An existence theorem for the magneto-viscoelastic problem

The dynamics of magneto-viscoelastic materials is 
described by a nonlinear system which couples the equation of the 
magnetization, given in Gibert form, and the viscoelastic 
integro-differential equation for the displacements. We study the 
general three-dimensional case and establish a theorem for the 
existence of weak solutions. The existence is proved by compactness 
of the approximated penalty problem.

1. Introduction. We consider the initial value problem The system (1) describes the evolution of a magneto-visco-elastic material. We denote by m and u the magnetization and the displacement respectively; by G and L the tensors of visco-elasticity and magneto-elasticity respectively; by ρ, γ and a three positive constants. The function f is given and it takes also into account the history. Let Ω an open bounded set of R 3 and ν the outer normal at the boundary ∂Ω, we assume the boundary conditions m ν | ∂Ω = 0, u| ∂Ω = 0. (3) The first equation (1) is the the well known Gilbert-Landau-Lifschitz equation (see [9], [12]) introduced for describing the dynamics of micro-magnetic processes. The magnetization is constant in modulus but not in direction and here we assume |m| = 1.
We study the existence of solutions to the nonlinear integro differential problem (1), (2), (3). Our reference here is the paper [14] where a theorem of existence for the general three dimensional magnetoelastic problem is proved.
In particular, we first introduce a penalty auxiliary problem depending on a small parameter ε, to remove the constraint |m| = 1, then we approximate the new problem by the Faedo-Galerkin method. The crucial point in the proof of the existence theorem is the characterization of some a-priori estimates which allow us to obtain convergence results. For this we need some constitutive assumptions on the quadratic form associated to the tensor G(s), that is positive, monotonically nonincreasing and convex. In the following section 2. we detail our assumptions and some notations; the section 3. is devoted to the proof of the existence result. Note that the existence and uniqueness of the solution to the one-dimensional magnetoviscoelastic problem is proved in [2] via the fixed point theory.
from (4), (5) and (10) we can get to the equations of the magneto-viscoelasticity in the form (1).
3.1. The approximated penalty problem. In this section we focus our attention to the penalty version of the previous system (1), (2), (3). First we introduce a small positive parameter ε and consider in Q the problem with initial and boundary conditions Then we consider the Faedo-Galerkin approximation for the problem (13), (14), (15) obtained taking where the functions φ j , ψ j both in H 2 (Ω) are the eigenfunctions of the eigenvalue problems and respectively and φ j L 2 (Ω) = ψ j L 2 (Ω) = 1, for j = 1, 2, ...N .
We get to the following approximated penalty problem with the components of the vector f N (x, t) defined by and the associated initial and boundary conditions defined by We in Ω. Since the sequence {φ j } j∈N forms a basis for L 2 (Ω) and the space of and analogously for the basis {ψ j } j∈N we have that space of finite linear combination of and If we multiply each scalar equation of the first vector equation (19) by ψ j (x) and the second equation (19) by φ j (x) and integrate in Ω we get to a system of integrodifferential differential equations in the unknown (a j (t), b j (t)), j = 1, 2, ..., N .
First of all we observe that that implies the matrix M is invertible and we can put our integro-differential system in the following integral system that is in a single integral equation where , and f(t) due to the contribution of the the function f . We denote by We claim that there exists a local solution to the above system (29). Indeed we put From the fixed point theorem we deduce the local existence of a unique solution to (29) and hence the local existence of a unique solution to the problem (19) (21), (22), (23).
In order to extend the solution to any finite time T > 0 we need the estimate established in the following lemma 3.2.
First of all we prove the following result in Ω and f ∈ L 2 (Q, R 3 ) then we have Proof. We put the above system (19) in the following equivalent form then we multiply the first equation (31) byṁ ε,N and the second one byu ε,N , integrating in Ω and omitting, for sake of simplicity, the indices ε and N we get Now we observe that Substituting in (33) we get to the proof of the lemma.

Now we introduce the functional
where the last four integrals are computed for t = T . We introduce the positive parameter λ such that in Ω, f ∈ L 2 (Q, R 3 ) and G verifying the assumptions (6), (7), (8), then there exists a positive constant C depending on the data m 0 , u 0 , u 1 , f and T but independent of N and ε, and a positive constant C ε,N such that for positive ε small enough (i.e. 16 ε ≤ βλ −2 ) the following estimate holds Proof. We multiply the first equation (19) byṁ ε,N and the second one byu ε,N , integrating in Ω × [0, T ] from Lemma 3.1 and taking into account the hypothesis on the operator G(t), we obtain the inequality Since for all b ∈ R + we have taking 2b = β/λ and recalling the inequality (7) we obtain and hence for ε small enough (i.e. ε < β/(16λ 2 )) one has using this last inequality in (36) we get From the Gronwall Lemma we obtain

Now writing
In the definition of the constantC andC ε,N we have assumed So we have from the convergence of the initial data (24), (25), (26), (27) and the convergence of the function f we have and setting the proof of the Lemma easily follows.
As a consequence of the estimate established in Lemma 3.2 we have where C 2 is a positive constant depending on the data of the problem but bounded for N → ∞ (see (37), (40)); then from compactness lemmas and passing to subsequences we have, as N → ∞ The convergence of the second and third term in the left side of (50) directly follows from (43), (44) thanks to the strong convergence of m N and the weak convergence ofṁ N and ∇m N . So we have For the first term of (50) we have where Since p vanishes at t=0 and at t=T, integrating by part the first integral we have from the regularity of the function p and from (43), (44), (45) we get P N → 0, as N → ∞ and hence The convergence of the last term in (50) follows from (49) and hence the proof of theorem 2.1 easily follows. Indeed from (50) and from (54), (51), (52), (55) we have that the couple (m, u) satisfies the weak form of the system (1). Moreover from the above convergence results we obtain F = a 2 Ω |∇m| 2 dΩ + β 4 Ω |∇(u)| 2 dΩ + ρ 2 Ω |u| 2 dΩ ≤ C where C (see (39)) is the constant independent of ε defined in lemma 3.1.