Weak Solutions to a Landau–Lifshitz Equation with a -Laplacian Operator as an Exchange Field

Abstract

We are dealing with the classical Landau–Lifshitz equation with an unusual exchange field expressed in terms of a -Laplacian operator for greater than 2. We obtain weak solutions to the model by using penalization, compactness arguments, and monotonicity method.

1. Introduction

In continuum physics, the Landau–Lifshitz (LL) equation describes the precession of the magnetization field inside ferromagnets. This equation is commonly used in several forms to modelize effects of magnetic field on magnetic materials. In particular, it can be used to describe the temporal behavior of magnetic elements under the magnetic field. In a ferromagnetic material, the magnetization vector can vary in space but keeps constant amplitude. The LL equation predicts the rotation of magnetization in response to different applied torques. Its general form writes

For more details on the LL equation, we refer for example to [1]. The LL equation has attracted much attention in recent years and many results are already obtained. Since the pioneering work [2, 3] where, it is generally proved that weak solutions are unique while exist globally, several theoretical works on LL (sometimes coupled to the Maxwell equations) have been conducted. For example in [2, 3], the existence and nonuniqueness of weak solution to Landau–Lifshitz equation is established. The work [4] addressed global existence of weak solution to a modern structure of the Landau–Lifshitz–Gilbert equation. The modification lies in occurrence of the effective field of a nonlinear term describing vertical spin stiffness and discuss the limit of got results as a vertical spin stiffness parameter converges to zero. The model of LLG considered in [5] contains a transport-type term in the effective field and global existence of weak solution is investigated. In the same context, a Landau–Lifshitz–Slonczewski equation is considered and global weak and classical solutions have been established in [6]. In [7], existence, regularity, and local uniqueness of the solutions to the Maxwell–Landau–Lifshitz system in three dimensions is proved. In the paper [8], a model of a ferromagnetic material governed by a nonlinear Landau–Lifshitz equation coupled with Maxwell equations is considered. Existence of weak solutions is proved and the -limit set of all trajectories is identified to the solutions of the stationary model. Another LLG model is considered [9] with inertial effects and the global existence of the proposed model is established. Here, the modification is taken in the presence of second order time derivative of magnetization in the effective field. In [10], global existence of the solutions for Landau–Lifshitz equation of the ferromagnetic spin chain from a -dimensional manifold into the unit sphere of is established and some links between harmonic maps and the solutions of the Landau–Lifshitz equation are studied. All of these proofs are based on some type of penalization and regularization. Finally, we note that significant progress has been made in developing techniques for constructing weak solutions to the generic LLG equation. The convergence of several techniques to weak solutions was demonstrated. For example, in the convergence theory of numerical schemes, a considerable step forward has been made, see for example [11–23].

In the present paper, we investigate the global existence of weak solutions for the following LL equation involving the -Laplacian operator.where and is a bounded regular subset of . The vector denotes the three-dimensional magnetization vector. The notation shows the vector cross product in . The positive constant parameterizes damping parameter and stands for the -Laplacian of which is defined by

This operator can be extended to a monotone, bounded, hemicontinuous, and coercive operator between the space and its dual as follows:for all and .

In (2), the initial condition is assumed to satisfy the saturation constraint a.e in .

Many researchers have studied several forms of the exchange field. We refer for example to the paper [24] where a model incorporating a singular kernel is proposed for the exchange field. Another form for the exchange field is considered in [25] where the exchange field is expressed by means of a function of with specific hypotheses on . For a third consideration for the exchange field we refer to the paper [26]. The main novelty of the present paper with respect to other related papers (dealing with the particular case ) is that . Since our problem involves a -Laplacian, related studies on the LL equation are not known yet, as far as the authors know. Thus, this paper aims to establish an existence result for (2) where by essentially using penalization, compactness arguments, and monotonicity method. The main difficulty in the proof of existence is to establish the convergence almost everywhere of the gradient where goes to 0 in order to pass to the limit in the nonlinear term (Section 3.2).

The current paper is structured as follows. In section 2, we present some notations and discuss our main result. Section 3 is dedicated to the proof of the main result. Finally, a conclusion is drawn is the last section.

2. Notations and the Main Result

Throughout, in current research, for , , are the usual Lebesgue and Sobolev spaces. The norm in is denoted by , for simplicity, we utilize the notation sometimes.

We begin by introducing the notion of weak solutions to the problem (2).

Definition 1. Let such that a.e, we say that the vector field is a weak solution of (2) if(i)For all , , and a.e;(ii)For all , there holds(iii)The following energy estimate holdsThe purpose of this paper is to prove the following main result.

Theorem 1. Let (with ) a bounded regular set, such that a.e, and satisfies the following conditions:

Then there exists a weak solution of (2) such that for all .

The proof of Theorem 1 will be given in the next section by using the penalization, compactness arguments, and monotonicity method. Let us recall some auxiliary lemmas required later.

Lemma 1 (See [26, 27]). Consider a reflexive Banach space and its dual space of . Suppose that an operator satisfies(i) is monotone on , i.e., for all .(ii) is semi-continuous, i.e., the map , is continuous on for all .(iii)If weakly in , weakly in as and .

Then .

Lemma 2 (See [26]). Let Note that this inequality is also true if and are two tensors. In our case we apply it for and where , to show the monotony of the operator . We end this section with this useful theorem for differential systems.

Theorem 2. Let where for some satisfies the following Caratheodory conditions:(i) is measurable for all and for all ,(ii) is continuous for almost all ,(iii)there exists an integrable function such that Then, there exists and a continues function such that.(iv) exists for almost all .(v) solves the problem

3. Proof of the Main Result

3.1. Existence for the Penalized Problem

Our aim is to prove Theorem 1. We begin by constructing weak solutions to a penalized problem associated with (2) where the constraint is relaxed, i.e., for we introduce

We have the following result.

Proposition 1. Let such that a.e and satisfies the hypothesis . Then for fixed , there exists a weak solution for the penalized problem (5) such that(i)For all , , ;(ii)For all , there holds(iii)The following energy estimate holds for a. e. .

Proof. The space is separable for , then there exists a sequence of finite dimensional subspaces of such that is dense in . For a fixed positive integer , we look for approximate solution for (5) in the formsuch thatfor withwhere is the approximations of the initial data withWe assume thatWe set and . Then the differential system (16) with initial condition (17) satisfying (18) and (19) can be written asIn fact, we havewheresuch that for are the components of the vector . In additionThen the matrix is invertible for all , then the system (20) becomesSince satisfies the Caratheodory conditions, the existence of a solution to the system (24) is ensured due to Theorem 2. The global existence then follows from the following a priori estimates.(i)A priori estimates. We multiply (16) by , add up the obtained equations for , we obtain Integrating (25) with respect to leads to(ii)Convergences and passage to the limit when goes to . Following hypothesis , we infer that. Thanks to (19), we deduce that the right-hand side of (26) is uniformly bounded with respect to . Then by Poincaré’s inequality, we getHence the following convergences hold up to a subsequenceSince the Sobolev embedding is compact and is continuous, we use Aubin–Lions lemma to obtain the strong convergence as

Proposition 2. If and satisfies the above convergences, then we have

Proof. Since , we get from (15)where is a positive constant independent of . Hence the sequence is bounded in , so there exists a subsequence such thatfor some . For the rest of the proof we use the following notation:where denote the duality pairing between and . As , from (33), we haveThenIn order to show that coincides with , we will use the fact that the operator is a maximal monotone. It follows from Lemma 1 that if we prove the inequality, we getIn order to verify (37) fix . Then, multiply (16) by , and sum over , to arrive atIntegrate identity (38) over to obtainThe weak convergence of in and the strong convergence of in imply thatNow from the estimate (26), we deduce that is bounded in . Then weakly in and using the fact that a.e, one can show that . Therefore, we getTaking the limit superior in (39) as goes to infinity, we obtain in particular thatIn the following, we show that is equal to the right-hand side of (42) for a.e , i.e.,We multiply (16) by an arbitrary function , and integrate it from 0 to obtainTaking the limit as on both sides of (44), we getSince (45) holds for each , we may replace by to getUsing the previous convergences, we can pass to the limit in (46) to obtain (43). Noting that for the convergence of the term , we haveWe know that is bounded in and then the sequence is bounded in . Thanks to the convergence a.e of to , we obtain that weakly in . Therefore, we getNow, it follows from (42) and (43) thatThen again by Lemma 1, we conclude thatwhich completes the proof of Proposition 2.
We need to show that satisfies equality (14) in Proposition 1. Let . Using the fact that in (45), we getLet . Since is dense in , then for any there exists a sequence such that in . Hence by replacing by and taking in (51), we obtainTake the limit as on both sides of (52) to verify that (71) holds for an arbitrary in . Inequality (14) is a consequence of (26) due to the lower semi-continuity, hence the proof of Proposition 1 is complete.

3.2. Existence Result for the Problem (2)

3.2.1. Some Properties of the Sequence

Our aim here is the passage to the limit when goes to 0. For this we need to prove the two following results. The first one is

Lemma 3. If is a solution of the penalized problem (5), then a.e in .

Proof. We test the first equation in (5) by to obtainA direct computation leads toWe conclude thatthen a.e in .
The second result lies with heat flow of -harmonic maps (see [26, 28] for ).

Proposition 3. Let and satisfying in the sense of distribution the following equation:Assume boundedness of in , in , and in . Then for a subsequence strongly in for all .

Proof. Taking the limit in (15) when by using the lower semi-continuity of the norm under the weak topology, we getThen up to a subsequence weakly in and strongly in and a.e in . For a fixed , we define the following set byLet be fixed such that and for all . Sobolev embedding and Hölder’s inequality imply thatfor .
Next, we shall estimate .
Let be the usual vector truncation function defined asSo, by Lemma 2, we obtainwhere(i)Estimate of . From (56), we have(ii)Estimate of . We have We remark that Using the fact that in , we get(iii)Estimate of . By Hölder’s inequality and the weak convergence in , we obtain whereFinally, we getUsing the fact that is absolutely continuous, we have  = 0 for enough small and from (59) and (70), we get the strong convergence of to in .

3.3. Passage to the Limit When Goes to 0

In order to pass to the limit in (12), let and . Since belong to , there holds

On the other hand, we get

From the estimate (57), we deduce that

Thanks to the convergence almost everywhere of to in Proposition 3, we obtain that . Since strongly in then

Hence we can pass to the limit in (71), to obtain (5). Inequality (6) is a consequence of (14) by using the lower semi-continuity of the norms and under the weak topology as . To get the Hölder regularity, we use (4) to getwhich complete the proof of our main Theorem 1.

4. Concluding Remarks

We were concerned with a LL equation involving a -Laplacian as the exchange field. Global existence of weak solutions is obtained and the key of the proof is penalization combined with some compactness arguments and monotonicity method. As a perspective to the present work, it will be interesting to investigate the limiting behaviors of the obtained solutions with respect to a small and a large damping parameter. Existence of weak solutions to the LL model in the case where the exchange field is expressed in term of a -Laplacian operator with is also a very challenging question.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The work was financially supported by the Anhui University of Finance and Economics, Bengbu, 233030, China.