- Email: [email protected]
On a phase transition model in ferromagnetism
Applied Mathematical Modelling 34 (2010) 3943–3948
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
On a phase transition model in ferromagnetism Mouhcine Tilioua Hassan I University, Polydisciplinary Faculty of Khouribga, P.O. Box 145, 25000 Khouribga, Morocco
a r t i c l e
i n f o
Article history: Received 14 June 2009 Received in revised form 13 March 2010 Accepted 30 March 2010 Available online 11 April 2010 Keywords: Ferromagnetic materials Non-isothermal phase transition Dimensional reduction
a b s t r a c t This paper deals with the limiting behavior of a phase transition model in ferromagnetism. The model describes the three-dimensional evolution of both thermodynamic and electromagnetic properties of the ferromagnetic material. We are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. We identify the limit problem by using the so-called energy method. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction and preliminary results Properties of matter at nanoscale may not be as predictable as those observed at larger scales. Important changes in behavior are caused not only by continuous modification of characteristics with diminishing size, but also by the emergence of totally new phenomena. Designed and controlled fabrication and integration of nanomaterials and nanodevices is likely to be revolutionary for sciences and technology. Among the materials being most actively studied, ferromagnetic materials stand out for being used in recording and electronics industries. In recent years the understanding of thin film behavior has been helped by the mathematical asymptotic analysis of energies defined on three-dimensional domains of vanishing thickness. So if this process can lead to 2D equations, it could be thought that such simplified equations are easier to solve than the original 3D ones in particular their numerical investigation through numerical simulations. This explains the amount of work devoted to dimensional reduction. In this paper, we are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. Our investigation has its starting point in the paper [1], where the authors propose a three-dimensional evolutive model and establish the existence and uniqueness of weak solutions. We intend to analyze the behavior of these solutions with one diminishing edge. The method consists in rescaling the e-thin domain into a reference body of unit thickness, so that the resulting equations will be defined on a fixed domain, while the dependence on e turns out to be explicit in the transversal derivatives which appear in the system. The second step is then to determine the limit of the rescaled problem as the thickness e tends to 0. This scaling technique is well known in elasticity, see for example Ciarlet [2]. Let us now describe the model equations. Let e be a real parameter taking values in a sequence of positive numbers converging to zero. We consider flat domains represented by the cylinder Xe = B (0, e) of R3 where B R2 a bounded and regular open set representing the cross section of Xe. The generic point of R3 is denoted by x ¼ ð^ x; x3 Þ with ^ x ¼ ðx1 ; x2 Þ 2 R2 . We assume that a ferromagnetic material occupies the domain Xe. Here and throughout the paper we use bold characters to denote the vector-valued functions. We set R3þ ¼ R2 Rþ and Q = (0, T) Xe for T > 0. The calculations combine phenomenological constitutive equations for the magnetization M and the absolute temperature h. According to Berti et al. [1], the system governing the evolution of the ferromagnetic material reads E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.03.030
3944
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
c@ t M ¼ mMM hc ðjMj2 1ÞM hM þ H in Q ;
ð1Þ
c1 @ t ðln hÞ þ c2 @ t h M @ t M ¼ k0 Mðln hÞ þ k1 Mh þ ^r inQ;
ð2Þ
where c, m, c1, c2, k0, k1 are strictly positive constants and hc is a certain temperature called Curie temperature. Here ^r is a known function of x, t. For simplicity we assume that ^r 0. The magnetic field H is the stray field that appears in the Maxwell’s equations. In the case where the wavelenghts are large compared to the size of the material, which is the context of the present work, the Maxwell system is usually replaced with the so-called quasi-static approximation. The magnetization M links the magnetic field H and the magnetic induction B by the relation B = lH + v(Xe)M where l is the magnetic permeability and v(Xe) is the characteristic function of Xe. The magnetic field H satisfies curl H = 0 by static Maxwell’s equations, and by the Faraday law we have div B = 0. Hence the magnetization M induces a magnetic field H given by
curl H ¼ 0;
ð3Þ
divðlH þ vðXe ÞMÞ ¼ 0; in Rþ R3þ with
ðlH þ vðXe ÞM Þ n ¼ 0 on @ Xe ;
ð4Þ
where n is the outer unit normal at the boundary oXe. Remark 1. Notice that the hypothesis curl H = 0 is not made in the differential system considered in [1]. From the first equation of (3), the magnetic field H may be derived from a scalar magnetic potential u, that is
H ¼ grad u; so that u satisfies
divðl grad u þ vðXe ÞMÞ ¼ 0 in Rþ R3þ ;
ð5Þ
ðl grad u þ vðXe ÞMÞ n ¼ 0 on @ Xe :
ð6Þ
with
Eqs. (1), (2) have to be fulfilled with initial and boundary conditions. Concerning the boundary conditions, we assume Neumann boundary condition both for magnetization and temperature
ðn rÞM ¼ 0;
ðn rÞh ¼ 0;
ð7Þ
on (0, T) oXe. For the initial data let
Mðx; 0Þ ¼ M 0 ðxÞ;
hðx; 0Þ ¼ h0 ðxÞ
ð8Þ
be given functions in Xe. Finally the initial magnetic field grad u0(x) satisfies the compatibility problem
(
divðl grad u0 þ vðXe ÞM 0 Þ ¼ 0 in R3þ ;
ð9Þ
ðl grad u0 þ vðXe ÞM 0 Þ n ¼ 0 on @ Xe : 0
Here we introduce some notation. For any Hilbert space X let kkX denote X-norm. Moreover we let X be the dual space of X. The letter C will denote various constants which are independent of e. We introduce the functional EðtÞ defined as
1 2
EðtÞ ¼
mkgrad Mk2L2 ðXe Þ þ
hc kMk4L4 ðXe Þ þ c2 khk2L2 ðXe Þ þ lkgrad uk2L2 ðR3 Þ þ 2c1 þ 2
hdx ;
Z
ð10Þ
Xe
and put
Eð0Þ ¼
1 2
mkgrad M 0 k2L2 ðXe Þ þ
hc kM 0 k4L4 ðXe Þ þ c2 kh0 k2L2 ðXe Þ þ lkgrad u0 k2L2 ðR3 Þ þ 2c1 þ 2
Z
h0 dx :
ð11Þ
Xe
We have the following energy estimate Lemma 1. For all t 2 (0, T) we have
EðtÞ þ
Z t c k@ t Mk2L2 ðXe Þ þ k1 kgrad hk2L2 ðXe Þ ds 6 b1 Eð0Þ þ b2 ; 2 0
where b1 and b2 are suitable positive constants depending on T.
ð12Þ
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
3945
Proof. The techniques to obtain (12) are analogous to those used in [1]. See also [3]. In order to get (12) we multiply (1), (2) respectively by @ tM, h, integrate by parts, sum the resulting equations and make use of Gronwall’s lemma. Notice that when multiplying (1) by @ tM we have to evaluate H@ tM. This may be done by multiplying (5) by @ tM and integrating by parts. h Remark 2. From (9) it follows that
l
Z R3þ
jgrad u0 j2 dx ¼
Z Xe
M 0 grad u0 dx;
ð13Þ
which implies that
kgrad u0 kL2 ðR3 Þ 6 þ
1
l
kM 0 kL2 ðXe Þ :
ð14Þ
In what follow, we assume that the initial data (M0, h0) are such that
M 0 2 H1 ðXe Þ;
h0 2 L2 ðXe Þ:
ð15Þ
Remark 3. Estimate (14) and hypothesis (15) ensure a bounded energy for T > 0 fixed and finite. The following existence and uniqueness result has been proved. Theorem 1. (Berti et al. [1]) Assume that (15) and (9) hold, then for every T > 0, problem (1)–(5) admits a solution (M, h, u) such that
M 2 L2 ð0; T; H2 ðXe ÞÞ \ H1 ð0; T; L2 ðXe ÞÞ; h 2 L2 ð0; T; H1 ðXe ÞÞ; h > 0; ln h 2 L2 ð0; T; H1 ðXe ÞÞ; c1 ln h þ c2 h 2 H1 ð0; T; H1 ðXe Þ0 Þ; gradu 2 L1 ð0; T; L2 ðR3þ ÞÞ: Moreover the energy estimate (12) holds. The rest of the paper is organized as follows. In the next section we introduce the natural scaling for the problem and prove uniform bounds for the solutions, with respect to vanishing parameter. Section 3 is devoted to the characterization of the limiting magnetic potential. We identify the limit problem in Section 4. Finally in the last section concluding remarks and suggestions for future work are given. 2. Scaling and uniform bounds Let (M, h, u) be the solution of the problem posed in Xe. We introduce the change of variables
ðx1 ; x2 ; x3 Þ ¼ ðx; y; ezÞ with ðX 0 ; zÞ ¼ X 2 X ¼ B ð0; 1Þ;
X 0 ¼ ðx; yÞ:
Let (me, he, /e) be the fields associated with (M, h, u). The magnetization field me satisfies in Q ¼ ð0; TÞ X the following equation
b me þ c@ t me ¼ m M
m 2 e d e þ 1 @ /e u ; @ m hc ðjme j2 1Þme he me þ grad/ e z 3 e2 z
ð16Þ
where we have set (u1, u2, u3) to represent the canonical basis of R3 . The scaled temperature satisfies the following equation
b ðln he Þ þ c1 @ t ðln he Þ þ c2 @ t he me @ t me ¼ k0 M
k0
@ 2 z
e
1 @ z he he
b he þ þ k1 M
k1
e2
@ 2z he :
ð17Þ
The magnetic potential /e satisfies in Rþ R3þ the equations
8 d /e þ vðXÞm ^e Þ þ @ z l2 @ z /e þ 1 vðXÞme u3 ¼ 0;
ð18Þ
d and M b represent divergence, gradient and Laplacian operators, respectively, with respect to the where the operators d div; grad d may also be considered as a ^ ^ variable x and m ¼ ðm1 ; m2 ; 0Þ with (m1, m2) are the first two components of m. The vector grad/ 2D vector or a 3D vector where the third component is 0. Global existence of solutions (me, he, /e) of the new system is guaranteed by Theorem 1. We shall describe the behavior of such solutions when e ? 0.
3946
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
By using the same method as in Lemma 2, the energy estimate satisfied by (me, he, /e) is
E e ðtÞ þ
Z t
c
2
0
e d e k22 þ k1 k@ z he k22 k@ t me k2L2 ðXÞ þ k1 k gradh L ðXÞ L ðXÞ ds 6 b1 E ð0Þ þ b2 ; 2
ð19Þ
e
where
E e ðtÞ ¼
1 2
d me k22 þ m k@ z me k22 þ hc kme k44 þ c2 khe k22 þ lk grad d /e k22 3 þ l k@ z /e k22 3 þ 2c1 mk grad L ðXÞ L ðXÞ L ðXÞ L ðXÞ L ðRþ Þ L ðRþ Þ 2 e2 e2
Z
he dX ;
X
ð20Þ and b1 and b2 are the constants of Lemma 1. Let us discuss the admissibility criterion for the initial data ðme0 ; he0 ; /e0 Þ. The condition that me0 is independent of the vard /e k22 3 þ 1 k@ z /e k22 3 6 C uniformly with respect iable z ensures that E e ð0Þ 6 C. In fact, under this condition we have k grad 0 L ðRþ Þ
to e. This follows from estimate (14).
e2
0 L ðRþ Þ
Lemma 2. Let me0 ðX 0 Þ such that me0 is uniformly bounded in H1(X) and he0 2 L2 ðXÞ. Then we have
E e ðtÞ þ
Z t 0
c
2
d he k22 þ k1 k@ he k22 k@ t me k2L2 ðXÞ þ k1 k grad z L ðXÞ L ðXÞ ds 6 C: 2
e
ð21Þ
We have the following uniform bounds. Lemma 3. Under the hypotheses of Lemma 2, the solutions (me, he, /e) satisfy the following estimates
8 e d e > > > k grad m kL1 ðRþ ;L2 ðXÞÞ þ k@ t m kL2 ðRþ ;L2 ðXÞÞ 6 C; > > > < k grad d e k 1 þ 2 3 6 C; d he k 2 þ 2 þ k grad/ L ðR ;L ðXÞÞ
L ðR ;L ðRþ ÞÞ
> > k@ z me kL1 ðRþ ;L2 ðXÞÞ þ k@ z /e kL1 ðRþ ;L2 ðR3 ÞÞ þ k@ z he kL1 ðRþ ;L2 ðXÞÞ 6 C e; > > þ > > e : kh kL1 ðRþ ;L2 ðXÞÞ þ khe kL1 ðRþ ;L1 ðXÞÞ 6 C:
ð22Þ
There exists a subsequence still denoted (me, he, /e) such that the following convergences hold
8 me * m weakly H in L1 ðRþ ; H1 ðXÞÞ \ L1 ðRþ ; L4 ðXÞÞ; > > > > < @ me * @ m weakly in L2 ðRþ ; L2 ðXÞÞ; t t 1 2 e > > @ z m ! 0 strongly in L ðRþ ; L ðXÞÞ; > > : e 2 þ 2 m ! m strongly in Lloc ðR ; L ðXÞÞ:
ð23Þ
The last convergence is a consequence of the classical use of Aubin’s compactness lemma, see for example [4]. For the temperature we have
8 e 2 1 þ > < h * h weakly in L ðR ; H ðXÞÞ; 1 e h * h weakly H in L ðRþ ; L2 ðXÞÞ; > : @ z he ! 0 strongly in L2 ðRþ ; L2 ðXÞÞ;
ð24Þ
and for magnetic potential we get
(
d d e * grad/ weakly H in L1 ðRþ ; L2 ðR3þ ÞÞ; grad/ @ z /e ! 0 strongly in L1 ðRþ ; L2 ðR3þ ÞÞ:
ð25Þ
Thus in the limit the admissible fields do not depend on the direction normal to the thin film. 3. Characterization of the limiting magnetic potential We have Lemma 4. The limiting magnetic potential / satisfies
b / þ vðXÞ d lM divm ¼ 0 in Rþ R3þ :
ð26Þ
3947
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
Proof. Let F 2 DðRþ R3þ Þ. The weak formulation of (18) is
Z
Rþ R3þ
d div
Z
d /e þ vðXÞc l grad m e F dX dt þ
Rþ R3þ
l e 1 @ / þ vðXÞme u3 F dX dt ¼ 0: e2 z e
@z
Arguing that in the limit, the magnetic potential is independent of the variable z (see (25)) we choose test functions F which are independent of z. This allows to get, by using boundary conditions (18), the Eq. (26). h The scaled Eq. (16) involves the quantity He :¼ 1e @ z /e for which we have the following convergence result. Lemma 5. We have
He * H weakly H in L1 ðRþ ; L2 ðR3þ ÞÞ;
ð27Þ
where H is given by
Hðt; X 0 ; zÞ ¼
vðXÞ mðt; X 0 Þ u3 in Rþ R3þ : l
ð28Þ
Proof. We refer to [3]. See also [5]. h
4. Passing to the limit We are now ready to state the main result of this paper. We have Theorem 2. Let (me, he, /e) be a global solution of problem (16)–(18). Let (m, h, /) be the weak-w limit of a subsequence of (me, he, /e). Then (m, h, /) is independent of the variable z and satisfies in QT ¼ ð0; TÞ B the coupled system
(
d / 1 ðm u3 Þu3 ; b m þ grad c@ t m þ hc ðjmj2 1Þm þ hm ¼ m M l
ð29Þ
b ðln hÞ þ k1 M b h; c1 @ t ðln hÞ þ c2 @ t h m @ t m ¼ k0 M where the magnetic potential / satisfies Eq. (26).
Proof. Let Fe and Ge be two test functions defined in DðQÞ. The weak formulation of the problem reads in Q as follows. For the magnetization field we have
Z
ðc@ t me þ hc ðjme j2 1Þme þ he me ÞF e dX dt ¼ m
Q
þ
Z
Z
d me grad d F e dX dt grad
Q
d /e F e dX dt þ 1 grad
e
Q
Z
m e2
Z
@ z me @ z F e dX dt
Q
@ z /e u3 F e dX dt;
ð30Þ
Q
while for the temperature one gets
Z
c1
@ t ðln he ÞGe dXdt þ c2
Q
¼ k0
Z Q
Z Q
Z
@ t he Ge dXdt
d d Ge dX dt k0 gradðln he Þ grad 2
e
Z Q
me @ t me Ge dXdt
Q
1 @ z he @ z Ge dX dt k1 he
Z Q
d he grad d Ge dX dt k1 grad 2
e
Z
@ z he @ z Ge dX dt:
ð31Þ
Q
In order to pass to the limit as e ? 0, we argue that, in the limit, (me, he) is independent of the variable z. We use test functions of the type
Z e ðt; X 0 ; zÞ ¼ Zðt; X 0 Þ þ eZ 0 ðt; X 0 ÞhðezÞ;
ð32Þ
where h 2 Dð½0; 1Þ and Z; Z 0 2 DðBÞ. Note that Ze satisfies
d e ¼ grad d Zðt; X 0 Þ þ ehðezÞ gradZ d 0 ðt; X 0 Þ; gradZ @ z Z e ¼ e2 h ðezÞZ 0 ðt; X 0 Þ: 0
ð33Þ
Let us pass to the limit into (31). Note that by Theorem 1, the solution of (17) is such that he 2 L2(0, T; H1(X)), c1lnhe + c2he 2 0 H1(0, T; H1(X) ) and lnhe 2 L2(0, T; H1(X)). Thus, all the terms in (31) can pass to the limit because they are written as weakstrong products. Recall the strong convergences me ? m in L2loc ðRþ ; L2 ðXÞÞ and @ zhe ? 0 in L2 ðRþ ; L2 ðXÞÞ. We have
Z ðc1 @ t ðln he Þ þ c2 @ t he ÞGe dX dt ! ðc1 @ t ðln hÞ þ c2 @ t hÞG0 dX 0 dt; Q QT Z Z me @ t me Ge dX dt ! m @ t mG0 dX 0 dt:
Z
Q
QT
3948
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
Next we have
k0
Z
d d Ge dX dt ! k0 gradðln he Þ grad
Q
k0
e
k1
Z
Q
1 @ z he @ z Ge dX dt ! 0; he
d he grad d Ge dX dt ! k1 grad
Q
k1
e
2
d d G0 dX 0 dt; gradðln hÞ grad
QT
Z
2
Z
Z
Z
d h grad d G0 dX 0 dt; grad
QT
@ z he @ z Ge dX dt ! 0:
Q
Similarly, passing to the limit in (30) by using convergences (23), Lemma 5 and properties (33) of test functions, yields (29). h
5. Concluding remarks In this paper we have derived a thin layer model in the context of phase transitions in ferromagnetism. The dimensional reduction 3D-2D is performed by using a scaling technique combined with the method of oscillating test functions. We have shown that for very thin samples the magnetization, the temperature and the magnetic potential satisfy a two-dimensional time-dependent coupled problem (Theorem 2). Inspection of (29) reveals the effect induced by the thin layer behavior. It takes the form l1 ðm u3 Þu3 which disfavors the out-of-plane component of the limiting magnetic field grad/. The reduced problem obtained in this work can be used as a toy model for introducing the mathematical approach which can be adapted to more realistic situations. Indeed, further progress is needed by investigating and setting up new models that describe, for example, the effect of the surface energy which plays a crucial role in thin film dynamics. Note that the effect of impurities and surface roughness is also of interest. Finally, we may mention that it becomes increasingly important to identify simpler models, valid in appropriate regimes, whose behavior is easier to understand or simulate in order to determine the microscopic configuration of the magnetization that is not directly accessible by experiment. Acknowledgements The author thanks the referees for careful reading and giving constructive comments to improve the paper. References [1] V. Berti, M. Fabrizio, C. Giorgi, A three-dimensional phase transition model in ferromagnetism: existence and uniqueness, J. Math. Anal. Appl. 355 (2009) 661–674. [2] P.G. Ciarlet, Introduction to Linear Shell Theory, North-Holland, 1998. [3] K. Hamdache, M. Tilioua, Interlayer exchange coupling for ferromagnets through spacers, SIAM J. Appl. Math. 64 (3) (2004) 1077–1097. [4] J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [5] G. Gioia, R.D. James, Micromagnetics of very thin films, Proc. R. Soc. Lond. A 453 (1997) 213–223.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
On a phase transition model in ferromagnetism Mouhcine Tilioua Hassan I University, Polydisciplinary Faculty of Khouribga, P.O. Box 145, 25000 Khouribga, Morocco
a r t i c l e
i n f o
Article history: Received 14 June 2009 Received in revised form 13 March 2010 Accepted 30 March 2010 Available online 11 April 2010 Keywords: Ferromagnetic materials Non-isothermal phase transition Dimensional reduction
a b s t r a c t This paper deals with the limiting behavior of a phase transition model in ferromagnetism. The model describes the three-dimensional evolution of both thermodynamic and electromagnetic properties of the ferromagnetic material. We are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. We identify the limit problem by using the so-called energy method. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction and preliminary results Properties of matter at nanoscale may not be as predictable as those observed at larger scales. Important changes in behavior are caused not only by continuous modification of characteristics with diminishing size, but also by the emergence of totally new phenomena. Designed and controlled fabrication and integration of nanomaterials and nanodevices is likely to be revolutionary for sciences and technology. Among the materials being most actively studied, ferromagnetic materials stand out for being used in recording and electronics industries. In recent years the understanding of thin film behavior has been helped by the mathematical asymptotic analysis of energies defined on three-dimensional domains of vanishing thickness. So if this process can lead to 2D equations, it could be thought that such simplified equations are easier to solve than the original 3D ones in particular their numerical investigation through numerical simulations. This explains the amount of work devoted to dimensional reduction. In this paper, we are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. Our investigation has its starting point in the paper [1], where the authors propose a three-dimensional evolutive model and establish the existence and uniqueness of weak solutions. We intend to analyze the behavior of these solutions with one diminishing edge. The method consists in rescaling the e-thin domain into a reference body of unit thickness, so that the resulting equations will be defined on a fixed domain, while the dependence on e turns out to be explicit in the transversal derivatives which appear in the system. The second step is then to determine the limit of the rescaled problem as the thickness e tends to 0. This scaling technique is well known in elasticity, see for example Ciarlet [2]. Let us now describe the model equations. Let e be a real parameter taking values in a sequence of positive numbers converging to zero. We consider flat domains represented by the cylinder Xe = B (0, e) of R3 where B R2 a bounded and regular open set representing the cross section of Xe. The generic point of R3 is denoted by x ¼ ð^ x; x3 Þ with ^ x ¼ ðx1 ; x2 Þ 2 R2 . We assume that a ferromagnetic material occupies the domain Xe. Here and throughout the paper we use bold characters to denote the vector-valued functions. We set R3þ ¼ R2 Rþ and Q = (0, T) Xe for T > 0. The calculations combine phenomenological constitutive equations for the magnetization M and the absolute temperature h. According to Berti et al. [1], the system governing the evolution of the ferromagnetic material reads E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.03.030
3944
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
c@ t M ¼ mMM hc ðjMj2 1ÞM hM þ H in Q ;
ð1Þ
c1 @ t ðln hÞ þ c2 @ t h M @ t M ¼ k0 Mðln hÞ þ k1 Mh þ ^r inQ;
ð2Þ
where c, m, c1, c2, k0, k1 are strictly positive constants and hc is a certain temperature called Curie temperature. Here ^r is a known function of x, t. For simplicity we assume that ^r 0. The magnetic field H is the stray field that appears in the Maxwell’s equations. In the case where the wavelenghts are large compared to the size of the material, which is the context of the present work, the Maxwell system is usually replaced with the so-called quasi-static approximation. The magnetization M links the magnetic field H and the magnetic induction B by the relation B = lH + v(Xe)M where l is the magnetic permeability and v(Xe) is the characteristic function of Xe. The magnetic field H satisfies curl H = 0 by static Maxwell’s equations, and by the Faraday law we have div B = 0. Hence the magnetization M induces a magnetic field H given by
curl H ¼ 0;
ð3Þ
divðlH þ vðXe ÞMÞ ¼ 0; in Rþ R3þ with
ðlH þ vðXe ÞM Þ n ¼ 0 on @ Xe ;
ð4Þ
where n is the outer unit normal at the boundary oXe. Remark 1. Notice that the hypothesis curl H = 0 is not made in the differential system considered in [1]. From the first equation of (3), the magnetic field H may be derived from a scalar magnetic potential u, that is
H ¼ grad u; so that u satisfies
divðl grad u þ vðXe ÞMÞ ¼ 0 in Rþ R3þ ;
ð5Þ
ðl grad u þ vðXe ÞMÞ n ¼ 0 on @ Xe :
ð6Þ
with
Eqs. (1), (2) have to be fulfilled with initial and boundary conditions. Concerning the boundary conditions, we assume Neumann boundary condition both for magnetization and temperature
ðn rÞM ¼ 0;
ðn rÞh ¼ 0;
ð7Þ
on (0, T) oXe. For the initial data let
Mðx; 0Þ ¼ M 0 ðxÞ;
hðx; 0Þ ¼ h0 ðxÞ
ð8Þ
be given functions in Xe. Finally the initial magnetic field grad u0(x) satisfies the compatibility problem
(
divðl grad u0 þ vðXe ÞM 0 Þ ¼ 0 in R3þ ;
ð9Þ
ðl grad u0 þ vðXe ÞM 0 Þ n ¼ 0 on @ Xe : 0
Here we introduce some notation. For any Hilbert space X let kkX denote X-norm. Moreover we let X be the dual space of X. The letter C will denote various constants which are independent of e. We introduce the functional EðtÞ defined as
1 2
EðtÞ ¼
mkgrad Mk2L2 ðXe Þ þ
hc kMk4L4 ðXe Þ þ c2 khk2L2 ðXe Þ þ lkgrad uk2L2 ðR3 Þ þ 2c1 þ 2
hdx ;
Z
ð10Þ
Xe
and put
Eð0Þ ¼
1 2
mkgrad M 0 k2L2 ðXe Þ þ
hc kM 0 k4L4 ðXe Þ þ c2 kh0 k2L2 ðXe Þ þ lkgrad u0 k2L2 ðR3 Þ þ 2c1 þ 2
Z
h0 dx :
ð11Þ
Xe
We have the following energy estimate Lemma 1. For all t 2 (0, T) we have
EðtÞ þ
Z t c k@ t Mk2L2 ðXe Þ þ k1 kgrad hk2L2 ðXe Þ ds 6 b1 Eð0Þ þ b2 ; 2 0
where b1 and b2 are suitable positive constants depending on T.
ð12Þ
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
3945
Proof. The techniques to obtain (12) are analogous to those used in [1]. See also [3]. In order to get (12) we multiply (1), (2) respectively by @ tM, h, integrate by parts, sum the resulting equations and make use of Gronwall’s lemma. Notice that when multiplying (1) by @ tM we have to evaluate H@ tM. This may be done by multiplying (5) by @ tM and integrating by parts. h Remark 2. From (9) it follows that
l
Z R3þ
jgrad u0 j2 dx ¼
Z Xe
M 0 grad u0 dx;
ð13Þ
which implies that
kgrad u0 kL2 ðR3 Þ 6 þ
1
l
kM 0 kL2 ðXe Þ :
ð14Þ
In what follow, we assume that the initial data (M0, h0) are such that
M 0 2 H1 ðXe Þ;
h0 2 L2 ðXe Þ:
ð15Þ
Remark 3. Estimate (14) and hypothesis (15) ensure a bounded energy for T > 0 fixed and finite. The following existence and uniqueness result has been proved. Theorem 1. (Berti et al. [1]) Assume that (15) and (9) hold, then for every T > 0, problem (1)–(5) admits a solution (M, h, u) such that
M 2 L2 ð0; T; H2 ðXe ÞÞ \ H1 ð0; T; L2 ðXe ÞÞ; h 2 L2 ð0; T; H1 ðXe ÞÞ; h > 0; ln h 2 L2 ð0; T; H1 ðXe ÞÞ; c1 ln h þ c2 h 2 H1 ð0; T; H1 ðXe Þ0 Þ; gradu 2 L1 ð0; T; L2 ðR3þ ÞÞ: Moreover the energy estimate (12) holds. The rest of the paper is organized as follows. In the next section we introduce the natural scaling for the problem and prove uniform bounds for the solutions, with respect to vanishing parameter. Section 3 is devoted to the characterization of the limiting magnetic potential. We identify the limit problem in Section 4. Finally in the last section concluding remarks and suggestions for future work are given. 2. Scaling and uniform bounds Let (M, h, u) be the solution of the problem posed in Xe. We introduce the change of variables
ðx1 ; x2 ; x3 Þ ¼ ðx; y; ezÞ with ðX 0 ; zÞ ¼ X 2 X ¼ B ð0; 1Þ;
X 0 ¼ ðx; yÞ:
Let (me, he, /e) be the fields associated with (M, h, u). The magnetization field me satisfies in Q ¼ ð0; TÞ X the following equation
b me þ c@ t me ¼ m M
m 2 e d e þ 1 @ /e u ; @ m hc ðjme j2 1Þme he me þ grad/ e z 3 e2 z
ð16Þ
where we have set (u1, u2, u3) to represent the canonical basis of R3 . The scaled temperature satisfies the following equation
b ðln he Þ þ c1 @ t ðln he Þ þ c2 @ t he me @ t me ¼ k0 M
k0
@ 2 z
e
1 @ z he he
b he þ þ k1 M
k1
e2
@ 2z he :
ð17Þ
The magnetic potential /e satisfies in Rþ R3þ the equations
8 d /e þ vðXÞm ^e Þ þ @ z l2 @ z /e þ 1 vðXÞme u3 ¼ 0;
ð18Þ
d and M b represent divergence, gradient and Laplacian operators, respectively, with respect to the where the operators d div; grad d may also be considered as a ^ ^ variable x and m ¼ ðm1 ; m2 ; 0Þ with (m1, m2) are the first two components of m. The vector grad/ 2D vector or a 3D vector where the third component is 0. Global existence of solutions (me, he, /e) of the new system is guaranteed by Theorem 1. We shall describe the behavior of such solutions when e ? 0.
3946
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
By using the same method as in Lemma 2, the energy estimate satisfied by (me, he, /e) is
E e ðtÞ þ
Z t
c
2
0
e d e k22 þ k1 k@ z he k22 k@ t me k2L2 ðXÞ þ k1 k gradh L ðXÞ L ðXÞ ds 6 b1 E ð0Þ þ b2 ; 2
ð19Þ
e
where
E e ðtÞ ¼
1 2
d me k22 þ m k@ z me k22 þ hc kme k44 þ c2 khe k22 þ lk grad d /e k22 3 þ l k@ z /e k22 3 þ 2c1 mk grad L ðXÞ L ðXÞ L ðXÞ L ðXÞ L ðRþ Þ L ðRþ Þ 2 e2 e2
Z
he dX ;
X
ð20Þ and b1 and b2 are the constants of Lemma 1. Let us discuss the admissibility criterion for the initial data ðme0 ; he0 ; /e0 Þ. The condition that me0 is independent of the vard /e k22 3 þ 1 k@ z /e k22 3 6 C uniformly with respect iable z ensures that E e ð0Þ 6 C. In fact, under this condition we have k grad 0 L ðRþ Þ
to e. This follows from estimate (14).
e2
0 L ðRþ Þ
Lemma 2. Let me0 ðX 0 Þ such that me0 is uniformly bounded in H1(X) and he0 2 L2 ðXÞ. Then we have
E e ðtÞ þ
Z t 0
c
2
d he k22 þ k1 k@ he k22 k@ t me k2L2 ðXÞ þ k1 k grad z L ðXÞ L ðXÞ ds 6 C: 2
e
ð21Þ
We have the following uniform bounds. Lemma 3. Under the hypotheses of Lemma 2, the solutions (me, he, /e) satisfy the following estimates
8 e d e > > > k grad m kL1 ðRþ ;L2 ðXÞÞ þ k@ t m kL2 ðRþ ;L2 ðXÞÞ 6 C; > > > < k grad d e k 1 þ 2 3 6 C; d he k 2 þ 2 þ k grad/ L ðR ;L ðXÞÞ
L ðR ;L ðRþ ÞÞ
> > k@ z me kL1 ðRþ ;L2 ðXÞÞ þ k@ z /e kL1 ðRþ ;L2 ðR3 ÞÞ þ k@ z he kL1 ðRþ ;L2 ðXÞÞ 6 C e; > > þ > > e : kh kL1 ðRþ ;L2 ðXÞÞ þ khe kL1 ðRþ ;L1 ðXÞÞ 6 C:
ð22Þ
There exists a subsequence still denoted (me, he, /e) such that the following convergences hold
8 me * m weakly H in L1 ðRþ ; H1 ðXÞÞ \ L1 ðRþ ; L4 ðXÞÞ; > > > > < @ me * @ m weakly in L2 ðRþ ; L2 ðXÞÞ; t t 1 2 e > > @ z m ! 0 strongly in L ðRþ ; L ðXÞÞ; > > : e 2 þ 2 m ! m strongly in Lloc ðR ; L ðXÞÞ:
ð23Þ
The last convergence is a consequence of the classical use of Aubin’s compactness lemma, see for example [4]. For the temperature we have
8 e 2 1 þ > < h * h weakly in L ðR ; H ðXÞÞ; 1 e h * h weakly H in L ðRþ ; L2 ðXÞÞ; > : @ z he ! 0 strongly in L2 ðRþ ; L2 ðXÞÞ;
ð24Þ
and for magnetic potential we get
(
d d e * grad/ weakly H in L1 ðRþ ; L2 ðR3þ ÞÞ; grad/ @ z /e ! 0 strongly in L1 ðRþ ; L2 ðR3þ ÞÞ:
ð25Þ
Thus in the limit the admissible fields do not depend on the direction normal to the thin film. 3. Characterization of the limiting magnetic potential We have Lemma 4. The limiting magnetic potential / satisfies
b / þ vðXÞ d lM divm ¼ 0 in Rþ R3þ :
ð26Þ
3947
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
Proof. Let F 2 DðRþ R3þ Þ. The weak formulation of (18) is
Z
Rþ R3þ
d div
Z
d /e þ vðXÞc l grad m e F dX dt þ
Rþ R3þ
l e 1 @ / þ vðXÞme u3 F dX dt ¼ 0: e2 z e
@z
Arguing that in the limit, the magnetic potential is independent of the variable z (see (25)) we choose test functions F which are independent of z. This allows to get, by using boundary conditions (18), the Eq. (26). h The scaled Eq. (16) involves the quantity He :¼ 1e @ z /e for which we have the following convergence result. Lemma 5. We have
He * H weakly H in L1 ðRþ ; L2 ðR3þ ÞÞ;
ð27Þ
where H is given by
Hðt; X 0 ; zÞ ¼
vðXÞ mðt; X 0 Þ u3 in Rþ R3þ : l
ð28Þ
Proof. We refer to [3]. See also [5]. h
4. Passing to the limit We are now ready to state the main result of this paper. We have Theorem 2. Let (me, he, /e) be a global solution of problem (16)–(18). Let (m, h, /) be the weak-w limit of a subsequence of (me, he, /e). Then (m, h, /) is independent of the variable z and satisfies in QT ¼ ð0; TÞ B the coupled system
(
d / 1 ðm u3 Þu3 ; b m þ grad c@ t m þ hc ðjmj2 1Þm þ hm ¼ m M l
ð29Þ
b ðln hÞ þ k1 M b h; c1 @ t ðln hÞ þ c2 @ t h m @ t m ¼ k0 M where the magnetic potential / satisfies Eq. (26).
Proof. Let Fe and Ge be two test functions defined in DðQÞ. The weak formulation of the problem reads in Q as follows. For the magnetization field we have
Z
ðc@ t me þ hc ðjme j2 1Þme þ he me ÞF e dX dt ¼ m
Q
þ
Z
Z
d me grad d F e dX dt grad
Q
d /e F e dX dt þ 1 grad
e
Q
Z
m e2
Z
@ z me @ z F e dX dt
Q
@ z /e u3 F e dX dt;
ð30Þ
Q
while for the temperature one gets
Z
c1
@ t ðln he ÞGe dXdt þ c2
Q
¼ k0
Z Q
Z Q
Z
@ t he Ge dXdt
d d Ge dX dt k0 gradðln he Þ grad 2
e
Z Q
me @ t me Ge dXdt
Q
1 @ z he @ z Ge dX dt k1 he
Z Q
d he grad d Ge dX dt k1 grad 2
e
Z
@ z he @ z Ge dX dt:
ð31Þ
Q
In order to pass to the limit as e ? 0, we argue that, in the limit, (me, he) is independent of the variable z. We use test functions of the type
Z e ðt; X 0 ; zÞ ¼ Zðt; X 0 Þ þ eZ 0 ðt; X 0 ÞhðezÞ;
ð32Þ
where h 2 Dð½0; 1Þ and Z; Z 0 2 DðBÞ. Note that Ze satisfies
d e ¼ grad d Zðt; X 0 Þ þ ehðezÞ gradZ d 0 ðt; X 0 Þ; gradZ @ z Z e ¼ e2 h ðezÞZ 0 ðt; X 0 Þ: 0
ð33Þ
Let us pass to the limit into (31). Note that by Theorem 1, the solution of (17) is such that he 2 L2(0, T; H1(X)), c1lnhe + c2he 2 0 H1(0, T; H1(X) ) and lnhe 2 L2(0, T; H1(X)). Thus, all the terms in (31) can pass to the limit because they are written as weakstrong products. Recall the strong convergences me ? m in L2loc ðRþ ; L2 ðXÞÞ and @ zhe ? 0 in L2 ðRþ ; L2 ðXÞÞ. We have
Z ðc1 @ t ðln he Þ þ c2 @ t he ÞGe dX dt ! ðc1 @ t ðln hÞ þ c2 @ t hÞG0 dX 0 dt; Q QT Z Z me @ t me Ge dX dt ! m @ t mG0 dX 0 dt:
Z
Q
QT
3948
M. Tilioua / Applied Mathematical Modelling 34 (2010) 3943–3948
Next we have
k0
Z
d d Ge dX dt ! k0 gradðln he Þ grad
Q
k0
e
k1
Z
Q
1 @ z he @ z Ge dX dt ! 0; he
d he grad d Ge dX dt ! k1 grad
Q
k1
e
2
d d G0 dX 0 dt; gradðln hÞ grad
QT
Z
2
Z
Z
Z
d h grad d G0 dX 0 dt; grad
QT
@ z he @ z Ge dX dt ! 0:
Q
Similarly, passing to the limit in (30) by using convergences (23), Lemma 5 and properties (33) of test functions, yields (29). h
5. Concluding remarks In this paper we have derived a thin layer model in the context of phase transitions in ferromagnetism. The dimensional reduction 3D-2D is performed by using a scaling technique combined with the method of oscillating test functions. We have shown that for very thin samples the magnetization, the temperature and the magnetic potential satisfy a two-dimensional time-dependent coupled problem (Theorem 2). Inspection of (29) reveals the effect induced by the thin layer behavior. It takes the form l1 ðm u3 Þu3 which disfavors the out-of-plane component of the limiting magnetic field grad/. The reduced problem obtained in this work can be used as a toy model for introducing the mathematical approach which can be adapted to more realistic situations. Indeed, further progress is needed by investigating and setting up new models that describe, for example, the effect of the surface energy which plays a crucial role in thin film dynamics. Note that the effect of impurities and surface roughness is also of interest. Finally, we may mention that it becomes increasingly important to identify simpler models, valid in appropriate regimes, whose behavior is easier to understand or simulate in order to determine the microscopic configuration of the magnetization that is not directly accessible by experiment. Acknowledgements The author thanks the referees for careful reading and giving constructive comments to improve the paper. References [1] V. Berti, M. Fabrizio, C. Giorgi, A three-dimensional phase transition model in ferromagnetism: existence and uniqueness, J. Math. Anal. Appl. 355 (2009) 661–674. [2] P.G. Ciarlet, Introduction to Linear Shell Theory, North-Holland, 1998. [3] K. Hamdache, M. Tilioua, Interlayer exchange coupling for ferromagnets through spacers, SIAM J. Appl. Math. 64 (3) (2004) 1077–1097. [4] J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [5] G. Gioia, R.D. James, Micromagnetics of very thin films, Proc. R. Soc. Lond. A 453 (1997) 213–223.