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Nonlinear Analysis 63 (2005) e199e207www.elsevier.com/locate/na

Maximal monotone model with history termJrme Bastiena, Claude-Henri Lamarqueb,

aLaboratoire Mcatronique 3M, quipe daccueil A 3318, Universit de Technologie de Belfort-Montbliard,90010 Belfort cedex, France

bURA 1652 CNRS, Dpartement Gnie Civil et Btiment, Laboratoire Gomatriaux, cole Nationale desTravaux Publics de lEtat, Rue Maurice Audin, 69518 Vaulx-en-Velin Cedex, France

Abstract

Models involving maximal monotone terms and history (delay) term are considered in a math-ematical and numerical point of view. Mechanical example of elasto-plastic material with creep ispresented. 2005 Elsevier Ltd. All rights reserved.

Keywords: Differential inclusion; Delay; Maximal monotone model; Numerical scheme

1. Introduction and assumptions

Models governed by differential inclusions have been studied in classical works byBrezis[35] (see also [10]). Numerical analysis has been considered in a paper by Lippold [8] forsubdifferential operators. In previous works results have been obtained for models governedby general differential inclusions. In [1,2], numerical scheme for differential inclusionswith maximal monotone operators have been studied including error estimates. In [6],these results have been extended to the case of models governed by differential inclusionsinvolving delay term. Initial data have been replaced by initial given functions. Delay term isexpressed as function of nite number of passed values of unknown function. In [7], resultshave been obtained for delay term expressed as convolution but for unknown functionsfrom interval of R to Rn. Here a general case is examined with unknown functions froman interval of R to innite dimensional Hilbert space and again delay term expressed as

Corresponding author.E-mail address: claude.lamarque@entpe.fr (C.-H. Lamarque).

0362-546X/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.03.103

e200 J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207

convolution. In this paper, results of [1,2,6,7] are extended: only mains results and ideas ofproofs are given. The complete proof will be given in a future work.

We use the same framework as in [2], which is briey recalled hereafter.Let be a positive number and T be a strictly positive number. Let V, H and V be three

separable Hilbert spaces, equipped with norms and scalar products denoted by ., ((., .)),|.|, (., .), . and ((., .)). We denote by ., . the duality bracket between V and V . Weassume that these three spaces constitute a Gelfand triple, i.e. V H V , where wedenote by a dense and continuous inclusion. V is a dense subspace of H. Let A be amaximal monotone operator from V to V , with non empty domain D(A); its properties aredescribed in detail in [35,10]. Let B be a Lipschitz continuous and coercive mapping fromV to V , i.e.

l0: x, y V, B(x) B(y) lx y, (1a)> 0: x, y V, B(x) B(y), x yx y2. (1b)

Let f be a function from [0, T ] H to V , Lipschitz continuous with respect to its secondargument and whose derivative maps the bounded sets of L2(0, T ;V ) into bounded sets ofL2(0, T ;V ), i.e.

L0: t [0, T ], x1, x2 H, f (t, x1) f (t, x2)L|x1 x2|, (1c)and

R0, (R) = sup{ft (., v)

L2(0,T ;V )

: vL2(0,T ;V )R}

< + . (1d)

Let z be a function from [, 0] to V satisfyingz C0([, 0];V ) and z L(, 0;H). (1e)

We setu0=z(0). LetL be the history term, i.e. it is a function from [0, T ]L2(, T ;H)to V such that, for all t [0, T ], for all u L2(, T ;H),L(t, u) is depending only ont and on values {u(s) : s [, t]} from u. We make the following regularity assumption:

A(u0): f (0, u0) +L(0, z) + + B(u0) H . (1f)Moreover, we make two regularity assumptions onL: we assume that there exits C suchthat, for all u1, u2 L2(, T ;H)

t [0, T ], t0

L(s, u1) L(s, u2)2 dsC t

|u1(s) u2(s)|2 ds. (1g)

We assume moreover that

v L2([, T ];H), t

(L(., v)) L2(0, T ;V ). (1h)

For all T ]0, T ], we consider the spaceWT = {u C0([, T ];V ) : u L(, T ;H)}. (2)

J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207 e201

In this paper, we study the differential inclusion

u(t) + A(u(t)) + B(u(t)) + f (t, u(t)) +L(t, u) 0, a.e. on ]0, T [, (3a)t [, 0], u(t) = z(t), (3b)

where u belongs to WT . Since we have C0([, T ];V ) C0([, T ];H) L2(,T ;H),L(., u) is correctly dened.

2. Existence and uniqueness results

Existence and uniqueness results are based on previous results [1,2]; thanks to Proposition2.6 of [2], to two technical lemmas and xed point theorem in Banach space, existence anduniqueness of the solution of (3) can be established. Technical lemmas based on Gronwallslemma are now provided:

Lemma 2.1. Let T ]0, T ] and v1, v2 C0([, T ];H). Under hypotheses (1), if u1and u2 belong toWT and satisfy, for all i {1, 2}

ui (t) + A(ui(t)) + B(ui(t)) + f (t, ui(t)) +L(t, vi) 0, a.e. on ]0, T [,t [, 0], ui(t) = vi(t),

then, we have u1 u2C0([0,T ];H)

2CT eT L2/v1 v2C0([,T ];H).

Lemma 2.2. Let T ]0, T ]. Under hypotheses (1), there exists M such that, if u belongsto WT and satises (3a) on ]0, T [ and (3b), then, we have for all t [0, T ], |u(t) u0|2T/Me(L2/+C)T .

Now, we can give the main results of existence and uniqueness:

Proposition 2.3. Assume that (1) holds. There exists a unique solution u of (3) inC0([, T ], V ) such that u belongs to L(, T ;H).

Main Ideas of the Proof. The proof has three steps:

Uniqueness of u comes from Lemma 2.1. Local existence is proved thanks to Lemma 2.1 and Proposition 2.6 of [2], by using the

xed point theorem. We choose T ]0, T ] such that 2CT eT L2/ < 1. For this valueof T , consider the Banach space E = C0([, T ];H), andF the closed subset of Edened byF = {v E : t [, 0], v(t) = z(t)}. Let v F. Thus, we considerthe application dened as follows: for each function v F, we consider u, the uniquesolution of{ u(t) + A(u(t)) + B(u(t)) F(t, u(t)), a.e. on ]0, T [,

u(0) = u0,

e202 J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207

where for all (t, w) [0, T ] H , F(t, w) = f (t, w) L(t, v). Existence anduniqueness of u comes from Proposition 2.6 of [2]. We dene u by: for all t [, 0],u(t)=z(t) and for all t [0, T ], u(t)= u(t) and we set(v)=u. Thus, is a Lipschitzcontinuous function fromF toF, whose Lipschitz constant is strictly smaller than 1.The xed point theorem in Banach space E provides the local existence of solution of(3).

In fact, this solution is dened on the whole interval [0, T ]. Indeed, according to Lemma2.2, the solution u is uniformly bounded by

2T/Me(L2/+C)T + |u0| on each subin-

terval [0, T ] of [0, T ]. We can consider the maximal solution, which is extended to theinterval [0, T ].

3. The numerical scheme

The proof of convergence and order is very close to the results of [2].We assume that there exists N and M satisfying

N1, h = TN

and{> 0 h =

M, M1,

= 0 M = 0. (4)

Thus, we set for all p {M, . . . , N}, tp = hp. For all h satisfying (4), we now discretizethe termL of (3a) by a discrete history termLh: for all h> 0, for all p {0, . . . , N},for all u L2(, T ;H), we assume that there existsLh(tp, u) V , depending only ontp and on the values of u in times tM, tM+1, . . . , tp1, tp. We set

p {0, . . . , N 1}, t [tp, tp+1[, Lh(t, u) =Lh(tp, u). (5)Thus,Lh(t, u) is a function dened from [0, T ] L2(, T ;H) to V such that, for allt [0, T ], for all u L2(, T ;H), Lh(t, u) is depending only on t and on values{u(s) : s [, t]} from u. Finally, we assume that

suph[0,max(,T )]

Lh(0, u0) < + (6a)

and there existsD such that, for all h> 0, for all function w of L2(, T ;H), constant oneach subinterval [tp, tp+1[ of [, T ]

Lh(., w) L(., w)L2(0,T ;V )hDwL2(,T ;H) (6b)and

p {1, . . . , N}, Lh(tp, w) Lh(tp1, w)hDwL(,T ;H). (6c)We are now able to discretize (3): ifUM,UM+1, . . . , UN1, UN are given, we denote

by uh the linear interpolation of the Ups at tp for p {M, . . . , N}. As in [6,7], we dene(Un)Mn0 VM+1 by

n {M, . . . , 0}, Un = z(nh). (7a)

J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207 e203

Let n belong to {0, . . . , N 1}; We assume that UM,UM+1, . . . , Un1, Un are givenin V. As in [2,7], we dene Un+1 belonging to V by

Un+1 Unh

+ A(Un+1) + B(Un+1) + f (tn, Un) +Lh(tn, uh) 0. (7b)We recall (see [2]) that this scheme possesses a unique solution: indeed, A is maximalmonotone and B is continuous and coercive; according to Zeidler [10], if we denote by jthe injection V V , then, for all > 0, the operator (j + A + B)1 is dened on allof V and single-valued from V to V; thus, (7b) is equivalent to

Un+1 = (j + hA + hB)1(hf (tn, Un) hLh(tn, uh) + j (Un)). (8)As in [2] (see Lemma 2.2), let us give now estimates on uh and uh independently of h.

Lemma 3.1. Under assumptions (1) and (6), there exists a constant M such that for allh> 0

uh u0C0([0,T ],H) + uh u0L2(0,T ;V ) + uhL(0,T ;H) + uhL2(0,T ;V )M .

We now give the main results of this paper. uh is dened by numerical schema (7).

Proposition 3.2. Under assumptions (1) and (6), there exists a constant C1 such that

h> 0, maxt[0,T ]

(|u(t) uh(t)|2 +

t0

u(s) uh(s)2 ds)1/2

C1h1/2. (9)

Main Ideas of the Proof. The proof is very close to the proof of Proposition 2.5 of [2]:we consider the difference between (3) and numerical (7b).

Proposition 3.3. Under assumptions (1) and (6), and if K is a non empty closed convexsubset of V and A is the maximal monotone operator K , equal to the subdifferential ofthe indicatrix of the convex K , then there exists a constant C2 such that

h> 0, maxt[0,T ]

(|u(t) uh(t)|2 +

t0

u(s) uh(s)2 ds)1/2

C2h. (10)

Main Ideas of the Proof. It is similar to the proof of Proposition 3.1 of [2].

4. Other results when V = H = V

As in Section 4 of [2], we quickly give existence, uniqueness and convergence results ifV = H = V . Let us assume B = 0 and = 0. Since proofs are similar to those given inSection 4 of [2], we only give results. We study convergence of numerical scheme denedby (7a) and

n {0, . . . , N 1}, Un+1 Un

h+ A(Un+1) + f (tn, Un) +Lh(tp, uh) 0

e204 J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207

to unique solution of (3b) andu(t) + A(u(t)) + f (t, u(t)) +L(t, u) 0, a.e. on ]0, T [. (11)

Proposition 4.1. There exists a unique solution u of (3b)(11) in W 1,([, T ], H).

Proposition 4.2. There exists a constant D1 such that

h> 0, maxt[0,T ] |u(t) uh(t)|D1h

1/2. (12)

Proposition 4.3. If K is a non empty closed convex subset of V and A is the maximalmonotone operator K , equal to the subdifferential of the indicatrix of the convex K , thenthere exists a constant D2 such that

h> 0, maxt[0,T ] |u(t) uh(t)|D2h. (13)

5. Applications

5.1. Previous frameworks

In this section, we prove that propositions of existence, uniqueness and convergence ofnumerical schema cover the previous framework of [1,2,6,7].

We cannot directly apply theoretical results of previous section to nd again the frame-work of [6] (with a delay term G(u(t )), in Hilbert space H); but, they can be easilyextended for a Gelfand triple V H V .

The framework of [1,2] is obtained by choosingL= 0 and = 0.The framework of [7] can be so obtained: we dene a Gelfand triple V H V ;

we choose > 0 and B, f, z and u0 satisfying (1a)(1f). We assume thatH is a functionfrom [0, T ] toL(H, V ), the Banach space of linear continuous functions from H to V ,equipped with the associated norm |||.|||; we set, by using the usual abuse of notations,

s [0, ], u L2(, T ;H), L(s, u) = 0H(x)u(s x) dx (14a)

and we assume thatFor all t [0, ], H(t) is linear continuous from H to V ,

with norm |||H(t)|||; (14b)The function t |||H(t)||| belongs to L2(0, ); (14c)For all t [0, ], dH/dt (t) exists

and the function t |||dH(t)/dt ||| belongs to L2(0, ). (14d)The framework of [7] can by obtained by choosing V = H = V = Rn.We can easily verify thatH is correctly dened and that assumptions (14) onH imply

that assumptions (1g) and (1h) holds.

J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207 e205

As in [7] (see (3.7)), we set, for all w L2(, T ;H), for all p {0, . . . , N}

Lh(tp, w) = hM1l=0H(lh)w(tp lh). (15)

We can easily prove that (6) holds.

5.2. Another example

In this section, we give an abstract example of application in the spirit of [9, Example66.1, pp. 348361]: slow deformation process, with a creep term. We only give an abstractframework and use theoretical results of this paper. The complete proof will be done in afuture work. We consider a model of the slow processes for elasto-viscoplastic bodies witha linear hardening. Moreover, we add a creep term, given of the form of a convolution term.LetT and be two non negative numbers. The observable stress tensor can be decomposedin two terms, q and r, the plastic stress tensor

(t) = q(t) + r(t). (16a)The strain tensor is the sum of three terms

(t) = e(t) + p(t) +F(t), (16b)where e and p are, respectively, the elastic part and the plastic part of andF a creep term.The viscoplastic constitutive law is

p(t) F(q(t)), (16c)where is the subdifferential of F, which is the plastic potential. The linear elastic consti-tutive laws are

(t) = Ae(t), r(t) = Bp(t). (16d)The linear relation between displacement and linearized strain tensor and the equilibriumcondition for the stress tensor and the outer force K are

(t) = Du(t), D(t) = K(t). (16e)The creep term is given under the form

F(t) =J(t)(t) + t

k(t, s)(s) ds, (16f)

whereJ and k are known. Moreover, we assume that

s [, 0], (s) = z(s), (16g)where z is given. The initial conditions at time t = 0 are

u(0)=u0, (0)=0, p(0)=p0, (0)=0, q(0)=q0, F(0)=F0. (16h)

e206 J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207

Let U and be two Hilbert spaces and their dual denoted by U and . We assume thatU is the space of displacements u, U the space of outer forces K, the space of strains and the space of stresses . We assume nally that F: ] ,] is convex,lower semicontinuous and proper. A and B: are linear, continuous, symmetric andstrongly positive;D:U is linear and continuous and satises the generalized inequalityof Korn: there exits d > 0 such that for all u U

DuduU . (16i)We can choose V = H = V = U , so that the results of Section 4 can be applied.As in [9, pp. 351352], we can eliminate some auxiliary functions of (16a) through (16f)

to obtain problem only related to u, p and q

ADu(t) (I + AJ)q(t) (A + B + AJB)p(t) G(t) = 0,D(q + Bq) = K(t),p(t) F(q(t)) 0,

where G(t) = A t kt (t, s)(q(s) + Bp(s)) ds + Ak(t, t)(q(t) + Bp(t)).By setting X = (u, p, q), this problem can be rewritten under the formAX(t) + A(X(t)) + f (t, X(t)) +L(t, X) 0 (17)

with initial conditions similar to (3b). As in the proof of [9, pp. 355358], we can provethatA is invertible linear operator and (17) can be rewritten under the form

X(t) +A1A(X(t)) +A1f (t, X(t)) +A1L(t, X) 0, a.e. on ]0, T [with initial conditions: for all t [, 0],X(t)=X0(t). SinceA is strongly positive,A1Aismaximalmonotone operator andwecanuse existence, uniqueness and convergence resultsof Section 4.

6. Conclusions

We presented theoretical results for maximal monotone model with history term. Nu-merical scheme has been constructed and studied. Abstract mechanical example has beengiven.

References

[1] J. Bastien, M. Schatzman, Schma numrique pour des inclusions diffrentielles avec terme maximalmonotone, C. R. Acad. Sci. Paris Sr. I Math. 330 (7) (2000) 611615.

[2] J. Bastien, M. Schatzman, Numerical precision for differential inclusions with uniqueness, M2AN Math.Model. Numer. Anal. 36 (3) (2002) 427460.

[3] H. Brezis, Perturbations non linaires doprateurs maximaux monotones, C. R. Acad. Sci. Paris Sr. AB269 (1969) A566A569.

[4] H. Brezis, Problmes unilatraux, J. Math. Pure. Appl. 51 (1972) 1168.

J. Bastien, C.-H. Lamarque / Nonlinear Analysis 63 (2005) e199e207 e207

[5] H. Brezis, Oprateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,North-Holland Mathematics Studies, No. 5. Notas de Matemtica, vol. 50, North-Holland,Amsterdam, 1973.

[6] C.-H. Lamarque, J. Bastien, M. Holland, Study of a maximal monotone model with a delay term, SIAM J.Numer. Anal. 41 (4) (2003) 12861300.

[7] C.-H. Lamarque, J. Bastien, M. Holland, Maximal monotone model with delay term of convolution, Math.Problems Eng. (2005), to appear.

[8] G. Lippold, Error estimates for the implicit Euler approximation of an evolution inequality, Nonlinear Anal.15 (11) (1990) 10771089.

[9] E. Zeidler, Nonlinear FunctionalAnalysis and itsApplications, vol. IV,Applications toMathematical Physics,Springer, NewYork, 1988 (Translated from German by the author and L.F. Boron).

[10] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Nonlinear Monotone Operators,Springer, NewYork, 1990 (Translated from German by the author and L.F. Boron).

Maximal monotone model with history termIntroduction and assumptionsExistence and uniqueness resultsThe numerical schemeOther results when V=2pt=H=2pt=VApplicationsPrevious frameworksAnother example

ConclusionsReferences

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