Time Average in Micromagnetism.

Transcription

1 Time Average in Micromagnetism. Gilles Carbou, Pierre Fabrie Mathématiques Appliquées de Bordeaux, Université Bordeaux, 35 cours de la libération, 3345 Talence cedex, France. Abstract - In this paper we study a model of ferromagnetic material governed by onlinear Laudau-Lifschitz equation coupled with Maxwell equations. We prove the existence of weak solutions. Then we prove that all points of the ω-limit set of any trajectories are solutions of the stationary model. Furthermore we derive rigourously the quasistatic model by an appropriate time average method. Introduction. In this paper we study the following system where H e = u + H ϕu, + u = 2u H e in IR +,. µ H + ū + curl E = in IR+,.2 E ε curl H + σ E + f = in IR +,.3 with the associated boundary conditions and initial data ν = on IR+, u, x = u x in, E, x = E x in, H, x = H x in..4 We assume that u x = in, div H + ū = in. In the above equations is a smooth bounded open domain of, ν the unit normal on, is the characteristic function of, ū is the extension of u by zero outside. This system of equations which couples the Landau-Lifschitz equation with Maxwell s equations describes electromagnetic waves propagation in a ferromagnetic medium confined to the domain. In the ferromagnetic model the magnetic moment denoted by u links the magnetic field H with the magnetic induction B through the relationship B = µ H + ū. Moreover u is a vector field which takes its values on S 2 the unit sphere of. The conductivity of the body is.5

2 denoted by σ IR +, the anisotropic term is patterned by ϕu where ϕ : is the gradient of Φ a positively defined quadratic form of, f is a source term supported by IR +. Finaly ε is the dielectric permittivity and µ is the magnetic permeability. This model is described in detail in [3], [] and [5]. Remark. When the solution of. is regular enough, this equation is equivalent to = u H e u u H e in IR +..6 In [4] A. Visintin establishes the existence of weak solutions of the system.6, When H e reduces to u, F. Alouges and A. Soyeur show in [2] the existence and the non uniqueness of the solutions of.. F. Labbé establishes in [] the non uniqueness of the solution for the quasistatic model. Numerical studies are carried on by P. Joly and O. Vacus in [9], and by F. Alouges in the steady state case in []. At least in the case when H e reduces to H and =, J.L. Joly, G. Métivier and J. Rauch obtain existence and uniqueness results for the solutions of.6,.2,.3,.4. Notations : in the sequel we denote IH = H 3 and IL 2 = L Statement of the results. Let us assume that u IH, H IL 2, E IL 2, f IL 2 IR +, u = a.e., div H + ū =. Definition 2. We say that u, E, H is a weak solution of.-.5 if. u, E, H verifies H u L IR + ; IH, IL2 IR +, ut, x = a. e., E L IR + ; IL 2, H L IR + ; IL For all Ψ C IR + ; IH, IR + t, x + ut, x t, x Ψt, xdx dt = 2 IR + i= ut, x t, x Ψ t, xdx dt x i x i +2 ut, x Ht, x ϕut, x Ψt, xdx dt. IR u, x = u x in the trace sense. 2

3 4. For all Ψ IH IR +, Ht, x + ūt, x IR + Ψ t, xdt dx + H x Ψ, xdx + Et, x curl Ψt, xdx dt = IR + u x Ψ, xdx For all Ψ IH IR +, Ψ Et, x IR + 3 IR t, xdx dt Ht, x curl Ψt, xdx dt IR + 3 IR +σ Et, x + ft, x Ψt, xdx dt = E x Ψ, xdx. IR + 6. For all t >, we have the following energy estimate : t Et + t, x 2 dx dt + σ t Et, x 2 dx dt E µ where Et = + σ t µ ft, x 2 dx dt ut, x 2 + 2Φut, x dx + Ht, x 2 + ε o Et, x 2 dx. µ Theorem 2. Under the assumption H, there exists at least one weak solution of.-.5. This theorem is established in section 3 using a Galerkine approximation for a relaxed problem. Definition 2.2 Let u be a weak solution of.-.5. We call ω-limit set of the trajectory u the following set { } ωu = v IH, t n, lim t n = +, ut n,. v in IH weakly From the energy estimate 2.5, for any u, ωu is non empty. Theorem 2.2 Under the assumption H, if u is a weak solution of.-.5, each point v in ωu is a weak solution of the steady state system v H, v = a.e., 2.6 v v + v H ϕv = in, 2.7 x i= i x i H IL 2, curl H = in D, 2.8 div H + v = in D. 3

4 Remark 2. As v lies in IH, v lies in IH so the product v v makes sense in W,t with t = 2 + see J. Simon [3]. Moreover from the equation 2.7 this product 6 belongs to IL 2. Theorem 2.2 is proved in section 4. The limit process for v is carried out thanks to the estimate t, x 2 dx dt < +. IR + On the other hand an averaging technique is used to justify the limit for H. The last part of this article is devoted to the validation when ε and µ go to zero of the quasi-stationary model. We suppose for this result that the source term f is zero. Let us assume that u IH, H IL 2, E IL 2, u = a.e., div H + ū =. H q Definition 2.3 We say that u is a weak solution of the quasi-stationary model if. u satisfies u L IR + ; IH, IL2 IR +, u = a.e For all Ψ C IR + ; IH, IR + t, x + ut, x t, x Ψt, xdx dt = 2 IR + i= ut, x x i t, x Ψ x i t, xdx dt +2 ut, x Ht, x ϕut, x Ψt, xdx dt, IR u, x = u x in the trace sense. 4. For all t IR +, Ht, x is the unique solution of curl Ht, =, div Ht, + ūt, =, 2. Ht,. IL For all t we have the following energy estimate t E q t + t, x 2 dx dt E q, 2.2 where E q t = ut, x 2 + 2Φut, x dx + Ht, x 2 dx. 4

5 Theorem 2.3 We consider two sequences ε n n and µ n n which tend to zero as n + and such that µ n /ε n remains bounded. Under the assumption H q if u n denote a weak solution of.-.5 with ε = ε n and µ = µ n, there exists a subsequence still denoted u n n such that u n tends to a limit u in L IR + ; IH weak where u is a solution of the quasi-stationary model This result is obtained via a time average process on H which avoid the high frequency oscillations of H. Proposition 2. Every point of the ω-limit set of any trajectory of is solution of the steady state model 2.7. This last result is straightforward from the estimate IR + t, x 2 dx dt < + and from the continuity of the map u H given by Proof of the existence. The main point is to establish that u = almost everywhere. In order to construct a solution which satisfies this condition we first solve a relaxed problem P λ where u λ takes its values in. The penalization term takes the form λ u 2 u, λ tends to. In fact instead of. we solve the following equation λ uλ λ 2 uλ 2ϕu λ + λ uλ 2 u λ = 2H. 3. By a Galerkine process we construct a solution of 3. satisfying an energy estimate, that allows us to pass to the limit as λ goes to zero. This limit u takes its values on S 2 and by a suitable test function we show that u satisfies.. First step. Resolution of 3.. Let us recall that the eigenfunctions of the operator A = + I with domain DA = {u IH 2, ν = on } build an orthonormal basis {ϕ k } k in IL 2 and an orthogonal basis in IH and IH 2. We denote V N the N dimensional vector space spaned by {ϕ k } k N. Now we introduce the Hilbert space IH curl = {ψ IL 2, curl ψ IL 2 } We denote {ψ k } k an hilbertian basis of IH curl orthonormal in IL 2 and W N dimensional vector space spaned by {ψ k } k N. the N 5

6 In the approximate problem we seek u N, H N, E N in V N W N W N such that N u N t, x = v k tϕ k x, k= N H N t, x = h k tψ k x, k= N E N t, x = e k tψ k x, k= which satisfies. For any Φ N in V N, N t, x u Nt, x N t, x Φ N xdx + 2 u N t, x Φ N xdx + 4 λ u N t, x 2 u N t, x Φ N xdx 2 H N t, x ϕu N t, x Φ N xdx = For any Ψ N in W N, µ H Nt, x + ū N t, x Ψ N xdx + E Nt, x curl Ψ N xdx = For any Θ N in W N E N ε t, x Θ Nxdx H Nt, x curl Θ N xdx +σ E N t, x + ft, x Θ N xdx = With the initial data u N = Π VN u, E N = Π WN E, 3.5 H N = Π WN H, where Π VN resp. Π WN denotes the orthogonal projection on V N resp. W N. Let us remark that v v u v is one to one in so the equation 3.2 can be solve for the derivative in time. Then by Cauchy Picard theorem there exists a local solution of The following a priori estimates show that, in fact, the approximate solution is global in time. 6

7 Taking Φ N = N in 3.2 one has N t, x 2 dx + d u N t, x 2 dx + d dt λ dt +2 d dt Φu N t, x = N t, x H Nt, x u N t, x 2 2 dx 3.6 µ 2 Now we put Ψ n = H N in 3.3 d dt H Nt, x 2 dx + curl E N N t, x H N t, xdx = µ t, x H Nt, xdx 3.7 In the same way taking Θ N = E N in 3.4, 2 ε d dt E N t, x 2 dx H Nt, x curl E N t, xdx +σ E N t, x 2 + ft, x E N t, x dx = Combining 3.6, 3.7 and 3.8 we derive the following estimate through Young inequality { d u N t, x 2 dx + } u N t, x 2 2 dx + Φu N t, xdx dt λ + 2 { d dt H Nt, x 2 + ε } E N t, x 2 dx µ + N t, x 2 dx + σ E N t, x 2 dx σ ft, x 2 dx µ µ As Φu N is non negative we obtain the following bound for u in IH, E and H in IL 2 and f in L 2 IR + : There exists constants k i independant of N and λ such that u N L IR + ; IL 2 k, N IL 2 IR + k 2, u N L IR + ;IL 4 k 3, E N L IR + ; IL 2 k 4, H N L IR + ; IL 2 k 5. So we can suppose that there exists a subsequence still denoted u N, H N, E N such that when N goes to +, u N u λ in L IR + ; IH weak, 3.8 N λ in L 2 IR + ; IL 2 weak, E N E λ in L IR + ; IL 2 weak, H N H λ in L IR + ; IL 2 weak. 7

8 And according to Aubin s Lemma u N u λ in L 4, T ; IL 4 strong for all T, Taking the limit in the equation we obtain. For any Φ in IH λ t, x Φxdx u λ t, x λ t, x Φxdx +2 u λ t, x Φxdx + 4 u λ t, x 2 u λ t, x Φxdx λ 2 H λ t, x ϕu λ t, x Φxdx = in L 2 IR + t For any Ψ in IH curl, µ < Hλ + ūλ, Ψ > + Eλ t, x curl Ψxdx = in D IR For any Θ in IH curl ε < Eλ, Θ > Hλ t, x curl Θxdx +σ E λ t, x + ft, x Θxdx = in D IR With the initial data E λ = E, u λ = u in IL 2, H λ = H in 3.2 H curl. As the L 2 resp. L norm is lower semi continuous for the weak resp. weak topology we obtain the energy estimate where t >, E λ t + E λ t = t σ 2µ λ t, x 2 dx dt + t u λ t, x 2 dx + λ + 2 Second step. Limit as λ tends to. σ 2µ t ft, x 2 dx dt + E λ, u λ t, x 2 2 dx + H λ t, x 2 + ε E λ t, x 2 dx. µ E λ t, x 2 dx dt Φu l t, xdx 3.3 8

9 We first note that as u =, E λ does not depend on λ. The estimate 3.3 allows us to suppose via the extraction of a subsequence that when λ goes to u λ u in L IR + ; IH weak, λ in IL 2 IR + weakly, u λ u in L 2, T ; IL 2 strongly for all T > and a.e., E λ E in L IR + ; IL 2 weak, H λ H in L IR + ; IL 2 weak. We remark, and it is the main point of the proof, that u = a.e. in IR +, as u λ u a.e. Now we derive the equation satisfied by u by taking in 3.9 Φ = u λ t, x ξt, x where ξ is any test function given in IL 2 loc IR+ ; IH 2. T λ t, x uλ t, x ξt, xdx dt T u λ t, x λ t, x uλ t, x ξxdx dt T +2 i= λ t, x u λ t, x ξt, x dx dt x i x i T 2 H λ t, x ϕu λ t, x u λ t, x ξt, x dx dt + 4 T u λ t, x 2 u λ t, x u λ t, x ξt, x dx dt = λ 3.4 The last term of the left-hand side of 3.4 vanishes identically. Furthermore we remark that λ u λ ξ = u λ λ ξ. x i x i x i x i Now we can take the limit when λ goes to to obtain T t, x ut, x t, x ut, x ξt, x dx dt T 2 i= ξ t, x ut, x t, x dx dt x i x i T 2 Ht, x ϕut, x ut, x ξt, x dx dt =, 9

10 that is T t, x + ut, x t, x ξt, xdx dt as T +2 since u = a.e. in IR +. ut, x t, x ξ t, xdx dt x i= i x i T 2 ut, x Ht, x ϕut, x ξt, xdx dt = u ξ = u ξ, and u u ξ = ξ 3.5 Moreover as the L 2 resp. L norm is lower semi continuous for the weak resp. weak the energy estimate 3.3 remains valid for u =. Next from 3.5 we derive that i= u belongs to L 2 x i x locir + ; IL 2 i so u ν makes sense in L2 loc IR+ ; IH /2. Moreover as u 2 =, one has u =. So from the equality ν ν = u ν u + u u ν = ν which is valid in H η for any η > according to the product of function in sobolev spaces see L. Hörmander [6] so in fact ν makes sense in L2 locir + ; H η for any η >. As the Maxwell equations are linear, it is straightforward to take the limit in 3. and 3. to obtain 2.3 and Description of the ω-limit set. Consider a weak solution u of.-.5. From the energy estimate 2.5, the ω-limit set ωu is not empty. We denote u a point of this set. Hence there exists a sequence t n n, with lim n + t n = + such that ut n,. tends to u in IH weak, in IL 2 strong, and almost everywhere in. In particular one has u = a.e. in. First step. Let be a on negative real number. For s in a, a and x in we define for n large enough U n s, x = ut n + s, x.

11 The sequence U n n tends to u in IL 2 a, a strongly and in L 2 a, a; IH weakly. In fact following [2], we have the estimate a U n s, x ut n, x 2 dx ds = a 2a a 2a a Now, as lies in IL2 IR +, one gets lim n + a s 2a a a + t n a s t 2 n + τ, xdτ dx ds + t n a τ, x a U n s, x ut n, x 2 dx ds =. 2a a τ, x 2 dx dτ. 2 dτdx ds Since ut n,. tends to u in IL 2 strongly, U n tends to u in L 2 a, a; IL 2 strongly. Moreover we obviously see that the sequence U n n is bounded in IL 2 a, a so there exists a subsequence still noted U n n such that U n tends to u in L 2 a, a; IH weakly, in L 2 a, a; IL 2 strongly and almost everywhere in. We set and Second step. We consider a C non negative function ρ a supported by a, a satisfying ρ a τ = for τ a +, a, ρ a τ, ρ aτ 2. H x = a Ht n + s, xρ a sds 2a a E x = a Et n + s, xρ a sds. 2a a From the estimate 2.5, E and H are bounded in L IR + ; IL 2. Then H n a and E n a are bounded in IL 2 independently of n and a. So by extracting a subsequence we may suppose that E n a, H n a n converges in IL 2 weakly to E a, H a when n goes to +. Third step. In the weak formulation 2.2 we take as test function ρ a t t n Ψx where Ψ is a function lying in D. We obtain after the change of chart s = t t n a Un 2a a s, x + U ns, x U n s, x Ψxρ a sdx ds +2 a 2a a i= U n s, x U n s, x Ψ ρ a sdx ds x i x i 2 a U n Ht n + s, x ϕu n s, x Ψxρ a sdx ds =. 2a a To pass through the limit in 4. we bound separately each term of

12 First term. a U n 2a a s, x Ψxρ asdx ds a ρ a s U n 2 /2 /2 2a s, x a dx Ψx dx 2 ds /2 tn+a Ψx dx 2 /2 2a s, x t n a dx ds Since belongs to IL2 IR +, this last term tends to zero as n goes to +. In the same way, as U n takes its values on S 2, one also has lim n + 2a a a U n s, x U n s, xρ as Ψxdx ds = Second term. As U n n tends to u strongly in IL 2 a, a, as U n n tends to weakly in x i x i IL 2 a, a and since Ψ ρ a belongs to IL a, a, the second term of 4. tends to x i Third term. 2 a ρ a sds 2a a i= u x x Ψ xdx. x i x i a U n s, x Ht n + s, x Ψxρ a sdx ds 2a a = a U n s, x u x Ht n + s, x Ψxρ a sdx ds 2a a + a u x Ht n + s, x Ψxρ a sdx ds. 2a a The first term of 4.2 goes to zero as U n u n tends strongly to zero in IL 2 a, a and as H is bounded in L IR + ; IL 2. The second term is equal to u x H x Ψxdx, and tends obviously to u x H a x Ψxdx. As ϕ is linear, it is straightforward to take the limit in the last term. So from equation 4. we derive that u solve the equation i= u x x Ψ x + u x ϕu x Ψxdx x i x i 2a a a ρsds u x H a x Ψxdx =

13 Forth step. In order to obtain the desired result it remains to take the limit in 4.3 when a tends to +. We first remark that lim a + a a 2a =. ρsds Through estimate 2.5 and by definition of H a, H a a is uniformly bounded in IL 2. Hence, by extraction we can suppose that H a tends to H weakly in IL 2. So at the limit one has i= u x x Ψ xdx + u x H x ϕu x Ψxdx = x i x i Fifth step. In order to derive the equation satisfied by H we first recall the equation verified by H and E. In equation 2.3 we take Ψt, x = θ a t t n ξx with ξ in D and θ a is defined by θ a t = t a ρ a sds. We obtain that for every ξ in D a Ht n + s, x + ūt n + s, x ξxρ a sds = a As div H + ū = in D, we obtain after dividing by 2a H x + a ūt n + s, xρ a sds ξxdx =. 2a When n goes to + we obtain that H a x + ū x ξxdx =, and so when a goes to infinity we get a div H + ū = in D. H x + ū x ξxdxθ a. Now we take Ψt, x = ρ a t t n ξx in 2.4. We obtain that a 2a Et n + s, x ρ asξxdx ds Hn a a x curl ξxdx +σ E a x ξxdx + σ ft n + s, x ρ a sξxdx ds 2a a = E x ξxdx ρ a t n. For n large enough, the righthand side of 4.4 vanishes identically. Let us bound the first term of 4.4. As ρ a is identically zero on a +, a and is bounded by 2, one has a 2a Et n + s, x ρ a a sξxdx a ξ IL 2 E L IR + ; IL

14 Moreover a 2a ft n + s, x ρ a sξxdx ds a that is 2a a a a+tn /2 a /2 fs 2 2a IL a+t 2 ρ a s ds 2 ξ IL 2, n a ft n + s, x ρ a sξxdx ds a+tn /2 fs 2 2a IL a+t 2 ξ IL n since ρ a s. When n goes to infinity, by extraction of a subsequence the first term of the left-hand side of 4.4 tends to a real α a satisfying α a 2a E L IR + ; IL 2 ξ IL Due to 4.6, the fourth term of the left-hand side of 4.4 goes to zero as ft, x 2 dx dt < +. Hence we obtain IR + α a H ax curl ξxdx + σ E a x ξxdx =. Then taking the limit as a goes to infinity, one has from 4.7 H x curl ξxdx = σ E x ξxdx. 4.8 In the same way, taking Ψt, x = ρ a t n tξx in 2.3 we derive that E x curl ξxdx =, that is curl E =. So it is valid to take ξx = E x in 4.8 which leads to σ E x 2 dx =. This 4.8 gives curl H =. Finaly H is uniquely determined by div H + ū = in, curl H = in, H IL 2. Therefore u is a solution of the stationary model Remark 4. Following an idea of G. Métivier, it is possible to prove Theorem 2.2 without average Maxwell Equations. This is due to the fact that Ht,. Hut tends to zero in L 2 loc when t tends to + see [8]. 4

15 5 Quasi-stationary model The last part of this paper is devoted to the justification of the quasi-stationary model. We recall that we suppose f. We consider ε n and µ n such that ε n, µ n and ε n /µ n tend to zero. In the sequel we denote u n, H n, E n a family of weak solutions of.-.5 with ε = ε n and µ = µ n. We recall the energy estimate satisfied by u n, H n, E n. where E n t = E n t + t n t, x 2 dx dt + σ µ n t E n t, x 2 dx dt E n 5. u n t, x 2 + 2Φu n t, x dx + H n t, x 2 + εn µ n En t, x 2 dx. Since ε n /µ n remains bounded, the right hand-side term of 5. remains bounded uniformly in n. Therefore, by the energy estimate 5., u n is bounded in L IR + ; IH and n is bounded in IL 2 IR + uniformly in n. Furthermore H n and ε n /µ n E n are uniformly bounded in L IR + ; IL 2. Extracting a subsequence we can suppose that u n u in L IR + ; IH weak, u n u in L 2, T ; IL 2 strong for all T >, n First step. For any a > we set in L 2, T ; IL 2 weak for all T >. u n at, x := a a u n t + s, xds, H n a t, x := a a H n t + s, xds, 5.2 E n a t, x := a a E n t + s, xds. Lemma 5. For each n IN and a >, u n a, Hn a, En a satisfies the following estimates. u n a L IR + ; IH un L IR + ; IH, 5.3 n a IL 2 IR + n IL 2 IR +, 5.4 H n a L IR + ; IL 2 Hn L IR + ; IL 2, 5.5 E n a L IR + ; IL 2 En L IR + ; IL

16 Proof. The estimates 5.3, 5.5 and 5.6 follow directly from the definition 5.2. For 5.4 we write so That is n a t, x = a un t + a, x u n t, x = IR + n a s, x 2 ds IR + IR + n t + θa, xdθ, n 2 t + θa, xdθ dt n IR+ s, x 2 ds. n a s, x 2 ds dx n IR + s, x 2 ds dx. Lemma 5.2 For every a > we have the following estimate u n a un L IR + ; IL 2 a n IL 2 IR +. Proof. From the definition 5.2 one gets u n a t, x un t, x = a a u n s + t, x u n t, xds so hence = a u n at, x u n t, x 2 a a s a a s n n t + τ, xdτ ds, n t + τ, x dτ ds t + τ, x dτ t+a a n t s, x 2 ds, u n at, x u n t, x 2 dx a n IR + s, x 2 ds dx. 2 2 Second step. We choose = ε n µ n 4, and we denote in the sequel u n := u n, H n := H n, and E n := E n. Thanks to the energy estimate 5. and Lemma 5., we can suppose after extraction of a subsequence that u n u in L IR + ; IH weak, u n u in L 2, T ; IL 2 strong for all T >, H n H in L IR + ; IL 2 weak, n in IL 2 IR + weak. 6

17 Furthermore Lemma 5.2 ensures that u = u and u n, u in IL 2 strong. Third step. For t given in IR + we take Ψs, x = [t,t+a[ sξx in 2.. After dividing by we obtain that n t, x ξxdx + an u n t + s, x δdtu n t + s, x ξxds dx 2 +2 an an i= u n t + s, x n t + s, x ξ xds dx x i x i u n t + s, x H n t + s, x ϕu n t + s, x ξxds dx =. Multiplying this last formula by a test function ρt, we obtain after integration n t, x ξxρtdx dt + IR + + IR + an an i= IR + u n t + s, x n t + s, x ξxρtds dx dt u n t + s, x n t + s, x ξ xρtds dx dt x i x i 2 t+an u n s, x H n s, x ϕu n s, x ξxρtds dx dt =. IR + t 5.7 As n in IL 2 IR + weakly, the first term of 5.7 tends to IR + t, x ξxρtdx dt. Let us now study the second term. an u n t + s, x n t + s, x ξxρtds dx dt = IR + an ρtξx u n n t, x IR + t + s, xds dx dt an n u n t + s, x u n t, x + ρtξx IR + The definition of u n shows that this is equal to IR + an IR + + ρtξx IR + u n t + s, x n t + s, x ρtξx an u n t, x n s, xds dt dx. ξxρtds dx dt = t, x dt dx u n t + s, x u n t, x n s, xds dt dx

18 as The first term of 5.8 tends to IR + ρtξx u t, x t, x dt dx u n u in L 2 loc IR+ ; IL 2 strongly and n in IL 2 IR + weakly. Now we prove that the second term goes to zero. We use the Cauchy-Schwarz inequality to obtain A := an ρtξx u n t + s, x u n t, x n t + s, xds dx dt IR + A ξ IL ρ IL IR + { IR + { an IR + an s } n t + s, 2 x 2 ds dx dt. Now by the Cauchy-Schwarz inequality and Fubini theorem we get { A ξ IL ρ IL IR + an So after integration IR + an n } 2 2 t + τ, xdτ dx dt ds an } s n t + τ, 2 n x 2 dτ ds dt dx IL 2 IR +. A 2 ξ IL ρ IL IR + n 2 IL 2 IR +. Hence by the energy estimate 5., A tends to zero as. In the same way as in the previous section we obtain finally IR + t, x + u t, x t, x ξxρtdx dt + 2 IR + i= u t, x t, x ξ xρtdx dt x i x i 2 u t, x H t, x ϕu t, x ξxρtdx dt =. IR Fourth step. As for the study of the ω-limit set we can prove that div H + ū =. Now it remains to obtain curl H =. 5. 8

19 We recall that for all ξ in D and ρ in D[, + we have according to 2.4 that εn E n s, x ρ IR + sξxds dx Hn s, x curl ξxρsdx ds IR + +σ E n s, x ρsξxds dx = E x ξxρdx. IR + 5. Formally, the identity 5. is obtained taking ρ = t,t+an in 5.. function is not regular enough, so we introduce a regularised function ρ δ. For each δ > given, < δ <, we denote δ s δ ρ δ s = s or s Unfortunately this linear s δ and δ s Now, for ρ = ρ δ s t equation 5. gives εn t+δ a En s, x ρ δ n t s t ξxds dx εn t+an a En s, x ρ δ n t+ δ s t ξxds dx Hn a x curl ξxdx + σ t+an t ρ δ t se n t, x ξxdx ds = t+an a Hn s, x ρ δ s tcurl ξxdx ds. n t 5.2 The two first terms of the left-hand side of 5.2 are bounded by The last term of the left-hand side of 5.2 is bounded by The right-hand side of 5.2 is bounded by According to the energy estimate 5. we have 2 εn E n L IR + ; IL 2 ξ IL σ E n L IR + ; IL 2 ξ IL δ H n L IR + ; IL 2 curl ξ IL E n L IR + ; IL 2 k µ n ε n E n L IR + ; IL 2 k µ n 9

20 for some constant k So by choosing = ε n µ n 4 and δ = a 2 n we get, for any test function ϕ H t, x curl ξxϕtdx dt =. Fifth step. Energy estimate. By convexity and thanks to the definition 5.2, one has a a On the other hand Hence u n at, x 2 dx + 2 Φu n at, x + Hn a t, x 2 dx u n t + s, x 2 dx + 2 Φu n t + s, x + Hn t + s, x 2 dx t a n a s, x 2 dx ds = a a a t+s E n t + sds. t a a n τ + s, xdτ n τ, x 2 dτ dx ds. u n at, x 2 dx + 2 Φu n at, x + Hn a t, x 2 dx + a a E n t + s + t+s n τ, x 2 dτ dx t 2 dx ds n a s, x 2 dx ds ds E n. Since ε n /µ n tends to zero, E n tends to E q. Therefore using the semi continuity of the norms for the weak topology, we derive the desired energy estimate 2.2. Acknowledgements: The authors wish to thank professors T. Colin, J.L. Joly, M. Langlais, and G. Métivier for many stimulating discussions. References [] F. Alouges, Private communication. [2] F. Alouges et A. Soyeur, On global weak solutions for Landau Lifschitz equations: existence and non uniqueness, Nonlinear Anal., Theory Methods Appl. 8, No., [3] W.F. Brown, Micromagnetics, Interscince publisher, John Willey & Sons, New York, 963. [4] G. Carbou, Modèle quasi-stationnaire en micromagnétisme. C.R. Acad. Sci. Paris, t. 325, Série, p , 997. [5] G. Carbou et P. Fabrie, Comportement asymptotique des solutions faibles des équations de Landau-Lifschitz. C.R. Acad. Sci. Paris, t 325, Série, p ,

21 [6] L. Hörmander, Progress in Nonlinear Differential Equations and their Applications, 2. Birkhuser Boston, Inc., Boston, MA, 996. [7] J.L. Joly, G. Métivier et J. Rauch, Solution globale du système de Maxwell dans un milieu ferromagnétique, Ecole Polytechnique, Séminaire EDP, , exposé N o. [8] J.L. Joly, G. Métivier et J. Rauch, Private communication. [9] P. Joly et O. Vacus, Mathematical and numerical studies of nonlinear ferromagnetic materials, à paraître M2AN. [] F. Labbé, Private communication. [] L. Landau et E. Lifschitz, Electrodynamique des milieux continues, cours de physique théorique, tome VIII ed. Mir Moscou 969. [2] M. Langlais et D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equation, Nonlinear Analysis, TMA, Vol. 9, n 4 p.p , 985. [3] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure, Siam. J. Math. Anal., Vol. 2, n 5 p.p; 93-7, 99. [4] A. Visintin, On Landau Lifschitz equation for ferromagnetism, Japan Journal of Applied Mathematics, Vol. 2, n, p.p , 985. [5] H. Wynled, Ferromagnetism, Encyclopedia of Physics, Vol. XVIII / 2. Springer Verlag, Berlin,