A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics

and Applied Analysis 3 linear equation, which requires special tricks and techniques. In the convergence process, a compensated compactness principle is applied. The rest of this paper is organized as follows. Section 2 is devoted to studying 1.5 . More precisely, we first study the penalized equation. In order to do this, we consider the corresponding linear equation and get its formal solution and well-posedness, then we get the existence of a unique mild solution of the penalized equation using semigroup theory. Second, we get the existence of a weak solution of 1.5 by passing to the limit in the penalized equation. The key point in the convergence process relies on a compensated compactness principle. In Section 3, we get existence of weak solution of 1.4 in Theorem 3.1 by passing to the limit in 1.5 as ε → 0. 2. Approximation Equations In this section, we always suppose that T2 R2/ 2πZ 2 is the flat torus. We prove existence of a weak solution of the following equations: ∂m ∂t εΔm ∇ −Δ −1/2divmε ε∣∇mε∣2mε − ∇ −Δ −1/2divmε ·mεmε, in T2 × 0, ∞ , 2.1 m x, 0 m0 x , on T2, 2.2 m : T2 × 0,∞ −→ R2, ∣mε∣ 1 a.e. in T2. 2.3 Denote Lm −εΔmε − ∇ −Δ −1/2divmε. Note that the corresponding energy is E m ε ∫ Ω|∇m|dx ∫ R2 |ξ · m̂χΩ|/|ξ| dξ. The variation of the self-induced energy is lim η→ 0 ∫ R2 ∣ ∣ξ · ̂ mχΩ ηv ∣ ∣2 − ∣ξ · m̂χΩ ∣ ∣2 |ξ|η dξ ∫ R2 2iξ · m̂χΩ |ξ|1/2 iξ · v̂ |ξ|1/2 2 ∫ R2 ( −Δ divmχΩ )( −Δ −1/4divvdx 2 ∫ R2 −Δ divmχΩ divv dx 2 ∫ R2 − ∇ −Δ divmχΩ ·v dx. Equation 2.1 can be written as ∂m ∂t −Lmε Lmε ·mεmε. 2.4 It is very easy to prove that 2.1 is equivalent to m × ∂m ε ∂t m × Lm 0. 2.5 4 Abstract and Applied Analysis The equivalence follows from the following. Lemma 2.1. In the classical sense, m is a solution of 2.1 – 2.3 if and only if m is a solution of 2.5 . Proof. Suppose thatm is a solution of 2.1 – 2.3 . By the vector cross product formula a × b × c a · c b − a · b c, 2.6 we have ∂m ∂t −Lmε Lmε ·mεmε ( Lm ·mεmε − mε ·mεLmε m × mε × Lmε. 2.7 By the cross product ofm and 2.7 , we have m × ∂m ε ∂t m × mε × mε × Lmε −mε × Lm. 2.8 This proves thatm satisfies 2.5 . Suppose that m is a solution of 2.5 . Then by the cross product of m and 2.5 , we obtain m × ( m × ∂m ε ∂t ) m × mε × Lmε 0. 2.9 Since |mε| 1, we havem · ∂m/∂t 0. Hence 2.9 implies ∂m ∂t −Lmε Lmε ·mεmε. 2.10 We define a local weak solution of 2.1 as follows. Definition 2.2. A vector-valued function m x, t is said to be a local weak solution of 2.1 , if m is defined a.e. in T2 × 0, T such that 1 m ∈ L∞ 0, T ;H1 T2 and ∂m/∂t ∈ L2 T2 × 0, T ; 2 |mε x, t | 1 a.e. in T2 × 0, T ; 3 2.1 holds in the sense of distribution; 4 m x, 0 m0 x in the trace sense. Abstract and Applied Analysis 5 We state our main result in this section as follows. Theorem 2.3. For everym0 x ∈ H1 T2 and |m0 x | 1, a.e. in T2, there exists a weak solution of 2.1 – 2.3 . To prove Theorem 2.3, we have to consider a penalized equation.and Applied Analysis 5 We state our main result in this section as follows. Theorem 2.3. For everym0 x ∈ H1 T2 and |m0 x | 1, a.e. in T2, there exists a weak solution of 2.1 – 2.3 . To prove Theorem 2.3, we have to consider a penalized equation. 2.1. The Penalized Equation In the spirit of 13 , we first construct weak solutions to a penalized problem, where the constraint |mε| 1 is relaxed: ∂m ∂t Lm − k2 ( 1 − ∣mk∣2 ) m 0, in T2 × 0, ∞ , 2.11 m x, 0 m0 x , on T2, 2.12 ∣ ∣m0 x ∣ ∣ 1, on T2. 2.13 Here m : T2 × 0,∞ → R2. In order to prove the existence of a mild solution of semilinear system 2.11 – 2.13 , we consider the corresponding linear equation. 2.1.1. The Corresponding Linear Equation First, we consider the corresponding linear equation of 2.11 – 2.13 in the whole space: ∂m ∂t εΔm ∇ −Δ −1/2divm k2m, in R2 × 0, ∞ , m x, 0 m0 x , on R2, 2.14 where m0 x m01 x , m02 x . While dealing with linear equation 2.14 , we just write m instead ofm unless there may be some confusion. By Fourier transform in the x-variable, 2.14 are turned into m̂t ε ∣ ∣ξ ∣ ∣m̂ ( ξ ⊗ ξ |ξ| ) m̂ − k2m̂ 0, in R2 × 0, ∞ , m̂ ξ, 0 m̂0 ξ , on R2. 2.15 For each fixed ξ, the problem has a unique solution m̂ ξ, t e−B ξ t · e−A ξ m̂0 ξ , 2.16 where A ξ 1 |ξ| ( ξ2 1 ξ1ξ2 ξ1ξ2 ξ 2 2 ) , B ξ −k2 ε|ξ|2 0 0 −k2 ε|ξ|2 ) . 2.17 6 Abstract and Applied Analysis So the problem has the solution m x, t 1 4π2 ( e−B ξ t ∗e−A ξ t∗m0 x . 2.18 Now the only problem left is to find the inverse Fourier transforms of e−A ξ t and e−B ξ . First, we need to find an orthogonal matrix O ξ such that O ξ A ξ Oτ ξ is the Jordan normal form of A ξ . In fact, O ξ 1 |ξ| ( ξ2 −ξ1 ξ1 ξ2 ) . 2.19 Now we begin to calculate the inverse Fourier transform of e−A ξ t ( e−A ξ t )∨ 1 2π ∫ R2 e · ξe−A ξ dξ 1 2π ∫ R2 e · ξOτ ξ O ξ e−A ξ tOτ ξ O ξ dξ 1 2π ∫ R2 e · ξOτ ξ ( ∞ ∑ n 0 −1 tn n! O ξ A ξ Oτ ξ n ) O ξ dξ 1 2π ∫ R2 e · ξOτ ξ ( ∞ ∑ n 0 −1 tn n! ( 0 0 0 |ξ| )n) O ξ dξ 1 2π ∫ R2 e · ξOτ ξ ( I ∞ ∑ n 1 −1 tn n! ( 0 0 0 |ξ|n )) O ξ dξ 1 2π ∫ R2 e · ξOτ ξ ( I ( 0 0 0 e−|ξ|t − 1 )) O ξ dξ 1 2π ∫ R2 e · ξ ( I 1 |ξ|2 ( ξ2 1 ξ1ξ2 ξ1ξ2 ξ 2 2 ) ( e−|ξ|t − 1 ) dξ δ x I 1 2π ∫ R2 e · ξ 1 |ξ|2 (( ξ2 1 ξ1ξ2 ξ1ξ2 ξ 2 2 ) ( e−|ξ|t − 1 ) dξ. 2.20


Introduction
Landau-Lifshitz equations are fundamental equations in the theory of ferromagnetism.They describe how the magnetization field inside ferromagnetic material evolves in time.The study of these equations is a very challenging mathematical problem, and is rewarded by the great amount of applications of magnetic devices, such as recording media, computer memory chips, and computer disks.The equations were first derived by Landau and Lifshitz on a phenomenological ground in 1 .They can be written as where × is the vector cross product in R n n ≥ 2 , m m 1 , m 2 , . . ., m n : Ω × 0, ∞ → R n is the magnetization and α 2 is a Gilbert damping constant.The system 1.1 is implied by the conservation of energy and magnitude of m.H m −δE/δm is the unconstrained first variation of the energy functional E m .The magnitude of the magnetization is finite, that is, Here Abstract and Applied Analysis is the free energy functional, and it is composed of three parts: i E ex m Ω |∇m| 2 dx is the exchange energy.It tends to align m in the same direction and prevents m from being discontinuous in space; ii E an m Ω φ m dx is the anisotropy energy.φ ∈ C ∞ R 3 , φ ≥ 0 depends on the crystal structure of the material.It arises from the fact that the material has some preferred magnetization direction, for example, if 1,0,0 is the preferred magnetization direction, φ m m 1.3 Equation 1.1 has been widely studied.In the case β 0, α / 0, 1.1 corresponds to the heat flow for harmonic maps studied in 2, 3 ; if β / 0, α / 0 which implies strong damping in physics , the interested readers can refer to 2, 4-7 for mathematical theory; while in the conservative case, that is, β / 0, α 0, 1.1 corresponds to Schr ödinger flow which represents conservation of angular momentum 8 .The numerical treatment to the problem can be found in 9, 10 .
Recently, the study of the theory of ferromagnetism, especially the theory on thin film, is one of the focuses for both physicists and mathematicians.In the asymptotic regime which is readily accessible experimentally, DeSimone and Otto, and so forth, deduced a thin film micromagnetics model in which self-induced energy is the leading term of the free energy functional see 11 .The physical consequences of the model are discussed further in 12 .The free energy functional is E m R 2 |ξ • mχ Ω | 2 /|ξ| dξ.We have δE/δm −∇ −Δ −1/2 div m see Section 2 for detailed computation and the Landau-Lifshitz equation β 0 becomes, To the best knowledge of ours, this is the first time a new model has been raised.Equation 1.4 is not easy to deal with because of lower order of differential operator with respect to x-variable and its strong nonlinear term.Inspired by physical prototype of the problem, we approximate it by a second-order equation, The rest of this paper is organized as follows.Section 2 is devoted to studying 1.5 .More precisely, we first study the penalized equation.In order to do this, we consider the corresponding linear equation and get its formal solution and well-posedness, then we get the existence of a unique mild solution of the penalized equation using semigroup theory.Second, we get the existence of a weak solution of 1.5 by passing to the limit in the penalized equation.The key point in the convergence process relies on a compensated compactness principle.In Section 3, we get existence of weak solution of 1.4 in Theorem 3.1 by passing to the limit in 1.5 as ε → 0.

Approximation Equations
In this section, we always suppose that T 2 R 2 / 2πZ 2 is the flat torus.We prove existence of a weak solution of the following equations: Equation 2.1 can be written as It is very easy to prove that 2.1 is equivalent to

Abstract and Applied Analysis
The equivalence follows from the following.
Lemma 2.1.In the classical sense, m ε is a solution of 2.1 -2.3 if and only if m ε is a solution of 2.5 .
Proof.Suppose that m ε is a solution of 2.1 -2.3 .By the vector cross product formula we have

2.7
By the cross product of m ε and 2.7 , we have This proves that m ε satisfies 2.5 .Suppose that m ε is a solution of 2.5 .Then by the cross product of m ε and 2.5 , we obtain Since |m ε | 1, we have m ε • ∂m ε /∂t 0. Hence 2.9 implies We define a local weak solution of 2.1 as follows.
Definition 2.2.A vector-valued function m ε x, t is said to be a local weak solution of 2.1 , if m ε is defined a.e. in T 2 × 0, T such that To prove Theorem 2.3, we have to consider a penalized equation.

The Penalized Equation
In the spirit of 13 , we first construct weak solutions to a penalized problem, where the constraint |m ε | 1 is relaxed:

2.13
Here In order to prove the existence of a mild solution of semilinear system 2.11 -2.13 , we consider the corresponding linear equation.

The Corresponding Linear Equation
First, we consider the corresponding linear equation of 2.11 -2.13 in the whole space:

2.14
where m 0 x m 01 x , m 02 x .While dealing with linear equation 2.14 , we just write m instead of m k unless there may be some confusion.
By Fourier transform in the x-variable, 2.14 are turned into

2.15
For each fixed ξ, the problem has a unique solution m ξ, t e −B ξ t • e −A ξ t m 0 ξ , 2.16 where

2.17
So the problem has the solution m x, t 1 4π 2 e −B ξ t ∨ * e −A ξ t ∨ * m 0 x .

2.18
Now the only problem left is to find the inverse Fourier transforms of e −A ξ t and e −B ξ t .First, we need to find an orthogonal matrix O ξ such that O ξ A ξ O τ ξ is the Jordan normal form of A ξ .In fact,

2.19
Now we begin to calculate the inverse Fourier transform of

2.21
By the property of the Fourier transform, we have

2.24
In harmonic analysis, 2.24 is known as Poisson kernel.Also by 14, page 107 , we have

2.28
Hence we obtain By standard procedure, we can get

2.33
Notice that For any ε > 0, choosing A large enough such that For above ε, there exists a By standard procedure see 14 , we can prove that lim Therefore the proof is completely finished.

2.36
where m 0 x 2π n m 0 x , ∀ n ∈ Z 2 and T 2 is a flat torus R 2 / 2πZ 2 .By extending the equations periodically with respect to variable x to the whole space, and using Fourier transform, we obtain

Existence of a Unique Mild Solution of the Penalized Equation
First, let us recall a classical theorem in the theory of semigroup.
Theorem 2.6 see 15 .Let N u : X → X be locally Lipschitz continuous in u.If L is the infinitesimal generator of a C 0 semigroup S t on X, then for every u 0 x ∈ X there is a T ≤ ∞ such that the initial value problem

2.39
has a unique mild solution u on 0, T .Moreover, if T < ∞, then lim t → T u t ∞.
By Theorem 2.4 and Remark 2.5, we know that L is the infinitesimal generator of a C 0 semigroup on H 1 T 2 .Next, we want to check the inequality

2.40
Letting B u, v, w k 2 uvw, we have

2.42
This last result is an easy consequence of Sobolev embedding theorem.Therefore, Theorem 2.6 gives us the desired result.2.1 -2.3

Existence of Weak Solution of Approximate Equation
In this section, we establish our main results about the approximate equations 2.1 -2.3 by passing to the limit in the penalized equation 2.11 as k → ∞.

2.43
We now take the limit as k goes to infinite: from 2.43 , we deduce that

2.44
Therefore, up to a subsequence, we have From 2.45 , 2.46 , and 2.47 , as k goes to infinite, we have

2.50
Namely, 2.49 is convergent to

2.51
Hence by Lemma 2.1, we know that 2.1 -2.3 has a weak solution.
Remark 2.8.From 2.43 and Theorem 2.6, we know that the unique mild solution of the penalized equation 2.11 globally exists.

Existence of Weak Solution of 1.4
From above section, we know that for each fixed ε > 0, 2.1 -2.3 admit weak solutions m ε ∈ L ∞ 0, T; H 1 T 2 .In this section, we will prove that there exists a subsequence of m ε still denoted by m ε strongly converging to m in L 2 0, T; L 2 T 2 , which is the weak solution of 1.4 .More precisely, we state our main result of this section in the following theorem.

3.4
Passing to the limit as ε goes to zero in 2.51 , we have,
|∇Φ| 2 dx is the energy of the stray field ∇Φ induced by m.By the magnetostatics theory For every m 0 x ∈ H 1 T 2 and |m 0 x | 1, a.e. in T 2 , there exists a weak solution of 2.1 -2.3 .
t | 1 a.e. in T 2 × 0, T ; 3 2.1 holds in the sense of distribution; 4 m ε x, 0 m 0 x in the trace sense.
For every m 0 ∈ H 1 T 2 , there exists a unique mild solution m k of 2.11 -2.13 .Proof.Here Lm k εΔm k 2strongly and a.e. in T 2 × 0, T , 2.48and |m ε | 1 a.e. in T 2 × 0, T .In order to pass to the limit in 2.11 , let Φ be in C ∞ T 2 × 0, T 3 , and let the test function ψ m k × Φ, there holds 2T 2 of 1.4 .Passing to the limit as k → ∞ and taking 2.45 , 2.46 into consideration, we have So we conclude that m ε is bounded in L ∞ 0, T; H 1 Ω , and ∂m ε /∂t is bounded inL 2 0, T; L 2 Ω .Therefore, up to subsequence,m ε m in L ∞ 0, T; H 1 T 2 weak * ,