2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(01)00086-5/FLA
QUASILINEAR HYPERBOLIC EQUATIONS WITH HYSTERESIS
Augusto VISINTIN
Universitá degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo di Trento, Italy
Received 13 February 2001
ABSTRACT. – Plasticity, ferromagnetism, ferroelectricity and other phenomena lead to quasi- linear hyperbolic equations of the form
∂2
∂t2
u+F(u)
+Au=f,
where F is a (possibly discontinuous) hysteresis operator, and A is a second order elliptic operator. Existence of a solution is proved for an associated initial- and boundary-value problem in the framework of Sobolev spaces. The argument is based on the dissipation properties of hysteresis, and can be applied to a large class of hysteresis operators, including the classic Preisach model.2002 Éditions scientifiques et médicales Elsevier SAS
RÉSUMÉ. – Plusieurs phénomènes (plasticité, ferro-magnétisme, ferro-électricité, etc.) condui- sent à la formulation d’une équation hyperbolique quasilinéaire de la forme
∂2
∂t2
u+F(u)
+Au=f;
iciFest un opérateur d’hystérésis éventuellement discontinu, etAest un opérateur elliptique du deuxième ordre. On démontre l’existence d’une solution faible pour un problème aux limites et aux valeurs initiales associé à cette équation. L’argument est basé sur les propriétés de dissipation de l’hystérésis, et s’applique à une large classe d’opérateurs, qui inclut le modèle classique de Preisach.2002 Éditions scientifiques et médicales Elsevier SAS
Introduction
In the last years the mathematical research on hysteresis models has been progressing, see, e.g., [1,4,6,10]. In particular results have been obtained for PDEs containing hysteresis nonlinearities, including quasilinear parabolic and semilinear hyperbolic
E-mail address: Visintin@science.unitn.it (A. Visintin).
equations. On the other hand, if F is a scalar hysteresis operator and A is a second order elliptic operator, the quasilinear hyperbolic equation
∂2
∂t2
u+F(u)+Au=f (1)
has still been looking rather challenging, if we exclude the especially easy case in which F can be represented by one or more variational inequalities, see [10, Chap. VII]. In a single space dimension, existence of a solution of the initial- and boundary-value problem associated to (1) was proved by Krejˇcí [5,6] assuming strict convexity of the hysteresis loops. Strict convexity is also required by known existence results for the analogous equation without hysteresis, see, e.g., Di Perna [3].
Although at first sight occurrence of hysteresis may look as an element of difficulty, its dissipative character suggests that it might allow to prove existence of a solution under weaker hypotheses than for the problem without hysteresis. This is shown in the present paper, in which existence of a solution is proved for the multidimensional problem, for a large class of (possibly discontinuous) nondegenerate scalar hysteresis operators. Discontinuities in the hysteresis relation account for the occurrence of free boundaries. Uniqueness of the solution remains an open question.
Equations of the form (1) occur in elastoplasticity, pseudoelasticity, ferromagnetism, ferrimagnetism, ferroelectricity, and in other physical phenomena; however, due to their scalar character, applications are essentially restricted to univariate systems. The study of Maxwell equations in three-dimensional domains requires the use of a vector hysteresis model. In [12] the hysteresis relation has been represented by a vector model of [2]; a weak formulation has been provided along the lines of the present paper, and existence of a solution has been proved for corresponding quasilinear hyperbolic and parabolic initial- and boundary-value problems, which are respectively obtained by including or neglecting displacement currents; see [14].
The plan of this paper is as follows. In Section 1 we outline the problem. In Section 2 we shortly illustrate how equations of the form (1) arise in mathematical physics.
In Section 3 we provide a precise definition of the relay and Preisach operators. In Sections 4, 5 (6, 7, respectively) we formulate our problem for A:= − , and prove existence of a solution for F equal to a relay operator (a Preisach operator, resp.). In Section 8 we discuss a regularity issue, and deal with a modified formulation in which the variableuis convoluted in space with a regularizing kernel. Finally, in Section 9 we point out some remarks, draw conclusions and discuss some open questions.
The results of the present paper have been announced in [13].
1. The problem
Hysteresis. A causal operator F which acts between Banach spaces of time- dependent functions is called a hysteresis operator whenever it is rate-independent, that is,[F(u)](ϕ(t))= [F(u◦ϕ)](t)for any increasing time-homeomorphismϕand for any instant t. We allow F to be multi-valued, and extend it to space- and time-dependent functions by setting[F(u)](x, t):= [F(u(x,·))](t); see [10] for details.
Fig. 1. Relay operator.
A large and important class of hysteresis operators is provided by the classic Preisach model; loosely speaking, this consists in a linear combination of more elementary operators, named (delayed) relay operators. Throughout this paper byρwe denote any pair (ρ1, ρ2)∈R2 such that ρ1< ρ2; the corresponding relay operator hρ is outlined in Fig. 1. For instance, let u(0) < ρ1; then w(0)= −1, and w(t)= −1 as long as u(t) < ρ2; if at some instant ureaches ρ2 then w jumps up to 1, where it remains as long asu(t) > ρ1; if laterureachesρ1, thenwjumps down to−1, and so on. Whenever ρ1< u(0) < ρ2, the initial valuew0(= ±1)must be prescribed. The operatorhρis causal and rate-independent; it is the most simple model of discontinuous hysteresis.
Let us now consider a finite Borel measure µ (called Preisach measure) on the half-plane P:= {ρ:=(ρ1, ρ2)∈R2: ρ1< ρ2}of admissible thresholds, and define the Preisach operatorHµ:=Phρdµ(ρ). See Fig. 2 for a simple example.
Let be a bounded domain of RN (N 1) of Lipschitz class, T >0 and set Q:= × ]0, T[. Let f:Q→R be a given function, F a (possibly multi-valued) scalar hysteresis operator, andAa second order elliptic operator; regularity requirements will be specified afterwards. In this paper we deal with the second order quasilinear hyperbolic equation
∂2
∂t2
u+F(u)+Au=f inQ, (1.1)
coupled with the initial and boundary conditions u+w=u0+w0, ∂
∂t(u+w)=u1+w1 in× {0}, (1.2)
u=0 on∂× ]0, T[; (1.3)
Fig. 2. Preisach model corresponding to a positive measureµconcentrated at three points ofP. hereu0, w0, u1, w1:→R are prescribed functions. Integrating (1.1) in time we get
∂
∂t
u+F(u)+A t
0
u(·, τ )dτ =u1+w1+ t 0
f (·, τ )dτ =:F inQ. (1.4)
If the operatorAis in divergence form, Eq. (1.1) is also equivalent to a first-order system.
2. Applications
In this section we briefly illustrate how equations like (1.1) arise in continuum me- chanics and in electromagnetism.
Elastoplasticity and pseudoelasticity. Let ⊂R3 represent a continuum body, and let us denote the displacement vector by u, the (linearized) strain tensor by ε, the stress tensor byσ, and a distributed load byh. Newton’s law and the definition ofεyield
∂2ui
∂t2 = 3 =1
∂σi
∂x +hi, εij :=1 2
∂ui
∂xj +∂uj
∂xi
inQ (i, j =1,2,3), (2.1) whence
∂2εij
∂t2 =1 2
3 =1
∂2σi
∂x∂xj
+ ∂2σj
∂x∂xi
+1
2 ∂hi
∂xj
+∂hj
∂xi
inQ (i, j=1,2,3). (2.2) In an elastoplastic material the dependence of the stress on the strain exhibits hysteresis.
The same applies to austenitic materials exhibiting so-called pseudoelasticity, i.e., shape
memory. In either case we may assume thatε=aσ+F1(σ ),abeing a positive constant andF1a tensor hysteresis operator. This yields an equation of the form (1.1) for tensor variables; for univariate systems, this is reduced to an equation for scalar variables.
Ferromagnetism and ferroelectricity. Letrepresent an electromagnetic mater- ial, and let us denote the electric field byE, the electric displacement by D, the electric current density byJ, the magnetic field byH, the magnetic induction byB, the electric conductivity byσ, the dielectric permittivity byε, the magnetic permeability byµ, the speed of light in vacuum byc, and an applied electromotive force byg. For the sake of simplicity, here we assume that the equations incan be decoupled from those outside.
However, this outline and the results of this paper can be extended to the case in which this restriction is dropped.
In Gauss units, the Ampère, Faraday and Ohm laws respectively read c∇ × H=4πJ+∂D
∂t inQ (∇× :=curl), (2.3) c∇ × E= −∂B
∂t inQ, (2.4)
J=σE+ g inQ. (2.5)
In a ferromagnetic material we can assume that D =εE. Applying the curl operator to (2.3), differentiating (2.4) in time, and eliminatingJ , D, E, we then get
ε∂2B
∂t2 +4π σ∂B
∂t +c2∇ × ∇ × H=4π cσ∇ × g inQ. (2.6) This equation applies to both ferromagnetic and ferrimagnetic materials. The former are all metals, and so for slow processes the displacement term ε∂∂t2B2 is dominated by the Ohmic term 4π σ∂∂tB. In this case (2.6) is then usually replaced by the so-called eddy- current equation
4π σ∂B
∂t +c2∇ × ∇ × H=4π cσ∇ × g inQ. (2.7) However, for fast processes (2.6) applies. On the other hand, ferrimagnetic materials may be insulators; (2.6) withσ=0 can then be used for either slow or fast processes.
In any of these cases B = H +4πM, and the magnetization M depends with hysteresis onH. We may then assume thatM =F2(H ), where F2is a vector hysteresis operator.
If instead we deal with a ferroelectric material, we may assume that B =µH;
Eqs. (2.3), (2.4) and (2.5) then yield µ∂2D
∂t2 +4π µσ∂E
∂t +c2∇ × ∇ × E= −4π µσ∂g
∂t inQ. (2.8)
HereD = E+4πP, and the electric polarizationP depends onE with hysteresis; we may then assume thatP=F3(E), where F3is a vector hysteresis operator.
Special geometries. For univariate systems we are reduced to an equation for scalar variables, with A= −∂x∂22. Under severe restrictions on the geometry and on the symmetry of the fields, the above setting can also be reduced to a planar problem for scalar variables. Let D be a domain of R2, :=D×R, and assume that, using orthogonal Cartesian coordinatesx, y, z,H is parallel to thez-axis and only depends on the coordinatesx, y, i.e.,H =(0,0, H (x, y)). Then
∇ × ∇ × H =(0,0,− x,yH )
x,y:= ∂2
∂x2 + ∂2
∂y2
in.
Dealing with a strongly anysotropic material, we can also assume that M =(0,0, M(x, y)). If∇ × g:=(0,0, r), Eq. (2.6) is then reduced to
ε∂2
∂t2(H+4π M)+4π σ ∂
∂t(H+4π M)−c2 x,yH=4π cσ r inD. (2.9) We can then assume thatM=F(H ), whereF is a scalar hysteresis operator. A similar discussion applies to (2.8).
Remark. – We have represented the above phenomena by equations to the form (1.1).
This has required differentiation operations, and indeed (1.1) can be expected to hold just in the sense of distributions. We might also derive equivalent systems. For instance, in continuum mechanics one might just couple the system (2.1) with the constitutive law ε=aσ +F1(σ ). Similarly, in electromagnetism one might couple the Maxwell–Ohm equations (2.3), (2.4), (2.5) with the appropriate constitutive relations.
3. Hysteresis
In this section we make the definition of the relay and Preisach operators precise, and specify the functional framework.
Relay operator. Let us fix any pair ρ :=(ρ1, ρ2)∈ R2, ρ1< ρ2. For any u∈ C0([0, T])and any ξ ∈ {−1,1}, we setXt := {τ ∈ ]0, t]: u(τ )=ρ1orρ2}and define the functionw=hρ(u, ξ ):[0, T] → {−1,1}as follows:
w(0):=
−1 ifu(0)ρ1, ξ ifρ1< u(0) < ρ2, 1 ifu(0)ρ2,
(3.1)
w(t):=
w(0) ifXt= ∅,
−1 ifXt= ∅andu(maxXt)=ρ1, 1 ifXt= ∅andu(maxXt)=ρ2,
∀t∈ ]0, T]. (3.2)
Any functionu∈C0([0, T])is uniformly continuous, hence it can only oscillate a finite number of times between ρ1 andρ2, if any. Thereforew can just have a finite number of jumps between−1 and 1, if any; hence the total variation ofwin[0, T]is finite, i.e., w∈BV(0, T ).
Completion. The relay operator is not closed in natural function spaces, because of its discontinuity, cf. [10, Chap. VI]. A suitable extension of this operator is then needed, in view of coupling with PDEs. Following [10], we introduce the (multi-valued) completed relay operator, kρ. For any u∈ C0([0, T]) and any ξ ∈ [−1,1], we set w∈kρ(u, ξ )if and only ifwis measurable in]0, T[,
w(0):=
−1 ifu(0) < ρ1, ξ ifρ1u(0)ρ2, 1 ifu(0) > ρ2,
(3.3)
and, for anyt∈ ]0, T],
w(t)∈
{−1} ifu(t) < ρ1, [−1,1] ifρ1u(t)ρ2, {1} ifu(t) > ρ2,
(3.4)
ifu(t)=ρ1, ρ2, then w is constant in a neighbourhood oft, ifu(t)=ρ1, then w is nonincreasing in a neighbourhood oft, ifu(t)=ρ2, then w is nondecreasing in a neighbourhood oft.
(3.5)
The graph of kρ in the(u, w)-plane invades the whole rectangle[ρ1, ρ2] × [−1,1], cf.
Fig. 3. Moreover,w∈BV(0, T )for anyu∈C0([0, T]), by the argument we used forhρ. The operatorkρ is the closure ofhρ in certain function spaces which are relevant for the analysis of PDEs, cf. [10, Chap. VI]. The definition of kρ can also be justified by means of the Preisach model; see below at the end of this section.
Reformulation of the relay operator. In view of the analysis of PDEs, it is convenient to provide an alternative formulation of the completed relay operator. It is easy to see that (3.4) is equivalent to the system
(w−1)(u−ρ2)0,
(w+1)(u−ρ1)0, a.e. in]0, T[.
|w|1,
(3.6)
Fig. 3. Completed relay operator. Here the pair(u, w)can also attain any value of the rectangle [ρ1, ρ2] × [−1,1].
Ifw∈W1,1(0, T ), the dynamics (3.5) is tantamount to t
0
dw dτ udτ =
t 0
dw dτ
+ ρ2−
dw dτ
− ρ1
dτ=:1ρ(w, t) ∀t∈ ]0, T]. (3.7) In continuum dynamics and in electromagnetism a quantity like1ρ(w, t)represents the dissipated energy.
If the pair(u, w)moves along a closed loop in[0, t], then 1ρ(w, t)equals the area of the region bounded by the loop. The condition (3.7) is extended to w∈BV(0, T ), provided that we set
1ρ(w, t):=
t 0
ρ2dw+− t 0
ρ1dw− ∀t∈ ]0, T],
and interpret the latter expression as a Stieltjes integral. Notice that 1ρ(w, t)=
t 0
ρ2dw+− t
0
ρ1dw−=ρ2+ρ1
2 t 0
dw+ρ2−ρ1
2 t
0
|dw|
=ρ2+ρ1
2
w(t)−w(0)+ρ2−ρ1
2 t
0
|dw| ∀t∈ ]0, T], ∀w∈BV(0, T ), (3.8) and that the total variation,0t|dw|, equals the total mass of the Borel measure dw/dτ in C0([0, t]); i.e.,0t|dw| = dw/dτC0([0,t]).
The condition (3.4) entails that0tudw1ρ(w, t), independently from the dynamics;
(3.7) is then equivalent to the opposite inequality. Therefore the system (3.4) and (3.5) is equivalent to the confinement condition (3.6) coupled with the dissipation inequality
t 0
udw1ρ(w, t) ∀t∈ ]0, T]. (3.9)
Wheneveru∈C0([0, T])∩BV(0, T ), (3.9) also reads t
0
ud(u+w)−1 2
u(t)2−u(0)21ρ(w, t) ∀t∈ ]0, T]. (3.10)
Notice that (3.9) is also equivalent to the variational inequality t
0
(u−v)dw0 ∀v∈ [ρ1, ρ2], ∀t∈ ]0, T]. (3.11) The above formulation of the relay operator can be extended to space-distributed systems, just assuming that u(x,·)∈C0([0, T]), w(x,·)∈BV(0, T ), and (3.3), (3.6) and (3.9) hold a.e. in. Let us set Qt :=× ]0, t[for any t >0. In alternative, (3.9) may also be extended by requiring that
C0(Q)¯
u,∂w
∂τ
C0(Q¯t)
ρ2+ρ1
2
Qt
w(x, τ )−w0(x)dxdτ +ρ2−ρ1
2
∂w
∂τ
C0(Q¯t)
=:
¯
1ρ(w, t) ∀t∈ ]0, T]. (3.12)
Here we denote the total mass of the Borel measure 1ρ(w, t) in ¯ by ¯1ρ(w, t) (without the dx), and reserve the notation 1ρ(w, t)dx (with the dx) for the case in which1ρ(w(x,·), t)is Lebesgue integrable.
Preisach integral. Let µ be a finite Borel measure on the half-plane P := {ρ:=
(ρ1, ρ2)∈R2:ρ1< ρ2}, and define the completed Preisach operatorKµ:=Pkρdµ(ρ).
More precisely, for anyu∈C0([0, T])and any measurable functionξ:P→ {−1,1}, we set w˜ ∈Kµ(u, ξ ) iff there exists a measurable function w:P × ]0, T[ → [−1,1]
such that wρ ∈kρ(u, ξρ) a.e. in P × ]0, T[ (w.r.t. the product of the Preisach and Lebesgue measures), andw˜ =Pwρdµ(ρ) a.e. in]0, T[. For anyρ∈P the inclusion wρ∈kρ(u, ξρ)can be represented by (3.3), (3.6) and (3.9).
If the measureµ has no masses concentrated either in points or on lines parallel to any coordinate axis, then Kµ operates in C0([0, T]) and is continuous with respect to the uniform topology. Under further hypotheses onµ,Kµoperates in smaller spaces, or
is invertible, or fulfils other properties; for instance, ifµ0 then any hysteresis branch is nondecreasing. For these and other properties of the Preisach model see, e.g., [10, Chap. IV]. Notice thatKµ=Hµ, whenever the measureµ is such thatHµ operates in C0([0, T]).
In view of latter application, we notice that, wheneverρ2+ρ1 isµ-integrable onP, by (3.8) we have
A
1ρ(wρ, t)dµ(ρ)=
A
ρ2+ρ1
2
wρ(t)−wρ(0)dµ(ρ)
+
A
ρ2−ρ1
2 t
0
|dwρ|
dµ(ρ)
∀t∈ ]0, T], ∀measurable setA⊂P. (3.13) Finally, we briefly outline how the definition of the completed relay operator, kρ, can be justified by means of the Preisach model. According to this model each relay hρ corresponds to a Dirac measure, µ, concentrated at the point ρ ∈P. In order to approximate the relay, it is then natural to smear out that measure by a sequence of absolutely continuous measures, {µn}, having a bell-shaped density w.r.t.
to the Lebesgue measure. It is not difficult to show that, as n → ∞, the Preisach operators corresponding to the approximating measures converge to the completed relay operator,kρ, pointwise inC0([0, T]).
4. Weak formulation of the PDE for the relay operator
We shall deal with Eq. (1.1) forA:= − (associated with the homogeneous Dirichlet condition). We assume that
u0, w0∈L2(), F ∈L20, T;H−1() ,
|w0|1, w0= −1 ifu0< ρ1, w0=1 ifu0> ρ2 a.e. in,
(4.1) and provide a weak formulation of the initial- and boundary-value problem associated with Eq. (1.4).
We denote the duality pairing between H−1() and H01() by ·,·, and the trace operator byγ0.
PROBLEM 1. – To find U ∈H1(Q) and w∈ L∞(Q) such that γ0U =0 a.e. in (× {0})∪(∂× ]0, T[),|w|1 a.e. inQ, and∂w/∂t∈C0(Q)¯ . Moreover, we set
u:=∂U
∂t a.e. inQ, w(·,0):=w0 a.e. in, (4.2) and require that
Q
u0+w0−u−w ∂η
∂t + ∇U·∇η
dxdt= T
0
F, ηdt
∀η∈C1(Q), η=0 a.e. in× {T} ∪∂× ]0, T[ , (4.3) (w−1)(u−ρ2)0, (w+1)(u−ρ1)0 a.e. inQ, (4.4) 1
2
u(x, t)2+∇U (x, t)2−u0(x)2dx+
¯
1ρ(w, t) t
0
F, udτ
for a.a.t∈ ]0, T[. (4.5)
See (3.12) for the definition of¯ 1ρ(w, t).
The initial condition in (4.2) is meaningful, on account of the regularity ofw.
Interpretation. (4.3) is equivalent to the system
∂
∂t(u+w)− U=F inH−1(Q), (4.6)
(u+w)|t=0=u0+w0 inH−1(). (4.7) Differentiating (4.6) in time, we get
∂2
∂t2(u+w)− u=f inD(Q). (4.8)
A comparison of the terms of (4.6) yields ∂t∂(u+w)∈L2(0, T;H−1()); the initial condition (4.7) is then meaningful.
For a moment let us assume that u∈L2(0, T;H01()). Multiplying (4.6) by uand integrating in time, we get
t 0
∂
∂τ(u+w)−F, u
dτ +1 2
∇U (x, t)2dx=0 for a.a.t∈ ]0, T[; (4.9)
(4.5) is then equivalent to t
0
∂
∂τ(u+w), u
dτ −1 2
u(x, t)2−u0(x)2dx
¯
1ρ
w(x,·), t
for a.a.t∈ ]0, T[. (4.10)
This inequality can be compared with (3.10), which jointly with (3.6) and with the second condition of (4.2) represents the hysteresis relation
w∈kρ(u, w0) a.e. in. (4.11) Therefore we can regard Problem 1 as a weak formulation of an initial- and boundary- value problem associated to the system (4.6) and (4.11).
Finally, we notice that, on account of the equivalence between (3.9) and (3.11), in Problem 1 (4.5) might be replaced by the following variational inequality, for any v∈H1(Q)such thatρ1vρ2a.e. inQ,
1 2
u(x, t)2+∇U (x, t)2−u0(x)2dx+
u(x, t)v(x, t)−u0(x)v(x,0)dx
+ t
0
dτ
u∂v
∂τ + ∇U· ∇v
dx
t 0
F, u−vdτ for a.a.t∈ ]0, T[, (4.12) which is formally equivalent to
t 0
dτ
∂u
∂τ(u−v)+∇U·∇(u−v)
dx t
0
F, u−vdτ for a.a.t∈ ]0, T[. (4.13)
5. Existence result for the relay operator
At first we recall a result of interpolation of spaces of vector-valued functions.
LEMMA 5.1 (see, e.g., Triebel [9, p. 128]). – Let A0 and A1 be Banach spaces, subspaces of the same separated topological vector space. Let
1p0, p1<+∞, 0< θ <1, 1 p =θ 1
p0
+(1−θ ) 1 p1
. (5.1)
Denoting by(A0, A1)[θ]the complex interpolation space, we then have
Lp0(;A0), Lp1(;A1) [θ]=Lp;(A0, A1)[θ] . (5.2) This statement allows us to derive the following compensated compactness result.
LEMMA 5.2. – Letz, wand the sequences{zm},{wm}be such that zm→z weakly inL2(Q)∩H−10, T;H1() , wm→w weakly star inL∞(Q),
wmL1(;BV(0,T ))Constant. (5.3)
Then
Q
wmzmdxdt→
Q
wzdxdt. (5.4)
Proof. – Letθ, p, rbe such that
0< θ <1, 2< p <+∞, 0< r <p1, pθ +1−θ=12.
Obviously these conditions are nonempty. By Sobolev inclusion and by interpolation, we have
BV(0, T )⊂Hr(0, T ), L2(0, T ), Hr(0, T ) [θ]=Hr(1−θ )(0, T ).
Lemma 5.1 then yields
L∞(Q)∩L1;BV(0, T ) ⊂L∞(Q)∩L1;Hr(0, T )
⊂Lp;L2(0, T ) , L1;Hr(0, T ) [θ]
=L2;L2(0, T ), Hr(0, T ) [θ] =L2;Hr(1−θ )(0, T ) . Note that 0< r(1 − θ ) < p1 < 12. Again by interpolation, see, e.g., Lions and Magenes [7, Chap. IV], we also have
L2(Q)∩H−10, T;H1() =L2;L2(0, T ) ∩H1;H−1(0, T )
⊂L2;H−r(1−θ )(0, T ) =L2;Hr(1−θ )(0, T ) with compact injection. Therefore (5.4) holds.
THEOREM 5.3 (Existence). – If(4.1)is fulfilled andF ∈W1,1(0, T;H−1()), then there exists a solution(U, w)of Problem 1 such that
U∈W1,∞0, T;L2() ∩L∞0, T;H01() . (5.5) Proof. – (i) Approximation. Let us fix anym∈N, setk:=T /m,u0m:=u0,wm0 :=w0, F1mn := 1k(nnk−1)kF1(·, t)dta.e. in,F2mn :=F2(nk),Fmn:=F1mn +F2mn forn=1, . . . , m, and
Gρ(v, ξ ):=
{−1} ifv < ρ1, [−1, ξ] ifv=ρ1, {ξ} ifρ1< v < ρ2, [ξ ,1] ifv=ρ2, {1} ifv > ρ2,
∀(v, ξ )∈R× [−1,1], cf. Fig. 4. (5.6)
We are now able to introduce a time-discretization scheme of implicit type for our problem.
Fig. 4. Graph of the multi-valued functionGρ(·, ξ ), for a fixedξ ∈ [−1,1].
PROBLEM1m. – To findunm∈H01()andwmn ∈L∞()forn=1, . . . , m, such that wnm∈Gρ
unm, wmn−1 a.e. in, forn=1, . . . , m, (5.7)
unm−unm−1
k +wnm−wmn−1
k −k
n j=1
ujm=Fmn inH−1(), forn=1, . . . , m. (5.8) For anynand for a.a. x∈, the maximal monotone function Gρ(·, wnm−1(x))is the subdifferential of a lower semicontinuous and convex function,gρ(·, x), which depends measurably onx. The functional
Jmn:v →
v2 2 +gρ
v(x), x +k2
2|∇v|2+k2
n−1
j=1
∇unm−1· ∇v−unm−1+wmn−1 v
dx
−kFmn, v (5.9)
is (strictly) convex, lower semicontinuous and coercive on H01(). Hence it has a (unique) minimizer unm, and ∂Jmn(unm)!0 in H−1(). This inclusion is equivalent to the system (5.7) and (5.8).
(ii) A priori estimates. Preliminarly, for any family{vmn}n=1,...,m of functions→R let us denote by vm the time interpolate of vm0 :=v0, vm1, . . . , vmm a.e. in , and set
¯
vm(·, t):=vmn a.e. in, if(n−1)k < tnk, forn=1, . . . , m.
Let us multiply Eq. (5.8) by kunm, and sum forn=0, . . . , , for any∈ {1, . . . , m}. By (5.7) we have
n=1
wmn −wnm−1 unm n=1
wmn −wmn−1 +ρ2−wnm−wnm−1 −ρ1
=1ρ(wm, k)
a.e. in, for=1, . . . , m. (5.10)
SettingUmn :=knj=1ujma.e. inforn=1, . . . , m, we then get 1
2
(um)2+∇Um2−u0 2dx+
1ρ(wm, k)dxk n=1
Fmn, unm
for=1, . . . , m. (5.11)
A standard calculation then yields
UmW1,∞(0,T;L2())∩L∞(0,T;H01()),1ρ(wm,·)L∞(0,T;L1())C1. (5.12) (Throughout this paper byC1, C2, . . .we denote suitable positive constants independent ofm.) Hence by (3.8) we get
wmL1(;BV(0,T ))C2. (5.13)
(iii) Limit procedure. By the above estimates, there existU, wsuch that, asm→ ∞ along a suitable sequence,
Um→U weakly star inW1,∞0, T;L2() ∩L∞0, T;H01() , (5.14)
wm→w weakly star inL∞(Q), (5.15)
∂wm
∂t →∂w
∂t weakly star inC0(Q)¯ . (5.16) SettingUm:=0tu¯m(·, τ )dτ a.e. inQ, (5.8) reads
∂
∂t(um+wm)− Um= ¯Fm inH−1(), a.e. in]0, T[, (5.17) and passing to the limit we get (4.6), whence (4.3) follows on account of the initial condition (4.7). (5.7) entails
(w¯m−1)(u¯m−ρ2)0, (w¯m+1)(u¯m−ρ1)0 a.e. inQ, (5.18) whence, for any nonnegativeϕ∈D(Q),
Q
(w¯m−1)(u¯m−ρ2)ϕ(x, t)dxdt0,
Q
(w¯m+1)(u¯m−ρ1)ϕ(x, t)dxdt0.
(5.19) Applying Lemma 5.1 to the sequences{zm} := { ¯umϕ}and{wm}, we have
Q
wmu¯mϕdxdt→
Q
wuϕdxdt.
Hence
Q
¯
wmu¯mϕdxdt→
Q
wuϕdxdt, as ¯wm−wmL2(Q)→0. Passing to the limit in (5.19) we then get
Q
(w−1)(u−ρ2)ϕ(x, t)dxdt0,
Q
(w+1)(u−ρ1)ϕ(x, t)dxdt0, for any nonnegativeϕ∈D(Q); this is equivalent to (4.4). (5.11) yields
1 2
u¯2m+∇ ¯Um2−(u0)2dx+
1ρ(wm, t)dx t 0
F¯m,u¯m
dτ
for a.a.t∈ ]0, T[. (5.20)
Integrating in]0, T[and passing to the inferior limit asm→ ∞, by lower semicontinuity we finally get an inequality equivalent to (4.5).
6. Weak formulation of the PDE for the Preisach operator We assume that
µis a finite positive Borel measure onP:=ρ:=(ρ1, ρ2)∈R2:ρ1< ρ2
, (6.1) and equip×P and Q×P with the product of the Lebesgue and Preisach measures, LN×µ(L(N+1)×µ, respectively). We also set
¯ ×P
1ρ(wρ, t):=
P
dµ(ρ)ρ2+ρ1
2
Qt
wρ(x, τ )−wρ0(x)dxdτ
+ρ2−ρ1
2
∂wρ
∂τ
C0(Q¯t×P)
∀t∈ ]0, T], (6.2) and reserve the notation×P1ρ(wρ, t)dxdµ(ρ)for the case in which1ρ(wρ(x,·), t) is Lebesgue integrable w.r.t.LN×µ. We assume that
u0∈L2(), F ∈L20, T;H−1() , w0∈L∞(×P),
|wρ0|1, w0ρ= −1 ifu0< ρ1, wρ0=1 ifu0> ρ2 a.e. in×P,
(6.3) and provide a weak formulation of the initial- and boundary-value problem associated with (1.4), forF equal to the Preisach operator.
PROBLEM 2. – To findU∈H1(Q)and w∈L∞(Q×P)such thatγ0U=0 a.e. in (× {0})∪(∂× ]0, T[),|w|1 a.e. inQ×P, and∂w/∂τ∈C0(Q¯ ×P). Moreover, we set
u:=∂U
∂t a.e. inQ, w˜0:=
P
wρ0dµ(ρ), w˜ :=
P
wρdµ(ρ)a.e. inQ, (6.4)
wρ(·,0):=wρ0 a.e. in×P, w˜0:=
P
w0ρdµ(ρ) a.e. in, (6.5) and require that
Q
(u0+ ˜w0−u− ˜w)∂η
∂t + ∇U· ∇η
dxdt= T 0
F, ηdt
∀η∈C1(Q), η¯ =0 a.e. in× {T} ∪∂× ]0, T[ , (6.6) (wρ−1)(u−ρ2)0, (wρ+1)(u−ρ1)0 a.e. inQ×P, (6.7) 1
2
u(x, t)2+∇U (x, t)2−u0(x)2dx+
¯ ×P
1ρ(wρ, t) t
0
F, udτ
for a.a.t∈ ]0, T[. (6.8)
Interpretation. (6.6) is equivalent to the system
∂
∂t(u+ ˜w)− U=F inH−1(Q), (6.9) (u+ ˜w)|t=0=u0+ ˜w0 inH−1(). (6.10) Similarly to what we saw in the interpretation of Problem 1, ifu∈L2(0, T;H01()) then (6.6) yields (4.8), here withw˜ in place ofw. (6.8) is then equivalent to
t 0
∂
∂τ(u+ ˜w), u
dτ1 2
u(x, t)2−u0(x)2dx+
¯ ×P
1ρ(wρ, t)
for a.a.t∈ ]0, T[. (6.11)
This can be regarded as a weak formulation of the inequality (3.10) a.e. in×P; the opposite inequality follows from the confinement condition (3.6). (By displaying the dx in the integral expression, we still distinguish the Lebesgue integral onfrom the total mass of a Borel measure, cf. Section 3.) (6.7) and (6.8) then account for the hysteresis relation
˜
w∈Kµ(u, w0):=
P
kρ
u, wρ0 dµ(ρ) in]0, T[, a.e. in. (6.12)
Therefore Problem 2 can be regarded as a weak formulation of an initial- and boundary-value problem associated to the system (6.9) and (6.12).
7. Existence result for the Preisach operator
We still assume thatµ is a finite positive Borel measure on the Preisach half-plane P := {ρ :=(ρ1, ρ2)∈R2: ρ1< ρ2}. The following extension of Lemma 5.2 can be justified by the same argument.
LEMMA 7.1. – . Let A be any measurable subset of P. Assume that z, w and the sequences{zm},{wm}are such that
zm→z weakly inL2(Q)∩H−10, T;H1() , uniformly w.r.t.ρ inA,
wm→w weakly star inL∞(Q×A), (7.1)
wmL1(×A;BV(0,T ))Constant.
Then
A
dµ
Q
wmzmdxdt→
A
dµ
Q
wzdxdt. (7.2)
THEOREM 7.2. – Assume that(6.1)and (6.3) hold, thatF ∈W1,1(0, T;H−1()), and that
ρ1+ρ2∈L1(P;µ). (7.3)
Then there exists a solution (U, w)of Problem 2 such that
U∈W1,∞0, T;L2() ∩L∞0, T;H01() . (7.4)
