The paper is concerned with a two-dimensional Landau-Lifshitz equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

1. 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 is the magnetization and is a Gilbert damping constant. The system (1.1) is implied by the conservation of energy and magnitude of is the unconstrained first variation of the energy functional The magnitude of the magnetization is finite, that is, Hereis the free energy functional, and it is composed of three parts:

(i) is the exchange energy. It tends to align in the same direction and prevents from being discontinuous in space;(ii) is the anisotropy energy. 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, for (iii) is the energy of the stray field induced by By the magnetostatics theory

Equation (1.1) has been widely studied. In the case (1.1) corresponds to the heat flow for harmonic maps studied in [2, 3]; if (which implies strong damping in physics), the interested readers can refer to [2, 47] for mathematical theory; while in the conservative case, that is, (1.1) corresponds to Schrödinger 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 We have (see Section 2 for detailed computation) and the Landau-Lifshitz equation () becomes,in which is in-plane component of the magnetization, is a flat torus. is the inner product. 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 -variable and its strong nonlinear term. Inspired by physical prototype of the problem, we approximate it by a second-order equation,Equation (1.5) is the Landau-Lifshitz equation corresponding to the free energy sum of exchange and self-induced energy. One difficulty in dealing with (1.5) lies in the nonconvex constraint which is overcame by considering a penalty approximation mimicking treatment of harmonic maps. To get existence of a unique mild solution of the penalized equation, we first give the formal solution of the corresponding 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

2. Approximation Equations

In this section, we always suppose that is the flat torus. We prove existence of a weak solution of the following equations: Denote Note that the corresponding energy is The variation of the self-induced energy is Equation (2.1) can be written asIt is very easy to prove that (2.1) is equivalent toThe equivalence follows from the following.

Lemma 2.1. In the classical sense, is a solution of (2.1)–(2.3) if and only if is a solution of (2.5).

Proof. Suppose that is a solution of (2.1)–(2.3). By the vector cross product formulawe haveBy the cross product of and (2.7), we haveThis proves that satisfies (2.5).
Suppose that is a solution of (2.5). Then by the cross product of and (2.5), we obtainSince we have Hence (2.9) implies

We define a local weak solution of (2.1) as follows.Definition 2.2. A vector-valued function is said to be a local weak solution of (2.1), if is defined a.e. in such that
(1) and (2)(3)(2.1) holds in the sense of distribution;(4) in the trace sense.
We state our main result in this section as follows.Theorem 2.3. For every and a.e. in 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 is relaxed: Here 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:where While dealing with linear equation (2.14), we just write instead of unless there may be some confusion.

By Fourier transform in the -variable, (2.14) are turned intoFor each fixed the problem has a unique solutionwhereSo the problem has the solutionNow the only problem left is to find the inverse Fourier transforms of and First, we need to find an orthogonal matrix such that is the Jordan normal form of In fact,Now we begin to calculate the inverse Fourier transform of DenoteBy the property of the Fourier transform, we haveDenote by Obviously, we haveBy [14, page 15-16], we know thatIn harmonic analysis, (2.24) is known as Poisson kernel.

Also by [14, page 107], we have

HenceTherefore,We continue to compute other terms,in which Hence we obtainBy standard procedure, we can getwhere Therefore,Theorem 2.4. Suppose that then there exists a solution of (2.14) andProof. From (2.21) and (2.30), we know so and is a Hörmander space (see [14], page 49-50). Moreover,Notice that
For any choosing large enough such that we have
For above there exists a such that as and Hence that isBy standard procedure (see [14]), we can prove thatTherefore the proof is completely finished.
Remark 2.5. Considerwhere and is a flat torus By extending the equations periodically with respect to variable to the whole space, and using Fourier transform, we obtainin which

2.1.2. 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 be locally Lipschitz continuous in If is the infinitesimal generator of a semigroup on then for every there is a such that the initial value problemhas a unique mild solution on Moreover, if then Applying Theorem 2.6 to (2.11)–(2.13), we get the following theorem.Theorem 2.7. For every there exists a unique mild solution of (2.11)–(2.13).Proof. Here By Theorem 2.4 and Remark 2.5, we know that is the infinitesimal generator of a semigroup on Next, we want to check the inequalityLetting we haveSo it is sufficient to proveThis last result is an easy consequence of Sobolev embedding theorem. Therefore, Theorem 2.6 gives us the desired result.

2.2. Existence of Weak Solution of Approximate Equation (2.1)–(2.3)

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

Proof of Theorem 2.3. Multiplying (2.11) with and integrating over we haveWe now take the limit as goes to infinite: from (2.43), we deduce that Therefore, up to a subsequence, we have and
In order to pass to the limit in (2.11), let be in and let the test function there holdsFrom (2.45), (2.46), and (2.47), as goes to infinite, we haveNamely, (2.49) is convergent toHence 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.

3. Existence of Weak Solution of (1.4)

From above section, we know that for each fixed (2.1)–(2.3) admit weak solutions In this section, we will prove that there exists a subsequence of (still denoted by ) strongly converging to in which is the weak solution of (1.4). More precisely, we state our main result of this section in the following theorem.Theorem 3.1. Suppose that a.e. in and there exists a weak solution and of (1.4).Proof. Form (2.43), we havePassing to the limit as and taking (2.45), (2.46) into consideration, we have So we conclude that is bounded in and is bounded in
Therefore, up to subsequence, By [16, Chapter 1, Theorem 5.1, pages 56–60], we know thatPassing to the limit as goes to zero in (2.51), we have,That is to say, is the weak solution ofBy an argument analogous to Lemma 2.1, (3.6) is equivalent to (1.4).


The project is supported by NNSFC (10171113) (10471156) and NSFGD (4009793).