DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.1815DYNAMICAL SYSTEMS SERIES BVolume 19, Number 7, September 2014 pp. 1815–1835

VISCOELASTIC FLUIDS: FREE ENERGIES, DIFFERENTIAL

PROBLEMS AND ASYMPTOTIC BEHAVIOUR

Giovambattista Amendola

Dipartimento di MatematicaUniversita di Pisa, Pisa, Italy

Sandra Carillo

Dipartimento di Scienze di Base e Applicate

per l’Ingegneria - Sezione MatematicaSapienza Universita di Roma, Rome, Italy

John Murrough Golden

School of Mathematical Sciences

Dublin Institute of Technology, Dublin, Ireland

Adele Manes

Dipartimento di Matematica

Universita di Pisa, Pisa, Italy

Dedicated to Mauro Fabrizio on his 70th Birthday.

Abstract. Some expressions for the free energy in the case of incompressible

viscoelastic fluids are given. These are derived from free energies already in-

troduced for other viscoelastic materials, adapted to incompressible fluids. Anew free energy is given in terms of the minimal state descriptor. The internal

dissipations related to these different functionals are also derived. Two equiv-

alent expressions for the minimum free energy are given, one in terms of thehistory of strain and the other in terms of the minimal state variable. This

latter quantity is also used to prove a theorem of existence and uniqueness

of solutions to initial boundary value problems for incompressible fluids. Fi-nally, the evolution of the system is described in terms of a strongly continuoussemigroup of linear contraction operators on a suitable Hilbert space. Thus, a

theorem of existence and uniqueness of solutions admitted by such an evolutionproblem is proved.

1. Introduction. Materials with memory typically have many free energy func-tionals associated with them. In recent years, various authors have studied theproblem of finding explicit formulae for the maximum recoverable work, that is, themaximum work which can be obtained from a specific state of a linear viscoelasticmaterial, under isothermal conditions [5, 6, 7, 12, 13, 14, 15, 17, 18, 20, 21, 23].This quantity is equal to the isothermal minimum free energy associated with thegiven viscoelastic state. A general closed expression for the minimum free energy,in terms of Fourier-transformed quantities, was given for compressible viscoelastic

2010 Mathematics Subject Classification. Primary: 74D05, 76A10, 35B40; Secondary: 30E20.Key words and phrases. Viscoelastic fluid, Fabrizio free energy, minimum free energy, materials

with memory, asymptotic behavior of solutions.

1815

1816 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

fluids in [16]. Also, expressions for other free energies have been proposed (see, forexample [24, 25]).

Here, we seek to derive analogous results for incompressible viscoelastic fluids.Their characteristic constitutive equation gives the stress tensor in terms of the rel-ative strain history, by means of a linear functional and also the isotropic stress dueto the incompressibility constraint. The restrictions imposed by Thermodynam-ics on the constitutive equation are derived in [19], where also a study of existence,uniqueness and stability of solutions to boundary initial value problems is presented.Some results for these fluids are given in [1, 3].

Other free energy functionals, already introduced in [15, 18, 23] for example, areadapted to these fluids; in addition, expressions for the rate of internal dissipationrelated to the different forms are derived. Formulae for the minimum free energyare given in terms of the relative strain history and also in terms of the variable It,which characterizes the minimal state of the fluid.

Using the quantity It, a theorem on existence and uniqueness of solutions isproved. Finally, energy exponential decay is demonstrated. It should be noted thatthe quantity It turns out to be a key tool to prove existence, uniqueness as well asexponential decay of solutions also in the framework of linear rigid heat conductorswith memory [8, 9, 10].

The layout of this paper is as follows. In Section 2, the basic equations are writtendown and the constraint imposed by thermodynamics is considered. In Section 3,definitions of states and processes are given together with the notion of equivalentstates. In Section 4, the thermodynamic laws, together with the properties whichdefine free energies, in particular those of the minimum free energy, are considered.Section 5 is devoted to deriving expressions for the work in terms of relative strainhistories. Then, forms of the free energy and, in particular, the Fabrizio free energyψF are given, together with the corresponding rates of internal dissipation. InSection 6, explicit formulae for the minimum free energy are presented. Initial andboundary value problems for incompressible viscoelastic fluids are considered inSection 7. Existence and uniqueness of solutions are proved and energy exponentialdecay is shown.

2. Basic equations. Viscoelastic fluids, assumed to be isotropic and incompress-ible, with constitutive equations given by

T(x, t) = −p(x, t)I + 2

∫ +∞

0

µ′(x, s)Etr(x, s)ds. (1)

are studied in this work. Thus, the symmetric stress tensor T is determined by the

infinitesimal strain E = 12

[∇u + (∇u)

T], where u is the displacement, through its

relative history Etr, defined by

Etr(x, s) := Et(x, s)− E(x, t) ∀s ∈ (0,+∞) = RI ++, (2)

where E(x, t) ∈ Sym and Et(x, s) := E(x, t − s), ∀s ∈ RI ++ denote, respectively,the instantaneous or present value of the strain and its past history; p is the scalarfunction, which yields the constitutively indeterminate reaction pressure due tothe constraint of incompressibility, I is the identity second order tensor, Ω ⊂ RI 3

denotes a smooth bounded domain, with smooth boundary ∂Ω, and x ∈ Ω ⊂ RI 3

is the position within Ω. The dependence on the spatial variable x is omitted, if

VISCOELASTIC FLUIDS & FREE ENERGIES 1817

unneeded. The memory kernel µ′ ∈ L1(RI +)∩L2(R+) is a material function, termedthe shear relaxation function. The quantity

µ(s) = −∫ +∞

s

µ′(ξ)dξ ∀ s ∈ [0,+∞) = RI + (3)

belongs to L1(RI +). It must satisfy constraints imposed by thermodynamics [19, 4],so that the relation

µc(ω) > 0 ∀ω ∈ RI (4)

holds, where µc(ω) is the cosine transform of µ(s), or the real part of µF (ω)1. Inparticular, ∫ +∞

0

µ(s)ds = µc(0) 6= 0. (5)

The regularity requirement on µ′, recalling Fourier transform properties, impliesthat the Fourier transform of µ′(s), i.e. µ′F (ω) = µ′c(ω) − iµ′s(ω), belongs toL2(RI ),∀ω ∈ RI . Moreover, the assumption of analyticity of the Fourier transformson RI yields that µ′F (z) is analytic in the lower half complex plane C−. Observethat µ′c(ω) is an even function of ω, while µ′s(ω), which is an odd function, vanisheslinearly near the origin.

Under the hypothesis µ′ ∈ L1(RI +) and µ′′ ∈ L1(RI +), the following relationshold:

µ′s(ω) = −ωµc(ω), ωµ′s(ω) = µ′(0) + µ′′c (ω). (6)

Also, the constitutive equation, on application of Fourier transforms, can be writtenas [21]

T(t) = T(t) + p(t)I =2i

π

∫ +∞

−∞µ′s(ω)Etr+(ω)dω

= −2i

π

∫ +∞

−∞ωµc(ω)Etr+(ω)dω. (7)

Here T represents the extra stress, that is the stress due to the relative strainhistory Etr(s), whose Fourier’s transform is denoted by Etr+(ω). We also introducethe notation

T(Etr) = T(t), T(Etr) = T(t). (8)

when it is necessary to specify the relative strain history generating the extra andtotal stress.

3. States, equivalent states and processes. Relation (1) allows us to considerthe fluid as a simple material, and therefore, its thermodynamic behaviour can bedescribed via the concepts of state and process [26].

The state σ of the fluid is represented by the relative strain history, that is

σ(t) = Etr(s), ∀s ∈ RI ++. (9)

We denote the state space by Σ, which is the set of all possible states of the fluid.A mechanical process is a piecewise continuous map P : [0, d)→ Lin, the set of

the second order tensors, given by

P (τ) = LP (τ), ∀τ ∈ [0, d). (10)

Thus, the process P denotes the values of the velocity gradient L = ∇v over thetime interval [0, d), where d is the duration of the process. Let Π be the set of all

1The convention used here for denoting Fourier transforms is that specified in [5], for example.

1818 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

mechanical processes. Given any process P ∈ Π of duration d, its restriction toany [τ1, τ2) ⊂ [0, d) is a process denoted as P[τ1,τ2) ∈ Π; also the composition of theprocesses Pj ∈ Π of duration dj (j = 1, 2), defined by

P1 ∗ P2(τ) =

P1(τ) ∀τ ∈ [0, d1)P2(τ − d1) ∀τ ∈ [d1, d1 + d2)

, (11)

belongs to Π. Finally, we consider the state transition function ρ : Σ × Π → Σ,which maps any initial state σi ∈ Σ and each process P ∈ Π into the final stateσf = ρ(σi, P ). Therefore, if σi = σ(0) is given together with a process P[0,τ), thefinal state is σ(τ) = ρ(σ(0), P[0,τ)); in particular, if the final state coincides withthe initial one, namely σ(d) = ρ(σ(0), P ) = σ(0), then the pair (σ, P ) is termed acycle.

Since (1) is determined by the strain history, a process P of duration d, definedby (10), can be expressed in terms of D = (L+LT )/2, the strain rate tensor. Hence,

we can introduce the function DP : [0, d)→ Sym, represented by DP (τ) = EP (τ),the derivative of the strain tensor. Thus, henceforth we consider

P (τ) = EP (τ) ∀τ ∈ [0, d). (12)

A process P ∈ Π can be applied at any time t ≥ 0. Two cases arise dependingon whether or not the initial time is assumed to be the zero time.

Let t > 0 denote the time when P is applied to the state. Then, σ(t) = Etr(s)and EP (τ) = E(t+ τ). The extra stress, introduced in (7), can be derived from (1)by considering the integral over (0,+∞) as the sum of two integrals, the first from0 to τ , and the second from τ to +∞. Integration by parts in the first of them,gives

T(t+ τ) = T(t+ τ) + p(t+ τ) I = 2

∫ τ

0

µ(s)Et+τ (s)ds− It(τ,Etr), (13)

wherein

It(τ,Etr) := −2

∫ +∞

0

µ′(ξ + τ)Etr(ξ)dξ, τ ≥ 0. (14)

When the initial time is t = 0, we alter notation by putting τ ≡ t ∈ [0, d). ThenP is applied to the state σ(0) = E0

r(s), ∀s ∈ RI ++, so that, on substitution in (13),it follows that

T(t) = T(t) + p(t)I = 2

∫ t

0

µ(η)Et(η)dη − I0(t,E0r), (15)

where I0(t,E0r) is given by (14), so that

I0(t,E0r) = −2

∫ +∞

0

µ′(ξ + t)E0r(ξ)dξ. (16)

Note that the extra stress, defined by (7), vanishes for any constant historyEt(s) = E†(s) = E, ∀s ∈ RI +, to which corresponds the relative strain historyEtr(s) = 0†r(s) = 0, ∀s ∈ RI ++. Thus,

T(0†r) = 0. (17)

Moreover, if we consider a static continuation of duration τ ∈ RI ++, of the relativestrain history, defined by

Et+τrc (s) =

0 ∀s ∈ (0, τ ]Etr(s− τ) ∀s > τ

, (18)

VISCOELASTIC FLUIDS & FREE ENERGIES 1819

then the extra stress, given by

T(t+ τ) = T(t+ τ) + p(t+ τ)I = 2

∫ +∞

τ

µ′(s)Et+τrc (s)ds

= 2

∫ +∞

0

µ′(ξ + τ)Etr(ξ)dξ, (19)

vanishes as τ diverges, i.e.

limτ→∞

T(Et+τrc ) = T(0†r) = 0. (20)

Referring to (19)3, we introduce the vector space of the relative strain histories

Γr =

Etr : RI ++ → Sym :

∣∣∣∣∫ +∞

0

µ′(ξ + τ)Etr(ξ)dξ∣∣∣∣ < +∞ ∀τ ≥ 0

. (21)

The quantity It, defined in (14), coincides with the negative of the extra stress (19)3

corresponding to the static continuation (18) of the relative strain history.

Incidently, (1) shows that T is a state function, in that it depends only on thestate variable given by (9).

Definition 3.1. Two states σj(t) = Etrj (j = 1, 2) are said to be equivalent, if the

subsequent states σj(t+ τ) = Et+τrj (j = 1, 2) satisfy the identity

T(Et+τr1 ) = T(Et+τr2 ) ∀τ ∈ [0, d) , (22)

for all P ∈ Π of duration d.

The following theorem, proved in [3], is recalled here for convenience.

Theorem 3.2. - For a viscoelastic fluid of type (1), two states σj(t) = Etrj (j = 1, 2)are equivalent if and only if∫ +∞

0

µ′(ξ + τ)[Etr1(ξ)− Etr2(ξ)

]dξ = 0 ∀τ > 0. (23)

Since those states which correspond to the same extra stress, are, according toDefinition 3.1, indistinguishable, then an equivalence relation R is established inΣ; hence, ΣR represents the quotient space whose elements are equivalence classesdenoted by σR. These are the minimal states of the fluid. Thus, (23) is equivalentto (22). Recalling (14), we see that (23) can be expressed by

It(τ,Etr1) = It(τ,Etr2) ∀τ > 0. (24)

Accordingly, the quantity It characterizes an equivalence class σR ∈ ΣR and henceit represents a minimal state.

4. Thermodynamics and maximum recoverable work. Some relevant resultsconcerning the maximum recoverable work for isothermal processes are now recalled.For such processes, the Second Law of Thermodynamics reduces to the DissipationPrinciple. Let the work done by a process P , of duration d and starting from aninitial state σ(t), be defined as

W (σ, P ) =

∫ t+d

t

T(Eξr) · E(ξ)dξ =

∫ d

0

T(Et+τr ) · EP (τ)dτ, (25)

1820 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

Then, this principle states that the work done on a cycle (σ, P ) is non-negative, i.e.

W (σ, P ) =

∮ d

0

T(Et+τr ) · EP (τ)dτ ≥ 0, (26)

where the equality sign corresponds to reversible processes [20, 23]. The relationshipbetween maximum recoverable work and minimum free energy is exploited in [17,22, 23].

Definition 4.1. A subset S ⊂ Σ is called invariant under ρ if, for every σi ∈ Σand P ∈ Π, the state σ = ρ(σi, P ) ∈ S.

We define a free energy by the following properties.

Definition 4.2. A function ψ : Sψ → RI + is a free energy if

(a) the domain Sψ ⊂ Σ is invariant under ρ,(b) for any pair σ1, σ2 ∈ Sψ and for every P ∈ Π, such that σ2 = ρ(σ1, P ), we

haveψ(σ2)− ψ(σ1) ≤W (σ1, P ). (27)

It is proved in [17] that

ψm(σ) = − infW(σ), W(σ) = W (σ, P );P ∈ Π, (28)

is the minimum free energy. The right-hand side represents the maximum recover-able work, which is the maximum quantity of work stored by the material at a givenstate, or the amount of energy available at the given state [17, 22, 23]. In [3] thenotion of equivalence between two states is expressed in terms of the work (see also[2] referring to heat conduction with memory). According to [3], any two differentstates σj(t) = Etrj (j = 1, 2) are said w-equivalent if, for every P : [0, τ)→ Sym andfor every τ > 0, they satisfy

W (σ1, P ) = W (σ2, P ). (29)

Thus, two w-equivalent states, when continued with the same process P , yield thesame work. Furthermore, again in [3], the two notions of equivalence are comparedto show that two states are w-equivalent if and only if they are equivalent in thesense of Definition 3.1. Thus, two equivalent states σj (j = 1, 2), continued withthe same process P , yield the same extra stress, and also give the same work, asexpressed by (29). It follows that W (σ, P ) = W (σR, P ) if σ ∈ σR for all P ∈ Π andalso

infW(σ) = infW(σR), (30)

so that, via (28), we haveψm(σ) = ψm(σR). (31)

This means that (9) and the minimal state both give the same minimum free energywhich can therefore be considered independently of the definition of state.

For any time t where the process is continuous, (25) and (27), referred to thecase when P is applied at time t = 0 and hence τ ≡ t, yield

ψ(t) ≤ T(Etr) · E(t). (32)

Finally, on introduction of D(t), termed the internal dissipation function, which isnon-negative by virtue of the Second Law, the necessary condition

ψ(t) +D(t) = T(t) · E(t) (33)

follows.

VISCOELASTIC FLUIDS & FREE ENERGIES 1821

5. Mechanical work & Free energies. This section is devoted to consideringvarious free energy functionals adapted to incompressible fluids. Firstly, we recallsome expressions for the work obtained in [3].

5.1. The work function. Consider the work done on the material by the relativestrain history up to time t when the final state is σ(t) = Etr. Taking into account (25)together with the expression (1) for the stress tensor, while recalling the constraint

of incompressibility, which implies I · E = ∇ · v = 0, the work is given by

W (Etr) =

∫ t

−∞T(τ) · E(τ)dτ = 2

∫ t

−∞

∫ +∞

0

µ′(s)Eτr (s)ds · E(τ)dτ < +∞, (34)

since only those relative strain histories which yield finite work are considered. Thiswork, using integration by parts in (34)2, and recalling (2), can be written in theform

W (Etr) =

∫ t

−∞

∫ t

−∞µ(|ρ− u|)E(u) · E(ρ)dudρ

=

∫ +∞

0

∫ +∞

0

µ(|η − ξ|)Et(ξ) · Et(η)dξdη,

(35)

by taking the even extension of µ(s) from RI + to RI . Moreover, two further integra-tions by parts give

W (Etr) =

∫ +∞

0

∫ +∞

0

µ12(|η − ξ|)Etr(ξ) · Etr(η)dξdη, (36)

where

µ12(|η − ξ|) =∂2

∂η∂ξµ(|η − ξ|) = −µ′′(|η − ξ|)− 2δ(η − ξ)µ′(|η − ξ|), (37)

whence, using Plancherel’s theorem and (6)2, the work (36) can be represented inthe frequency domain by

W (Etr) = − 1

π

∫ +∞

−∞ωµ′s(ω)Etr+(ω) · (Etr+(ω))∗dω. (38)

As a consequence, since ωµ′s(ω) < 0, ∀ω 6= 0 by virtue of (6)1, the following lemmaholds [3].

Lemma 5.1. - The work done on the material by the relative strain history, Etr(s),∀s ∈ RI ++ is non-negative.

In addition, taking into account (25) and (13), the work done by the process

P (τ) = EP (τ), ∀τ ∈ [0, d), applied at time t when the state is σ(t) = Etr(s),∀s ∈ RI ++, can be written as

W (σ, P ) =

∫ d

0

[2

∫ τ

0

µ(τ − ξ)EP (ξ)dξ − It(τ,Etr)]· EP (τ)dτ. (39)

The process P , defined for any τ ∈ [0, d), can be extended to R+ by means ofits trivial extension, P (τ) = 0, ∀τ ∈ [d,+∞). Hence, if P is applied at time t = 0,

when the initial state is σ(0) = E0r(s), T has the form (15) and the work W is given

by

W (E0r, E) =

∫ +∞

0

[∫ +∞

0

µ(|t− τ |)E(τ)dτ − I0(t,E0r)

]· E(t)dt. (40)

1822 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

In the particular case when E0r(s) = 0†r(s) = 0 ∀s ∈ RI ++, we have, by virtue of

(16),

W (0†r, E) =

∫ +∞

0

∫ +∞

0

µ(|t− τ |)E(τ) · E(t)dτdt, (41)

which represents the work due only to the process. Gentili, in [22], proves that thiswork is positive and bounded and gives, also, the following definition.

Definition 5.2. A process P , of any duration, is said to be a finite work process if

W (0†r, EP ) < +∞. (42)

In [3], it is proved in particular for incompressible fluids that

W (0†r, EP ) > 0. (43)

According to Gentili [22], the process space can be identified with the Hilbert space[3]

Hµ(RI +) =

EP : RI + → Sym :

∫ +∞

−∞µc(ω)EP+(ω) · (EP+(ω))∗dω < +∞

,

equipped with the norm induced by the inner product

(g1,g2)µ :=

∫ +∞

−∞µc(ω)g1+(ω) · (g2+(ω))∗dω.

In the general case, when P is applied at time t > 0 and extended to RI + using itstrivial extension, the work done on P is given by (39), where (σ, P ) = (Etr, EP ) can

be replaced by (It, EP ). Thus, we obtain

W (It, EP ) =

∫ +∞

0

[∫ +∞

0

µ(| τ − ξ |)EP (ξ)dξ − It(τ,Etr)]· EP (τ)dτ =

=1

π

∫ +∞

−∞µc(ω)EP+(ω) · (EP+(ω))∗dω − 1

2π

∫ +∞

−∞It+(ω) · (EP+(ω))∗dω,

(44)

where the last equality follows from Plancherel’s theorem, and

It+(ω) =

∫ +∞

0

It(τ,Etr)e−iωτdτ.

The dual space of Hµ(RI +), that is

H ′µ(RI +)=

It : RI +→ Sym :

∫ +∞

−∞It+(ω) · (EP+(ω))∗dω < +∞,∀EP ∈ Hµ(RI +)

,

(45)provides the admissible states.

For the remainder of this work, excluding Section 6, we adopt assumptions thatcoincide with those given by Slemrod [28] in his study of the stability question,namely

lims→+∞

µ(s) = 0 , µ(s) > 0 , µ′(s) < 0 , µ′′(s) ≥ 0, ∀s ∈ RI +. (46)

Note that (4) yields the negativity of µ′(0) by virtue of the property

limω→∞

ωµ′s(ω) = − limω→∞

ω2µc(ω) = µ′(0) < 0. (47)

VISCOELASTIC FLUIDS & FREE ENERGIES 1823

5.2. The Graffi-Volterra free energy. An important example of the free energyis given by the Graffi-Volterra functional, which, in the case of the fluids studiedhere, reads:

ψG(t) = −∫ +∞

0

µ′(s)Etr(s) · Etr(s)ds. (48)

This functional is a free energy if (46)3,4 hold. If we evaluate the time derivative of(48), taking account of (1)-(2) and the constraint of incompressibility, we obtain

ψG(t) = −2

∫ +∞

0

µ′(s)Etr(s) · Etr(s)ds = 2

∫ +∞

0

µ′(s)d

dsEt(s) · Etr(s)ds+ T(t) · E(t),

(49)which, on integrating by parts, gives

2

∫ +∞

0

µ′(s)d

dsEt(s) · Etr(s)ds =

∫ +∞

0

µ′(s)d

ds[Etr(s) · Etr(s)]ds

= −∫ +∞

0

µ′′(s)Etr(s) · Etr(s)ds. (50)

Thus, substitution of (50) into (49) shows that (48) obeys the relation (33) if

D(t) :=

∫ +∞

0

µ′′(s)Etr(s) · Etr(s)ds, (51)

which is non-negative for all histories.

5.3. General representation of free energies. The general representation offree energies considered in [23] and other references can be adapted easily to incom-pressible fluids. In this case, according to [23], the memory kernel is assumed todepend on two different time variables, that is,

ψ(t) =

∫ +∞

0

∫ +∞

0

µ12(s, u)Etr(s) · Etr(u)dsdu, (52)

where numerical subscripts on µ denote differentiation with respect to the corre-sponding variables,

µ12(s, u) =∂2

∂s∂uµ(s, u). (53)

We require that µ12(+∞, u) = µ12(s,+∞) = 0, to guarantee boundedness of theintegral in (52) for any finite relative histories. Moreover, for all histories, µ12

should be such that the integral in (52) is non-negative. Then, from (52), takinginto account (2) and integrating by parts, we obtain

ψ(t) =

∫ +∞

0

∫ +∞

0

µ12(s, u)Etr(u) · [Et(s)− E(t)] + Etr(s) · [Et(u)− E(t)]

dsdu

=

∫ +∞

0

∫ +∞

0

[µ1(s, u) + µ2(s, u)]Et(u) · Et(s)dsdu+ 2

∫ +∞

0

µ′(s)Etr(s)ds · E(t),

(54)

provided that

µ1(s,+∞) = µ2(+∞, u) = 0, µ1(s, 0) = µ2(0, s) = µ′(s), (55)

for all s, u ∈ RI +, where µ′(·) is the derivative of µ(·), defined by (60) below. In(54) the last term, by virtue of (1) and the constraint of incompressibility, gives

1824 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

T(t) · E(t); therefore (33) is satisfied if

D(t) := −∫ +∞

0

∫ +∞

0

K(s, u)Et(s) · Et(u)dsdu

= −∫ +∞

0

∫ +∞

0

K12(s, u)Etr(s) · Etr(u)dsdu ≥ 0. (56)

The second form follows from integration by parts; moreover

K(s, u) = µ1(s, u) + µ2(s, u), K12(s, u) =∂2

∂s∂uK(s, u). (57)

The expression (52) for ψ(t) can also be written as

ψ(t) =

∫ +∞

0

∫ +∞

0

µ(s, u)Et(s) · Et(u)dsdu, (58)

via some integrations by parts together with the relation

µ(s, u) =

∫ +∞

s

∫ +∞

u

µ12(s′, u′)ds′du′, (59)

which satisfies (53). The quantity µ(s, u) has the properties

µ(s) = µ(0, s) = µ(s, 0), µ(s,+∞) = µ(+∞, u) = 0, (60)

for all s, u ∈ RI +.

5.4. The Fabrizio free energy. Finally, consider the functional, denoted by ψFin [15], which will be referred to in this work as the Fabrizio free energy. Forincompressible fluids, it has the form

ψF (It) = −1

4

∫ +∞

0

1

µ′(τ)It(1)(τ) · It(1)(τ)dτ, (61)

where It(1)(τ) denotes the derivative with respect to τ of It(τ,Etr) defined by (14),

i.e.

It(1)(τ) =d

dτIt(τ,Etr) = −2

∫ +∞

0

µ′′(ξ + τ)Etr(ξ)dξ. (62)

The kernel is required to satisfy the conditions given by (46)3,4. By virtue of (46)3,ψF is non-negative. The quantity 1/µ′(τ) typically diverges as τ →∞, but ψF (t),given by (61), remains finite. This is clear from (64) below. The domain of thefunctional ψF is given by

H ′F (RI +) =

It :

∣∣∣ ∫ +∞

0

1

µ′(τ)It(1)(τ) · It(1)(τ)dτ

∣∣∣ < +∞,

which properly includes the domain of the functional ψG. In addition, the Fabriziofree energy ψF can be expressed in terms of the relative strain history on substitutionof (62) into (61), giving

ψF (t) = −∫ +∞

0

∫ +∞

0

∫ +∞

0

1

µ′(τ)µ′′(s1 + τ)µ′′(s2 + τ)Etr(s1) ·Etr(s2)ds1ds2dτ. (63)

Keeping in mind (52), let us take

µ(s1, s2) = −∫ +∞

0

1

µ′(τ)µ′(s1 + τ)µ′(s2 + τ)dτ = µ(s2, s1). (64)

VISCOELASTIC FLUIDS & FREE ENERGIES 1825

Relation (63) can be rewritten as

ψF (t) =

∫ +∞

0

∫ +∞

0

µ12(s1, s2)Etr(s1) · Etr(s2)ds1ds2. (65)

Observe that (64) satisfies (60). The quantity K in (57)1 becomes

K(s1, s2) = −∫ +∞

0

1

µ′(τ)[µ′′(s1 + τ)µ′(s2 + τ) + µ′(s1 + τ)µ′′(s2 + τ)]dτ

=1

µ′(0)µ′(s1)µ′(s2) +

∫ +∞

0

d

dτ

[1

µ′(τ)

]µ′(s1 + τ)µ′(s2 + τ)]dτ, (66)

after a partial integration with respect to τ . The second expression shows that Kis negative, since in the integral term of (66)2,

d

dτ

[1

µ′(τ)

]= − µ′′(τ)

[µ′(τ)]2≤ 0 (67)

which follows from the sign requirements imposed on µ′ and µ′′. Therefore, we seethat the dissipation is non-negative.

A more direct proof that (61) is a free energy can be given. Consider the deriv-ative, with respect to time, of ψF in the form (61):

ψF = −1

2

∫ +∞

0

1

µ′(τ)˙It(1)(τ) · It(1)(τ)dτ. (68)

The superimposed dot denotes the derivative with respect to time t. Using (62)and (2), we find that

˙It(1)(τ) =

d

dtIt(1)(τ) = −2

∫ +∞

0

µ′′(ξ + τ)Et(ξ)dξ − 2µ′(τ)E(t)

= It(2)(τ)− 2µ′(τ)E(t),

(69)

where

It(2)(τ) =d2

dτ2It(τ) =

d

dτIt(1)(τ). (70)

Note that∫ +∞

0

It(1)(τ)dτ = −It(0,Etr) = 2

∫ +∞

0

µ′(ξ)Etr(ξ)dξ = T(t) + p(t)I,

where (62) and (1) have been used. Substitution of (69) into (68) gives, by virtueof the constraint of incompressibility,

ψF (t) = −1

4

∫ +∞

0

1

µ′(τ)

d

dτ

[It(1)(τ)

]2dτ + T(t) · E(t). (71)

Carrying out an integration by parts, we find that

ψF (t) = T(t) · E(t) +1

4

1

µ′(0)It(1)(0) · It(1)(0) +

1

4

∫ +∞

0

d

dτ

[1

µ′(τ)

]It(1)(τ) · It(1)(τ)dτ.

(72)Hence, by virtue of (33) and (67),

D(t) = −1

4

1

µ′(0)It(1)(0) · It(1)(0)− 1

4

∫ +∞

0

d

dτ

[1

µ′(τ)

]It(1)(τ) · It(1)(τ)dτ ≥ 0. (73)

1826 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

The following inequality is thus obtained

D(t) ≥ −1

4

∫ +∞

0

d

dτ

[1

µ′(τ)

]It(1)(τ) · It(1)(τ)dτ ≥ 0. (74)

In the particular case when there exists α ∈ RI ++ such that

µ′′(τ) + αµ′(τ) ≥ 0 ∀τ ∈ RI +, (75)

it follows that

− µ′′(τ) ≤ αµ′(τ), − d

dτ

[1

µ′(τ)

]≥ − α

µ′(τ)(76)

and, therefore, (74) yields

D(t) ≥ αψF (t). (77)

6. The minimum free energy. The minimum free energy, defined in Section 4,coincides with the maximum recoverable work that can be obtained from a givenstate σ; it is represented by (28) or equivalently by

ψm(σ) = WR(σ) = sup −W (σ, P ) : P ∈ Π , (78)

where Π is the set of finite work processes. Let us consider the work done on theprocess P (τ) = EP (τ), ∀τ ∈ [0, d), which is applied at time t and extended to RI + by

means of P (τ) = 0, ∀τ ∈ [d,+∞). We can identify EP (τ) with E(t+ τ) = Et(−τ).This work is given by (44)1, from which we obtain, by reversing the sign of theintegration variables,

W (σ, P ) =

∫ 0

−∞

∫ 0

−∞µ(|s−u|)Et(s) ·Et(u)dsdu = −

∫ 0

−∞It(−s,Etr) ·Et(s)ds. (79)

The quantity It(−s,Etr), ∀s ∈ RI − is given by (14).Thus, we can consider the set of those processes Hµ(RI −), defined by means of

the space Hµ(RI +) introduced in Section 5. Consider the processes

Et(s) = Et(m)(s) + γ E(s) ∀s ∈ RI −, (80)

where γ is a real parameter, E(s) ∈ Hµ(RI −) is an arbitrary smooth function and

P (m)(s) = Et(m)(s) is the process which corresponds to the maximum recoverable

work. Substitution of (80) in (79) gives

d

dγ[−W (σ, P )]

∣∣∣γ=0

= −∫ 0

−∞

[2

∫ 0

−∞µ(|s− u|)Et(m)(u)du− It(−s,Etr)

]·E(s)ds = 0.

(81)The arbitrariness of E(s) implies that

2

∫ 0

−∞µ(|s− u|)Et(m)(u)du = It(−s,Etr) ∀s ≤ 0. (82)

This is a Wiener-Hopf equation of the first kind, whose solution gives the requiredmaximum recoverable work. Thus, (79) and (82) allow to write the free energy (78)as

ψm(Etr) = ψm(t) =

∫ 0

−∞

∫ 0

−∞µ(|s− u|)Et(m)(u) · Et(m)(s)duds. (83)

VISCOELASTIC FLUIDS & FREE ENERGIES 1827

To solve (82), let us introduce into it the relative strain history, as defined by (2),

related to the optimal process P (m)(u) = Et(m)(u) and given by

Et(m)r(u) = Et(m)(u)− E(t). (84)

It follows that

Et(m)(u) = − ∂

∂uEt(m)(u) = − ∂

∂uEt(m)r(u). (85)

Then, integration by parts in (82), gives

2

∫ 0

−∞

∂

∂sµ(|s− u|)Et(m)r(u)du = −Jt(s), ∀s ≤ 0, (86)

whereJt(s) := It(−s,Etr) ∀s ≤ 0. (87)

The quantity Jt(s), taking account of (14), can be written in the form

Jt(s) = −2

∫ +∞

0

µ′(u− s)Etr(u)du

= 2

∫ +∞

0

∂

∂sµ(|s− u|)Etr(u)du ∀s < 0. (88)

This quantity can be extended to RI , giving

Jt(τ) = 2

∫ +∞

0

∂

∂τµ(|τ − u|)Etr(u)du ∀τ ∈ RI . (89)

We introduce a function Rt : RI → Sym with the property that

Rt(s) = 0 ∀τ ∈ RI −−, (90)

which allows us to extend (86) to RI , writing it in the form

2

∫ 0

−∞

∂

∂sµ(|s− u|)Et(m)r(u)du+ Jt(s) = Rt(s) ∀s ∈ RI . (91)

We assume that Et(m)r(u) = 0 on RI ++. Taking the Fourier transform of (91), using

the convolution theorem, gives

4i

ωH(ω)Et(m)r−(ω) + JtF (ω) = Rt

+(ω), (92)

where [14, 17, 20],

H(ω) := −ωµ′s(ω) or, equivalently, H(ω) = ω2µc(ω) ∀ω ∈ RI , (93)

on use of (6)1. In (92), the Fourier transform Et(m)r−(ω) of the optimal relative

strain history (84) is analytic in C(+) since Et(m)r(u) is zero on RI ++, while Rt+(ω)

is analytic in C(−) since Rt(s) satisfies condition (90).The quantity H(ω) is non-negative for all ω ∈ RI by virtue of (93)2 and the

thermodynamic restriction (4). The even signature and the analyticity of H(ω)on the real axis imply that H(ω) vanishes at the origin quadratically, by virtue ofthe assumption (5). It follows [23] that H(ω), and indeed µc(ω), can always befactorized as follows:

H(ω) = H+(ω)H−(ω), µc(ω) = µ(+)(ω)µ(−)(ω), µ(±)(ω) =H±(ω)

ω, (94)

where H+(ω) has no singularities and zeros in C(−). It is assumed to be analyticin an open set containing RI so that its region of analyticity includes C−. Similarly

1828 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

H−(ω) has no singularities and zeros in C+, by virtue of the same assumption forH−(ω). Relation (47) yields

H(∞) = H+(∞)H−(∞) = −µ′(0) > 0. (95)

The factorization (94) of H(ω), together with (92) gives, by virtue of (47),

H−(ω)Et(m)r−(ω) +ω

4i[H+(ω)]−1JtF (ω) =

ω

4i[H+(ω)]−1Rt

+(ω), (96)

wherein the quantity [H+(ω)]−1 is defined ∀ω ∈ RI \ 0. The solution of thisequation gives the optimal relative history and the minimum free energy. Usinga similar argument to that given in for example [18, 5], the free energy can bewritten as [1, 3]

ψm(t) =1

π

∫ +∞

−∞|qt(−)(ω)|2dω. (97)

where

qt(−)(ω) = limξ→ω+

1

2πi

∫ +∞

−∞

−ω′µ(−)(ω′)Etr+(ω′)

ω′ − ξdω′. (98)

The minimum free energy can also be expressed in terms of It for incompressiblefluids in a manner entirely analogous to that given in [15, 5]. One obtains

ψm(t) = 2

∫ +∞

0

∫ +∞

0

K(ξ, η)It(2)(ξ,Etr) · It(2)(η,E

tr)dξdη, (99)

where

K(ξ, η) =

∫ min(ξ,η)

0

M(ξ − ρ)M(η − ρ)dρ ,

M(s) =1

2π

∫ +∞

−∞

[H+(ω)]−1

4iω−eiωsdω, ∀s ∈ RI

(100)

7. Application to initial and boundary value problems. It is interesting tostudy the dynamical behaviour of linear viscoelastic fluids using the new state func-tion It. Consider the partial differential equations given by the linear approximationof the equations of motion together with the constraint of incompressibility, that is

v(x, t) = −∇p(x, t) +∇ ·∫ t

0

µ(x, s)∇vt(x, s)ds−∇ · I0(x, t),

∇ · v(x, t) = 0,

(101)

where the right-hand side of (101)1 is the divergence of T, given by (15)2, taking

into account (101)2. Also, I0(x, t) is defined by (16). The kernel µ is assumed todepend on x ∈ Ω, while the constant mass density is assumed equal to 1.

This differential problem is set up in the space-time domain Ω × RI + and iscompleted when the initial and boundary conditions are specified. In order to givea precise formulation of such a problem we consider the following spaces

H10 (Ω) =

v ∈ H1

0 (Ω) : ∇ · v = 0

(102)

VISCOELASTIC FLUIDS & FREE ENERGIES 1829

and

Hµ(RI +,Ω) =

v ∈ L2loc(RI

+ : H10 (Ω));

1

π

∫ +∞

−∞

∫Ω

µc(x, ω)∇v+(x, ω) · (∇v+(x, ω))∗dxdω

=

∫ +∞

0

∫ +∞

0

∫Ω

µ(x, |τ − τ ′|)∇v(x, τ) · ∇v(x, τ ′)dxdτdτ ′ < +∞,

F(RI +,Ω) = H1/2(RI +;L2(Ω)) ∩Hµ(RI +,Ω).

(103)

It is easy to show, as in [15], that Hµ and F are Hilbert spaces. Let us alsointroduce the Hilbert spaces

Sµ(RI +,Ω) =

I0 ∈ L2loc(RI

+;L2(Ω)) :∫ +∞

−∞

∫ +∞

−∞

∫Ω

µ(x, |τ − τ ′|)I0(x, τ) · I0(x, τ ′)dxdτdτ ′

=1

2π

∫ +∞

−∞µ−1c (x, ω)I0

+(x, ω) ·(I0+(x, ω)

)∗dxdω < +∞

,

(104)

where µ is defined by ∫ +∞

−∞µ(x, |τ − τ ′|)µ(x, |τ ′|)dτ ′ = δ(τ), (105)

and

Sµ(RI +,Ω)=

I0 ∈ L2loc(RI

+;L2(Ω)) :∫ +∞

−∞

∫Ω

(1 + |ω|)µ−1c (x, ω)|I0

+(x, ω)|2dxdω < +∞ (106)

Definition 7.1. A function v ∈ F(RI +,Ω) represents a weak solution of the prob-lem (101) subject to the initial and boundary conditions

v(x, 0) = v0(x), It=0(x, τ) = I0(x, τ) , ∀τ ≥ 0, v(x, t)|∂Ω = 0, (107)

such that v0 ∈ L2(Ω), ∇ · v0 = 0, I0 ∈ Sµ(RI +,Ω), if it satisfies the equation∫ +∞

0

∫Ω

[v(x, t) ·wt(x, t)−

∫ t

0

µ(x, t− s)∇v(x, s)ds · ∇w(x, t)

]dxdt

=−∫

Ω

v(x, 0) ·w(x, 0)dx−∫ +∞

0

∫Ω

µ(x, t− s)I0(x, t) · ∇w(x, t)dxdt

(108)

for any w ∈ F(R+,Ω) such that limt→+∞w(x, t) = 0.

Observe that, if v is a solution of the system (101) with (107), then the functionV = v + y, such that y ∈ F

(RI +,Ω

)and y(x, 0) = −v0(x), satisfies the same

system with the initial condition V(x, 0) = 0. Therefore, without loss of generality,we suppose

v(x, 0) = 0; and also w(x, 0) = 0. (109)

Theorem 7.2. If the kernel µ ∈ L1(RI +) satisfies the thermodynamic condition(4), then there exists a unique solution v ∈ F(RI +,Ω) of the problem (101), subject

to ((107)2 and (109), and I0 ∈ Sµ(RI +,Ω) .

1830 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

Proof of Theorem 7.2. Application of a Fourier transformation to the system (101)subject to homogeneous initial conditions (109), gives

−iωv+(x, ω) +∇ · [µ+( x, ω)∇v+(x, ω)]−∇ · p( x, ω) = ∇ · I0(x, ω),

∇ · v+(x, ω) = 0,

v+(x, ω)|Ω = 0.

If the bilinear form

b(v+(x, ω),w+ (x, ω)) :=

∫Ω

iωv+(x, ω) · (w+(x, ω))∗dx

+

∫Ω

µc(x, ω)∇v+(x, ω)(∇w+(x, ω))∗dx

(110)

for any fixed ω ∈ RI , is bounded and coercive in H10 (Ω), then, application of

the Lax-Milgram theorem shows that the problem (110) admits a unique solutionv+(x, ω) ∈ H1

0 . The definition (110) easily yields the boundedness of b(·, ·). Toprove coercivity, observe that (110) implies that

b(v+(x, ω),w+(x, ω)) ≥ k(ω)

∫Ω

∇v+(x, ω) · (∇v+(x, ω))∗dx, (111)

where k(ω) is the minimum of µc(x, ω) on Ω. From this inequality, the Poincaretheorem shows that

b(v+(x, ω),w+(x, ω)) ≥ h(ω)‖v+(x, ω)‖2H10, (112)

where h(ω) is a suitable function of ω, which depends also upon the domain Ω.Therefore, for any ω ∈ RI , the problem (110) admits a solution v+(·, ω) ∈ H1

0 (Ω)

whenever the supply ∇ · I0(x, ω) ∈ H−10 (Ω), the dual of H1

0 (Ω).

Now, to study the asymptotic behaviour of the solution v(x, t), consider (108)which, by virtue of Plancherel’s theorem, can be written as follows∫ +∞

−∞

∫Ω

[−iωv+(x, ω) · (w+(x, ω))∗ − µ+(x, ω)∇v+(x, ω) · (∇w+(x, ω))∗]dxdω

= −∫ +∞

−∞

∫Ω

I0+(x, ω) · (∇w+(x, ω))∗dxdω.

(113)

Substitution of v(x, t) = w(x, t) gives, on application of Schwartz inequality,∫ +∞

−∞

∫Ω

µc(x, ω) |∇v+(x, ω)|2 dxdω

=

∫ +∞

−∞

∫Ω

µ−1/2c (x, ω)I0

+(x, ω) · µ1/2c (x, ω)(∇v+(x, ω))∗dxdω

≤[∫ +∞

−∞

∫Ω

µ−1c (x, ω)

∣∣∣I0+(x, ω)

∣∣∣2 dxdω]1/2 [∫ +∞

−∞

∫Ω

µc(x, ω) |∇v+(x, ω)|2 dxdω]1/2

.

(114)

Hence,∫ +∞

−∞

∫Ω

µc(x, ω)|∇v+(x, ω)|2dxdω ≤∫ +∞

−∞

∫Ω

µ−1c (x, ω)|I0

+(x, ω)|2dxdω. (115)

VISCOELASTIC FLUIDS & FREE ENERGIES 1831

Therefore, if I0 ∈ Sµ(RI +,Ω), then v+ ∈ Hµ(RI +,Ω). Now, choose w+ = i(signω)v+

in (113): this choice yields∫ +∞

−∞

∫Ω

[|ω| |v+(x, ω)|2 − (signω)µs(x, ω)|∇v+(x, ω)|2

]dxdω

= =∫ +∞

−∞

∫Ω

(signω)I0+(x, ω) · (∇v+(x, ω))∗dxdω.

(116)

Observe that

µs(x, ω) =1

ω[µ(x, 0) + µc(x, ω)] , (117)

whose asymptotic behaviour is

µs(x, ω) ∼ µ(x, 0)

ω; (118)

moreover, recalling (6)1, which implies µ′s(x, ω) ∼ µ′(x, 0)

ωso that, if µ(x, 0) 6= 0, it

follows that

µc(x, ω) = − 1

ωµs(x, ω) ∼ − µ(x, 0)

ω2; (119)

and hence

µs(x, ω) = −ωµ(x, 0)

µ(x, 0)+ o

(1

ω

). (120)

Next, note that

=∫ +∞

−∞

∫Ω

(signω)I0+(x, ω) · (∇v+(x, ω))∗dxdω

≤∣∣∣ ∫ +∞

−∞

∫Ω

(signω)µ−1/2c (x, ω) I0

+(x, ω) · µ1/2c (x, ω)(∇v+(x, ω))∗dxdω

∣∣∣≤ 1

2

∫ +∞

−∞

∫Ω

[µ−1c (x, ω)|I0

+(x, ω)|2 + µc(x, ω)|∇v+(x, ω)|2]dxdω,

(121)

which, taking account of (120), allows us to obtain from (115)∫ +∞

−∞

∫Ω

|ω| · |v+(x, ω)|2dxdω ≤

≤ C∫ +∞

−∞

∫Ω

[(1 + |ω|)µc(x, ω)|∇v+(x, ω)|2 + µ−1c (x, ω)|I0

+(x, ω)|2]dxdω,

(122)

where C is a constant. Furthermore, let us chose the arbitrary function in (113) tobe w+(x, ω) = |ω|v+(x, ω). Then, as in the case of (115), the following expressioncan be derived:∫ +∞

−∞

∫Ω

|ω|µc(x, ω)|∇v+(x, ω)|2dxdω

=

∫ +∞

−∞

∫Ω

µ−1/2c (x, ω)|ω|I0

+(x, ω) · µ1/2c (x, ω)(∇v+(x, ω))∗dxdω

≤[∫ +∞

−∞

∫Ω

|ω|µ−1c (x, ω)|I0

+(x, ω)|2dxdω]1/2[∫ +∞

−∞

∫Ω

|ω|µc(x, ω)|∇v+(x, ω)|2dxdω]1/2

,

(123)

1832 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

whence∫ +∞

−∞

∫Ω

|ω|µc( x, ω)|∇v+(x, ω)|2dxdω ≤∫ +∞

−∞

∫Ω

|ω|µ−1c (x, ω)|I0

+(x, ω)|2dxdω.

(124)Combining (122), (124) and (115) gives∫ +∞

−∞

∫Ω

|ω|(|v+(x, ω)|2 + µc(x, ω)|I0

+(x, ω)|2)dxdω

≤ C∫ +∞

−∞

∫Ω

(1 + |ω|)µ−1c (x, ω)|I0

+(x, ω)|2dxdω,(125)

where C is a constant. Thus, if I0+ ∈ Sµ(RI +,Ω), the function v(x, t) ∈ F(RI +,Ω)

and it represents a weak solution of problem (101), when v(x, 0) = 0.

8. Asymptotic behaviour of solutions. The initial boundary value problem forthe incompressible fluid we are considering is expressed by the differential equations(101), together with the initial and boundary conditions (107). This system can bewritten as follows

vt(x, t) = −∇p(x, t) +∇ ·∫ +∞

0

µ(x, s)∇vt(x, s)ds,

∇ · v(x, t) = 0,

v(x, 0) = v0(x), It=0(x, τ) = I0(x, τ) ∀τ ≥ 0,

v(x, t)|∂Ω = 0,

(126)

where I0 is given by (16). The kernel µ in (126) is assumed to satisfy the restrictions

(46). The differential system, on introduction of the function It(·, ·), can be writtenin the form

vt(x, t) = −∇p(x, t)−∇ · It(x, 0),

d

dtIt(x, τ) = − d

dτIt(x, τ)− µ′(τ)∇v(x, t),

∇ · v(x, t) = 0,

v(x, t)|∂Ω = 0.

(127)

Now, the solution of this problem is sought in the function space

G = H10 (Ω)×HF (Ω; RI +), (128)

where H10 (Ω) is defined in (102), while

HF (Ω; RI +) =It ∈ L2

loc(RI+;L2(Ω)) :

∫ +∞

0

∫Ω

µ−1(x, τ)It(x, τ) · It(x, τ)dxdτ < +∞..

(129)

The space G is a Hilbert space when equipped with the following scalar product

(v1(t), It1(t); v2(t), It2(t)) :=∫Ω

v1(x, t) · v2(x, t)dx−∫ +∞

0

∫Ω

µ−1(x, s)It1(x, s) · It2(x, s)dxds,(130)

which will be denoted as < y1, y2 >, where y(x, t) = (v(x, t), It(x, τ)).

VISCOELASTIC FLUIDS & FREE ENERGIES 1833

Theorem 8.1. If the kernel µ satisfies conditions (46) and if there exists α ∈ RI +

such that for any x ∈ Ω

µ′′(x, s) + αµ′(x, s) ≥ 0 ∀s ≥ 0, (131)

then, for any initial condition y0 = (v(0), I0(τ)) ∈ G, there exists a solution

y(t) = (v(t), It(τ)) : [0,+∞)→ G such that

1

2‖ v(t) ‖2L2 +ψF (It) ≤Me−νt

(1

2‖ v(0) ‖2L2 +ψF (I0)

), (132)

where Mand νare suitable positive constants, with given ψF by (61).

Proof of Theorem 8.1. Let us introduce the functional

e(x, t) =1

2v2(x, t) + ψF (It(x, τ)),

which satisfies the following inequality∫Ω

e(x, t)dx ≤ α

4

∫Ω

∫ +∞

0

1

µ′(x, s)It(1)(x, τ) · It(1)(x, τ)dτdx ≤ 0, (133)

by virtue of (74). Let E denote the total energy

E(t) =

∫Ω

e(x, t)dx. (134)

Integration over (0, t) of (133), gives

0 ≤ E(t) = E(0) ∀t ≥ 0, (135)

while, via integration over (0,+∞), we obtain

E(0) ≥ −α4

∫ +∞

0

∫ +∞

0

∫Ω

1

µ′(x, s)It(1)(x, τ) · It(1)(x, τ)dτdxdt. (136)

Moreover, the solutions must satisfy the equality∫ +∞

0

E(t)dt =

∫ +∞

0

∫Ω

[1

2v2(x, t) + ψF (It(x, τ))

]dxdt (137)

=1

2

∫ +∞

−∞

∫Ω

[v2(x, t)− 1

2

∫ +∞

0

1

µ′(x, s)|It(1)(x, τ)|2dτ

]dxdt.

The system (127), can be written in the abstract form

y = Ay, (138)

where A is the operator such that, applied to y(t) = (v(t), It(τ)), gives the right-hand sides of (127). The operator A is defined in the space G.

Now, as in [11], the following lemma can be proved.

Lemma 9.1 - Under the hypotheses (46) and (131) for the kernel µ, the operatorA : G → K = L2(Ω)×HF (Ω; RI +) is a maximal operator, that is

a) < Ay, y >≤ 0 for any y ∈ G,b) the range of A− I is K, where I is the identity operator.

Lemma 9.1 allows us to apply the Lumer-Phillips theorem [27]. Therefore, Agenerates a strongly continuous semigroup of linear contraction operators S(t) onK. Thus, the solutions of the system (127) can be written as

y(t) = S(t) y0. (139)

1834 G. AMENDOLA, S. CARILLO, J. M. GOLDEN AND A. MANES

Hence, according to [5], introduction of a suitable Hilbert space allows to proveexponential decay of solutions to system (138).

Acknowledgments. Work supported by the Dublin Institute of Technology, theItalian GNFM-INdAM, Universita di Pisa, Italy, Sapienza Universita di Roma,Italy.

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Received March 2013; revised March 2014.

E-mail address: amendola@dma.unipi.it

E-mail address: sandra.carillo@sbai.uniroma1.it

E-mail address: murrough.golden@dit.ie

E-mail address: manes@dm.unipi.it