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

and Applied Analysis 3 ≤ a2 ∫Ω 󵄨󵄨󵄨󵄨Λαm0󵄨󵄨󵄨󵄨2 dx + ρ2 ∫Ω 󵄨󵄨󵄨󵄨ω1󵄨󵄨󵄨󵄨2 dx + 1 4 ∫Ω 󵄨󵄨󵄨󵄨ωx (0)󵄨󵄨󵄨󵄨2 dx + C (Ω, λ) , (14) where C(Ω, λ) is a positive constant which depends only on Ω and λ. The main result of this paper is the following. Theorem 2. Let α ∈ (1/2, 1),m0 ∈ Hα(Ω) such that |m0| = 1 a.e., ω0 ∈ H1 0 (Ω), and ω1 ∈ L2(Ω). 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 A, B, and C are three Banach spaces and satisfy A ⊂ B ⊂ C with compact embedding A 󳨅→ B. Let Θ be bounded in L∞(0, T; A) and Θt fl {ft; f ∈ Θ} be bounded in Lp(0, T; C), p > 1. ThenΘ is relatively compact inC([0, T]; B). There is another lemmawhose proof can be found in [[9], page 12]. Lemma 4. Let Θ be a bounded open set of Rdx × Rt, hn and h in Lq(Θ), 1 < q < ∞ such that ‖hn‖Lq(Θ) ≤ C, hn → h a.e. in Θ; then hn ⇀ h weakly in Lq(Θ). The following lemma will ensure a compact embedding for the space Ws,p. Lemma5. LetΘ be a bounded open set ofR, which is uniform Lipschitz. Let s ∈ [0, 1[, p > 1, d ≥ 1. If sp < d then the injection of Ws,p(Θ) in Lk(Θ) is compact, for any k < dp/(d − sp). The proof can be found in [[10], Theorem 4.54., 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 s > 0 and p ∈ (1, +∞). If f, g ∈ S (the Schwartz class) then 󵄩󵄩󵄩󵄩Λs (fg) − fΛsg󵄩󵄩󵄩󵄩Lp ≤ C (󵄩󵄩󵄩󵄩∇f󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩?̇?s−1,p2 + 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩?̇?s,p3 󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩Lp4 ) , (15) 󵄩󵄩󵄩󵄩Λs (fg)󵄩󵄩󵄩󵄩Lp ≤ C (󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩?̇?s,p2 + 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩?̇?s,p3 󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩Lp4 ) (16) with p2, p3 ∈ (1, +∞) such that 1/p = 1/p1 + 1/p2 = 1/p3 + 1/p4. 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 p > q > 1 and 1/p+s = 1/q. Assume that f ∈ Lq; then Λ−sf ∈ Lp and there is a constant C > 0 such that 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩?̇?−s,p fl 󵄩󵄩󵄩󵄩Λ−sf󵄩󵄩󵄩󵄩Lp ≤ C 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lq . (17) We finish this section with the following result (the proof can be found in [2]). Lemma 8. If f and g belong to H2α per(Ω) fl {f ∈ L2(Ω)/Λ2αf ∈ L2(Ω)}, then ∫ Ω Λ2αf ⋅ g dx = ∫ Ω Λαf ⋅ Λαgdx. (18) 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 ε > 0 be a fixed parameter. We construct approximated solutions m converging, as ε → 0, to a solution m of the problem. System (7) is reduced to the following problem: γ−1mεt × m + mεt + aΛ2αmε + l (mε, ωε) + 󵄨󵄨󵄨󵄨mε󵄨󵄨󵄨󵄨2 − 1 ε m = 0 ρωε tt − ωε xx − λ (mε 1mε 3)x = 0 (19) in Q = Ω × (0, T), where the vector l(m, ω) is given by l(m, ω) = (λm3ωx, 0, λm1ωx), λ1 = λ2 = 0, λ3 = λ, and σ1313 = 1. System (19) is supplemented with initial and boundary conditions ωε (⋅, 0) = ω0, ωε t (⋅, 0) = ω1, m (⋅, 0) = m0, 󵄨󵄨󵄨󵄨m0󵄨󵄨󵄨󵄨 = 1 a.e. in Ω, ωε = 0, m (x, t) = m (x + 2π, t) on Σ. (20) 4 Abstract and Applied Analysis We apply Faedo-Galerkin method: let {fi}i∈N be an orthonormal basis of L2(Ω) consisting of all the eigenfunctions for the operator Λ2α (the existence of such a basis can be proved as in [14], Ch. II), Λ2αfi = αifi, i = 1, 2, . . . , fi (0) = fi (2π) , (21) and let {gi}i∈N be an orthonormal basis of L2(Ω) consisting of all the eigenfunctions for the operator −Δ: −Δgi = βigi, i = 1, 2, . . . , gi = 0 on ∂Ω. (22) We then consider the following problem in Q = Ω × (0, T): γ−1mε,N t × m + m t + aΛ2αmε,N + l (mε,N, ωε,N) + 󵄨󵄨󵄨󵄨󵄨mε,N󵄨󵄨󵄨󵄨󵄨2 − 1 ε m = 0 ρωε,N tt − ωε,N xx − λ (mε,N 1 mε,N 3 )x = 0 (23) with initial and boundary conditions, ωε,N (⋅, 0) = ωN (⋅, 0) , ωε,N t (⋅, 0) = ωN t (⋅, 0) , m (⋅, 0) = m (⋅, 0) , in Ω, ωε,N = 0, m (x, t) = m (x + 2π, t) on Σ = ∂Ω × (0, T) , ∫ Ω ωN (x, 0) gi (x) dx = ∫ Ω ω0 (x) gi (x) dx, ∫ Ω ωN t (x, 0) gi (x) dx = ∫ Ω ω1 (x) gi (x) dx, ∫ Ω m (x, 0) fi (x) dx = ∫ Ω m0 (x) fi (x) dx. (24) We are looking for approximate solutions (mε,N, ωε,N) to (23) under the form m = N ∑ i=1 ai (t) fi (x) , ωε,N = N ∑ i=1 bi (t) gi (x) . (25) If we multiply each scalar equation of the first equation of (23) by fi and the second by gi and integrate in Ω we get to a system of ordinary differential equations in the unknown (ai(t), bi(t)), i = 1, 2, . . . , N. We observe that we can write the first equation in the form − aΛ2αmε,N − l (mε,N, ωε,N) − 󵄨󵄨󵄨󵄨󵄨mε,N󵄨󵄨󵄨󵄨󵄨2 − 1 ε m = A (mε,N)mε,N t (26)


Introduction
The nonlinear parabolic hyperbolic coupled system describing magnetoelastic dynamics in  = (0, ) × Ω ( > 0 and Ω is a bounded open set of R  ,  ⩾ 1) is given by (see [1]) Equation ( 1), well known in the literature, is the Landau-Lifshitz-Gilbert (LLG) equation.The unknown m, the magnetization vector, is a map from Ω to  2 (the unit sphere of R 3 ) and m  is its derivative with respect to time.The symbol × denotes the vector cross product in R 3 .Moreover we denote by   ,  = 1, 2, 3, the components of m.The constant  represents the damping parameter.H eff represents the effective field which is given by where  is a positive constant and the components of the vector ℓ(m, u) are given by Here   (u) = (1/2)(    +     ) stand for the components of the linearized strain tensor ,   =  1   +  2     +  3 (    +     ) with   = 1 if  =  =  =  and   = 0 otherwise.
Equation (2) describes the evolution of the displacement u,  is a positive constant, and the tensors S(u), L(m) are given by S  =     (u) , = (  ) 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.

The Model and Main Result
We assume that Ω is a subset of R and the displacement is only in one direction.Specifically, we consider a simple variable space  and assume that Ω = (0, 2).We take the following system: with associated initial and boundary conditions The effective field is given by where Λ = (−Δ) 1/2 denotes the square root of the Laplacian which can be defined via Fourier transformation [7].In this paper we are interested in the case  ∈ (1/2, 1).For the vector u, we assume that u = (0, 0, ) and we keep the three components of the vector m = ( 1 ,  2 ,  3 ).
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 m  : Ω → R 3 .penalization in (11) replaces the magnitude constraint |m| = 1.
Throughout, we make use of the following notation.For Ω, an open bounded domain of R 3 , we denote by L  (Ω) = (  (Ω)) 3 and H 1 (Ω) = ( 1 (Ω)) 3 the classical Hilbert spaces equipped with the usual norm denoted by ‖ ⋅ ‖ L  (Ω) and ‖ ⋅ ‖ H 1 (Ω) (in general, the product functional spaces () 3 are all simplified to X).For all  > 0,  , denotes the usual Sobolev space consisting of all  such that          , fl where F denotes the Fourier transform and F −1 its inverse.Let Ẇ, denote the corresponding homogeneous Sobolev space.When  = 2,  , corresponds to the usual Sobolev space   .Now we give a definition of the solution in the weak sense of problem ( 7)-( 8)- (9).
The main result of this paper is the following.
The proof of Theorem 2 will be given in Section 4.

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]).
The following lemma will ensure a compact embedding for the space  , .
The proof can be found in [ [10], Theorem 4.54., p 216].We give now a lemma that will play a very important role in the convergence of approximate solutions (see [11][12][13] for a proof).

Lemma 6 (commutator estimates
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  >  > 1 and 1/ +  = 1/.Assume that  ∈   ; then Λ −  ∈   and there is a constant  > 0 such that We finish this section with the following result (the proof can be found in [2]). (18)

Proof of Theorem 2
Our goal is to show global existence of weak solutions for the fractional problem ( 7)-( 8)-( 9).

The Penalty Problem.
Let  > 0 be a fixed parameter.We construct approximated solutions m  converging, as  → 0, to a solution m of the problem.System ( 7) is reduced to the following problem: in  = Ω × (0, ), where the vector ℓ(m, ) is given by ℓ(m, ) = ( 3   , 0,  1   ),  1 =  2 = 0,  3 = , and  1313 = 1.System (19) is supplemented with initial and boundary conditions a.e. in Ω, We apply Faedo-Galerkin method: let {  } ∈N be an orthonormal basis of  2 (Ω) consisting of all the eigenfunctions for the operator Λ 2 (the existence of such a basis can be proved as in [14], Ch.II), and let {  } ∈N be an orthonormal basis of  2 (Ω) consisting of all the eigenfunctions for the operator −Δ: We then consider the following problem in  = Ω × (0, ): with initial and boundary conditions, We are looking for approximate solutions (m , ,  , ) 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 Ω we get to a system of ordinary differential equations in the unknown (a  (),   ()),  = 1, 2, . . ., .We observe that we can write the first equation in the form with ) .
It is clear that the matrix A 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 [0, ] using a priori estimates.For this, we multiply the first equation of ( 23 Omitting superscripts, we obtain for all  > 0,        ∫ thanks to the strong convergence   (⋅, 0) →  0 in  1 0 (Ω).For the other term (   (0)), the estimate can be carried out in an analogous way using the strong convergence    (⋅, 0) →  1 in  2 (Ω).Moreover, noting that (for a constant  independent of  and ) therefore, for fixed  > 0, 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 (m , ) and ( , ) such that for fixed  > 0 and for any 1 <  < ∞ m , ⇀ m  weakly in   (0, ; Convergence ( 38) is due to Lemma 3 and thanks to Lemma 4 it can be shown that  = |m  | 2 − 1. Moreover from the Sobolev embedding (Lemma 5)   () →  4 (), 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  ∈  2 (0, ; H  (Ω)) and  ∈  1 0 ().Taking  → ∞ in (40), we find for any  ∈  2 (0, ; H  (Ω)) and  ∈  1 0 ().We proved the following result.As  is in  2 (0, ; H  (Ω)), the following holds: Note that for this choice we have Λ  (m  × ) ∈ L 2 (), indeed applying the multiplicative estimates ( 16 ≤ ; (46) since 2 > 1 (1 here is the dimension) then H  (Ω) → L ∞ (Ω) and consequently (m  )  is bounded in L ∞ ().