Global Existence of Weak Solutions to a Fractional Model in Magnetoelastic Interactions

Abstract

The paper deals with global existence of weak solutions to a one-dimensional mathematical model describing magnetoelastic interactions. The model is described by a fractional Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We prove global existence by using Faedo-Galerkin/penalty method. Some commutator estimates are used to prove the convergence of nonlinear terms.

1. Introduction

The nonlinear parabolic hyperbolic coupled system describing magnetoelastic dynamics in ( and is a bounded open set of , ) is given by (see [1])

Equation (1), well known in the literature, is the Landau-Lifshitz-Gilbert (LLG) equation. The unknown , the magnetization vector, is a map from to (the unit sphere of ) and is its derivative with respect to time. The symbol denotes the vector cross product in . Moreover we denote by , , the components of . The constant represents the damping parameter. represents the effective field which is given bywhere is a positive constant and the components of the vector are given by

Here stand for the components of the linearized strain tensor , with if and otherwise.

Equation (2) describes the evolution of the displacement , is a positive constant, and the tensors , are given by

is the elasticity tensor satisfying the following symmetry property:

Many studies have been done on the fractional Landau-Lifshitz equation; we quote here, for example, [2], where the existence of weak solutions under periodical boundary condition has been proven for equation of a reduced model for thin-film micromagnetics. In [3], the main purpose is to consider the well-posedness of the fractional Landau-Lifshitz equation without Gilbert damping. The global existence of weak solutions is proved by vanishing viscosity method. Note that the existence and asymptotic behaviors of global weak solutions to the one-dimensional periodical fractional Landau-Lifshitz equation modeling the soft micromagnetic materials are studied in [4]. For the magnetoelasticity coupling, in [1], the authors study the three-dimensional case and establish the existence of weak solutions taking into account three terms of the total free energy. Existence and uniqueness of solutions have been proven in [5] for a simplified model and in [6] a one-dimensional penalty problem is discussed and the gradient flow of the associated type Ginzburg-Landau functional is studied. More precisely the authors prove the existence and uniqueness of a classical solution which tends asymptotically for subsequences to a stationary point of the energy functional. Our aim here is to study the coupled system of magnetoelastic interactions with fractional LLG equation.

The rest of the paper is divided as follows. In the next section we present the model equation we will be interested in. Section 3 recalls some useful lemmas. Finally in Section 4 we prove a global existence result of weak solutions to the considered model.

2. The Model and Main Result

We assume that is a subset of and the displacement is only in one direction. Specifically, we consider a simple variable space and assume that . We take the following system:with associated initial and boundary conditions

The effective field is given bywhere denotes the square root of the Laplacian which can be defined via Fourier transformation [7]. In this paper we are interested in the case . For the vector , we assume that and we keep the three components of the vector .

It is a common practice (see [5]) to replace the first equation of system (7) by the quasilinear parabolic equation (Ginzburg-Landau type equation):

Here is a positive parameter and . -penalization in (11) replaces the magnitude constraint .

Throughout, we make use of the following notation. For , an open bounded domain of , we denote by and the classical Hilbert spaces equipped with the usual norm denoted by and (in general, the product functional spaces are all simplified to ). For all , denotes the usual Sobolev space consisting of all such thatwhere denotes the Fourier transform and its inverse. Let denote the corresponding homogeneous Sobolev space. When , corresponds to the usual Sobolev space .

Now we give a definition of the solution in the weak sense of problem (7)-(8)-(9).

Definition 1. Let , a.e., , and . One says that the pair is a weak solution of problem (7)-(8)-(9) if(i)for all , , , a.e., , and ;(ii)for all and one has(iii) and in the trace sense;(iv)for all we havewhere is a positive constant which depends only on and .

The main result of this paper is the following.

Theorem 2. Let , such that a.e., , and . Then there exists at least a weak solution for problem (7)-(8)-(9) in the sense of Definition 1.

The proof of Theorem 2 will be given in Section 4.

3. Some Technical Lemmas

In this section we present some lemmas which will be used in the rest of the paper. We start by recalling the following lemma due to Simon (see [8]).

Lemma 3. Assume , , and are three Banach spaces and satisfy with compact embedding . Let be bounded in and be bounded in . Then is relatively compact in .

There is another lemma whose proof can be found in [[9], page 12].

Lemma 4. Let be a bounded open set of , and in such that , a.e. in ; then weakly in .

The following lemma will ensure a compact embedding for the space .

Lemma 5. Let be a bounded open set of , which is uniform Lipschitz. Let , . If then the injection of in is compact, for any .

The proof can be found in [[10], Theorem ., p 216]. We give now a lemma that will play a very important role in the convergence of approximate solutions (see [11–13] for a proof).

Lemma 6 (commutator estimates). Suppose that and . If (the Schwartz class) thenwith , such that .

Here is another lemma which can be viewed as a result of the Hardy-Littlewood-Sobolev theorem of fractional integration; see [7] for a detailed proof.

Lemma 7. Suppose that and . Assume that ; then and there is a constant such that

We finish this section with the following result (the proof can be found in [2]).

Lemma 8. If and belong to , then

4. Proof of Theorem 2

Our goal is to show global existence of weak solutions for the fractional problem (7)-(8)-(9).

4.1. The Penalty Problem

Let be a fixed parameter. We construct approximated solutions converging, as , to a solution of the problem. System (7) is reduced to the following problem:in , where the vector is given by , , , and .

System (19) is supplemented with initial and boundary conditions

We apply Faedo-Galerkin method: let be an orthonormal basis of consisting of all the eigenfunctions for the operator (the existence of such a basis can be proved as in [14], Ch. II),and let be an orthonormal basis of consisting of all the eigenfunctions for the operator :We then consider the following problem in :with initial and boundary conditions,

We are looking for approximate solutions to (23) under the form

If we multiply each scalar equation of the first equation of (23) by and the second by and integrate in we get to a system of ordinary differential equations in the unknown , We observe that we can write the first equation in the formwith

It is clear that the matrix is invertible which implies the system of first-order ordinary differential equations is Lipschitz locally; then there exists a local solution to the problem that we can extend on using a priori estimates. For this, we multiply the first equation of (23) by and the second by ; integrating in , we obtain (by using Lemma 8)

We find after summing

On the other hand

Now integrating in time,

Omitting superscripts, we obtain for all ,

Hence, taking into account (31),and hence for small enough () and one has

The right-hand side is uniformly bounded. Indeed, for , ; furthermore (for constants , , and independent of )thanks to the strong convergence in . For the other term (), the estimate can be carried out in an analogous way using the strong convergence in . Moreover, noting that (for a constant independent of and )

therefore, for fixed , we have

Note that (37) is due to the Poincaré lemma. Now, from classical compactness results there exist two subsequences which we still denote by and such that for fixed and for any

Convergence (38) is due to Lemma 3 and thanks to Lemma 4 it can be shown that . Moreover from the Sobolev embedding (Lemma 5) , the further compactness result follows:

The above estimates allow us to pass to the limit as goes to infinity and to get the desired result. Indeed consider the variational formulation of (23):for any and . Taking in (40), we findfor any and . We proved the following result.

Proposition 9. Given such that a.e., , and , then there exists a solution to problem (19) in the sense of distributions. Moreover we have the following energy estimate:

4.2. Convergence of Approximate Solutions

Our aim here is to pass to the limit as . For this, we will use estimate (42), from which we have the following:

Then there exist two subsequences which we still denote by and such that for any

It can be shown from convergence (44) that a.e.

Now in order to pass to the limit in (41), let , and let . We first recall the identity for all , , , and in .

As is in , the following holds:

Note that for this choice we have , indeed applying the multiplicative estimates (16) in Lemma 6 to and (for , , , , , and ); we find for different constants independent of :since (1 here is the dimension) then and consequently is bounded in .

Taking and following the idea introduced in [4] we have

Now for the fourth term of the first equation, we introduce the commutator (see [4]):

Let and . We will show that .

First, note that . Indeed, applying (15) for , and in , we find (for different constants ) thanks to Lemma 7. Once againwhere the right-hand side of the last inequality tends to 0.

Finally, we have

Therefore

This being true for every and by a standard density argument, it is true for any in . Note that, from (42), one can easily get (14). Hence is a solution of problem (7)-(8)-(9) in the sense of Definition 1. The proof of Theorem 2 is complete.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by the PHC Volubilis Program MA/14/301 “Elaboration et analyse de modèles asymptotiques en micro-magnétisme, magnéto-élasticité et électro-élasticité” with joint financial support from the French Ministry of Foreign Affairs and the Moroccan Ministry of Higher Education and Scientific Research.