Asymptotic convergence results for a system of partial differential equations with hysteresis

A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero. Classification: 35K55, 47J40, 35B40.


Introduction
The aim of this paper is to study the following system of partial differential equations in Ω × (0, T ), (1.1) where Ω is an open bounded set of R N , N ≥ 1, F is a continuous rate independent invertible hysteresis operator, f is a given function, γ , α and β are given positive constants.
This system can be obtained by coupling the Maxwell equations, the Ohm law and a constitutive relation between the magnetic field and the magnetic induction, provided we neglect the displacement current.A detailed derivation will be given in Section 3 below.The meaning of the parameter γ is to take into account in the constitutive relation also a rate dependent component of the memory.A similar system has been considered recently in [1] in the context of soil hydrology, with γ fixed and with a more general form of the elliptic part.The main goal of this paper, instead, is to investigate the behaviour of the solution as γ → 0 .Our main result consists in proving that the solutions to (1.1) converge as γ → 0 to the (unique) solution (see [5]) of the system as an extension of the results contained in Chapter 4 of [4].For γ positive, the second equation in (1.1) defines a constitutive operator S : R × C 0 ([0, T ]) → C 1 ([0, T ]) which with each u ∈ C 0 ([0, T ]) and each initial condition w 0 ∈ R associates w = S(w 0 , u) .Then (1.1) has the form 3) The regularizing properties of S enable us to solve the problem by means of a simple application of the Banach contraction mapping principle.The passage to the limit as γ → 0 is achieved in several steps, using in particular a lemma constructed ad hoc which allows us to pass to the limit in the nonlinear hysteresis term.The outline of the paper is the following: after some remarks concerning Preisach operators (Section 2), we explain the physical interpretation of our model system in Section 3. Then we present in Section 4 the existence and uniqueness result while Section 5 is devoted to the asymptotic convergence of the solution as γ → 0 .
Here we use the one-parametric representation of the Preisach operator which goes back to [10].The starting point of our theory is the so called play operator.This operator constitutes the simplest example of continuous hysteresis operator in the space of continuous functions; it has been introduced in [9] but we can also find equivalent definitions in [2] and [18]; for its extension to less regular inputs, see also [12] and [13].
Let r > 0 be a given parameter.For a given input function u ∈ C 0 ([0, T ]) and initial condition x 0 ∈ [−r, r] , we define the output ξ = P r (x 0 , u) ∈ C 0 ([0, T ]) ∩ BV (0, T ) of the play operator as the solution of the variational inequality in Stieltjes integral form (2.1) Let us consider now the whole family of play operators P r parameterized by r > 0 , which can be interpreted as a memory variable.Accordingly, we introduce the hysteresis memory state space together with its subspaces where x 0 r is given by the formula is Lipschitz continuous in the sense that, for every u, v ∈ C 0 ([0, T ]), λ, µ ∈ Λ and r > 0 we have For more details, see Sections II.3, II.4 of [11].Now we introduce the Preisach plane as follows and consider a function ϕ ∈ L 1 loc (P) such that there exists Then the Preisach operator generated by the function g is defined by the formula for any given λ ∈ Λ ∞ , u ∈ C 0 ([0, T ]) and t ∈ [0, T ] .The equivalence of this definition and the classical one in [15], [18], e.g., is proved in [10].
Using the Lipschitz continuity (2.3) of the operator ℘ r , it is easy to prove that also W is locally Lipschitz continuous, in the sense that, for any given R > 0 , for every The first result on the inverse Preisach operator was proved in [3].We make use of the following formulation proved in [11], Section II.3.
) is invertible and its inverse is Lipschitz continuous.
Finally we have the following local monotonicity result for the Preisach operator W .
As we are dealing with partial differential equations, we should consider both the input and the initial memory configuration λ that additionally depend on x .If for instance λ(x, •) belongs to Λ ∞ and u(x, •) belongs to C 0 ([0, T ]) for (almost) every x , then we define 3 Physical interpretation of the model system (1.1) Let a ferromagnetic material occupy a bounded region D ⊂ R 3 ; we set D T := D ×(0, T ) for a fixed T > 0 , and we assume that the body is surrounded by vacuum.We denote by g a prescribed electromotive force; then Ohm's law reads where σ is the electric conductivity, J is the electric current density and E is the electric field; we also prescribe J = 0 outside D .
In D , we consider the Ampère and the Faraday laws in the form where c is the speed of light in vacuum, H is the magnetic field, D is the electric displacement and B is the magnetic induction.
In case of a ferromagnetic metal, σ is very large, hence we can assume provided that the field g does not vary too rapidly.
Then we neglect the displacement current ∂ D ∂t in Ampère's law; this is the so-called eddy current approximation.By coupling this reduced law with Faraday's and Ohm's laws, in Gauss units we get We consider the constitutive equation between H and B in the form where M is the magnetization, so we can rewrite (3.1) as For more details on this topics, we refer to a classical text of electromagnetism, for example [6].
We now reduce this system to a scalar one describing planar waves.More precisely, let Ω be a domain of R 2 .We assume (using the orthogonal Cartesian coordinates x, y, z ) that H is parallel to the z− axis and only depends on the coordinates x, y, i.e.
H = (0, 0, H(x, y)). Then We also assume that The purely rate independent hysteretic constitutive relation between H and M is considered in the form M = W(H), (3.4) where W is a Preisach operator.Since W itself is in typical cases not invertible, we introduce a new variable V = M + δ H with some δ ∈ (0, 1/4π) to be specified below, and rewrite (3.3), (3.4) as The rate dependent relaxed constitutive law leading to (1.1) reads 4 Existence and uniqueness In the setting (1.1) or (1.2), the space dimension is not relevant.We therefore consider an open bounded set of Lipschitz class Ω ⊂ R N , N ≥ 1 , set Q := Ω × (0, T ) , and fix an initial memory configuration where Λ K is introduced in (2.2).Let M(Ω; C 0 ([0, T ])) be the Fréchet space of strongly measurable functions Ω → C 0 ([0, T ]) , i.e. the space of functions v : Ω → C 0 ([0, T ]) such that there exists a sequence v n of simple functions with v n → v in C 0 ([0, T ]) a.e. in Ω .
We fix a constant b F > 0 and introduce the operator )) in the following way here W is the scalar Preisach operator defined in (2.4).Now Theorem 2.1 yields that F is invertible and its inverse is a Lipschitz continuous operator in C 0 ([0, T ]) .Let us set G = F −1 and let L G be the Lipschitz constant of the operator G .At this point we introduce the operator 3) It turns out that it follows from Theorem 2.1 that G is Lipschitz continuous in the following sense hold.Then (1.1) with homogeneous Dirichlet boundary conditions and initial conditions admits a unique solution Proof.The proof is divided into two steps.
• step 1: the solution operator S .We neglect for the moment the dependence on the space parameter x within the constitutive relation This means that we deal here with the following problem: for a given u Clearly problem (4.9) admits a unique solution w ∈ C 1 ([0, T ]) , for every u ∈ C 0 ([0, T ]) , due to the Lipschitz continuity of G .In this manner we can define the solution operator Let us show now that S is Lipschitz continuous in the sense that we prove that there exists a constant L S such that Let us consider u 1 , u 2 ∈ C 0 ([0, T ]) and let w 1 , w 2 ∈ C 1 ([0, T ]) be such that w i = S(u i ) , i = 1, 2 .The initial data are fixed, that is, w 1 (0) = w 2 (0) = w 0 .For any t ∈ [0, T ] we have Hence, by Gronwall's argument, for every t ∈ [0, T ] .Hence (4.10) holds with We easily extend this estimate to the space dependent problem a.e. in Q , (4.11) with given functions u ∈ L 2 (Ω; C 0 ([0, T ])) , w 0 ∈ L 2 (Ω) .It immediately follows from (4.10) that the solution mapping associated with (4.11) is well defined and Lipschitz continuous, with Lipschitz constant L S .step 2: fixed point.Our model problem can be rewritten now as with u(•, 0) = u 0 (•) and homogeneous Dirichlet boundary conditions.The unique solution will be found by the Banach contraction mapping principle.Let us fix z ∈ H 1 (0, T ; L 2 (Ω)) ; then z ∈ L 2 (Ω; C 0 ([0, T ])) and therefore S(z) is welldefined and belongs to L 2 (Ω; C 1 ([0, T ])) .Instead of (4.13), we consider the equation which is nothing but the linear heat equation.As f ∈ L 2 (Q) , this means that (4.14) admits a unique solution u ∈ H 1 (0, T ; We now introduce the set and the operator J : B → B : z → u, which with every z ∈ B associates the solution u ∈ B of (4.14).In order to prove that J is a contraction, consider now two elements z 1 , z 2 ∈ B , and set u 1 := J(z 1 ), u 2 := J(z 2 ).Then we have We test this equation by ∂ ∂t (u 1 − u 2 ) and obtain where L S is the Lispchitz constant of the operator S.This implies that We set θ := L 2 S β 2 α 2 and we introduce the following equivalent norm on H 1 (0, T ; L 2 (Ω)) If now we multiply (4.15) by e −θ t 2 and integrate over t ∈ (0, T ) , we obtain that and thus J is a contraction on the closed subset B of H 1 (0, T ; L 2 (Ω)) , which yields the existence and uniqueness of the solution u ∈ H 1 (0, T ;

Asymptotic convergence
In this section we investigate the behaviour of the solution of our model problem if the parameter γ goes to zero.We prove the following theorem.
Theorem 5.1.Under the assumptions of Theorem 4.1, let (u γ , w γ ) be the unique solution of (1.1) corresponding to γ > 0 with initial conditions (4.7) and homogeneous Dirichlet boundary conditions.Then there exists as γ → 0 , and u is the unique solution of the equation with initial condition u(x, 0) = u 0 (x) and homogeneous Dirichlet boundary condition.
Proof (5.4) Summing up (5.3), (5.4) and using (4.6), we obtain This allows us to obtain the following estimates ) and, by comparison, || u γ || L 2 (Q) ≤ C 5 .This entails that there exists a function u and a sequence γ n → 0 such that On the other hand, by interpolation and after a suitable choice of representatives, we deduce that (see [14], Chapter 4) with continuous and compact injection; this ensures that in particular (passing to subsequences if necessary), u γn → u uniformly in [0, T ] , a.e. in Ω . (5.6) On the other hand, the constitutive relation (4.8) yields and this, together with (5.5c), entails that From now on, we keep the sequence γ n → 0 fixed as in (5.6).Our aim is now to show that there exists a function w such that w γ n → w uniformly in [0, T ] , a.e. in Ω . (5.7) In fact, this will allow us to pass to the limit in the nonlinear hysteresis term.We show that (5.7) is obtained from (5.6) by using the following lemma: Let now ε > 0 be given.Using the Ascoli-Arzelà theorem, we find δ > 0 independent of n such that Let Ω ⊂ Ω be a set of full measure ( meas(Ω \ Ω ) = 0 ) such that, by virtue of (5.6), u γ n (x, •) → u(x, •) converges uniformly for all x ∈ Ω .Keeping now x ∈ Ω fixed, set u γ (x, •) := ũγ (•), w γ (x, •) := wγ (•).
We have that ũγn → ũ uniformly in C 0 ([0, T ]) as γ n → 0 , We thus checked that u is a solution of (5.1) with the required boundary and initial condition.Since this solution is unique by the argument of [5], we conclude that u γ converges to u independently of how γ tends to 0. This completes the proof of Theorem 5.1.
a.e. in Ω. Moreover Theorem 2.2 entails that there exist two constants c F and C F such that Theorem 4.1.(Existence and uniqueness) Let α, β, γ be given positive constants.Suppose that the following assumptions on the data . The regularity of u γ and w γ allows us to differentiate(4.11)intime and obtainIn the series of estimates below, we denote by C 1 , C 2 , . . .any positive constant depending only on the data of the problem, but independent of γ .