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On the ferromagnetism equations in the non static case.
Gilles Carbou, Pierre Fabrie, Olivier Guès
To cite this version:
Gilles Carbou, Pierre Fabrie, Olivier Guès. On the ferromagnetism equations in the non static case.. Communications on Pure and Applied Mathematics, Wiley, 2004, 3 (3), pp.367-393.
�10.3934/cpaa.2004.3.367�. �hal-00295062�
On the ferromagnetism equations in the non static case
Gilles Carbou†, Olivier Gu`es‡ and Pierre Fabrie†
† Math´ematiques Appliqu´ees de Bordeaux, UMR 5466 et Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 Talence cedex, France.
‡ LATP, UMR 6632 CNRS, Universite de Provence, 39 rue Joliot-Curie, 13453 MARSEILLE Cedex 13
Contents
1 Introduction 2
2 A local existence result for a fixed ε >0 3
3 Asymptotic analysis as ε→0 3
4 Reduction of the problem 5
4.1 The modified equation foru. . . . 5
4.2 The wave equation forH⊥. . . . 6
4.3 Initial data and compatibility conditions . . . . 6
4.4 An existence result and the proof of theorem (2.1). . . . 7
5 Proof of Theorem 4.1 8 5.1 Technical lemmas and notations . . . . 9
5.2 Estimates on h . . . . 9
5.3 Estimation on ku(t)kL2 . . . . 10
5.4 Estimation on k∇u(t)kL2 . . . . 10
5.5 Estimation on k∆u(t)kL2 . . . . 11
5.6 Estimations on∂tu . . . . 12
5.7 End of the proof . . . . 13
6 BKW method for Theorem 3.1 14 6.1 Formal asymptotic expansion . . . . 14
6.2 Existence and regularity of the terms of the ansatz . . . . 18
7 Proof of Theorem 3.1 19 7.1 Equation satisfied by the remainder term . . . . 20
7.2 Estimates for the remainder terms . . . . 21
7.3 End of the proof of Theorem 3.1 . . . . 22
8 Proof of the estimates 22 8.1 Proof of Proposition 7.1 . . . . 23
8.2 Proof of Proposition 7.2 . . . . 29
8.3 Proof of Proposition 7.3 . . . . 29
1 Introduction
In this paper we are interested in an initial boundary value problem for a mathematical model in ferromagnetism. The physical context is the following. A piece of ferromagnet is supposed to be a regular bounded open set Ω inR3. The magnetic state at a point x∈Ω at time tis given by a vector u(t, x) ∈R3 which belongs to the unit sphere of R3, called the magnetic moment.
The evolution of u is coupled to the evolution of the electromagnetic field E(t, x), H(t, x) in the whole space R3, by a system of nonlinear partial differential equations.
The first equation is the following Landau-Lifschitz equation inR+
t ×Ωx, whereε2 is supposed to be a constant :
∂tu=u∧(H+ε2∆u)−u∧ u∧(H+ε2∆u)
in [0,+∞[×Ω
∂nu= 0 in [0,+∞[×∂Ω u|t=0=u0 .
(1.1)
wheren is the unitary outward normal at the boundary∂Ω.
This equation is coupled with the Maxwell system in R+t ×R3
∂t(H+ ¯u) + curlE= 0
∂tE−curlH= 0 (E, H)|t=0 = (E0, H0).
(1.2)
where ¯u means the extension of u by 0 outside ofR×Ω.
Remark 1.1 In all the paper we take all the physical constants equal to 1, excepted the exchange coefficient, since their value don’t change the mathematical analysis of the equations.
Furthermore, the solution must satisfy the divergence condition
div (H+ ¯u) = 0, (1.3)
and the constraint
|u(t, x)|= 1, x∈Ω, t≥0. (1.4) A basic observation is that these two last conditions are propagated by the full system, from the initial conditions. The condition (1.3) is given by the Maxwell equations (1.2), since the first equation of (1.2) implies that ∂tdiv (H + ¯u) = 0. The condition (1.3) is then satisfied for all t≥0 if and only if it is satisfied for t = 0. In other words, condition (1.3) means exactly that the initial dataH0 and u0 satisfy
div (H0+u0) = 0. (1.5)
The same remark is true for the condition (1.4), assuming however that u is regular enough, since the equation (1.1) implies∂t(|u(t, x)|2) = 0.
The existence ofglobal weak solutionsfor the system (1.1),(1.2) was established by A. Visintin in [32], and for another form of the system (equivalent for regular enough solutions) by G. Carbou and P. Fabrie in [8].
In this paper, we first prove the existence and uniqueness of regular enough solutions for the system (1.1), (1.2). The solutions obtained are local in time. This result is stated in section 2.
The section 3 is concerned with the question of the asymptotic behavior of the solution of (1.1)- (1.2) as ε > 0 tends to 0. ¿From a formal point of view, the system obtained when ε= 0, is equivalent to a first order semilinear symmetric hyperbolic system, which is known to admit local piecewise regular solutions (Sobolev regularity) discontinuous across the boundaryR×∂Ω, which is a characteristic hypersurface of constant multiplicity for this hyperbolic system. This hyperbolic system has a very particular structure and admits global solutions as proved by Joly, M´etivier and Rauch in [19]. We prove here two new results. First, a solution (u0, E0, H0) of the limit hyperbolic system being given on [0, T], we show that under some natural assumptions, this solution is limit of a family of solutions (uε, Eε, Hε) of (1.1)-(1.2). The other result is that if u0 satisfy the additional condition ∂nu0(0, .)|∂Ω = 0, the solution of (1.1)-(1.2) with initial data (u0|t=0, E0|t=0, H0|t=0) converges to (u0, E0, H0) asεgoes to 0. To obtain this results, we perform an asymptotic expansion in ε and bring to the fore a boundary layer of characteristic size ε, and amplitude ε, localized closed to ∂Ω. As it is classical in BKW method, we have to suppose that the limit solution (u0, E0, H0) is very regular on each side of Ω (Sobolev piecewise regularity).
Notation. In all the paper, we will note Hm := (Hm)3 = (Wm,2)3 the usual Sobolev spaces of functions with values inR3, andLp:= (Lp)3 the usual Lebesgue spaces with values inR3.
2 A local existence result for a fixed ε > 0
Let us introduce some notations. ForT >0, let us callA(T) the set of functions u∈L2 [0, T];H3(Ω)
∩ C [0, T];H2(Ω)
∩ C1 [0, T];H1(Ω) such that∂tu∈L2 [0, T];H2(Ω)
and ∂t2u∈L2([0, T]×Ω).
Concerning the regularity of the electromagnetic field, we will use the following classical space Hcurl :={v∈L2(R3;R3) such that curl v∈L2(R3;R3)}
equiped with the natural normkvkL2+kcurl vkL2. The main result of the section is the following.
Theorem 2.1 Let ε > 0 be fixed. Let u0 ∈ H3(Ω) satisfying |u0| = 1, ∂nu0|∂Ω = 0. Let (E0, H0)∈Hcurl×Hcurl. Assume thatdiv (H0+ ¯u0) = 0. Then there existsT >0 and a unique solution (u, E, H) to the problem (1.1), (1.2), such thatu∈ A(T), and
E, H ∈ C1 [0, T];L2(R3)
∩ C [0, T] :Hcurl . Furthermore, |u|= 1 in [0, T]×Ω anddiv (H+ ¯u) = 0 for all t∈[0, T].
This theorem will be deduced from theorem 4.1 below, which is proved in section 5.
3 Asymptotic analysis as ε → 0
In this section, we are interested in the behavior of the local solution described in theorem 2.1, as εtends to 0. This is a natural question of current interest in the modelisation of micromagnetism.
Let us consider the system formally obtained whenε= 0, on a time interval ]0, T[, which writes
∂tu0=u0∧H0−u0∧(u0∧H0) in ]0, T[×Ω
∂t(H0+u0) + curlE = 0 in ]0, T[×R3
∂tE0−curlH0 = 0 in ]0, T[×R3
(3.1)
Note that the first equation holds in ]0, T[×Ω, and that no boundary condition is needed on ]0, T[×∂Ω foru0. This system satisfies as the original (1.1)-(1.2) system, the propagation prop- erties of|u0(t, x)| and div (H0+u0) in the sens that the relations
|u0(t, x)|2 = 1,∀x∈Ω,∀t∈[0, T]
div (H0+u0) = 0,∀t∈[0, T] (3.2) hold if and only if they are satisfied at t= 0.
Now, since the principal part of the first equation is the field∂t, it follows that (u0, H0, E0)∈L∞loc satisfies system (3.1) in the sens of distributions if and only if (V0 := u0, H0, E0) satisfies the following semilinear first order symmetric hyperbolic system in the domain ]0, T[×R3:
∂tV0 =V0∧H0−V0∧(V0∧H0)
∂tH0+ curlE =−V0∧H0+V0∧(V0∧H0)
∂tE0−curlH0= 0
(3.3)
For this system, the hypersurface R×∂Ω is characteristic (of constant multiplicity). Hence, it admits classical piecewise regular (Sobolev) solutions discontinuous across R×∂Ω ([26], [28], [29], [27]). More precisely, ifm ∈N, and if we call Ω0:=R3\Ω, let us denote by p−Hm(Ω) the space of functionsv∈L2(R3) such that v|Ω∈Hm(Ω) andv|Ω0 ∈Hm(Ω0). The spacep−Hm(Ω) is endowed with the natural norm kv|ΩkHm(Ω)+kv|Ω0kHm(Ω0). As before, we use the notation p−Hm(Ω) when the function is valued inR3. For any givenm, it is a consequence of the theory of discontinuous solutions of hyperbolic semilinear systems ([26], [28], [29], [27]) that the system (3.1) has solutions which satisfy
u0, E0, H0∈ C1 [0, T], p−Hm(Ω)
, (3.4)
for some T >0. For m big enough (m >3/2), and inside this class of functions, it is equivalent to solve system (3.1) with initial datas
u0|t=0 =u00, E|t=00 =E00, H|t=00 =H00, (3.5) or to solve the system (3.3) with initial conditions
V|t=00 =u00, E|t=00 =E00, H|t=00 =H00. (3.6)
In this paper we consider such solutions of system (3.1) which satisfy u0, E0, H0 ∈ C1 [0, T], p−H5(Ω)
, (3.7)
for some T > 0. Our result in this section is that such a solution is actually the limit of a sequence of solutions of original system (1.1)-(1.2) asεgoes to zero. In order to state the result, let us introduce a functionϕ∈ C∞(R3,R) such that Ω ={ϕ >0}, Ω0 ={ϕ <0}, ∂Ω ={ϕ= 0} and normalized such that |∇ϕ(x)| = 1 for all x in a neighborood V of ∂Ω. This implies that ϕ(x) = dist(x, ∂Ω) on V ∩Ω.
Theorem 3.1 Assume that(u0, H0, E0)satisfies the system(3.1),(3.2) and the condition(3.7), for some T >0. Then the following holds.
1. There exists a family of initial datas (uε0, H0ε, E0ε)ε>0 satisfying the assumptions of theorem 2.1 such that the corresponding solution(uε, Hε, Eε) of (1.1)−(1.2) given by theorem 2.1 exists on [0, T] and converges to (u0, H0, E0) in C [0, T],L2(Ω)×L2(R3)×L2(R3)
, as ε→0.
2. If u0 := u0|t=0 satisfies ∂nu0|∂Ω = 0. Then, the solution (uε, Hε, Eε) of (1.1) − (1.2) given by theorem 2.1with initial data (u0|t=0, H|t=00 , E0|t=0)exists on [0, T] and converges to (u0, H0, E0) in C [0, T],L2(Ω)×L2(R3)×L2(R3)
, as ε→0.
3. In both cases (1 et 2) there exists a boundary layer profile V(t, x, z) ∈ C([0, T], H4(Ω)⊗ H4([0,+∞[) such that:
uε(t, x) =U0(t, x) +εV(t, x,ϕ(x)
ε ) +εrε(t, x) Hε(t, x) =H0(t, x) +εRHε (t, x)
Eε(t, x) =E0(t, x) +εRεE(t, x) with the following uniform estimate
krεkL∞(0,T;H1)+kεrεkL∞(0,T;H2)+kRHεkL∞(0,T;Hcurl)+kRεEkL∞(0,T;Hcurl) ≤ C.
Note that in the point1. the functionu0 is not supposed to satisfy any boundary condition, and in particular the trace u0 :=u0|t=0 is not supposed to satisfy ∂nu0|[0,T]×∂Ω = 0. This comment is to emphasize the fact that one cannot apply the existence theorem (2.1) with the initial values u0|t=0, E|t=00 , H|t=00 . On the other hand, this is a natural motivation for the point2..
4 Reduction of the problem
4.1 The modified equation for u.
For regular solutions, the equation (1.1) is equivalent to the following equation (see [9]):
∂tu−ε2∆u−ε2u∧∆u=ε2|∇u|2u+u∧H−u∧(u∧H) ( inR×Ω) . (4.1) Let us introduce some notations. We will note P⊥ the orthogonal projector of L2(R3;R3) onto the subspace of divergence free vector fields, and Pk := Id− P⊥. For convenience we will also
use the notationsv⊥:=P⊥v and vk :=Pkv ifv∈L2(R3;R3). Using the Fourier transformbin L2(R3;R3) gives the following expressions
c
vk(ξ) =|ξ|−2 hξ.v(ξ)b iξ , vc⊥(ξ) =−|ξ|−2 ξ∧(ξ∧bv(ξ)) (4.2) whereh.i is the scalar product and∧the vectorial product in R3.
The relation (1.3) means that Pk(H) +Pk(¯u) = 0. Replacing then H = H⊥− Pk(¯u) in the Landau-Lifschitz equation (4.1), gives the following equation
∂tu−ε2 ∆u=ε2 ∧∆u+ε2|∇u|2u
−u∧ Pk(¯u) +u∧ u∧ Pk(¯u) +u∧H⊥−u∧(u∧H⊥) .
(4.3)
4.2 The wave equation for H⊥.
In a classical way, we use the Maxwell system to get a scalar wave equation onH⊥, with a right hand side depending on ¯u: we apply ∂t to the first equation in (1.2) and take the curl of the second equation to get
∂t2H⊥− ∆H⊥=−∂2tP⊥(¯u). (4.4) We are then interested in solving the following non linear system of equations
∂tu−ε2 ∆u=ε2 u∧∆u+ε2|∇u|2u
−u∧ Pk(¯u) +u∧ u∧ Pk(¯u) +u∧h−u∧(u∧h) in ]0,∞[×Ω,
(4.5)
∂t2h− ∆h=−∂t2P⊥(¯u) in ]0,∞[×R3, (4.6) with boundary condition
∂nu|]0,∞[×∂Ω= 0 (4.7)
and initial conditions foru
u|t=0=u0 in Ω, (4.8)
and for h
h|t=0=h0, ∂th|t=0 =h1 inR3. (4.9) 4.3 Initial data and compatibility conditions
A natural question is to express the initial data forH⊥ and ∂tH⊥ in terms of the original data u0, E0, H0. ConcerningH⊥ we just have:
(H⊥)|t=0=P⊥H0. (4.10)
For∂tH⊥ we must use the equations. The Maxwell equations imply
∂tH⊥=−curlE−∂tP⊥(¯u). (4.11) The modified Landau-Lifschitz equation (4.5) writes
∂tu=F H⊥, u,∇u,∆u,Pk(¯u)|Ω
with obvious notations. Let us call F0 :=
F H⊥, u,∇u,∆u,Pk(¯u)|Ω
|t=0
=F P⊥H0, u0,∇u0,∆u0,Pk( ¯u0)|Ω . It follows that
∂tP⊥(¯u)
|t=0 =P⊥(F0).
Coming back to equation (4.11) we find the following expression for the initial value of ∂tH⊥, expressed with the original datasu0, E0, H0:
(∂tH⊥)t=0 =−curlE0− P⊥(F0). (4.12) We will solve the wave equation for H⊥ in the space
C 0, T;H1(R3)
∩ C1 0, T;L2(R3) ,
so we need an initial data for ∂tH⊥ in L2(R3). This requirement will be our first ”compati- bility condition”. Since u0 ∈ H2(Ω), we see that F0 and also P⊥(F0) are in L2(R3). In view of relation(4.11) in follows that the condition (∂tH⊥)|t=0 ∈ L2(R3) reduces to the following necessary compatibility condition
curlE0 ∈L2(R3). (4.13)
This is the reason why we assume that our initial data E0 belongs to Hcurl.
Let us turn now to the compatibility conditions for u0. The point is that the function ∂tu has to be inC [0, T] :H1(Ω)
. A necessary compatibility condition is then
∇F0 ∈L2(Ω) . (4.14)
This condition is always fulfilled whenu0 belongs to H3(Ω).
4.4 An existence result and the proof of theorem (2.1).
Let us first state the main theorem of this section.
Theorem 4.1 Let u0 ∈ H3(Ω) such that ∂nu0|∂Ω = 0 and let h0 ∈H1(R3) and h1 ∈L2(R3).
There exists T > 0 and a unique solution (u,h) to the system (4.5)· · ·(4.9) on ]0, T[×Ω such that u∈ A(T) and
h∈ C1 [0, T];L2(R3)
∩ C [0, T];H1(R3)
. (4.15)
Moreover, if Pkhj = 0 for j = 0,1,then Pkh(t, .) = 0 for all t∈[0, T].
Assuming for a moment theorem 4.1, we can now give the proof of theorem 2.1. Apply theorem 4.1 with initial datasu0 andh0:=P⊥(H0) and
h1 := curlE0− P⊥(F0) ∈L2(R3).
This gives a function u ∈ A(T) and a function h, with regularity (4.15) satisfying Pkh = 0, because of the last observation in theorem 4.1. Now, one can solve the Maxwell hyperbolic system (1.2) with∂tu¯in the right hand side
∂tH+ curlE=∂tu¯
∂tE−curlH= 0 (E, H)|t=0 = (E0, H0).
(4.16)
Since ∂t(¯u) = (∂tu) is in L2([0, T] ×R3) it follows that this equation has a unique solution (H, E) ∈ C [0, T];L2(R3;R6)
.
Let us consider now the wave equation (4.4) satisfied by H⊥. Observing that the initial values (H⊥)|t=0 and (∂tH⊥)|t=0 are the same as the initial data h0 and h1 (since of relation (4.12)), we deduce that H⊥ = h. Now, the fact that H belongs to C [0, T];Hcurl
is a consequence of the following lemma.
Lemma 4.1 A function v given in L2(R3) belongs to Hcurl if and only if v⊥ is in H1(R3). In such a case, it satisfies the inequality
c−1 kcurl vkL2(R3)≤ k∇v⊥kL2(R3)≤c kcurl vkL2(R3) , for some c >0 independent ofv.
Proof.
We write v = vk +v⊥ and by Fourier transform on R3, (curl\v)(ξ) = iξ∧v(ξ) =b iξ∧vc⊥(ξ).
Now, since ξ and v⊥(ξ) are orthogonal vectors, noting |.| the Euclidean norm in R3 we have:
|(curl\v)(ξ)|=|ξ| |bv(ξ)|. Now, using the Parseval-Plancherel equality we obtain the lemma.
Then, as we already observed in section 2, we have
Hk=−Pk(¯u)∈ C1 [0, T];L2(R3) .
which implies that H=Hk+H⊥ is also C1 from [0, T] toL2(R3). Applying the time derivative
∂t to the Maxwell system (4.16) we see that H0 :=∂tH and E0 :=∂tE, are solutions of
∂tH0+ curlE0 =∂t2u∈L2([0, T]×R3;R3)
∂tE0−curlH0 = 0
(E0, H0)|t=0= (−curlE0−F0,curlH0)∈L2(R3;R6),
(4.17)
which implies that∂tE(and also∂tH, which is already known) is inC [0, T];L2(R3)
. It remains to prove that curl E belongs to C [0, T];L2(R3)
. This follows from the first equation of the Maxwell system
curlE=∂tu¯−∂tH because of the regularity ofH andu. This proves theorem 2.1.
The next section is devoted to the proof of theorem 4.1.
5 Proof of Theorem 4.1
For the proof of theorem 4.1 we use a priori estimates on a Galerkin approximation. The approximation space is based on the eigenspaces of the Laplacian on the domain
D(∆) ={u∈H2(Ω) such that ∂nu|∂Ω = 0}.
Let’s call Πn the usual orthogonal projector on the finite dimensional invariant subspace built on the firstn eigenspaces.
Our goal is to establisha prioriestimates, uniform innon the solution (un,hn) of the following non linear problem (where we note simply (u,h) instead of (un,hn)):
∂tu−ε2 ∆u= Πn
ε2 ∧∆u+ε2|∇u|2u
−Πn
u∧ Pk(¯u) +u∧ u∧ Pk(¯u) + Πn
u∧h−u∧(u∧h)
in ]0,∞[×Ω ,
(5.1)
∂t2h− ∆h=−∂t2P⊥(¯u) in ]0,∞[×R3 , (5.2) with boundary condition
∂nu|]0,∞[×∂Ω= 0 (5.3)
and initial conditions
u|t=0 = Πnu0 in Ω, (h, ∂th)|t=0 = (h0,h1) inR3. (5.4) 5.1 Technical lemmas and notations
Form≥0, We will noteHm(Ω) =Wm,2(Ω) the usual Sobolev space, and we will notek.km the usual norm
kvkm:= X
|α|≤m
k∂xαvkL2(Ω) .
We will denote by Hm(Ω) :=Hm(Ω;R3) and will still denote k.km the corresponding norm on Hm(Ω). We will also use the corresponding notations withR3 in place of Ω. We will use many times the following lemma (see [1], [2], [31]).
Lemma 5.1 Let Ω be a regular open subset ofR3. On the linear space V:={u∈H2(Ω) such that ∂nu|∂Ω= 0},
the norms kukH2 and kukL2 +k∆ukL2 are equivalent. On the subspace H3(Ω)∩V, the norms kukH3 and kukH2+k∇∆ukL2 are equivalent.
The following result is also very useful in the study of ferromagnetism equations, and a proof can be found in [9] and [10].
Lemma 5.2 Let m ≥ 0 and p ∈]1,∞[. The mapping u → Pk(¯u)
|Ω is continuous from Wm,p(Ω) into Wm,p(Ω). The same is true with P⊥.
Let us also recall that H1(Ω) is continuously embedded in L6(Ω) ([1], [2]).
5.2 Estimates on h
Let us begin with the classical estimate for the wave equation, obtained by taking the scalar product of the equation with ∂th.
We get:
1 2
d
dt k∂thk2L2 +k∇hk2L2
≤ kP⊥(∂t2u)¯ kL2 k∂thkL2
≤ k∂t2ukL2 k∂thkL2.
(5.5)
