4.1.1. Scaling and Uniform Bounds
Let (M,W) be a solution of the problem (3.1)–(3.5) posed in Ωɛ. We introduce the change of variables (x1,x2,x3)=(x,y,ɛz) with (x,y,z)=X∈Ω=ω×(0,1). For functions R(x1,x2,x3) and S(x1,x2,x3) defined in Ωɛ we introduce the functions rɛ(x,y,z) and sɛ(x,y,z) defined on Ω by setting
(4.1)R(x1,x2,x3)=rɛ(x,y,z); S(x1,x2,x3)=sɛ(x,y,z).
Let (mɛ,wɛ) be the fields associated with (M,W). The scaled equations satisfied by (mɛ) are the following:
(4.2)γ-1∂tmɛ-mɛ×(a(Δ^mɛ+1ɛ2∂zzmɛ)-∂tmɛ-λV~ɛ)=0.
The vector V~ɛ is defined by
(4.3)V~ɛ=(m3ɛ∂xwɛ+1ɛm1ɛ∂zwɛ,m3ɛ∂ywɛ+1ɛm2ɛ∂zwɛ,m1ɛ∂xwɛ+m2ɛ∂ywɛ+3ɛm3ɛ∂zwɛ).
For the scaled displacement uɛ=(0,0,wɛ) we have
(4.4)ρ∂ttwɛ-τ(Δ^wɛ+2ɛ2∂zzwɛ)-λ(∂x(m1ɛm3ɛ)+∂y(m2ɛm3ɛ)-λɛ∂z(m3ɛ)2)=0,2τɛ∂xzwɛ-λ∂x(m1ɛ)2-λ∂y(m1ɛm2ɛ)-τɛ∂zxwɛ-λɛ∂z(m1ɛm3ɛ)=0,-λ∂x(m1ɛm2ɛ)+2τɛ∂yzwɛ-λ∂y((m2ɛ)2)-τɛ∂zxwɛ-λɛ∂z(m2ɛm3ɛ)=0.
The associated energy ℰɛ(t), defined in (2.7), becomes
(4.5)Eɛ(t)=a2∫Ω|grad^ mɛ|2dΩ+a2ɛ2∫Ω|∂z mɛ|2dΩ +β4∫Ω|grad^ wɛ|2dΩ+β4ɛ2∫Ω|∂zwɛ|2dΩ+ρ2∫Ω|∂twɛ|2dΩ.
The energy equation remains unchanged as well as the saturation constraint on magnetization (see (2.5)) which is written as
(4.6)|mɛ(t,X)|2=|m0ɛ(X)|2=1
for almost every (t,X). The following estimates hold true for all t≥0(4.7)Eɛ(t)+∫0t∫Ω|∂tmɛ|2dΩ dt≤c1E0ɛ+c2,
where ℰ0ɛ is given by
(4.8)E0ɛ=a2∫Ω|grad^ m0ɛ|2dΩ+a2ɛ2∫Ω|∂zm0ɛ|2dΩ+τ4∫Ω|grad^ w0ɛ|2dΩ+τ4ɛ2∫Ω|∂zw0ɛ|2dΩ+ρ2∫Ω|w1ɛ|2dΩ.
To get uniform bounds for the solutions we discuss the admissibility criterion for the initial data. An initial datum (m0ɛ,w0ɛ) is said to be admissible if we have
(4.9)E0ɛ<+∞.
The admissibility criterion means
(4.10)a2∫Ω|grad^ m0ɛ|2dΩ+a2ɛ2∫Ω|∂zm0ɛ|2dΩ +τ4∫Ω|grad^ w0ɛ|2dΩ+τ4ɛ2∫Ω|∂zw0ɛ|2dΩ+ρ2∫Ω|w1ɛ|2dΩ<+∞.
Thus, since |m0ɛ|2=1 a.e., to satisfy the criterion, we assume that there exists C>0 independent of ɛ such that
(4.11)|grad^ m0ɛ|L2(Ω)≤C, |∂zm0ɛ|L2(Ω)≤Cɛ, |m0ɛ(x,y)|2=1 a.e.,|grad^ w0ɛ|L2(Ω)≤C, |∂zw0ɛ|L2(Ω)≤Cɛ, |w1ɛ|L2(Ω)≤C.
Condition (4.11) means that the couple (m0ɛ,w0ɛ) is essentially independent of the variable z and its strong limit (m0,w0) is independent of z.
Remark 4.1.
If the initial data are not admissible, then we expect an initial layer to occur when ɛ tends to zero.
4.1.2. Passing to the Limit
Let (mɛ,wɛ) be a solution of the problem associated to an admissible initial datum (m0ɛ,w0ɛ). We have
(4.12)m0ɛ⇀m0 weakly-⋆ in L∞(Ω) and weakly in H1(Ω),w0ɛ⇀w0 weakly in H1(Ω).
Moreover m0(x^,z)=m0(x^) is independent of z. For subsequences, the solutions verify the convergences
(4.13)mɛ⇀m weakly-⋆ in L∞(R+×Ω)∩L∞(R+,H1(Ω)),wɛ⇀w weakly in L2(0,T,H01(Ω)),(4.14)∂zmɛ⟶0 strongly in L∞(R+,L2(Ω)),∂zwɛ⟶0 strongly in L∞(R+,L2(Ω)),∂tmɛ⇀∂tm weakly in L2(R+,L2(Ω)),∂twɛ⇀∂tw weakly in L2(0,T;L2(Ω)).
Hence, the couple (m,w) is independent of the variable z. By Aubin’s compactness results, we have
(4.15)(mɛ,wɛ)⟶(m,w) strongly in Lloc2(R+,L2(Ω)).
Moreover from the Sobolev embedding theorem W1,2(Q)→Lq(Q) (2≤q≤6), the further compactness result follows
(4.16)miɛmjɛ⟶mimj strongly in L2(Q),i,j=1,2,3.
Recall that Q=(0,T)×Ω with Ω=ω×(0,1).
In order to pass to the limit we look at the variational formulation of the scaled problem (4.2)–(4.4) by using an oscillating test functions. Let ψɛ(t,x^,z) and gɛ(t,x^,z) be a regular test functions depending on ɛ. Multiplying (4.2) by ψɛ, each Equation (4.4) by gɛ and integrating by parts, we get the weak formulations
(4.17)γ-1∫Q∂tmɛ⋅ψɛ dΩ dt+∫Qmɛ×∂tmɛ⋅ψɛ dΩ dt =-λ∫Qmɛ×V~ɛ⋅ψɛ dΩ dt-a∫Qmɛ×grad^ mɛ⋅grad^ ψɛ dΩ dt-aɛ2∫Qmɛ×∂zmɛ⋅∂zψɛ dΩ dt, (4.18)-ρ∫Q∂twɛ∂tgɛ dΩ dt+τ∫Qgrad^ wɛgrad^ gɛ dΩ dt+2τɛ2∫Q∂zwɛ∂zgɛ dΩ dt +λ∫Qm1ɛm3ɛ∂xgɛ dΩ dt+λ∫Qm2ɛm3ɛ∂ygɛ dΩ dt+λɛ∫Q(m3ɛ)2∂zgɛ dΩ dt=0,-2τɛ∫Q∂zwɛ∂xgɛ dΩ dt+λ∫Q(m1ɛ)2∂xgɛ dΩ dt+λ∫Qm1ɛm2ɛ∂ygɛ dΩ dt+τɛ∫Q∂xwɛ∂zgɛ dΩ dt+λɛ∫Qm1ɛm2ɛ∂zgɛ dΩ dt=0, λ∫Qm1ɛm2ɛ∂xgɛ dΩ dt-2τɛ∫Q∂zwɛ∂ygɛ dΩ dt+λ∫Q(m2ɛ)2∂ygɛ dΩ dt+τɛ∫Q∂xwɛ∂zgɛdΩ dt+λɛ∫Qm2ɛm3ɛ∂zgɛdΩ dt=0.
To pass to the limit in these equations we need the following convergence result.
Lemma 4.2.
Defining Θɛ:=(1/ɛ)∂zwɛ, then
(4.19)Θɛ⇀Θ=-λ2τ(m3)2+K
weakly
-⋆
in
L∞(R+,L2(Ω)),
where K is a function of the variable x^.
Proof.
We multiply the first equation of (4.18) by ɛ and choose gɛ=g∈𝒟(Q) independent of ɛ. We get
(4.20)ɛ(-ρ∫Q∂twɛ∂tg dΩ dt+τ∫Qgrad^ wɛgrad^ g dΩ dt+λ∫Qm1ɛm3ɛ∂xg dΩ dt +λ∫Qm2ɛm3ɛ∂yg dΩ dt)+2τɛ∫Q∂zwɛ∂zg dΩ dt+λ∫Q(m3ɛ)2∂zg dΩ dt=0.
Hence, passing to the limit, by using convergences (4.14), (4.15), and (4.16), we deduce that the weak-⋆ limit Θ of the sequence Θɛ satisfies ∂z(2τΘ+λm32)=0 which allows to get (4.19).
Remark 4.3.
In the sequel and without loss of generality we will assume that K≡0.
Now we are able to pass to the limit. We set Q𝒯=ℝ+×ω. We choose in the above weak formulations test functions of the form
(4.21)ψɛ(t,x^,z)=ψ0(t,x^)+ɛψ(t,x^,ɛz),gɛ(t,x^,z)=g(t,x^)+ɛg0(t,x^)h(ɛz).
We pass to the limit in each term of (4.17) by using the convergence results (4.14), (4.15), (4.16) and the following facts, ∂zψɛ=ɛ2(∂zψ)(ɛz) and ∂x^ψɛ=∂x^ψ0+ɛ∂x^ψ(ɛz). Hence we first get
(4.22)∫Q∂tmɛ⋅ψɛ dΩ dt⟶∫QT∂tm⋅ψ0 dx^ dt,∫Qmɛ×∂tmɛ⋅ψɛ dΩ dt⟶∫QTm×∂tm⋅ψ0 dx^ dt.
Next, we have
(4.23)∫Qmɛ×grad^ mɛ⋅grad^ ψɛ dΩ dt⟶∫QTm×grad^ m⋅grad^ ψ0 dx^ dt.
We also get
(4.24)1ɛ2∫Qamɛ×∂zmɛ⋅∂zψɛ dΩ dt⟶0.
Recall that
(4.25)V~ɛ=(m3ɛ∂xwɛ+1ɛm1ɛ∂zwɛ, m3ɛ∂ywɛ+1ɛm2ɛ∂zwɛ, m1ɛ∂xwɛ+m2ɛ∂ywɛ+3ɛm3ɛ∂zwɛ).
To pass to the limit in the term with V~ɛ we make use of the convergence of Lemma 4.2.
Similarly we pass to the limit in the weak formulation (4.18). The convergences (4.14) and (4.15) allow to get
(4.26)∫Q∂twɛ∂tgɛ dΩ dt⟶∫QT∂tw∂tg dx^ dt,∫Qgrad^ wɛgrad^ gɛ dΩ dt⟶∫QTgrad^ w grad^ g dx^ dt.
Next, we have
(4.27)∫Qm1ɛm3ɛ∂xgɛ dΩ dt⟶∫QTm1m3∂xg dx^ dt,∫Qm2ɛm3ɛ∂ygɛ dΩ dt⟶∫QTm2m3∂yg dx^ dt.
We also get
(4.28)1ɛ2∫Q∂zwɛ∂zgɛ dΩ dt⟶0,1ɛ∫Q(m3ɛ)2∂zgɛ dΩ dt⟶0.
It remains to identify the limit of the two last equations of (4.18). Let us pass to the limit in the second equation of (4.18). We make use of the Lemma 4.2 to get
(4.29)1ɛ∫Q∂zwɛ∂xgɛ dΩ dt⟶-∫QTλ2τ(m3)2∂xg dx^ dt.
Similarly, by the same arguments above, we also get the limit both for the other terms and for the last equation of (4.18).
We proved the result.
Theorem 4.4.
Let (mɛ,wɛ) be a solution of the problem associated with the admissible initial datum (m0ɛ,w0ɛ). Then, one has (mɛ,wɛ)→(m,w) strongly in Lloc 2(ℝ+,L2(Ω)), mɛ⇀m weakly-⋆ in L∞(ℝ+,H1(Ω)) and wɛ⇀w weakly in L2(ℝ+,H01(Ω)). The couple (m,w) is independent of the variable z and satisfies in ℝ+×ω, |m(t,x^)|2=1 and the following two-dimensional coupled system
(4.30)γ-1∂tm-m×(aΔ^m-∂tm-λV~)=0,ρ∂ttw-τΔ^w-λ∂x(m1m3)-λ∂y(m2m3)=0,∂x(m12+m32)+∂y(m1m2)=0,∂y(m22+m32)+∂x(m1m2)=0,
where
(4.31)V~=(m3∂xw-λ2τm1m32, m3∂yw-λ2τm2m32, m1∂xw+m2∂yw-3λ2τm33).
The associated initial and boundary conditions are given by
(4.32)w(0,x^)=w0, ∂tw(0,x^)=w1, m(0,x^)=m0, |m0|=1
in
ωw=0, ∂νm=0 on ∂ω,
where w1 is the weak limit of w1ɛ in L2(Ω).
Remark 4.5.
Note that if the function K introduced in Lemma 4.2 is such that K≢0, then the vector V~ in (4.30) becomes
(4.33)V~=(m3∂xw-(λ2τm32+K)m1,m3∂yw-(λ2τm32+K)m2,m1∂xw +m2∂yw-3(λ2τm32+K)m3).