A dynamic model for viscoelasticity in domains with time-dependent cracks

In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracking domains.


Introduction
In this paper we study the dynamic crack growth in viscoelastic materials with long memory. When no crack is present, important contributions in the theory of linear viscoelasticity are due to such scientists as Maxwell, Kelvin, and Voigt. Their names are associated with two well-known models of dissipative solids which can be described in terms of a spring and a dash-pot in series (Maxwell's model) or in parallel (Kelvin-Voigt's model), see [16]. Boltzmann was the first to develop a three-dimensional theory of isotropic viscoelasticity in [2], and later Volterra in [17] obtained similar results for anisotropic solids.
In literature we can find two different classes of materials in the case of viscoelastic deformations: materials with short memory and materials with long memory. The first case is associated to a local model, which means that the state of stress at the instant t only depends on the strain at that instant. In the second case, instead, the associated model is non-local in time, in the sense that the state of stress at the instant t depends also on the past history up to time t of the strain. According to [11,12], in the case of viscoelastic materials with long memory the general stress-strain relation is the following for a suitable choice of the memory kernel G, and with some prescribed boundary conditions. To describe our model we start with a short description of the standard approach to dynamic fracture in the case of linearly elastic materials with no viscosity. In this situation, the deformation of the elastic part of the material evolves according to elastodynamics; for an antiplane displacement, elastodynamics together with the stress-strain relation σ(t, x) = ∇u(t, x), leads to the following wave equation with some prescribed boundary and initial conditions. Here, Ω ⊂ R d is a bounded open set, which represents the cross-section of the body in the reference configuration, Γ t ⊂ Ω models the cross-section of the crack at time t, u(t, ·) : Ω \ Γ t → R is the antiplane displacement, and f (t, ·) : Ω \ Γ t → R is a forcing term. From the mathematical point of view, a first step towards the study of the evolution of fractures is to solve the wave equation (1.1) when the time evolution of the crack is assigned, see for example [3,7,8,14].
In this paper, we consider Maxwell's model in the case of dynamic fracture, when the crack evolution t → Γ t is prescribed. In this case, the memory kernel G has an exponential form (see for example [16]), and the displacement satisfies the following equation to ∂Ω, which is defined H d−1 -a.e. on the boundary. The Lebesgue and Sobolev spaces on Ω are defined as usual; the boundary values of a Sobolev function are always intended in the sense of traces. The norm of a generic Banach space X is denoted by · X ; when X is a Hilbert space, we use (·, ·) X to denote its scalar product. We denote by X ′ the dual of X and by ·, · X ′ the duality product between X ′ and X. Given two Banach spaces X 1 and X 2 , the space of linear and continuous maps from X 1 to X 2 is denoted by L (X 1 ; X 2 ); given A ∈ L (X 1 ; X 2 ) and u ∈ X 1 , we write Au ∈ X 2 to denote the image of u under A.
Moreover, given an open interval (a, b) ⊂ R and p ∈ [1, ∞], we denote by L p (a, b; X) the space of L p functions from (a, b) to X; we use H k (a, b; X) to denote the Sobolev space of functions from (a, b) to X with k derivatives in L 2 (a, b; X). Given u ∈ H 1 (a, b; X), we denote byu ∈ L 2 (a, b; X) its derivative in the sense of distributions. When dealing with an element u ∈ H 1 (a, b; X) we always assume u to be the continuous representative of its class, and therefore, the pointwise value u(t) of u is well defined for every t ∈ [a, b]. We use C 0 w ([a, b]; X) to denote the set of weakly continuous functions from [a, b] to X, namely, the collection of maps u : [a, b] → X such that t → x ′ , u(t) X ′ is continuous from [a, b] to R, for every x ′ ∈ X ′ . We adopt the notation Lip([a, b]; X) to denote the space of Lipschitz functions from the interval [a, b] into the Banach space X.

Formulation of the evolution problem, notion of solution
Let T be a positive real number and d ∈ N. Let Ω ⊂ R d be a bounded open set (which represents the reference configuration of the body) with Lipschitz boundary. Let ∂ D Ω be a (possibly empty) Borel subset of ∂Ω, on which we prescribe the Dirichlet condition, and let ∂ N Ω be its complement, on which we give the Neumann condition. Let Γ ⊂ Ω be the prescribed crack path. We assume the following hypotheses on the geometry of the cracks: (E1) Γ is a closed set with L d (Γ) = 0 and H d−1 (Γ ∩ ∂Ω) = 0; (E2) for every x ∈ Γ there exists an open neighborhood U of x in R d such that (U ∩ Ω) \ Γ is the union of two disjoint open sets U + and U − with Lipschitz boundary; (E3) {Γ t } t∈(−∞,T ] is a family of closed subsets of Γ satisfying Γ s ⊂ Γ t for every −∞ < s ≤ t ≤ T . Notice that the set Γ t represents the crack at time t. Thanks to (E1)-(E3) the space L 2 (Ω \ Γ t ; R d ) coincides with L 2 (Ω; R d ) for every t ∈ (−∞, T ]. In particular, we can extend a function u ∈ L 2 (Ω \ Γ t ; R d ) to a function in L 2 (Ω; R d ) by setting u = 0 on Γ t . Since H d−1 (Γ ∩ ∂Ω) = 0 the trace of u ∈ H 1 (Ω \ Γ; R d ) is well defined on ∂Ω. Indeed, we may find a finite number of open sets with Lipschitz boundary U j ⊂ Ω \ Γ, j = 1, . . . k, such that ∂Ω \ Γ ⊂ ∪ k j=1 ∂U j . There exists a positive constant C, depending only on Ω and Γ, such that u L 2 (∂Ω;R d ) ≤ C u H 1 (Ω\Γ;R d ) for every u ∈ H 1 (Ω \ Γ; R d ). (3.1) Similarly, we can find a finite number of open sets V j ⊂ Ω \ Γ, j = 1, . . . l, with Lipschitz boundary, such that Ω \ Γ = ∪ l j=1 V j . By using the second Korn's inequality in each V j (see, e.g., [15,Theorem 2.4]) and taking the sum over j we can find a positive constant C K , depending only on Ω and Γ, such that ; the symbols (·, ·) and · denote the scalar product and the norm in H or in H d s , according to the context. Moreover, we define the following spaces Notice that in the definition of V t and V , we are considering only the distributional gradient of u in Ω \ Γ t and in Ω \ Γ, respectively, and not the one in Ω. Taking into account (3.2), we shall use on the set V t (and also on the set V ) the equivalent norm Furthermore, by (3.1), we can consider for every t ∈ (−∞, T ] the set We assume that the elasticity and viscosity tensors A and B satisfy the following assumptions: x ∈ Ω and for every η 1 , for a.e. x ∈ Ω and for every η ∈ R d×d sym , (3.5) for some positive constants C A and C B independent of x, and the dot denotes the Euclidean scalar product of matrices. Let β a positive real number. We wish to study the following viscoelastic dynamic system together with the boundary conditions where the data satisfy Notice that in (3.6)-(3.9) the explicit dependence on x is omitted to enlighten notation.
As usual, the Neumann boundary conditions are only formal, and their meaning will be specified in  Now we are in position to explain in which sense we mean that u ∈ V loc (−∞, T ) is a solution to the viscoelastic dynamic system (3.6)-(3.9). Roughly speaking, we multiply (3.6) by a test function, we integrate by parts in time and in space, and taking into account (3.7)-(3.9) we obtain the following definition. Definition 3.1 (Weak solution). We say that u ∈ V loc (−∞, T ) is a weak solution to system (3.6) with boundary conditions (3.7)-(3.9) if u(t) − z(t) ∈ V D t for a.e. t ∈ (−∞, T ), and ]}, and we have the following lemma. ) is a Hilbert space with respect to the following norm ) is a dense subset of the space of functions belonging to V D (a, b) which vanish on a and b.
Proof. It is clear that · V(a,b) is a norm induced by a scalar product on the set V(a, b). We just have to check the completeness of this space with respect to this norm. Let {ϕ k } k ⊂ V(a, b) be a Cauchy sequence. Then, {ϕ k } k and {φ k } k are Cauchy sequences in L 2 (a, b; V ) and L 2 (a, b; H), respectively, which are complete Hilbert spaces. Thus, there exists For the proof of the last statement we refer to [9,Lemma 2.8].
Now, suppose we know the past history of the system up to time 0. In particular, let u p ∈ V loc (−∞, 0) be a weak solution to (3.6)-(3.9) on the interval (−∞, 0) in the sense of Definition 3.1, in such a way that 0 is a Lebesgue's point for both u p andu p . This implies that there exist u 0 ∈ V 0 , with u 0 − z(0) ∈ V D 0 , and From this assumption, by defining we can reformulate (3.6)-(3.9) on the interval [0, T ] in the following way: with boundary and initial conditions 14) More in general, given F ∈ H 1 (0, T ; H d s ) we will study the following viscoelastic dynamic system with boundary and initial conditions (3.20) Notice that system (3.11)-(3.15) is a particular case of system (3.16)-(3.20). As we have already specified for system (3.6)-(3.9), also for (3.16)-(3.20) the Neumann boundary conditions are only formal, and their meaning is clarified by the following definition.
for every v ∈ D D (0, T ), and

Existence by using Dafermos' method
In this section we present an existence result which is to be considered in the framework of functional analysis; in particular it derives from an idea of C. Dafermos (see [5]) based on a generalization of Lax-Milgram's Theorem by J.L. Lions (see [13]). We start by stating the main result of this section.
for every ψ ∈ D D (0, T ), and Moreover, if u * satisfies for some positive constants C * the following estimate then u satisfies (4.1). Indeed, since Based on Remark 4.2, we now assume that the Dirichlet datum and the initial displacement are identically equal to zero. To prove the theorem in this case, we first prove that our weak formulation (3.21) with initial conditions (3.22) is equivalent to another one, which we call Dafermos' Equality. After that, by means of a Lions' theorem we prove that there exists an element which satisfies this equality. Namely, by defining for every a, b ∈ [0, T ] such that a < b the space    At this point, we state and prove some lemmas and propositions needed for the proof of Proposition 4.3. In particular, in the following lemma, we highlight a useful relation between D D (0, T ) and E D 0 (0, T ).
is well defined and satisfies ϕ v ∈ E D 0 (0, T ). Proof. Firstly, we can notice that ϕ v is well defined because v is a function with compact support, hence it vanishes in a neighborhood of T . Moreover, ϕ v (0) = 0 by definition and ϕ v ∈ C ∞ ([0, T ]; V ) because it is a primitive of a function with the same regularity. Now, we can observe that for every τ ≤ t, and by the properties of Bochner's integral we get In the next proposition we show that the distributional second derivative in time of a weak solution is an element of the space . Therefore, such a solution has an initial velocity in the space where ·, · represents the duality product between (V D 0 ) ′ and V D 0 . Let us consider a test function ϕ ∈ C ∞ c (0, T ), then for every v ∈ V D 0 the function ψ(t) := ϕ(t)v belongs to the space C ∞ c (0, T ; V 0 ), and consequently ψ ∈ D D (0, T ). Now we multiply both sides of (4.4) by ϕ(t) and we integrate it on (0, T ). Thanks to (3.21) we can write as elements of (V D 0 ) ′ , which concludes the proof. Remark 4.6. Proposition 4.5 implies thatu ∈ H 1 (0, T ; (V D 0 ) ′ ), hence it admits a continuous representative. Therefore, we can say that there In the next proposition we show how the weak formulation (3.21) changes if we use test functions which do not vanish at zero. In particular, we use the notation η(T ) to refer to the family of open neighborhoods of T , and we consider the following spaces for every Ψ ∈ Lip D T (0, T ). Proof. Let us consider Ψ ∈ Lip D T (0, T ) and define for every ε ∈ (0, T 3 ) the function It is easy to see that ψ ε ∈ Lip D 0,T (0, T ), and by using ψ ε as test function in (3.21) we get I ε + I m ε + J m ε = 0, where the three terms I ε , I m ε , and J m ε are defined in the following way: Let us study the convergence of I ε , I m ε , and J m ε as ε → 0 + . First of all, we notice that from the definition of ψ ε and the Lipschitz continuity of Ψ we have From (3.3), (4.7), and the absolute continuity of Lebesgue's integral, we have In the same way we can prove that Notice that, by virtue of the continuity of the translation operator in L 2 , and again by the absolute continuity of Lebesgue's integral, we can write Taking into account (4.8)-(4.12) we conclude that Now we analyze the limit of I m ε as ε → 0 + . By (4.5) we obtain In the same way, we can prove that Finally, we study the behaviour of J m ε as ε → 0 + . Since Ψ(T ) = 0, we can write By following the same strategy used in (4.19), we can prove that Thanks to (4.18)-(4.22) we can say that J m ε → 0 as ε → 0 + , and this concludes the proof.
We can now prove the equivalence result between the viscoelastic dynamic system is well defined and belongs to the space E D 0 (0, T ). By taking ϕ v as a test function in (4.3) we obtain , by the definition of ϕ v itself. This, together with (4.24), allows us to conclude that u ∈ V D (0, T ) satisfies (3.21) for every v ∈ D D (0, T ). Now we prove that u 1 coincides withu(0). Since the function u satisfies (3.21) for every v ∈ D D (0, T ), in particular, from Remark 3.4, it satisfies the same equality for every v ∈ Lip D 0,T (0, T ). Thanks to Proposition 4.7, the function u satisfies (4.6) for every v ∈ Lip D T (0, T ), and therefore for every function in the space By taking the difference between (4.6) and (4.25) we get This proves the first part of the proposition. Vice versa, let u ∈ V D (0, T ) be a weak solution in the sense of Definition 3.3. Therefore, u satisfies (3.21) for every v ∈ D D (0, T ), and as we have already shown before, u satisfies (4.6), with u 1 in place ofu(0), for every function v ∈ Lip D T (0, T ). Let us consider ϕ ∈ E D 0 (0, T ), then v ϕ (t) = (t − T )φ(t) ∈ Lip D T (0, T ), and so it can be used as a test function in (4.6). By noticing thatv ϕ (t) =φ(t) + (t − T )φ(t) and v ϕ (0) = −Tφ(0) we obtain the thesis.
In view of the previous proposition, it will be enough to prove the existence of a solution to Dafermos' Equality (4.3). In particular, we shall prove the existence of t 0 ∈ (0, T ] and of a function u ∈ V D (0, t 0 ) such that u(0) = 0, and which satisfies Dafermos' Equality on the interval [0, t 0 ]. In order to do this, we use an abstract result due to Lions (see [13, Chapter 3, Theorem 1.1 and Remark 1.2]). We first introduce the necessary setting. Let X be a Hilbert space and Y ⊂ X be a linear subspace, endowed with the scalar product (·, ·) Y which makes it a pre-Hilbert space. Suppose that the inclusion of Y in X is a continuous map, i.e., there exists a positive constant C such that (4.26) Let us consider a bilinear form B : for every ϕ ∈ Y , for some positive constant α.
Moreover, the solution u satisfies After defining for every a, b ∈ [0, T ] with a < b the space : u(a) = 0}, we can state the following proposition. Proposition 4.9. There exists t 0 ∈ (0, T ] and a function u ∈ V D 0 (0, t 0 ) which satisfies Dafermos' Equality (4.31) For simplicity of notation, we denote the spaces V D 0 (0, t 0 ) and E D 0 (0, t 0 ) with the symbols V t0 and E t0 , respectively. On the space V t0 we take the usual scalar product, instead on the space E t0 we consider the following one for every φ, ϕ ∈ E t0 , and we denote by · Et 0 the norm associated. Let us consider the bilinear form B : and the linear operator L : E t0 → R represented by Notice that, from these definitions, Dafermos' Equality By definition we have then (4.32) can be reworded as , eφ(t)) + (Bψ(t), eφ(t))]dt.  By substituting this information in (4.33), we get after some integration by parts From the coerciveness in (3.5) and the definition of the V -norm, we have Moreover, since which corresponds to the hypothesis (4.28), with We now show the validity of assumption (4.27). We have to prove that for every ϕ ∈ E t0 the functional B(·, ϕ) is continuous on V t0 , and that L : E t0 → R is a linear continuous operator on the space E t0 . To this aim, we fix ϕ ∈ E t0 and we consider {u k } k ⊂ V t0 such that By using Cauchy-Schwarz's inequality we get whence, by considering (4.38), we can say that there exist two positive constants C 1 = C 1 (ϕ, t 0 ) and Now it remains to show that L is a continuous operator on E t0 , and since it is linear it is enough to show its boundedness. Let ϕ ∈ E t0 , then In particular there exists a positive constant    Proof. We begin by proving (4.42). We consider ψ ∈ V D (0, b) such that ψ(0) = 0, and we define for ε ∈ (0, b) the function Since ψ ε ∈ V D (0, b) and ψ ε (0) = ψ ε (b) = 0, we can use it as a test function in (3.21) to obtain I ε + J ε = K ε , where Thanks to the absolute continuity of Lebesgue's integral and to Remark 4.6 we get which concludes the proof of (4.42).
To prove (4.43), it is enough to consider for ε ∈ (0, c − b) the function where Ψ ∈ V D (b, c) such that Ψ(c) = 0, and to repeat similar argument before performed.
Taking into account the previous lemma we can state and prove the following proposition. Proof. We divide the proof into two steps. In the first one, we show how to extend the solution. After this, in the second step, we prove (4.45). We firstly chooseb ∈ (b − t0 2l , b) in such a way that Notice that (4.46)-(4.48) are possible becauseũ ∈ V(0, b).
Step 1. Sinceũ is a weak solution on the interval [0, b], then We define the function G ∈ H 1 (b,b + t0 l ; H d s ) in the following way Since t0 l ≤ t 0 ,ũ(b) ∈ V , andu(b) ∈ H, we can apply Remark 4.2, Propositions 4.3 and 4.9 on the interval and also the following limits and we claim that it is a weak solution on the interval [0,b + t0 l ]. Notice that, sinceb ≥ b − t0 2l thenb + t0 l > b. To prove this, let us fix ψ ∈ D D (0,b + t0 l ). Clearly ψ ∈ V D (0,b) and ψ(0) = 0, and sinceũ is a weak solution on [0,b], we can use (4.42) of Lemma 4.11 to get (4.53) From (4.47) and (4.50), by summing (4.52) and (4.53), we obtain the following equality Step 2. Now, we want to prove (4.45). We can write is a function which satisfies Dafermos' Equality (4.3) on the interval [b, c] with the right-hand side equal to Therefore, by following the estimates in (4.39)-(4.41), we can apply (4.29) of Theorem 4.8, with Now notice that and Taking into account the information provided by (4.46)-(4.48), we can use estimates (4.56)-(4.58) to deduce the existence of a positive constantC =C(t 0 , l, A, B, β) such that

Existence: A coupled system equivalent to the viscoelastic dynamic system
In this section, we illustrate a second method to find solutions to the viscoelastic dynamic system (3.16)-(3.20) according to Definition 3.3. This method is based on a minimizing movement approach deriving from the theory of gradient flows, and it is a classical tool used to prove the existence of solutions in the context of fractures, see, e.g., [4], [7], [9]. By means of this method, we are also able to provide an energy-dissipation inequality satisfied by the solution, and consequently, thanks to this inequality, we prove that such a solution satisfies the initial conditions (3.20) in a stronger sense than the one stated in (3.22).
To this aim, let us define the following coupled system with the following boundary and initial conditions   where the equalities are to be understood in the sense of the Hilbert space H d s ; • the initial conditions (3.22) are satisfied.
The following result proves that the new problem is equivalent to the first one. Proof. Let us consider a weak solution (u, w) ∈ V × H 1 (0, T ; H d s ) to the coupled system (5.1)-(5.5) according to Definition 5.1. In view of the theory of ordinary differential equations valued in Hilbert spaces, by (5.7) we can write for every ϕ ∈ D D . Let w 0 ∈ H d s and let w be the function defined in (5.8). It is easy to see that w ∈ H 1 (0, T ; H d s ) and by summing to both hand sides of (5.9) the term The proof of this result will be given at the end of this section.

Discretization in time.
In this subsection we prove Theorem 5.3 by means of a time discretization scheme in the same spirit of [7]. Let us fix n ∈ N and set For k = 1, ..., n let (u k n , w k n ) be the minimizer in V k n × H d s of the functional Using the coerciveness (3.5), it is easy to see that the functional in (5.11) is convex and bounded from below by for a suitable positive constant C k n . The existence of a minimizer then follows from the lower semicontinuity of the functional with respect to the strong (and hence to the weak) convergence in V k n × H d s . To simplify the exposition, for k = 0, ..., n we define for every (ϕ, ψ) ∈ V k n × H d s , where δw k n is defined for every k = 1, . . . , n as in (5.12), and δu 0 n = u 1 by (5.10). Notice that by choosing as a test function the pair (ϕ, 0) with ϕ ∈ V k n , we get (δ 2 u k n , ϕ) + ((A + B)eu k n − Bw k n , eϕ) = (f k n , ϕ) + (F k n − h k n , eϕ), which is a discrete-in-time approximation of (5.6). On the other hand, if we use as a test function in (5.13) the pair (0, ψ) with ψ ∈ H d s , we have (βδw k n + w k n − eu k n , ψ) = 0, whence βδw k n + w k n − eu k n = 0 (as element of H d s ), which is an approximation in time of (5.7). In the next lemma we show an estimate for the family {(u k n , w k n )} n k=1 , which is uniform with respect to n, and it will be used later to pass to the limit in the discrete equation (5.13). Proof. To simplify our computations, we define the following two bilinear symmetric forms Thanks to (3.5) we have that a((ϕ, ψ), (ϕ, ψ)) ≥ 0 and b(ψ, ψ) ≥ 0 for every ϕ ∈ V and ψ ∈ H d s . Now we set ω k n := (u k n , w k n ) for k = 0, . . . , n, and we take (ϕ, ψ) = τ n (δu k n − δz k n , δw k n ) ∈ V k n × H d s as a test function in (5.13), where δz 0 n :=ż(0) and δz k n is defined as in (5.12). Therefore, we obtain δu k where W k n := (f k n , δu k n − δz k n ) + (F k n , eδu k n − eδz k n ) − (h k n , eδu k n − eδz k n ) + (δ 2 u k n , δz k n ) + a(ω k n , (δz k n , 0)). We fix i ∈ {1, . . . , n} and we sum in (5.16) over k = 1, . . . , i to obtain the following discrete energy inequality where Let us now estimate the right-hand side of (5.17) from above. By means of Cauchy-Schwarz and Young's inequalities we can write  τ n a(ω k n , (δz k n , 0)) ≤ Notice that the following discrete integrations by parts hold where δh k n , δF k n , and δ 2 z k n are defined as in (5.12). By (5.22) and where C ε3 is a positive constant depending on ε 3 . Now we consider (5.17)-(5.29). By choosing ε 1 = 1 2 , ε 2 = ε 3 = C A 4 and using (3.4) and (3.5) we obtain the existence of two positive constants C 1 and C 2 such that We now want to pass to the limit into the discrete equation (5.13) to obtain a solution to the coupled system (5.1)-(5.5) according to Definition 5.1. We start by defining the following interpolation sequences of our limit solution u n (t) := u k n + (t − kτ n )δu k n ,ũ n (t) := δu k n + (t − kτ n )δ 2 u k n t ∈ [(k − 1)τ n , kτ n ], k = 1, . . . , n, u + n (t) := u k n ,ũ + n (t) := δu k n t ∈ ((k − 1)τ n , kτ n ], k = 1, . . . , n, u − n (t) := u k−1 n ,ũ − n (t) := δu k−1 n t ∈ [(k − 1)τ n , kτ n ), k = 1, . . . , n, and the same approximations w n , w + n , w − n for the function w. By using this notation, we can state the following convergence lemma.
By Banach-Alaoglu's Theorem there exist some functions Moreover, given that for a.e. t ∈ (0, T ), with (5.35) and the continuity of the translations in L 2 we deduce that Now let us check that u ∈ V. To this aim, we define the following sets . Notice thatṼ is a (strong) closed convex subset of L 2 (0, T ; V ), and so by Hahn-Banach Theorem the setṼ is weakly closed. In the same way we can prove thatṼ D is also a weakly closed set. Notice that {u − n } n ⊂Ṽ, indeed which we have u ∈ V. Finally, to show that u − z ∈ V D we observe that for t ∈ [(k − 1)τ n , kτ n ), k = 1, . . . , n, we get u − z ∈ V D . This concludes the proof.
With the next lemma we show that the limit identified by Lemma 5.5 is actually a weak solution to the coupled system (5.1)-(5.5) according to Definition 5.1.
It remains to show that the limit previously found assumes the initial data in the sense of (3.22). Before doing this, let us recall the following result, whose proof can be found for example in [10]. Proof. From the discrete equation (5.13) we deduce Therefore, taking the supremum over (ϕ, ψ) ∈ V D 0 × H d s with (ϕ, ψ) V ×H d s ≤ 1, we obtain the existence of a positive constant C ′ such that By multiplying this inequality by τ n and then by summing over k = 1, . . . , n, we get In particular {ũ n } n ⊂ H 1 (0, T ; (V D 0 ) ′ ) is uniformly bounded (notice thatu n (t) = δ 2 u k n for t ∈ ((k−1)τ n , kτ n ) and k = 1, . . . , n). Hence, up to extracting a further (not relabeled) subsequence from the one of Lemma 5.5, we haveũ so that u(0) = u 0 andu(0) = u 1 , since u n (0) = u 0 andũ n (0) = u 1 . By Lemma 5.6 we get the thesis.