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Existence results for the $A-\varphi$ magnetodynamic formulation of the Maxwell system

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Submitted on 12 Dec 2013

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Existence results for the A− φ magnetodynamic formulation of the Maxwell system

Serge Nicaise

To cite this version:

Serge Nicaise. Existence results for the Aφ magnetodynamic formulation of the Maxwell system.

2013. �hal-00917597�

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Existence results for the A −ϕ magnetodynamic formulation of the Maxwell system

Serge Nicaise December 11, 2013

Abstract

The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popular model considered in the engineering community. We establish exitence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems; using energy estimates, existence of weak solutions are also deduced.

Key Words : Maxwell equations, potential formulation, degenerate parabolic problems.

AMS (MOS) subject classification35H65; 35Q60; 78A25.

1 Introduction

Let T > 0 and Ω R3 be an open connected bounded domain with a lipschitz boundary Γ that is also connected. In this work, we consider the Maxwell system given in Ω×[0, T] by :

curlE=−∂tB, (1)

curlH=tD+J, (2)

with initial and boundary conditions to be specified. Here, E stands for the electrical field, H for the magnetic field,Bfor the magnetic flux density,Jfor the current flux density (or eddy current) andDfor the displacement flux density. In the low frequency regime, the quasistatic approximation can be applied, which consists in neglecting the temporal variation of the displacement flux density with respect to the current density [1], so that the propagation phenomena are not taken into account. Consequently, equation (2) reduces to

curlH=J. (3)

The current density J can be decomposed in two terms such thatJ=Js+Je. Js is a known distribution current density generally generated by a coil. Je represents the unknown eddy current. Both equations (1) and (3) are linked by the material constitutive laws :

B=µH, (4)

Je=σE, (5)

whereµstands for the magnetic permeability andσfor the electrical conductivity of the material. Figure 1 displays the domain configuration we are interested in. It is composed of an open connected conductor domain c Ω which boundary B=∂Ωc supposed to be lipschitz, connected and such thatBΓ =∅. In Ωc, the electrical conductivityσis not equal to zero so that eddy currents can be created. The domain Ωe= Ω\Ωc is defined as the part of Ω where the electrical conductivityσis identically equal to zero. Boundary conditions associated with the previous system are given byB·n= 0 on Γ andJe·n= 0 onB, wherendenotes the unit

LAMAV, FR CNRS 2956, Universit´e de Valenciennes et du Hainaut Cambr´esis, Institut des Sciences et Techniques de Valenciennes, F-59313 - Valenciennes Cedex 9, France, serge.nicaise@univ-valenciennes.fr.

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!

!

Γ c

B

µ >0 σ >0

Js

e µ >0 σ= 0

Figure 1: Domains configuration.

outward normal to Ω and Ωcrespectively. In order to solve the problem with the quasistatic approximation, a formulation which is able to take into account the eddy current in Ωc and which verifies in Ωe Maxwell’s equations must be developed. This can be obtained by chosing the potential formulation often used for electromagnetic problems [10]. Indeed from the fact that divB= 0 in Ω and that its boundary is connected, by Theorem 3.12 of [3], a magnetic vector potentialAcan be introduced such that

B= curlAin Ω, (6)

with the boundary condition A×n= 0 on Γ allowing to guarantee B·n = 0 on Γ. Like B, the vector potential A exists in the whole domain Ω. To ensure the uniqueness of the potential, it is necessary to impose a gauge condition. The most popular one is divA= 0 (the so-called Coulomb gauge). Moreover, from equations (1) and (6), an electrical scalar potentialϕcan be introduced in Ωcso that the electrical field takes the form :

E=−∂tA− ∇ϕin Ωc. (7) As for the vector potential, it must be gauged. To obtain uniqueness, the averaged value of the potentialϕ on Ωc is taken equal to zero. From (4),(5),(6) and (7), equation (3) leads to the so-calledA−ϕformulation :

curl µ−1curlA +σ

tA+∇ϕ

=Js. (8)

The great interest of this formulation relies in its effectivity in both domain Ωc and Ωe. Indeed, in Ωewhere σ is zero the second term vanishes and theAϕ formulation becomes the classicalA formulation used in the magnetostatic case.

Our main goal is to prove existence results for problem (8) completed with appropriated boundary con- ditions. More particularly, we have in mind to derive a weak formulation that will be used for the numerical resolution of (8) by the Finite Element Method in the context of electromagnetic problems [7].

Concerning the harmonic formulation of some Maxwell problems, several contributions have been proposed in the last decade. In that case, since no time derivatives are involved, we have only to deal with spatial problems for which existence results are easier to obtain (see for instance Theorem 2.1 in [6]).

Recent contributions on evolution Maxwell equations of degenerate parabolic type can be found in [2, 4, 5, 8, 9, 11]. A characteristic feature of these papers is the presence of conducting and nonconducting regions in the spatial domain. While [5] and [9] consider the model in bounded regions and [5] also sketches a quasilinear system, in [2] and [4] the problem is discussed in the whole space. The paper [8] deals with induction heating and considers a coupled system of the evolution Maxwell and heat equations. Finally in [11], an integrodifferential system was studied that accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil.

Let us finish this introduction by some notation used in the whole paper. On a given domainD, the L2(D) norm is denoted byk · kD, and the correspondingL2(D) inner product by (·,·)D. The usual norm and semi-norm onH1(D) are respectively denoted byk · k1,D and| · |1,D. In the caseD= Ω, we drop the index Ω. Recall thatH01(D) is the subspace ofH1(D) with vanishing trace on∂D. Finally, the notationa.band

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ab means the existence of positive constantsC1 andC2, which are independent of the quantitiesaandb under consideration such thataC2b andC1baC2b, respectively.

The paper is organized as follows. In section 2, the strong and weak formulations of the problem are presented and the existence result is stated. Then, section 3 is devoted to the proof of some preliminary results in order to apply a result of Showalter on degenerated parabolic problems. Finally, in section 4 we prove energy estimates and the main result of our paper.

2 Formulation of the problem and the main result

Assuming that divJs= 0, theA−ϕformulation of the magnetodynamic problem with given initial conditions onAcan be written as

curl µ−1curlA +σ

tA+∇ϕ

= Js in Ω×(0, T), (9)

div σ

tA+∇ϕ

= 0 in Ωc×(0, T), (10)

A×n = 0 on Γ×(0, T), (11)

σ(∂tA+∇ϕ) ·n = 0 onB×(0, T), (12) A(t= 0,·) = 0 in Ωc. (13) We suppose that µ L(Ω) and that there existsµ0 R

+ such that µ > µ0 in Ω. We also assume that σL(Ω),σ|Ωe 0, and that there existsσ0R

+ such thatσ > σ0 in Ωc. At last, we recall the Gauge conditions. Like mentioned in section 1, we choose the Coulomb one divA = 0 in Ω, and we ask for the averaged value ofϕin Ωc to be equal to zero.

We now define

X(Ω) = H0(curl,Ω) =n

AL2(Ω) ; curlAL2(Ω) andA×n=0on Γo , XN(Ω) = {AX(Ω) : divA L2(Ω)},

X0(Ω) = {AX(Ω) ; div A= 0 in Ω}, Hf1(Ωc) = n

ϕH1(Ωc) ; Z

c

ϕ dx= 0o ,

equipped with their usual norm

kAk2X(Ω) = kAk2+kcurlAk2,∀AX(Ω), kAk2X0(Ω) = kAk2+kcurlAk2,∀AX0(Ω), kAk2XN(Ω) = kAk2X(Ω)+kdivAk2,∀AXN(Ω),

kϕkHf1(Ωc) = |ϕ|1,Ωc,∀ϕHf1(Ωc).

Similarly, we set

H(div = 0,Ω) ={AL2(Ω)3 : div A= 0 in Ω}, that is a closed subspace of L2(Ω)3.

Note that, here and below, divA= 0 in Ω means equivalently that (A,∇ξ) = 0ξH01(Ω).

The variational (or weak) formulation associated with (9)-(13) is obtained in a usual way, multiplying (9) by a test function AX0(Ω) (resp. (10) by a test functionϕ Hf1(Ωc)), integrating the results in Ω, formal integrations by parts and taking the sum we find

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(µ−1 curlA,curlA)+(σ(∂tA+∇ϕ),A+∇ϕ)c = (Js, A),∀(A, ϕ)X0(Ω)×Hf1(Ωc), a.a. t(0, T).

(14) An existence result for this problem can be stated as follows

Theorem 2.1. Let us assume that Js H1((0, T);H(div = 0,Ω)) and setJs,0=Js(t= 0). Assume that Js,0·n= 0on B, (15) and that there exists A0X0(Ω) satisfying

A0= 0 inc, and

(µ−1curlA0,curlA) = (Js,0,A)e,A XN(Ω).

Then problem (14) has a unique solution (A, ϕ)inH1(0, T;X0(Ω))×L2(0, T;Hf1(Ωc))withA(t= 0) =A0. Proof. The proof is postponed to section 4.

By the uniqueness of the solution, this local existence result directly allows to obtain a global one.

3 Preparations for the application of a theorem by Showalter

Our results on existence and uniqueness rely on the following theorem:

Theorem 3.1 ([12], Theorem V4.B). Let Vm be a seminorm space obtained from a symmetric and non- negative sesquilinear form m(·,·), and let M ∈ L(Vm, Vm)be the corresponding operator given by Mx(y) = m(x, y),for allx, yVm. LetV be a Hilbert space which is dense and continuously embedded intoVm. Leta be a continuous, sesquilinear and elliptic form onV and denote byAthe corresponding isomophism fromV ontoV. LetD={uV :AuVm}. Then for any f C1([0,∞), Vm) andy0Vm, there exists a unique

solution y to

(My)t(t) +Ay(t) = f(t) in Vm, ∀t >0,

My(0) = My0 in Vm, (16)

with the regularity

MyC([0,∞), Vm)C1((0,∞), Vm) and such that

y(t)D,∀t >0.

In order to apply this theorem, we show that problem (9)-(13) fits in the associated framework. Before we need some preliminary results. First forAL2(Ω)3, we consider the unique solutionϕAHf1(Ωc) of

Z

c

σ∇ϕA· ∇χ dx¯ = Z

c

σA· ∇χ dx,¯ ∀χHf1(Ωc). (17) Such a solution exists by Lax-Milgram lemma and furthermore by Cauchy-Schwarz’s inequality, we have

1/2∇ϕAkc≤ kσ1/2Akc. (18)

¿From this problem, we deduce that the field

σ(A+∇ϕA) is divergence free in Ωc, i.e.,

div (σ(A+∇ϕA)) = 0 in Ωc, (19)

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and satisfies the boundary condition

σ(A+∇ϕA)·n= 0 on∂Ωc. (20)

As the boundary of Ωcis connected, by Theorem 3.12 of [3], there exists a vector potentialBAX0(Ωc) such that

σ(A+∇ϕA) = curlBA in Ωc. (21)

Note that the previous properties also show that for anyH1 extension ˜ϕA ofϕA to Ω, one has div (σ(A+ϕ˜A)) = 0 in Ω.

Recall that forBX0(Ωc), we have

kcurlBkc ∼ kBkXN(Ωc). Hence

σ−1curlX0(Ωc) :=−1curlB:BX0(Ωc)}

is a closed subspace ofL2(Ωc)3.

Note that (21) shows thatσ−1curlBAis the orthogonal projection ofAonσ−1curlX0(Ωc) for the inner product

(A,B)c:=

Z

c

σA·Bdx,∀A,BL2(Ωc)3. (22) Now we consider the mapping

Mc:L2(Ωc)3curlX0(Ωc)L2(Ωc)3:Aσ(A+∇ϕA). (23) By (18), this mapping is linear and continuous. Furthermore it is symmetric and non negative (for the standard inner product). Indeed forA,BL2(Ωc)3, we first show that

Z

c

σ∇ϕA·Bdx= Z

c

σ∇ϕA· ∇ϕ¯Bdx. (24)

Indeed by (17), we have

Z

c

σ∇ϕA·Bdx = Z

c

σB∇ϕ¯Adx

=

Z

c

σ∇ϕBϕ¯Adx

=

Z

c

σ∇ϕAϕ¯Bdx.

This proves (24).

With this identity we directly deduce that

(McA,B)c= (A,McB)c, and the symmetry ofMc is proved.

For the non negativeness, we first notice that (17) is equivalent to Z

c

McA· ∇χ dx¯ = 0,∀χHf1(Ωc). (25) This identity directly implies that

(McA,A)c = Z

c

σ(A+∇ϕA)·A¯ dx

= Z

c

σ(A+∇ϕA)·(A+∇ϕA)dx

= 1/2(A+∇ϕA)k2c

∼ kMcAk2c.

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We now equipL2(Ω)3 with the semi-inner product

m(A,B) := (Mc(A|Ωc),B)c,

that is denoted by Vm (and is not closed). Before going on, we directly see that m is a symmetric and non-negative sesquilinear form on Vm since

m(A,B) := (Mc(A|Ωc),B|Ωc)c.

Furthermore let Mbe the linear and continuous operator from Vm into its dualVm defined by (MA)(B) =m(A,B),∀A,BVm.

Now recall that (see for instance [12]) the dual spaceVm ofVmis a Hilbert space that is characterized in the next lemma.

Lemma 3.2.

Vm = curlX0(Ωc).

In other words,lVm if and only if there existBX0(Ωc)such that l(A) :=

Z

c

curlB·A¯ dx,AL2(Ω)3, and

klkVm ∼ kcurlBkc.

Proof. The inclusion curlX0(Ωc)Vm is direct since forBX0(Ωc) andAVmwe can define lB(A) :=

Z

c

curlB·A¯dx, that is a continuous linear form on Vm since

lB(A) = Z

c

σ−1/2curlB·σ1/2( ¯A+ϕ¯A)dx.

Therefore,

|lB(A)|.−1/2curlBkc1/2(A+∇ϕA)kc . kBkX0(Ωc)m(A,A)1/2. For the converse inclusion, let us fixlVm, which means that

|l(A)|.m(A,A)1/2,∀AVm. (26) We first notice that forχHf1(Ωc), we have

M∇χ= 0, hencem(∇χ,∇χ) = 0 and therefore

l(∇χ) = 0.

This identity implies that

l(A) =l(A+∇ϕA) =l(σ−1curlBA). (27)

As said before we have

σ−1curlX0(Ωc)Vm, and for anyCX0(Ωc), we have

m(σ−1curlC, σ−1curlC) =−1/2curlCk2c ∼ kCk2X0(Ωc).

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Therefore the mapping

l0:Cl(σ−1curlC),

is linear and continuous fromX0(Ωc) toC. Hence sinceX0(Ωc) is a Hilbert space for the inner product (B,C) :=

Z

c

σ−1curlB·curl ¯Cdx,∀B,CX0(Ωc), there exists BX0(Ωc) such that

l0(C) = Z

c

σ−1curlB·curl ¯Cdx,∀CX0(Ωc).

By (27), we deduce that

l(A) = Z

c

σ−1curlB·curl ¯BAdx,∀AL2(Ω)3, and by (21), we conclude that

l(A) = Z

c

curlB·( ¯A+ϕ¯A)dx= Z

c

curlB·A¯ dx,∀AL2(Ω)3. The first step shows that the mapping

curlX0(Ωc)Vm : curlBlB,

is linear and continous. Moreover the second step shows this mapping is surjective hence by the closed graph theorem, it is an isomorphism.

Before going on let us notice that for anyAL2(Ω)3, we see that MA=σ(A|Ωc− ∇ϕA|Ωc), and therefore by the previous consideration, we have

kMAk2Vm ∼ kMcA|Ωck2c(MA,A). (28) At this stage in order to apply Theorem 3.1, we define

a(A,A) = Z

µ−1 curlA·curl ¯A dx+ Z

divAdiv ¯Adx,∀A,A XN(Ω), that is clearly continuous and coercive onXN(Ω) due its compact embedding intoL2(Ω)3.

FurthermoreXN(Ω) is clearly dense inL2(Ω)3 and continuously embedded into Vm. Therefore we have checked all assumptions of Theorem 3.1 (with V =XN(Ω)). Before stating a consequence of this theorem, we recall that

D:={AXN(Ω) :∃lVm :a(A,B) =l(B),∀BXN(Ω)}, and show that elements of such a set are divergence free.

Lemma 3.3. Any AD is divergence free inΩ.

Proof. FixAD, then there existslVm such that

a(A,B) =l(B),∀BXN(Ω). (29)

For anyf L2(Ω), we take the unique solutionuf H01(Ω) of

∆uf=f in Ω,

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or equivalenty the unique solution of Z

∇uf · ∇¯v dx= Z

f¯v dx,∀vH01(Ω).

By takingB=∇uf in (29) (allowed since∇uf XN(Ω)), we find Z

divAf dx¯ =l(∇uf) = 0.

Sincef is arbitrary inL2(Ω), we conclude that divA= 0 in Ω.

4 Some existence results

Let us now give a first consequence of Theorem 3.1.

Theorem 4.1. Let FC1([0, T], Vm)andB0Vm, then there exists a unique solutionB to t(MB)(t) +AB(t) = F(t) inVm ,∀t(0, T),

(MB)(t= 0) =MB0 inc, (30)

with the regularity MBC([0, T], Vm)C1((0, T], Vm), and B(t)D,∀t(0, T].

Furthermore one has (where h··imeans the duality pairing betweenVm andVm)

h∂t(MB)(·, t),Zi+ Z

µ−1 curlB·curl ¯Zdx= (F(t),Z),ZXN(Ω),∀t(0, T), (31) and the next estimate holds

kBkL2(0,T;X0(Ω))+kMBkC([0,T];Vm)+k∂tMBkL1((0,T);XN(Ω)).kFkL2(0,T;Vm)+kB0kVm. (32) Proof. The existence and uniqueness result directly follows from Theorem 3.1, by taking an appropriate extension ofFin the whole [0,∞). Hence it remains to prove (31) and the estimate (32).

For that purpose, we closely follow the proof of Corollary 3.8 of [11]. AsXN(Ω)Vm, the first identity of (30) implies that

h∂t(MB)(·, t),Zi+a(B,Z) = (F(t),Z),∀ZXN(Ω),∀t(0, T).

By Lemma 3.3 we directly arrive at (31).

To prove (32), we first takeZ=B in (31), to get 1

2t(MB,B) +−1/2 curlBk2 = (F,B).

Hence by Cauchy-Schwarz’s inequality we get 1

2t(MB,B) +−1/2 curlBk2≤ kFkkBk,

and with the estimatekBk ≤Ckµ−1/2curlBk, for someC >0 and Young’s inequality we obtain 1

2t(MB,B) +−1/2curlBk2C2

2 kFk2+1

2−1/2 curlBk2. This shows that

t(MB,B) +−1/2curlBk2C2kFk2.

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Integrating this estimate int(η, u), withη, u(0, T] arbitrary but such thatη < u, we get

(MB(·, u),B(·, u)) + Z u

η

−1/2 curlB(·, t)k2dtC2 Z u

η

kF(·, t)k2dt+ (MB(·, η),B(·, η)).

Lettingη tends to zero and reminding the regularityMBC([0, T], Vm) (see also the estimates (28)), we find that

(MB(·, u),B(·, u)) + Z u

0

−1/2 curlB(·, t)k2dt. Z T

0

kF(·, t)k2dt+kB0k2Vm. (33) In a first step by takingu=T, this shows that

kBkL2(0,T;X0(Ω)).kFkL2(0,T;L2(Ω))+kB0kVm. (34) In a second step, for anyu(0, T], we also have

|(MB(·, u),B(·, u))|. Z T

0

kF(·, t)k2dt+kB0k2Vm+ Z u

0

−1/2 curlB(·, t)k2dt ,

and therefore by (34)

|(MB(·, u),B(·, u))|. Z T

0

kF(·, t)k2dt+kB0k2Vm,∀u[0, T]. (35) It remains to estimate the last term of the left-hand side of (32).

For that purpose, we come back to (31) that can be equivalently written h∂t(MB)(·, t),Zi=

Z

µ−1 curlB·curlZdx+ Z

F(x, t)·Z(x)dx,∀ZXN(Ω).

Now notice that it implies

|h∂t(MB)(·, t),Zi| ≤ Z

µ−1 curlB·curlZdx +

Z

F(x, t)·Z(x)dx

,∀ZXN(Ω).

Integrating this identity int(0, T) and applying Cauchy-Schwarz’s inequality we get Z T

0

|h∂t(MB)(·, t),Zi|dtT(kFkL2(0,T;L2(Ω))+−1 curlB(·, t)kL2(0,T;L2(Ω)))kZkXN(Ω), for allZXN(Ω).By (34), this implies that

Z T 0

|h∂t(MB)(·, t),Zi|dt.(kFkL2(0,T;L2(Ω))+kB0kVm)kZkXN(Ω), for allZXN(Ω).This estimate leads to

k∂t(MB)kL1(0,T;XN(Ω)).kFkL2(0,T;L2(Ω))+kB0kVm, (36) due to the definition of the normXN(Ω).

The proof of the estimate (49) is then complete.

Now we can prove an existence result of a strong solution to problem (9)-(13).

Theorem 4.2. Let us suppose that Js C3([0, T], H(div = 0,Ω)) with Js,0 = Js(t = 0) satisfying the assumptions of Theorem 2.1. Then, there exists one and only one solution (A, ϕ) H1(0, T;X0(Ω))× L2(0, T;Hf1(Ωc))to problem (9)-(13) with the (additional) initial condition

A(t= 0) =A0. (37)

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Proof. Assume that a solution (A, ϕ) of problem (9)-(13) exists and is sufficiently regular, then comparing (10) and (13) with (19) and (20), we can say that ϕ=tϕA, ϕA being the unique solution of (17). With this property, problem (9)-(13) is then (formally) equivalent to

t(MA) + curl(µ−1curlA) =Js in Ω×(0, T),

A(t= 0) = 0 in Ωc. (38)

Now in order to apply Theorem 3.1 (or equivalently Theorem 4.1), we need that the above right-hand sideJs is inVm (in particular it should be zero in Ωe) which is not the case in physical applications. Hence we perform an elliptic lifting, namely for allt[0, T], we consider the unique solutionLj(t)X0(Ω) of

curl(µ−1 curlLj(t)) =Js(t) in Ω,

Lj(t)×n= 0 on ∂Ω, (39)

or in a weak form Z

µ−1 curlLj(t)·curl ¯Wdx= Z

Js(t)·W¯ dx,∀WX0(Ω). (40) This last problem has a unique solution due to Lax-Milgram lemma. Now any WXN(Ω) can be written in the form

W=W0+∇ψ, withψH01(Ω) and W0X0(Ω) and since

Z

Js· ∇ψ dx= 0,∀ψH01(Ω), (40) implies that Z

µ−1curlLj·curl ¯Wdx= Z

Js·W¯ dx,∀WXN(Ω), (41) and therefore the unique solutionLj X0(Ω) of (40) is a solution of (39).

¿From (40), we also have

kLjkX0(Ω).kJskL2(Ω)3.

Furthermore due to this estimate, if Js Ck([0, T], H(div = 0,Ω)), for some k N, then Lj will be in Ck([0, T], X0(Ω)) with

kLjkCk([0,T],X0(Ω)).kJskCk([0,T],H(div =0,Ω)). (42) At this stage, by setting

B=ALj, we arrive at the problem

tMB+ curl(µ−1 curlB) =−∂tMLj in Ω,

B(t= 0) =−Lj(t= 0) in Ωc. (43)

Now asJsC2([0, T], H(div = 0,Ω)),MLj C2([0, T), Vm) (because the range ofMis exactly Vm).

Hence applying Theorem 4.1, we find a unique (strong) solution of

t(MB) +AB=−∂tMLj inVm, t(0, T],

(MB)(t= 0) =−MLj(t= 0) in Ωc, (44)

with the regularityMBC([0, T], Vm)C1((0, T], Vm), and B(t)D,∀t(0, T].

Furthermore for anyt >0 we have h∂t(MB)(·, t),Zi+

Z

µ−1 curlB·curl ¯Zdx=−(∂t(MLj)(·, t),Z),¯ ∀ZXN(Ω). (45)

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Nondestructive inspection, eddy currents, imaging probe, finite element modelling, pickup coil array, printed-circuit- board coil, micro-moulded coils, bobbin coil,

This paper has presented a cheap way of reducing eddy current losses in unconventional low cost actuators. This method has been applied to a linear multi-rod actuator. The final