Abstract
We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.
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1 Introduction
This paper deals with the mathematical analysis of the dynamics of elastic damping materials in the presence of external forces and time-dependent brittle fracture. In this framework, it is important to find the behavior of the deformation when the crack evolution is known. This is the first step toward the development of a complete model of dynamic crack growth in viscoelastic materials. From a mathematical point of view, this means solving the following dynamic system
In the equation above, \(\Omega \subset {\mathbb {R}}^d\) represents the reference configuration of the material, the set \(\Gamma _t\subset \Omega \) models the crack at time t (which is prescribed), \(u(t):\Omega \setminus \Gamma _t\rightarrow {\mathbb {R}}^d\) is the displacement of the deformation, \(\sigma (t)\) the stress tensor, and f(t) is the forcing term.
In the classical theory of linear viscoelasticity, the constitutive stress–strain relation of the so-called Kelvin–Voigt’s model is given by
where \({\mathbb {C}}\) and \({\mathbb {B}}\) are two positive tensors acting on the space of symmetric matrices, and ev denotes the symmetric part of the gradient of a function v (which is defined as \(ev:=\frac{1}{2}(\nabla v+\nabla v^T)\)). The local model associated with (1.2) has already been widely studied and we can find several existence results in the literature; we refer to [2, 3, 6, 7, 17, 24] for existence and uniqueness results in the pure elastodynamics case (\({\mathbb {B}}=0\)) and in the classic Kelvin–Voigt’s one.
In recent years, materials whose constitutive equations can be described by non-local models are of increasing interest. In this context, by non-local we mean that the state of the stress at instant t depends not only on that instant, but also on the previous ones (long memory). For solid viscoelastic materials, some experiments are particularly in agreement with models using fractional derivative, see for example [10, 11, 23, 25] and the references therein.
In this paper, we focus on the fractional Kelvin–Voigt’s model, i.e., we consider the following constitutive stress–strain relation
where \(D_t^\alpha \) denotes a fractional derivative of order \(\alpha \in (0,1)\). In the literature, we can find several definitions for the fractional derivative of a function \(g:(a,b)\rightarrow {\mathbb {R}}\); here, we focus on the most used ones which are Riemann–Liouville’s derivative of order \(\alpha \) at starting point a
and Caputo’s derivative of order \(\alpha \) at starting point a
We recall that \({\varvec{\Gamma }}\) denotes Euler’s Gamma function; notice that in order to define Caputo’s derivative the function g must be differentiable, while this is not necessary for Riemann–Liouville’s derivative. Given \(g\in AC([a,b])\), and \(t\in (a,b)\), we have the following relation between Riemann–Liouville’s and Caputo’s derivative (see, e.g., [13]):
In particular, when \(g(a)=0\), these two notions coincide. For more properties regarding these two fractional derivatives, we refer for example to [4, 16, 20, 21] and the references therein.
In this paper, we use Caputo’s derivative, which means we consider the dynamic system
One of the qualities of this definition for the fractional derivative is that the initial conditions can be imposed in the classical sense, see for example [16, 20]. The choice of 0 as a starting point is due to the fact that we want to couple dynamic system (1.1) with the initial conditions at time \(t=0\).
Dealing with (1.4) is very difficult, since in the definition of \({}_{0}^{C}{D}_t^\alpha e u(t)\) we need that eu is differentiable, which is a very strong request. Hence, we rephrase Caputo’s derivative in a more suitable way. Thanks to (1.3) for \(g\in AC([0,T])\), we can write
This formulation of Caputo’s derivative is well-posed in the distributional sense also when the function g is only integrable. We point out that formula (1.5) can be found in the recent literature on fractional derivatives, where it is used to define the notion of weak Caputo’s derivative for less regular functions, see for example [9, 15].
Thanks to formula (1.5), we can write system (1.4) in a weaker form (see Definition 2.2) as
where
Notice that the scalar function \(\rho \) appearing in \({\mathbb {F}}\) is positive, decreasing, and convex on \((0,\infty )\). Moreover, \(\rho \in L^1(0,T)\) for every \(T>0\), but it is not bounded on (0, T). In particular, we cannot compute the derivative in front of the convolution integral in (1.6).
In the literature, we can find several existence and uniqueness results for fractional type systems related to (1.6), but only when \(\Omega \) is a smooth domain without cracks. For example in [5], the authors studied an integral version of (1.6) with eu replaced by \(\nabla {u}\), and in [1, 14, 19] other fractional viscoelastic models are considered and the existence of solutions is obtained via Laplace’s transform. However, in the case of dynamic fracture, there are no existence results for the problem (1.6), since most of the previous techniques fail given that the set \(\Omega \setminus \Gamma _t\) is irregular and time dependent.
To prove the existence of a solution to (1.6), we proceed into two steps, taking inspiration by [5]. First, we consider a regularized version of (1.6), where we replace the kernel \({\mathbb {F}}\) by a regular kernel \({\mathbb {G}}\in C^2([0,T])\). Then, we prove the existence of a solution to the more regular system
and we show that this solution satisfies a uniform bound depending on the \(L^1\)-norm of \({\mathbb {G}}\). Finally, we consider a sequence of regular tensors \({\mathbb {G}}^\epsilon \) converging to \({\mathbb {F}}\) in \(L^1\) and we take the solutions to (1.8) with \({\mathbb {G}}:={\mathbb {G}}^\epsilon \). By a compactness argument, we show that the sequence \(u^\epsilon \) converge to a function \(u^*\) which solves (1.6). Moreover, we prove that this solution satisfies an energy-dissipation inequality. We conclude this paper by showing that, when the crack is not moving, the fractional Kelvin–Voigt’s system (1.6) admits a unique solution.
The paper is organized as follows: in Sect. 2, we fix the notation and the framework of our problem. Moreover, we give the notion of solution to the fractional Kelvin–Voigt’s system involving Caputo’s derivative (1.6) and we state our main existence result (see Theorem 2.4). Section 3 deals with the regularized system (1.8). First, by a time-discretization procedure in Theorem 3.13 we prove the existence of a solution to (1.8). Then, in Lemma 3.14 we derive the uniform energy estimate which depends on the \(L^1\)-norm of \({\mathbb {G}}\). In Sect. 4 we consider Kelvin–Voigt’s system (1.6): we prove the existence of a generalized solution to system (1.6) and in Theorem 4.2 we show that such a solution satisfies an energy-dissipation inequality. Finally, in Sect. 5 we prove that, for a not moving crack, the solution to (1.6) is unique.
2 Notation and framework of the problem
The space of \(m\times d\) matrices with real entries is denoted by \({\mathbb {R}}^{m\times d}\); in case \(m=d\), the subspace of symmetric matrices is denoted by \({{\mathbb {R}}}^{d\times d}_{\mathrm{sym}}\). Given a function \(u:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^m\), we denote its Jacobian matrix by \(\nabla u\), whose components are \((\nabla u)_{ij}:= \partial _j u_i\) for \(i=1,\dots ,m\) and \(j=1,\dots ,d\); when \(u:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\), we use eu to denote the symmetric part of the gradient, namely \(eu:=\frac{1}{2}(\nabla u+\nabla u^T)\). Given a tensor field \(A:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^{m\times d}\), by \(\mathop {\mathrm{div}}\nolimits A\) we mean its divergence with respect to rows, namely \((\mathop {\mathrm{div}}\nolimits A)_i:= \sum _{j=1}^d\partial _jA_{ij}\) for \(i=1,\dots ,m\).
We denote the d-dimensional Lebesgue measure by \({{\mathcal {L}}}^d\) and the \((d-1)\)-dimensional Hausdorff measure by \({{\mathcal {H}}}^{d-1}\); given a bounded open set \(\Omega \) with Lipschitz boundary, by \(\nu \) we mean the outer unit normal vector to \(\partial \Omega \), which is defined \({{\mathcal {H}}}^{d-1}\)-a.e. on the boundary. The Lebesgue and Sobolev spaces on \(\Omega \) 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 \(\Vert \cdot \Vert _X\); when X is a Hilbert space, we use \((\cdot ,\cdot )_X\) to denote its scalar product. We denote by \(X'\) the dual of X and by \(\langle \cdot , \cdot \rangle _{X'}\) 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 \(\mathscr {L}(X_1;X_2)\); given \({{\mathbb {A}}}\in {{\mathscr {L}}}(X_1;X_2)\) and \(u\in X_1\), we write \({{\mathbb {A}}} u\in X_2\) to denote the image of u under \({{\mathbb {A}}}\).
Moreover, given an open interval \((a,b)\subset {\mathbb {R}}\) and \(p\in [1,\infty ]\), we denote by \(L^p(a,b;X)\) the space of \(L^p\) functions from (a, b) to X; we use \(W^{k,p}(a,b;X)\) and \(H^k(a,b;X)\) (for \(p=2\)) to denote the Sobolev space of functions from (a, b) to X with k derivatives. Given \(u\in W^{1,p}(a,b;X)\), we denote by \({\dot{u}}\in L^p(a,b;X)\) its derivative in the sense of distributions. When dealing with an element \(u\in W^{1,p}(a,b;X)\) we always assume u to be the continuous representative of its class; in particular, it makes sense to consider the pointwise value u(t) for every \(t\in [a,b]\). We use \(C_w^0([a,b];X)\) to denote the set of weakly continuous functions from [a, b] to X, namely, the collection of maps \(u:[a,b]\rightarrow X\) such that \(t\mapsto \langle x', u(t)\rangle _{X'}\) is continuous from [a, b] to \({\mathbb {R}}\) for every \(x'\in X'\).
Let T be a positive real number and let \(\Omega \subset {\mathbb {R}}^d\) be a bounded open set with Lipschitz boundary. Let \(\partial _D\Omega \) be a (possibly empty) Borel subset of \(\partial \Omega \) and let \(\partial _N\Omega \) be its complement. Throughout the paper we assume the following hypotheses on the geometry of the cracks:
-
(H1)
\(\Gamma \subset {\overline{\Omega }}\) is a closed set with \({{\mathcal {L}}}^d(\Gamma )=0\) and \({{\mathcal {H}}}^{d-1}(\Gamma \cap \partial \Omega )=0\);
-
(H2)
for every \(x\in \Gamma \) there exists an open neighborhood U of x in \({\mathbb {R}}^d\) such that \((U\cap \Omega )\setminus \Gamma \) is the union of two disjoint open sets \(U^+\) and \(U^-\) with Lipschitz boundary;
-
(H3)
\(\{\Gamma _t\}_{t\in [0,T]}\) is an increasing family in time of closed subsets of \(\Gamma \), i.e., \(\Gamma _s\subset \Gamma _t\) for every \(0\le s\le t\le T\).
Thanks (H1)–(H3), the space \(L^2(\Omega \setminus \Gamma _t;{\mathbb {R}}^m)\) coincides with \(L^2(\Omega ;{\mathbb {R}}^m)\) for every \(t\in [0,T]\) and \(m\in {{\mathbb {N}}}\). In particular, we can extend a function \(u\in L^2(\Omega \setminus \Gamma _t;{\mathbb {R}}^m)\) to a function in \(L^2(\Omega ;{\mathbb {R}}^m)\) by setting \(u=0\) on \(\Gamma _t\). To simplify our exposition, for every \(m\in {{\mathbb {N}}}\) we define the spaces \(H:=L^2(\Omega ;{\mathbb {R}}^m)\), \(H_N:=L^2(\partial _N\Omega ;{\mathbb {R}}^m)\) and \(H_D:=L^2(\partial _D\Omega ;{\mathbb {R}}^m)\); we always identify the dual of H by H itself, and \(L^2((0,T)\times \Omega ;{\mathbb {R}}^m)\) by the space \(L^2(0,T;H)\). We define
Notice that in the definition of \(U_t\) we are considering only the distributional gradient of u in \(\Omega \setminus \Gamma _t\) and not the one in \(\Omega \). By (H2) we can find a finite number of open sets \(U_j\subset \Omega \setminus \Gamma \), \(j=1,\dots m\), with Lipschitz boundary, such that \(\Omega \setminus \Gamma =\cup _{j=1}^m U_j\). By using second Korn’s inequality in each \(U_j\) (see, e.g., [18, Theorem 2.4]) and taking the sum over j we can find a constant \(C_K\), depending only on \(\Omega \) and \(\Gamma \), such that
where eu is the symmetric part of \(\nabla u\). Therefore, we can use on the space \(U_t\) the equivalent norm
Furthermore, the trace of \(u\in H^1(\Omega \setminus \Gamma ;{\mathbb {R}}^d)\) is well defined on \(\partial \Omega \). Indeed, we may find a finite number of open sets with Lipschitz boundary \(V_k\subset \Omega \setminus \Gamma \), \(k=1,\dots l\), such that \(\partial \Omega \setminus (\Gamma \cap \partial \Omega )\subset \cup _{k=1}^l\partial V_k\). Since \({{\mathcal {H}}}^{d-1}(\Gamma \cap \partial \Omega )=0\), there exists a constant C, depending only on \(\Omega \) and \(\Gamma \), such that
Hence, we can consider the set
which is a closed subspace of \(U_t\). Moreover, there exists a positive constant \(C_{tr}\) such that
Now, we define the following sets of functions
in which we develop our theory. Moreover, we consider the Banach space
where \({{\mathcal {L}}}_{\mathrm{sym}}({\mathbb {R}}^{d\times d}_{\mathrm{sym}},{\mathbb {R}}^{d\times d}_{\mathrm{sym}})\) represents the space of symmetric tensor fields, i.e., the collections of linear and continuous maps \({{\mathbb {A}}}:{\mathbb {R}}^{d\times d}_{\mathrm{sym}}\rightarrow {\mathbb {R}}^{d\times d}_{\mathrm{sym}}\) satisfying
We assume that the Dirichlet datum z, the Neumann datum N, the forcing term f, the initial displacement \(u^0\), and the initial velocity \(u^1\) satisfy
We consider a coercive tensor \({\mathbb {C}}\in B\), which means that there exists \(\gamma >0\) such that
Moreover, let us take a time-dependent tensor \({\mathbb {F}}:(0,T+\delta _0)\rightarrow B\), with \(\delta _0>0\), satisfying
Remark 2.1
The tensor \({\mathbb {F}}\) may be not defined at \(t=0\) and unbounded on \((0,T+\delta _0)\). In the case of (1.7), the function \({\mathbb {F}}\) associated with the fractional Kelvin–Voigt’s model involving Caputo’s derivative, satisfies (2.5)–(2.8) provided that \({\mathbb {B}}\in B\) is nonnegative, that is
Since in our existence result we first regularize the tensor \({\mathbb {F}}\) by means of translations (see Sect. 4), we need that \({\mathbb {F}}\) is defined also on the right of T. This is not a problem, because our standard example for \({\mathbb {F}}\), which is (1.7), is defined on the whole \((0,\infty )\).
In this paper, we want to study the following problem
We give the following notion of solution to system (2.9):
Definition 2.2
(Generalized solution) Assume (2.1)–(2.8). A function \(u\in \mathcal C_w\) is a generalized solution to system (2.9) if \(u(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\), \(u(0)=u^0\) in \(U_0\), \({\dot{u}}(0)=u^1\) in H, and for every \(\varphi \in {{\mathcal {C}}}_c^1\) the following equality holds
Remark 2.3
The Neumann conditions appearing in (2.9) are only formal; they are used to pass from the strong formulation in (2.9) to the weak one (2.10).
The main existence result of this paper is the following theorem:
Theorem 2.4
Assume (2.1)–(2.8). Then, there exists a generalized solution \(u\in {{\mathcal {C}}}_w\) to system (2.9).
The proof of this theorem requires several preliminary results. First, in the next section, we prove the existence of a generalized solution when the tensor \({\mathbb {F}}\) is replaced by a tensor \({\mathbb {G}}\in C^2([0,T];B)\). Then, we show that such a solution satisfies an energy estimate, which depends via \({\mathbb {G}}\) only by its \(L^1\)-norm. In Sect. 4, we combine these two results to prove Theorem 2.4.
3 The regularized model
In this section, we deal with a regularized version of the system (2.9), where the tensor \({\mathbb {F}}\) is replaced by a tensor \({\mathbb {G}}\) which is bounded at \(t=0\). More precisely, we consider the following system
and we assume that \({\mathbb {G}}:[0,T]\rightarrow B\) satisfies
As before, on N, \(u^0\), \(u^1\), and \({\mathbb {C}}\) we assume (2.2)–(2.4), while for the Dirichlet datum z we can require the weaker assumption
The notion of generalized solution to (3.1) is the same as before.
Definition 3.1
(Generalized solution) Assume (2.2)–(2.4) and (3.2)–(3.6). A function \(u\in {{\mathcal {C}}}_w\) is a generalized solution to system (3.1) if \(u(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\), \(u(0)=u^0\) in \(U_0\), \({\dot{u}}(0)=u^1\) in H, and for every \(\varphi \in {{\mathcal {C}}}_c^1\) the following equality holds
Since the time-dependent tensor \({\mathbb {G}}\) is well defined in \(t=0\), we can give another notion of solution. In particular, the convolution integral is now differentiable, and we can write
Definition 3.2
(Weak solution) Assume (2.2)–(2.4) and (3.2)–(3.6). A function \(u\in {{\mathcal {C}}}_w\) is a weak solution to system (3.1) if \(u(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\), \(u(0)=u^0\) in \(U_0\), \({\dot{u}}(0)=u^1\) in H, and for every \(\varphi \in {{\mathcal {C}}}_c^1\) the following equality holds
In this framework, the two previous definitions are equivalent.
Proposition 3.3
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, \(u\in {{\mathcal {C}}}_w\) is a generalized solution to (3.1) if and only if u is a weak solution.
Proof
We only need to prove that (3.8) is equivalent to (3.7). This is true if and only if the function \(u\in {{\mathcal {C}}}_w\) satisfies for every \(\varphi \in {{\mathcal {C}}}_c^1\) the following equality
Let us consider for \(t\in [0,T]\) the function
We claim that \(p\in \mathop {\mathrm{Lip}}\nolimits ([0,T])\). Indeed, for every \(s,t\in [0,T]\) with \(s<t\) we have
Since
we deduce that \(p\in \mathop {\mathrm{Lip}}\nolimits ([0,T])\). In particular, there exists \(\dot{p}(t)\) for a.e. \(t\in (0,T)\). Given \(t\in (0,T)\) and \(h>0\), we can write
Let us compute these three limits separately. We claim that for a.e. \(t\in (0,T)\) we have
Indeed, by the Lebesgue’s differentiation theorem, for a.e. \(t\in (0,T)\) we get
Moreover, for every \(t\in (0,T)\) we have
since
Finally, for every \(t\in (0,T)\) we get
because
Therefore, by the identity
and the previous computations we deduce (3.9). \(\square \)
In the particular case in which the tensor \({\mathbb {G}}\) appearing in (3.1) is the one associated with the Standard viscoelastic model, i.e.,
with \(\beta >0\) and \({\mathbb {B}}\in B\) nonnegative tensor, then the existence of weak solutions (and so generalized solutions) was proved in [22]. Here, we adapt the techniques of [22] to a general tensor \({\mathbb {G}}\) satisfying (3.2)–(3.5).
3.1 Existence and energy-dissipation inequality
In this subsection, we prove the existence of a generalized solution to system (3.1), by means of a time-discretization scheme in the same spirit of [6]. Moreover, we show that such a solution satisfies the energy-dissipation inequality (3.40).
We fix \(n\in {{\mathbb {N}}}\) and we set
Let us define
Regarding the forcing term and the Neumann datum, we pose
For every \(j=1,\dots ,n\), let us consider the unique \(u_n^j\in U_T\) with \(u^j_n-z_n^j\in U_n^j\), which satisfies
for every \(v\in U_n^j\), where
The existence and uniqueness of \(u_n^j\) are a consequence of Lax–Milgram’s lemma. Notice that equation (3.10) is a sort of discrete version of (3.8), which we already know that is equivalent to (3.7).
We now use equation (3.10) to derive an energy estimate for the family \(\{u_n^j\}_{j=1}^n\), which is uniform with respect to \(n\in {{\mathbb {N}}}\).
Lemma 3.4
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, there exists a constant C, independent of \(n\in {{\mathbb {N}}}\), such that
Proof
First, since
we have
Therefore, equation (3.10) can be written as
for every \(v\in U_n^j\). We fix \(i\in \{1,\dots ,n\}\). By taking \(v:=\tau _n(\delta u_n^j-\delta z_n^j)\in U_n^j \) and summing over \(j=1,\dots , i\), we get the following identity
where
By using the identity
we deduce
Therefore,
Similarly, we have
Moreover, we can write
As consequence of this, we obtain
Finally, let us consider the term
We can write
because \(\delta {\mathbb {G}}_n^0=0\). Therefore, we deduce
By combining together (3.12)–(3.16), we obtain for \(i=1,\dots ,n\) the following discrete energy equality
By our assumptions on \({\mathbb {G}}\), we deduce that for a.e. \(x\in \Omega \) and every \(\xi \in \mathbb R^d\)
Hence, thanks to (3.17), for every \(i=1,\dots ,n\) we can write
Let us estimate the right-hand side in (3.18) from above. We set
Therefore, we have the following bounds
Notice that the following discrete integrations by parts hold
By means of (3.23), we can write
Moreover, thanks to
and to (3.24) we obtain
Similarly, by (3.25) we obtain
Finally, we have
By considering (3.18)–(3.30) and using (2.4), we obtain the existence of a constant \(C_1=C_1(z,N,f,u^0,{\mathbb {C}},{\mathbb {G}})\) such that for \(i=1,\dots ,n\)
In particular, since the right-hand side is independent of i, \(u_n^0=u^0\) and \(\delta u_n^0=u^1\), there exists another constant \(C_2=C_2(z,N,f,u^0,u^1,{\mathbb {C}},{\mathbb {G}})\) for which we have
This implies the existence of a constant \(C=C(z,N,f,u^0,u^1,{\mathbb {C}},{\mathbb {G}})\) independent of \(n\in {\mathbb {N}}\) such that
which gives (3.11). \(\square \)
A first consequence of Lemma 3.4 is the following uniform estimate on the family \(\{\delta ^2 u_n^j\}_{j=1}^n\).
Corollary 3.5
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, there exists a constant \({{\tilde{C}}}\), independent of \(n\in {{\mathbb {N}}}\), such that
Proof
Thanks to equation (3.10) and to Lemma 3.4, for every \(j=1,\dots ,n\) and \(v\in U^D_0 \subset U_n^j\) with \(\Vert v\Vert _{U_0}\le 1\) we have
By taking the supremum over \(v\in U^D_0\) with \(\Vert v \Vert _{U_0}\le 1\), we obtain
We multiply this inequality by \(\tau _n\) and we sum over \(j=1,\dots ,n\) to get (3.31). \(\square \)
We now want to pass to the limit into equation (3.10) to obtain a generalized solution to system (3.1). Let us recall the following result, whose proof can be found for example in [8].
Lemma 3.6
Let X, Y be two reflexive Banach spaces such that \(X\hookrightarrow Y\) continuously. Then,
Let us define the following sequences of functions which are an approximation of the generalized solution:
Moreover, we consider also the sequences
which approximate the first time derivative of the generalized solution. In a similar way, we define also \(f_n^+\), \(N_n^+\), \(\tilde{N}_n^+\), \(z_n^\pm \), \({{\tilde{z}}}_n\), \({{\tilde{z}}}_n^+\), \({\mathbb {G}}_n^\pm \), \({\tilde{{\mathbb {G}}}}_n\), \({\tilde{{\mathbb {G}}}}_n^+\). Thanks to the uniform estimates of Lemma 3.4, we derive the following compactness result:
Lemma 3.7
Assume (2.2)–(2.4) and (3.2)–(3.6). There exists a function \(u\in {{\mathcal {C}}}_w\cap H^2(0,T;(U_0^D)')\) such that, up to a not relabeled subsequence
and for every \(t\in [0,T]\)
Proof
Thanks to Lemma 3.4 and the estimate (3.31), the sequences
are uniformly bounded with respect to \(n\in {{\mathbb {N}}}\). By Banach–Alaoglu’s theorem and Lemma 3.6 there exist two functions \(u\in C^0_w([0,T];U_T)\cap H^1(0,T;H)\) and \(v\in C^0_w([0,T];H)\cap H^1(0,T;(U_0^D)')\), such that, up to a not relabeled subsequence
Thanks to (3.31), we get
therefore we deduce that \(v={\dot{u}}\). Moreover, by using (3.11) and (3.31) we have
We combine the previous convergences with (3.34) to derive
By (3.34) for every \(t\in [0,T]\), we have
Again, thanks to (3.11) and (3.31), for every \(t\in [0,T]\) we get
which imply (3.33). Finally, observe that for every \(t\in [0,T]\)
Therefore, \(u(t)\in U_t\) for every \(t\in [0,T]\) since \(U_t\) is a closed subspace of \(U_T\). Hence, \(u\in {{\mathcal {C}}}_w\). \(\square \)
Let us check that the limit function u defined before satisfies the boundary and initial conditions.
Corollary 3.8
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, the function \(u\in \mathcal C_w\) of Lemma 3.7 satisfies for every \(t\in [0,T]\) the condition \(u(t)-z(t)\in U_t^D\), and it assumes the initial conditions \(u(0)=u^0\) in \(U_0\) and \({\dot{u}}(0)=u^1\) in H.
Proof
By (3.32), we have
Hence, \(u\in {{\mathcal {C}}}_w\) satisfies \(u(0)=u^0\) in \(U_0\) and \({\dot{u}}(0)=u^1\) in H. Moreover, since \(z\in C^0([0,T];U_0)\) and thanks to (3.33), we have for every \(t\in [0,T]\)
Thus, \(u(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\) because \(U_t^D\) is a closed subspace of \(U_T\).
\(\square \)
Lemma 3.9
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, the function \(u\in \mathcal C_w\) of Lemma 3.7 is a generalized solution to system (3.1).
Proof
We only need to prove that the function \(u\in {{\mathcal {C}}}_w\) satisfies (3.7). We fix \(n\in {{\mathbb {N}}}\) and a function \(\varphi \in {{\mathcal {C}}}_c^1\). Let us consider
and, as we did before for the family \(\{u_n^j\}_{j=1}^n\), we define the approximating sequences \(\{\varphi _n^+\}_n\) and \(\{{\tilde{\varphi }}_n^+\}_n\). If we use \(\tau _n\varphi _n^j\in U_n^j\) as a test function in (3.10), after summing over \(j=1,\ldots ,n\), we get
By means of a time discrete integration by parts, we obtain
and since \(\delta {\mathbb {G}}_n^0=0\) and \(\varphi _n^0=\varphi _n^n=0\) we get
where \(t_n:=\left\lceil \frac{t}{\tau _n}\right\rceil \tau _n\) for \(t\in (0,T)\) and \(\lceil x\rceil \) is the superior integer part of the number x. Thanks to (3.35), we deduce
We use (3.32) and the following convergences
to derive
Moreover, for every fixed \(t\in (0,T)\)
which together with (3.32) gives
By (3.11) for every \(t\in (0,T)\) we deduce
Therefore, we can use the dominated convergence theorem to pass to the limit in the double integral of (3.36), and we obtain that u satisfies (3.7) for every function \(\varphi \in {{\mathcal {C}}}_c^1\). \(\square \)
Now we want to deduce an energy-dissipation inequality for the generalized solution \(u\in {{\mathcal {C}}}_w\) of Lemma 3.7. Let us define for every \(t\in [0,T]\) the total energy \(\mathcal E(t)\) and the dissipation \({{\mathcal {D}}}(t)\) as
Notice that \({{\mathcal {E}}}(t)\) is well defined for every time \(t\in [0, T]\) since \(u\in C_w^0([0, T];U_T)\) and \({\dot{u}}\in C_w^0([0, T];H)\). Moreover, by the initial conditions we have
Proposition 3.10
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, the generalized solution \(u\in {{\mathcal {C}}}_w\) to system (3.1) of Lemma 3.7 satisfies for every \(t\in [0,T]\) the following energy-dissipation inequality
where the total work is defined as
Proof
Fixed \(t\in (0, T]\) and \(n\in {{\mathbb {N}}}\) there exists a unique \(i=i(n)\in \{1,\dots ,n\}\) such that \(t\in ((i-1)\tau _n,i\tau _n]\). In particular, \(i(n)=\left\lceil \frac{t}{\tau _n}\right\rceil .\) After setting \(t_n:=i\tau _n\) and using that \(\delta {\mathbb {G}}_n^0=0\), we rewrite (3.17) as
where \(r_n:=\left\lceil \frac{r}{\tau _n}\right\rceil \tau _n\) for \(r\in (\tau _n,t_n)\), and the approximate total work \(\mathcal W^+_n(t)\) is given by
By (2.4), (3.3), and (3.33), we derive
Moreover, the estimate (3.11) imply
which together with inequality (3.45) gives
By (3.4) and (3.33), for every \(r\in (0,t)\) we have
Moreover,
because \(t_n-r_n\rightarrow t-r\). Hence, we can argue as before to deduce
In particular, we can use Fatou’s lemma and the fact that \(t\le t_n\) to obtain
By arguing in a similar way, we can derive
Let us consider the double integral in the left-hand side. We fix \(r\in (0,t)\) and by (3.5) for every \(s\in (0,r)\) we have
Moreover, for a.e. \(s\in (0,r_n-\tau _n)\) by defining \(s_n:=\left\lceil \frac{s}{\tau _n}\right\rceil \tau _n\) we deduce
Therefore, for a.e. \(s\in (0,r)\) we get
since \(s\in (0,r_n-\tau _n)\) for n large enough. If we apply again Fatou’s lemma, we conclude
By (3.11), we get
from which we derive
Since this is true for every \(r\in (0,t)\), arguing as before we obtain
Let us study the right-hand side of (3.42). Given that
we can deduce
By using the same argumentations of (3.37)–(3.39), together with the dominate convergence theorem, we can write
Thanks to the discrete integration by parts formulas (3.23)–(3.25), we have
By arguing as before, we deduce
thanks to Lemma 3.7 and to the following convergences:
and
By combining (3.42) with (3.43)–(3.52), we deduce the energy-dissipation inequality (3.40) for every \(t\in (0,T]\). Finally, for \(t=0\) the inequality trivially holds since \(u(0)=u^0\) in \(U_0\) and \({\dot{u}}(0)=u^1\) in H. \(\square \)
Remark 3.11
From the classical point of view, the total work on the solution u at time \(t\in [0, T]\) is given by
where \({{\mathcal {W}}}_{\mathrm{load}}(t)\) is the work on the solution u at time \(t\in [0, T]\) due to the loading term, which is defined as
and \({{\mathcal {W}}}_{\mathrm{bdry}}(t)\) is the work on the solution u at time \(t\in [0, T]\) due to the varying boundary conditions, which one expects to be equal to
Unfortunately, \({{\mathcal {W}}}_{\mathrm{bdry}}(t)\) is not well defined under our assumptions on u. In particular, the term involving the Dirichlet datum z is difficult to handle since the trace of the function \({\mathbb {C}}eu(r)\nu +\frac{\mathrm {d}}{\mathrm {d}r}\left( \int _0^r {\mathbb {G}}(r-s)eu(s)\mathrm {d}s\right) \nu \) on \(\partial _D\Omega \) is not well defined. If we assume that \(u\in L^2(0,T;H^2(\Omega \setminus \Gamma ;{\mathbb {R}}^d))\cap H^2(0,T;L^2(\Omega \setminus \Gamma ;{\mathbb {R}}^d))\) and that \(\Gamma \) is a smooth manifold, then the first term of \({{\mathcal {W}}}_{\mathrm{bdry}}(t)\) makes sense and satisfies
Moreover, we have
therefore \(\left( \frac{\mathrm {d}}{\mathrm {d}r}\int _0^r {\mathbb {G}}(r-s)(eu(s)-eu^0)\mathrm {d}s\right) \nu \in L^2(0,T;H_D)\). By using (3.1), together with the divergence theorem and the integration by parts formula, we derive
Therefore, by (3.54) and (3.55) we deduce the definition of total work given in (3.41) is coherent with the classical one (3.53).
We conclude this subsection by showing that the generalized solution of Lemma 3.7 satisfies the initial conditions in a stronger sense than the ones stated in Definition 2.2.
Lemma 3.12
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, the generalized solution \(u\in {{\mathcal {C}}}_w\) to system (3.1) of Lemma 3.7 satisfies
In particular, the functions \(u:[0, T]\rightarrow U_T\) and \({\dot{u}}:[0, T]\rightarrow H\) are continuous at \(t=0\).
Proof
By sending \(t\rightarrow 0^+\) into the energy-dissipation inequality (3.40) and using that \(u\in C_w^0([0, T];U_T)\), \({\dot{u}}\in C_w^0([0, T];H)\), and the lower semicontinuity of the real functions
we deduce
because the right-hand side of (3.40) is continuous in t, \(u(0)=u^0\) in \(U_0\) and \({\dot{u}}(0)=u^1\) in H. This gives
and in a similar way, we can also obtain
Since
and \(u\in C^0([0, T];H)\), we deduce (3.56). \(\square \)
By combining the previous results together, we obtain the following existence result for the system (3.1).
Theorem 3.13
Assume (2.2)–(2.4) and (3.2)–(3.6). Then, there exists a generalized solution \(u\in {{\mathcal {C}}}_w\) to system (3.1). Moreover, we have \(u\in H^2(0,T;(U_0^D)')\) and it satisfies the energy-dissipation inequality (3.40) and
Proof
It is enough to combine Lemma 3.7, Corollary 3.8, Lemma 3.9, Proposition 3.10, and Lemma 3.12. \(\square \)
3.2 Uniform energy estimates
In this subsection, we show that, under the stronger assumption (2.1) on z, the generalized solution to (3.1) of Theorem 3.13 satisfies some uniform estimates which depends on \({\mathbb {G}}\) only via \(\Vert {\mathbb {G}}\Vert _{L^1(0,T;B)}\).
Lemma 3.14
Assume (2.1)–(2.4) and (3.2)–(3.5). Let u be the generalized solution to system (3.1) of Theorem 3.13. Then, there exists a constant \(M=M(z,N,f,u^0,u^1,{\mathbb {C}},\Vert {\mathbb {G}}\Vert _{L^1(0,T;B)})\) such that
Proof
We define
Notice that K and E are well-posed since \(u\in C_w^0([0,T];U_T)\) and \({\dot{u}}\in C_w^0([0,T];H)\). Let us estimate the total work \({{\mathcal {W}}}_{tot}(t)\) in (3.40) by means of K and E. Since
we have
It remains to study the last two terms, which are
Since \(z\in W^{2,1}(0,T;U_0)\), arguing as in Proposition 3.3 we can deduce that the function
is absolutely continuous on [0, T]. In particular
which gives
Hence, we deduce
Therefore, since
by (3.40) we deduce the following estimate
where
In particular, being the right-hand side independent of \(t\in [0,T]\), we conclude
This implies the existence of a constant \(M=M(C_0,C_1,C_2)\) for which (3.57) is satisfied. \(\square \)
Remark 3.15
By the previous estimate, we can easily derive a uniform bound also for \({\dot{u}}\) in \(H^1(0,T;(U_0^D)')\), which unfortunately depends on \({\mathbb {G}}\) via \(\Vert {\mathbb {G}}(0)\Vert _B\). Indeed, let us assume that z, N, f, \(u^0\), \(u^1\), \({\mathbb {C}}\), and \({\mathbb {G}}\) satisfy (2.1)–(2.4) and (3.2)–(3.5) and let u be the generalized solution of Theorem 3.13. Thanks to (3.40) and (3.57), there exists a constant \({\overline{M}}=\overline{M}(z,N,f,u^0,u^1,{\mathbb {C}},\Vert {\mathbb {G}}\Vert _{L^1(0;T;B)})\) such that for every \(t\in [0,T]\)
By equation (3.7), it is easy to see that \({\dot{u}}\in H^1(0,T;(U_0^D)')\) and that \({\ddot{u}}\) satisfies for a.e. \(t\in (0,T)\) and for every \(v\in U_0^D\)
Hence, we derive
which gives
Therefore, the bounds on \({\ddot{u}}\) depends on \(\Vert {\mathbb {G}}(0)\Vert _B\) even when \(z\in W^{2,1}(0,T;U_0)\).
As explained in the previous remark, we cannot deduce a uniform bound for \({\dot{u}}\) in \(H^1(0,T;(U_0^D)')\) depending on \({\mathbb {G}}\) only via its \(L^1\)-norm. On the other hand, the bound on \({\dot{u}}\) in \(H^1(0,T;(U_0^D)')\) is useful if we want to prove the existence of a generalized solution \(u^*\) to the fractional Kelvin–Voigt system (2.9), especially to show that \({\dot{u}}^*\in C_w^0([0,T];H)\). To overcome this problem, we introduce another function that is related to \({\dot{u}}\) and for which is possible to derive a uniform bound. Let us consider the auxiliary function for every \(v\in U_0^D\) and \(t\in [0,T]\) as
Notice that \( \alpha \in C_w^0([0,T];(U_0^D)')\). Indeed, given \(t^*\in [0,T]\) and
we have for every \(v\in U_0^D\) the following convergence
since
The second convergence is true because
Clearly,
while \( e u\in L^\infty (0,t^*;H)\). Therefore,
Moreover,
For this function, \(\alpha \) is possible to find a uniform bound in \(H^1(0,T;(U_0^D)')\) which depends on \(\Vert {\mathbb {G}}\Vert _{L^1(0,T;B)}\).
Corollary 3.16
Assume (2.1)–(2.4) and (3.2)–(3.5). Then, the function \(\alpha \in H^1(0,T;(U_0^D)')\) and there exists a constant \({{\tilde{M}}}=\tilde{M}(z,N,f,u^0,u^1,{\mathbb {C}},\Vert {\mathbb {G}}\Vert _{L^1(0,T;B)})\) such that
Proof
First, by Lemma 3.14 we have
Moreover, by the definition of generalized solution, we deduce that for every \(\psi \in C_c^1(0,T)\) and \(v\in U_0^D\) it holds
This gives that there exists \({\dot{\alpha }} \in L^2(0,T;(U_0^D)')\) and for every \(v\in U_0^D\) and for a.e. \(t\in (0,T)\) we derive
In particular, \( \alpha \in C^0([0,T];(U_0^D)')\) and
which gives (3.59). \(\square \)
4 The fractional Kelvin–Voigt’s model
In this section, we prove the existence of a generalized solution to (2.9) for a tensor \({\mathbb {F}}\) which is not necessary bounded at \(t=0\), as it happens in (1.7). Here, we assume that our data \(z,N,f,u^0,u^1,{\mathbb {C}}\), and \({\mathbb {F}}\) satisfy the conditions (2.1)–(2.8). To prove the existence of a generalized solution to (2.9) under these assumptions, we first regularize \({\mathbb {F}}\) by a parameter \(\epsilon >0\) and we consider system (3.1) associated with this regularization. Then, we take the solution \(u^\epsilon \) given by Theorem 3.13, and thanks to Lemma 3.14 and Corollary 3.16, we obtain a generalized solution to (2.9).
Let us regularize \({\mathbb {F}}\) by defining
Clearly, \({\mathbb {G}}^{\epsilon }\) satisfies (3.2)–(3.5). Moreover, we have \({\mathbb {G}}^{\epsilon }\rightarrow {\mathbb {F}}\) in \(L^1(0,T;B)\) since \({\mathbb {F}}\in L^1(0,T+\delta _0;B)\). For every fixed \(\epsilon \in (0,\delta _0)\) we can consider the generalized solution \(u^\epsilon \) to system (3.1) with \({\mathbb {G}}\) replaced by \({\mathbb {G}}^\epsilon \) of Theorem 3.13. By Lemma 3.14 and Corollary 3.16, we deduce the following compactness result:
Lemma 4.1
Assume (2.1)–(2.8). For every \(\epsilon \in (0,\delta _0)\) let \(u^\epsilon \) be the generalized solution associated with system (3.1) with \({\mathbb {G}}\) replaced by \({\mathbb {G}}^\epsilon \) given by Theorem 3.13. Then, there exists a function \(u^*\in {{\mathcal {C}}}_w\) and a subsequence of \(\epsilon \), not relabeled, such that
and for every \(t\in [0,T]\)
Moreover, \(u^*(0)=u^0\) in \(U_0\), \({\dot{u}}^*(0)=u^1\) in H, and \(u^*(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\).
Proof
Thanks to Lemma 3.14, we deduce
with a constant M independent of \(\epsilon \) since \(\Vert {\mathbb {G}}^\epsilon \Vert _{L^1(0,T;B)}\le \Vert {\mathbb {F}}\Vert _{L^1(0,T+\delta _0;B)}\). Hence, by Banach–Alaoglu’s theorem and Lemma 3.6 there exists
and a not relabeled subsequence of \(\epsilon \) such that for every \(t\in [0,T]\)
In particular, we deduce that \(u^*(0)=u^0\) in \(U_0\), \(u^*(t)\in U_t\) and \(u^*(t)-z(t)\in U_t^D\) for every \(t\in [0,T]\).
It remains to show that \({\dot{u}}^*\in C_w^0([0,T];H)\), \({\dot{u}}^*(0)=u^1\) in H, and that for every \(t\in [0,T]\)
To this aim, we consider the auxiliary function defined at the end of the previous section. More precisely, for every \(\epsilon \in (0,\delta _0)\) let \( \alpha ^{\epsilon }:[0,T]\rightarrow (U_0^D)'\) be defined for every \(v\in U_0^D\) and \(t\in [0,T]\) as
In view of Corollary 3.16, we have
with \({{\tilde{M}}}\) independent of \({\epsilon }>0\) being \(\Vert {\mathbb {G}}^\epsilon \Vert _{L^1(0,T;B)}\le \Vert {\mathbb {F}}\Vert _{L^1(0,T+\delta _0;B)}\). Hence, up to extract a further subsequence, there exists \( \alpha ^*\in H^1(0,T;(U_0^D)')\) such that
In particular, since \( \alpha ^{\epsilon }(0)=u^1\) in \((U_0^D)'\) we conclude that \( \alpha ^* (0)=u^1\) in \((U_0^D)'\). We claim that for every \(v\in U_0^D\) and for a.e. \(t\in (0,T)\)
Indeed, for every \(\varphi \in C^\infty _c(0,T;U_0^D)\) we have
Notice that this convergence is true thanks to (4.3) and
which gives
Hence, by the dominated convergence theorem we have
Therefore, for a.e. \(t\in (0,T)\) we deduce that for every \(v\in U_0^D\)
Notice the function on the right-hand side is well defined in \((U_0^D)'\) for every \(t\in [0,T]\). Therefore, we can extend \({\dot{u}}^*\) to a function defined in the whole interval [0, T] with values in \((U_0^D)'\). In particular, we deduce \({\dot{u}}^*\in C_w^0([0,T];(U_0^D)')\), arguing in a similar way as we did in the previous section for \( \alpha \), and thanks to the fact that \({\dot{u}}^*(0)= \alpha ^*(0)=u^1\) in \((U_0^D)'\). Therefore, since \({\dot{u}}^*\in C_w^0([0,T];(U_0^D)')\cap L^\infty (0,T;H)\) we derive that \({\dot{u}}^*\in C_w^0([0,T];H)\) (thanks to Lemma 3.6), and that \({\dot{u}}^*(0)=u^1\) in H. Finally, we have
by definition of \({\dot{u}}^*\) and by (4.3) and (4.4). The convergence (4.5) combined with
gives us the last convergence required. \(\square \)
We can now prove the main existence result of Theorem 2.4 for the fractional Kelvin–Voigt’s system involving Caputo’s derivative.
Proof of Theorem 2.4
It is enough to show that the function \(u^*\) given by Lemma 4.1 is a generalized solution to (2.9). To this aim, it remains to prove that \(u^*\) satisfies (2.10). For every \(\varphi \in {{\mathcal {C}}}_c^1\), we know that the function \(u^\epsilon \in {{\mathcal {C}}}_w\) satisfy for every \(\epsilon \in (0,\delta _0)\) the following equality
Let us pass to the limit as \(\epsilon \rightarrow 0^+\). Clearly, by (4.1) we have
It remains to study the behavior as \(\epsilon \rightarrow 0^+\) of
We define for every \(\epsilon \in (0,\delta _0)\) the function
By (3.57) for every \(t\in [0,T]\), it holds
with M independent of \(\epsilon \) being \(\Vert {\mathbb {G}}^\epsilon \Vert _{L^1(0,T;B)}\le \Vert {\mathbb {F}}\Vert _{L^1(0,T+\delta _0;B)}\). Notice that
and thanks to (4.6) and to the fact that \({\mathbb {G}}^\epsilon \rightarrow {\mathbb {F}}\) in \(L^1(0,T;B)\) as \(\epsilon \rightarrow 0^+\), we get
On the other hand, since \( r \mapsto \int _ r ^T{\mathbb {F}}(t- r) e {\dot{\varphi }}(t)\,\mathrm {d}t\) belongs to \(L^\infty (0,T;H)\), we can write
As a consequence, \(u^*\) is a generalized solution to system (2.9). \(\square \)
We conclude this section by showing that for the fractional Kelvin–Voigt’s model, the generalized solution \(u^*\in {{\mathcal {C}}}_w\) to (2.9) found before satisfies an energy-dissipation inequality. As before, for \(t\in (0,T]\) we define the functions \({{\mathcal {E}}}^*(t)\) and \({{\mathcal {D}}}^*(t)\) as
Notice that the integrals in \({{\mathcal {E}}}^*\) and \({{\mathcal {D}}}^*\) are well-posed, eventually with values \(\infty \). Furthermore, we define the total work \({{\mathcal {W}}}_{tot}^*(t)\) for \(t\in [0,T]\) as
We point out the total work \({{\mathcal {W}}}_{tot}^*\) is continuous in [0, T] and that the definition given in (4.7) is coherent with the one of (3.41) thanks to identity (3.58).
Theorem 4.2
Assume (2.1)–(2.8). Then, the generalized solution \(u^*\in {{\mathcal {C}}}_w\) to system (2.9) of Theorem 2.4 satisfies for every \(t\in (0,T]\) the following energy-dissipation inequality
In particular, \({{\mathcal {E}}}^*(t)\) and \({{\mathcal {D}}}^*(t)\) are finite for every \(t\in (0,T]\).
Proof
Let us fix \(t\in (0,T]\). For every \(\epsilon \in (0,\delta _0)\) let \(u^{\epsilon }\in {{\mathcal {C}}}_w\) be the generalized solution to system (3.1) with \({\mathbb {G}}\) replaced by \({\mathbb {G}}^\epsilon \) given by Lemma 4.1. Thanks to Proposition 3.10, we know that the function \(u^{\epsilon }\) satisfies the energy-dissipation inequality (3.40) and we can rewrite the total work (3.41) as in (4.7) since \(z\in W^{2,1}(0,T;U_0)\) (as suggested by formula (3.58)). The convergences (4.2) of Lemma 4.1, and the lower semicontinuous property of the maps \(v\mapsto \Vert v\Vert ^2_H\), \(w\mapsto ({\mathbb {C}}w,w)_H\) (by (2.4)), and \(w\mapsto ({\mathbb {F}}(t) w,w)_H\) (by (2.6)), imply
Moreover, by (2.5) we have
being M independent of \(\epsilon \). Hence, (4.11) reads as
Similarly, by (2.5), (2.7), and (4.2), for every \(r\in (0,t)\) we have
In particular, we can use Fatou’s lemma to obtain
By arguing in a similar way, we can derive
For the term involving \({\ddot{{\mathbb {F}}}}\), we argue as we already did for \({\dot{{\mathbb {F}}}}\) and by using two times Fatou’s lemma we get
It remains to study the right-hand side of (3.40) with the formulation of the total work as in (4.7). Thanks to Lemma 4.1 and the fact that \({\mathbb {G}}^\epsilon \rightarrow {\mathbb {F}}\) in \(L^1(0,T;B)\), we deduce
It remains to study the term
For a.e. \(r\in (0,t)\), we have
with M independent of \(\epsilon \). By the dominated convergence theorem, we conclude
By combining (4.9)–(4.18), we deduce the energy-dissipation inequality (4.8) for every \(t\in (0,T]\). \(\square \)
Remark 4.3
Although we do not have any information about \(L^1\)-integrability of \({\dot{{\mathbb {F}}}}\) and \({\ddot{{\mathbb {F}}}}\) in \(t=0\), for the generalized solution \(u^*\) of Theorem 2.4 we obtain that the energy terms \({{\mathcal {E}}}^*\) and \({{\mathcal {D}}}^*\) are finite.
Corollary 4.4
Assume (2.1)–(2.8). Then, the generalized solution \(u^*\in {{\mathcal {C}}}_w\) to system (2.9) of Theorem 2.4 satisfies
In particular, (4.8) holds true also in \(t=0\) and
Proof
By (4.8) for every \(t\in (0,T]\), we have
Since \(u^*\in C_w^0([0,T];U_T)\) and \({\dot{u}}^*\in C_w^0([0,T];H)\), we get
Therefore, we get (4.19). As consequence of this, we derive
and this conclude the proof. \(\square \)
For the fractional Kelvin–Voigt’s model (2.9), we expect to have uniqueness of the solution, as it happens in [6, 24] for the classic Kelvin–Voigt’s one. Unfortunately, the technique used in the cited papers cannot be applied here, and we are able to prove it only when the crack is not moving (see Sect. 5). We point out that the uniqueness of the solution is still an open problem even for the pure elastic case (\({\mathbb {B}}=0\)), unless the family of cracks is sufficiently regular (see [2, 7]).
Moreover, according to the theory of dynamic fracture, we do not expect to have the equality in (4.8). Indeed, we should add also the energy used to the increasing crack, which is postulated to be proportional to the area increment of the crack itself, in line with Griffith’s criterion [12]. More precisely, we would like to have for every \(t\in [0,T]\)
However, with our approach we are not able to show the previous identity, which again is unknown even in the pure elastic case. We underline that there are no results regarding the validity of (4.20) for the fractional Kelvin–Voigt’s model (2.9) even when the crack is not moving.
5 Uniqueness for a not moving crack
Let us consider the case of a domain with a fixed crack, i.e., \(\Gamma _T=\Gamma _0\) (possibly \(\Gamma _T=\emptyset \)). In this case, we can show that the generalized solution to (2.9) is unique. As we explained in the introduction, uniqueness results for fractional type systems can be found in the literature, but they are proved only for regular sets \(\Omega \) (without cracks) and in particular cases (for \({{\mathbb {F}}}\) given by (1.7) or when eu is replaced by \(\nabla u\)).
The proof of the uniqueness is based on a particular energy estimate which holds for the primitive of a generalized solution. To this aim, we need to estimate
and we start with the following identity which is true for a regular tensor \({\mathbb {K}}\) (see also [26, Lemma 2.1]).
Lemma 5.1
Let \({\mathbb {K}}\in C^1([0,T];B)\) and \(v\in L^2(0,T;U_0)\). Then, for every \(t\in [0,T]\)
Proof
Let us fix \(t\in [0,T]\) and let us analyze the right-hand side of (5.1). We have
Notice that
and that for a.e. \(r\in (0,t)\)
from which we deduce
and thanks to the following relation
we can conclude the proof. \(\square \)
Lemma 5.2
Let \({\mathbb {F}}\) be satisfying (2.5)–(2.8) and \(u\in C_w^0([0,T];U_0)\). Then, for every \(t\in [0,T]\) it holds
Proof
First, we fix \(\epsilon \in (0,\delta _0)\) and we consider for every \(t\in [0,T]\) the following regularized kernel
Moreover, we fix \(t\in [0,T]\) and we define for every \(r\in [0,t]\) a primitive of u in the following way
Clearly, \({\mathbb {G}}^\epsilon \in C^2([0,T];B)\) and after an integration by parts, since \(ev(t)=0\), we obtain
Moreover, we have
Therefore, by (5.1) we can write
which implies
By sending \(\epsilon \rightarrow 0^+\), we conclude. \(\square \)
We can now state our uniqueness result.
Theorem 5.3
Assume (2.1)–(2.8) and \(\Gamma _T=\Gamma _0\). Then, there exists at most one generalized solution to system (2.9).
Proof
Let \(u_1,u_2\in {{\mathcal {C}}}_w\) be two generalized solutions to (2.9). Then, \(u:=u_1-u_2\) satisfies equality (2.10) with \(z=N=f=u^0=u^1=0\). Consider the function \(\beta :[0,T]\rightarrow (U_0^D)'\) defined for every \(r\in [0,T]\) as
for every \(v\in U^D_0\). Clearly, \(\beta \in C_w^0([0,T];(U^D_0)')\), \(\beta (0)=0\) since \({\dot{u}}(0)=0\) in \((U^D_0)'\), and by (2.10) we derive
Therefore, \(\beta \) is constant in [0, T], which gives \(\beta (t)=0\) in \((U^D_0)'\) for every \(t\in [0,T]\), namely for every \(v\in U_0^D\) and \(r\in [0,T]\)
In particular, for every \(t\in [0,T]\) we deduce
Hence, by (5.5) we conclude
Therefore, since both terms are nonnegative, we get that \(u(t)=0\) for every \(t\in [0,T]\).
\(\square \)
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Acknowledgements
The authors wish to thank Professors Gianni Dal Maso for the many useful discussions on the topic. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Caponi, M., Sapio, F. An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains. J. Evol. Equ. 21, 4095–4143 (2021). https://doi.org/10.1007/s00028-021-00713-2
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DOI: https://doi.org/10.1007/s00028-021-00713-2
Keywords
- Linear second-order hyperbolic systems
- Dynamic fracture mechanics
- Cracking domains
- Viscoelasticity
- Fractional Kelvin–Voigt
- Caputo’s fractional derivative