(IR) are equivalent;
• the norms kLf kL2
(IR) and kf kH2
(IR) are equivalent;
• the norms k(−L)3/2f k
L2(IR) and kf kH3(IR) are equivalent.
Writing A = JL, with
J = 1 1 −1 1
,
it is clear that the kernel of A is ker A = ker L × ker L; it is the two dimensional space of L2(IR2) generated by
a1(x) = 0 1 ch x and a2(x) = 1 ch x 0 .
Moreover, combining the facts that L|(ker L)⊥ is negative and that Spec J = {1 +
i, 1 − i}, it follows that the operator A, restricted to the subspace E = (ker A)⊥, is
negative.
In what follows, solutions r of (12) are written as the sum of an element of ker A and of an element of E. Since Equation (11) is invariant with respect to translations in x and rotations around the axis e1, for every Λ = (θ, σ) ∈ IR2,
MΛ(x) = RθM0(x − σ) is solution of (11). Define
RΛ(x) =hMhMΛ(x), M1(x)i Λ(x), M2i
, the coordinates of MΛ(x) in the mobile frame (M1(x), M2(x)).
The mapping
Ψ : IR2× E −→ H2(IR)
(Λ, W ) 7−→ r(x) = RΛ(x) + W (x)
is a diffeomorphism from a neighborhood U of zero in IR2× E into a neighborhood V of zero in H2(IR). Indeed, if r = R
Λ+ W with W ∈ E, then, by definition,
hr, a1iL2= hRΛ, a1iL2 and hr, a2iL2 = hRΛ, a2iL2. (15)
Conversely, if Λ ∈ IR2 satisfies ((15)), then W = r − RΛ ∈ E. The mapping
h : IR2−→ IR2, defined by h(Λ) = (hRΛ, a1iL2, hRΛ, a2iL2) is smooth and satisfies
dh(0) = −2 Id, thus is a local diffeomorphism at (0, 0). It follows easily that Ψ is a local diffeomorphism at zero.
Therefore, every solution r of (12), as long as it stays1 in the neighborhood V,
can be written as
r(t, ·) = RΛ(t)(·) + W (t, ·), (16)
where W (t, ·) ∈ E and Λ(t) ∈ IR2, for every t ≥ 0, and (Λ(t), W (t, ·)) ∈ U. In these new coordinates2, Equation (12) leads to (see [7] for the details of computations)
Wt(t, x) = AW (t, x) + R(t, δ, ε, Λ(t), x, W (t, x), Wx(t, x), Wxx(t, x)), Λ′(t) = M(Λ(t), W (t, ·), Wx(t, ·)), (17) where R : IR × IR × IR × IR2× IR × H2(IR)2 × H1(IR)2 × L2(IR)2 −→ E and M : IR2× H1(IR)2 × L2(IR)2
−→ IR2 are nonlinear mappings, for which there exist constants K > 0 and η > 0 such that
kR(t, δ, ε, Λ, ·, W, Wx, Wxx)k(H1 (IR))2 ≤ KkΛkIR2+ |δ| + t|ε| + kW k (H2(IR))2 kW k(H3(IR))2+ Kt|ε|, (18) |M(Λ, W, Wx)| ≤ K kΛkIR2+ kW k (H1(IR))2 kW k(H1(IR))2, (19)
for every W ∈ E, every δ ∈ IR, every t ≥ 0, and every Λ ∈ IR2satisfying kΛkIR2 ≤ η.
Note that, since L is selfadjoint, it follows that AW ∈ E, for every W ∈ E, and thus (17) makes sense.
2.3. Asymptotic estimates. Denoting W =WW1
2 , define on H2(IR)2 × IR2 the function V(W ) = 1 2 L 0 0 L W 2 (L2(IR))2 =1 2kLW1k 2 L2(IR)+ 1 2kLW2k 2 L2(IR). (20)
Remark 2. It follows from Remark1that, on the subspace E = (ker A)⊥,pV(W ) is a norm, which is equivalent to the norm kW k2
(H2(IR2)).
1This a priori estimate will be a consequence of the stability property derived next.
2This decomposition is actually quite standard and has been used e.g. in [14] to establish
stability properties of static solutions of semilinear parabolic equations, and in [2,19] to prove stability of travelling waves.
SMOOTH CONTROL OF NANOWIRES 7
Consider a solution (W, Λ) of (17), such that W (0, ·) = W0(·) and Λ(0) = Λ0.
Since L is selfadjoint, one has d dtV(W (t, ·)) = AW,L 2W 1 L2W 2 (L2(IR))2 + (−L) 1/2 0 0 (−L)1/2 R(t, δ, ε, Λ, ·, W, Wx, Wxx),(−L) 3/2W 1 (−L)3/2W 2 (L2(IR))2 . (21) Concerning the first term of the right-hand side of (21), one computes
AW,L 2W 1 L2W 2 (L2(IR2))2 = −k(−L)3/2W1k(L2(IR))2− k(−L)3/2W2k (L2(IR))2,
and, using Remark1, there exists a constant C1> 0 such that
AW,L 2W 1 L2W 2 (L2(IR))2 ≤ −C1kW k2(H3(IR))2. (22)
Concerning the second term of the right-hand side of (21), one deduces from the Cauchy-Schwarz inequality, from Remark1, and from the estimate (18), that
(−L)1/2 0 0 (−L)1/2 R(t, δ, ε, Λ, ·, W, Wx, Wxx),(−L) 3/2W 1 (−L)3/2W 2 (L2(IR))2 ≤ kR(t, δ, ε, Λ, ·, W, Wx, Wxx)k(H1(IR))2kW k (H3(IR))2 ≤ KkΛkIR2+ |δ| + t|ε| + kW k (H2(IR))2 kW k2(H3(IR))2+ t|ε|kW k(H3(IR))2 ≤ K kΛkIR2+ |δ| + t|ε| + kW k (H2(IR))2+ 1 2ξ2 kW k2(H3(IR))2+ ξ2 2 t 2ε2, (23) where, to get the last line, we used the inequality
t|ε|kW k(H3 (IR))2 ≤ ξ2 2t 2ε2+ 1 2ξ2kW k 2 (H3(IR))2;
here, ξ denotes some real number to be chosen later. One infers from (21), (22) and (23) that
d dtV(W ) ≤ −C1+K kΛkIR2+ |δ| + t| ˙δ| + kW k (H2 (IR))2+ 1 2ξ2 kW k2 (H3(IR))2 +ξ 2 2t 2˙δ2.
Fix ǫ > 0; then, under the a priori estimates kΛ(t)kIR2+ |δ| + t| ˙δ| + kW (t, ·)k (H2(IR))2+ 1 2ξ2 ≤ C1 2K and ξ2 2 t 2˙δ2≤ ǫ,
there holds d dtV(W (t, ·)) ≤ − C1 2 kW (t, ·)k 2 (H3 (IR))2+ ǫ ≤ −C1 2 kW (t, ·)k 2 (H2(IR))2+ ǫ ≤ −C2V(W (t, ·)) + ǫ.
The existence of a constant C2 > 0 follows from Remark 2. Therefore,
choos-ing ξ > 0 large enough, there exist constants C3 > 0 and C4 > 0 such that, if
kW (0, ·)k(H2(IR))2 ≤ C1
6K, if the a priori estimate
max
0≤s≤tkΛ(s)kIR
2≤
C1
6K (24)
holds, and if the control function δ(·) is chosen so that |δ(t)| + t| ˙δ(t)| ≤ C1
6K (25)
and
t2˙δ(t)2≤ 2ǫ/ξ2 (26) for every t ≥ 0, then
kW (s, ·)k(H2(IR))2 ≤ C3e−C4skW (0, ·)k
(H2(IR))2+ C3ǫ, (27)
for every s ∈ [0, T ], and moreover, one deduces from (17), (19), and (27) that, if the a priori estimate (24) holds, then
kΛ(t)kIR2≤ kΛ(0)kIR2+ C1C3 4 kW (0, ·)k(H2(IR))2 Z t 0 e−C4s ds + KC32kW (0, ·)k2(H2(IR))2 Z t 0 e−2C4s ds ≤ kΛ(0)kIR2+ C1C3 4C4 kW (0, ·)k(H2 (IR))2+ K C2 3 2C4 kW (0, ·)k2 (H2(IR))2. (28)
From the above a priori estimates, we infer that, if the quantity kΛ(0)kIR2 +
kW (0, ·)k(H2(IR))2 is small enough, and if the control function δ fits the conditions
(25) and (26), then kΛ(t)kIR2 remains small, for every t ≥ 0, and kW (t, ·)k
(H2(IR))2
is exponentially decreasing to 0.
Finally we must choose a smooth control function such that u(t, x) is close to uδ1,θ1,σ1(t, x) at initial time, and close to uδ2,θ2,σ2(t, x) for large times. Hence, we
can choose the function δ such that δ(t) = δ1 for t ≤ 0. Then, with the reasoning
above, we enforce v(t, x) to remain close to M0(x), that is, the solution u(t, x)
follows the profile uδ(t),θ1,σ1(t, x). At times t ≥ T , we require u(t, x) to be close to
uδ2,θ2,σ2(t, x) for some θ
2; one must have, for t ≥ T ,
−σ1+ δ(t)t = −σ2+ δ2t,
and hence,
δ(t) = δ2+σ1− σ2
t .
To conclude, observe that it is possible to choose a function δ and a time T > 0 large enough, such that δ is smooth on IR and satisfies the above requirements and the estimates (25) and (26).
SMOOTH CONTROL OF NANOWIRES 9
The first part of the theorem, on the interval [0, T ], then follows from the above considerations.
For the second part, we use a stronger version of the estimate (27), namely, kW (s, ·)k(H2
(IR))2 ≤ C3e−C4skW (0, ·)k
(H2
(IR))2+ ξ2t2˙δ(t)2.
Since t2˙δ(t)2 is integrable, it follows from the above estimate, and from (17) and
(19), that kΛ′(t)k
IR2is integrable on [0, +∞). Hence, Λ(t) has a limit in IR2, denoted
Λ∞ = (θ∞, σ∞), as t tends to +∞. The theorem follows with θ′2 = θ2+ θ∞ and
σ′
2= σ2+ σ∞.
REFERENCES
[1] F. Alouges, T. Rivi`ere and S. Serfaty, N´eel and cross-tie wall energies for planar micromag-netic configurations, ESAIM Cont. Optim. Calc. Var., 8 (2002), 31–68.
[2] A. L. Bertozzi, A. M¨unch, M. Shearer and K. Zumbrun, Stability of compressive and under-compressive thin film travelling waves. The dynamics of thin fluid films, European J. Appl. Math., 12 (2001), no. 3, 253–291.
[3] F. Brown, ”Micromagnetics”, Wiley, New York (1963).
[4] G. Carbou and P. Fabrie, Time average in micromagnetism, J. Differential Equations, 147 (1998), no. 2, 383–409.
[5] G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded do-main, Differential Integral Equations, 14 (2001), no. 2, 213–229.
[6] G. Carbou, P. Fabrie and O. Gu`es, On the ferromagnetism equations in the non static case, Comm. Pure Appli. Anal., 3 (2004), no. 3, 367–393.
[7] G. Carbou and S. Labb´e, Stability for static walls in ferromagnetic nanowires Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), no. 2, 273–290.
[8] G. Carbou and S. Labb´e, Stability for walls in Ferromagnetic Nanowires, in ”Numerical mathematics and advanced applications” (2006), Springer, Berlin,539–546.
[9] G. Carbou, S. Labb´e and E. Tr´elat, Control of travelling walls in a ferromagnetic nanowire, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), no. 1, 51–59.
[10] A. DeSimone, R. V. Kohn, S. M¨uller, and F. Otto, Magnetic microstructures—a paradigm of multiscale problems, ICIAM 99 (2000), Edinburgh, Oxford Univ. Press, Oxford, 175–190. [11] H. Haddar and Patrick Joly, Stability of thin layer approximation of electromagnetic waves
scattering by linear and nonlinear coatings, J. Comput. Appl. Math., 143 (2002), no. 2, 201–236.
[12] L. Halpern and S.Labb´e, La th´eorie du micromagn´etisme. Mod´elisation et simulation du comportement des mat´eriaux magn´etiques, Matapli, 66 (2001), 77–92.
[13] J.-L. Joly, G. M´etivier and J. Rauch, Global solutions to Maxwell equations in a ferromagnetic medium, Ann. Henri Poincar´e, 1 (2000), no. 2, 307–340.
[14] T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349(1997), no. 1, 257–269.
[15] S. Labb´e, ”Simulation num´erique du comportement hyperfr´equence des mat´eriaux ferro-magn´etiques”, Ph.D thesis, Univ. Paris 13 (1998).
[16] S. Labb´e and P.-Y. Bertin, Microwave polarisability of ferrite particles with non-uniform magnetization, Journal of Magnetism and Magnetic Materials, 206 (1999), 93–105.
[17] L.D. Landau and E.M. Lifschitz, ”Electrodynamics of continuous media”, Course of Theo-retical Physics, Vol. 8, Translated from the Russian by J. B. Sykes and J. S. Bell, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1960 x+417 pp.
[18] T. Rivi`ere and S. Serfaty, Compactness, kinetic formulation, and entropies for a problem related to micromagnetics, Comm. Partial Differential Equations, 28 (2003), no. 1-2, 249– 269.
[19] V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 21 (2004), no. 3, 341–379.
[20] D. Sanchez, Behaviour of the Landau-Lifschitz equation in a periodic thin layer, Asymptot. Anal., 41 (2005), no. 1, 41–69.
[21] A. Visintin, On Landau Lifschitz equation for ferromagnetism, Japan J. Appl. Math., 2 (1985), no. 1, 69–84.
[22] H. Wynled, ”Ferromagnetism”, Encyclopedia of Physics, Vol. XVIII / 2, Springer Verlag, Berlin (1966).
E-mail address: carbou@math.u-bordeaux1.fr E-mail address: stephane.labbe@imag.fr
