Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.


Introduction
In micro-electronic materials such as solder alloys, different physical processes are shaping the micro-structure. For a realistic description of these structures, phase separation, coarsening and elasticity as well as damage phenomena have to be taken into account. A fully coupled system has been originally studied in [HK11] and further developed in [HK13b] allowing, for instance, inhomogeneous elastic energy densities. The corresponding degenerating case has been analyzed in [HK12]. To the authors' best knowledge, before these works, phase separation and damage processes have only been investigated independently of each other in the mathematical literature.
Phase separation and coarsening phenomena are usually described by phase-field models of Cahn-Hilliard type. The evolution is modeled by a parabolic diffusion equation for the phase fractions. To include elastic effects, resulting from stresses caused by different elastic properties of the phases, Cahn-Hilliard systems are coupled with an elliptic equation in the case of a quasi-static balance of forces. Such coupled Cahn-Hilliard systems with elasticity are also called Cahn-Larché systems. Since in general the mobility, stiffness and surface tension coefficients depend on the phases (see for instance [BDM07] and [BDDM07] for the explicit structure deduced by the embedded atom method), the mathematical analysis of the coupled problem is very complex. Existence results were derived for special cases in [CMP00,Gar00,BP05] (constant mobility, stiffness and surface tension coefficients), in [BCD + 02] (concentration dependent mobility, two space dimensions), [SP13b,SP13a] (concentration dependent surface tension and nonlinear diffusion) and in [PZ08] in an abstract measure-valued setting (concentration dependent mobility and surface tension tensors).
Damage behavior, however, originates from breaking atomic links in the material from a microscopic point of view whereas a macroscopic theory may specify damage in the isotropic case by a scalar-valued variable related to the proportion of damaged bonds in the micro-structure of the material with respect to the undamaged ones. According to the latter perspective, phasefield models are quite common to model smooth transitions between damaged and undamaged material states. Such phase-field models have been mainly investigated for incomplete damage which means that damaged material cannot loose all its elastic energy.
Existence and uniqueness results for damage models of viscoelastic materials are proven in [BSS05] for scalar-valued displacements. Higher dimensional damage models are analytically investigated in [BS04,MR06,MT10,KRZ13,RR12] and, there, existence and regularity properties are shown. A coupled system describing incomplete damage, linear elasticity and phase separation appeared in [HK11,HK13b]. There, existence of weak solutions has been proven under mild assumptions, where, for instance, the stiffness tensor may be material-dependent and the chemical free energy may be of polynomial or logarithmic type. All these works are based on the gradient-of-damage model proposed by Frémond and Nedjar [FN96] (see also [Fré02]) which describes damage as a result from microscopic movements in the solid. The distinction between a balance law for the microscopic forces and constitutive relations of the material yield a satisfying derivation of an evolution law for the damage propagation from the physical point of view. In particular, the gradient of the damage variable enters the resulting equation and serves as a regularization term for the mathematical analysis as well as it ensures the structural size effect. Internal constraints are ensured by the presence of non-smooth operators (subdifferential operators) in the evolution system. Hence, in the case that the evolution of the damage is assumed to be uni-directional, i.e. the damage process is irreversible, the microforce balance law becomes a doubly-nonlinear differential inclusion.
The main aim of this paper is to generalize the results for hyperbolic-parabolic damage systems introduced in [HK13a] to coupled phase-field systems describing phase separation and damage processes in the presence of inertial terms with mixed boundary conditions on nonsmooth (Lipschitz) domains. The novelty of this contribution is to obtain existence results for phase separation with elasticity including inertial effects and damage processes on Lipschitz domains. We first utilize and adjust the notion of weak solutions introduced in [HK11]. Then, we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques. To this end, an energy estimate has, for instance, to be established and several convergence properties are shown.

Energies and evolutionary equations
Here, we qualify our model formally and postpone a rigorous treatment to Section 4. The presented model is based on two functionals, i.e. a generalized Ginzburg-Landau free energy functional E and a damage pseudo-dissipation potential R (in the sense by Moreau). The free energy density ϕ of the system is given by where the gradient terms penalize spatial changes of the variables c and z. W denotes the elastically stored energy density accounting for elastic deformations and damage effects, f is the damage dependent potential and Ψ stands for the chemical energy density.
The overall free energy E of Ginzburg-Landau type has the following structure: In this context, We assume that the energy dissipation for the damage process is triggered by a rate-dependent dissipation potential R of the form The governing evolutionary equations for a system state q = (u, c, z) can be expressed by virtue of the functionals (2) and (3). More precisely, the evolution is driven by the following hyperbolic-parabolic system of differential equations and differential inclusions: balance of forces: The Cahn-Hilliard system (4a)-(4b) describes phase separation phenomena in alloys, the hyperbolic equation (4c) formulates the balance of forces including inertial effects and the inclusion (4d)-(4g) is an evolution law for the damage processes. The sub-gradients correspond to the constraints that the damage is non-negative and irreversible. Let us note that linear contributions in f model damage activation thresholds. We choose Dirichlet conditions for the displacements u on a subset Γ of the boundary ∂Ω with H n−1 (Γ) > 0. Let b : [0, T ] × Γ → R n be a function which prescribes the displacements on Γ for a fixed chosen time interval [0, T ]. The imposed boundary and initial conditions and constraints are as follows: initial concentration : initial displacements : initial damage : Moreover, we use natural boundary conditions for the remaining variables on (parts of) the boundary: where ν stands for the outer unit normal to ∂Ω.
We like to mention that mass conservation of the system follows from the diffusion equation (4a) and (6b), i.e.
In the next section, we state the precise assumptions that are needed for a rigorous analysis. Section 3 presents the main results. We give a notion of weak solutions evolved from [HK13a] and state the existence theorem in Subsection 3.1. Since the proof is based on regularization techniques, we also give the weak notion and the associated existence result for the regularized system in Subsection 3.2. In the main part, Section 4, the existence proof is carried out first for the regularized case and then for the limiting case.

Notation and assumptions
Throughout this work, let p > n be a constant and let Ω ⊆ R n (n = 1, 2, 3) be a bounded Lipschitz domain. For the Dirichlet boundary Γ D and the Neumann boundary Γ N of ∂Ω, we adopt the assumptions from [Ber11], i.e., Γ D and Γ N are non-empty and relatively open sets in ∂Ω with finitely many path-connected components such that Γ D ∩ Γ N = ∅ and Γ D ∪ Γ N = ∂Ω.
The considered time interval is denoted by [0, T ] and Ω t := Ω× [0, t] for t ∈ [0, T ]. The partial derivative of a function h with respect to a variable s is abbreviated by h ,s . The set {v > 0} for a function v ∈ W 1,p (Ω) has to be read as {x ∈ Ω | v(x) > 0} by employing the embedding W 1,p (Ω) ֒→ C(Ω) (because p > n).
The elastic energy density W is assumed to be of the form where e * denotes the eigenstrain and C the material stiffness tensor which depends on the damage variable. For e * , we assume the linear relation e * (c) = cê withê ∈ R n×n sym (Vegard's law). We choose the stiffness tensor function C ∈ C 1 ([0, 1]; L sym (R n×n )), where L sym (R n×n ) denotes the linear mappings from R n×n into R n×n which are symmetric. We also assume the properties C(z)e : e ≥ η|e| 2 , C ′ (z)e : e ≥ 0 for all e ∈ R n×n sym , z ∈ [0, 1] and a constant η > 0 independent of e and z. Furthermore, we choose the mobility m ∈ C(R × [0, 1]; R + ) and suppose that the chemical energy density Ψ ∈ C 1 (R) can be decomposed into where Ψ 1 , Ψ 2 ∈ C 1 (R) with Ψ 1 convex and Ψ 1 ≥ 0.
In addition, we assume the following growth conditions: for all c ∈ R. Moreover, the mobility function should satisfy for all c ∈ R, z ∈ [0, 1]. Here, C 1 , C 2 > 0 denote constants independent of c and z, and 2 ⋆ is the Sobolev critical exponent.
The damage dependent potential f entering equation (4d) is assumed to be a function of C 1 ([0, 1]; R + ).

Notion of weak solutions and existence results
In what follows we define for k ≥ 1 the spaces Let the following initial-boundary data and volume forces be given: initial values: in Ω, external volume forces: l ∈ L 2 (0, T ; L 2 (Ω; R n )).
A weak formulation of system (4)-(6) is given in the following definition.
The main aim of this work is to prove existence of weak solutions in the sense above.
Theorem 3.3 Let the assumptions in Section 2 be satisfied. To the given data l, b, c 0 , u 0 , v 0 , z 0 , there exists a weak solution of system (4)-(6) in the sense of Definition 3.1.

Notion of weak solutions for a regularized system and existence results
We will first study a regularized version of our phase separation-damage model. The passage to the limit is performed in Section 4.2. The regularization is needed in the existence proof in the first instance to pass from the time-discrete to the time-continuous system. The regularized PDE system for δ > 0 is given by where the linear operator A : A weak formulation of the regularized system such as in Definition 3.1 can be obtained with the corresponding modifications including the δ-terms.
Definition 3.4 (Weak solution of the regularized system) A weak solution of the regularized PDE system for the data (l, b, c 0 , u 0 , v 0 , z 0 ) is a 5-tuple (c, u, z, µ, ξ) satisfying the following properties: • spaces: • for all ζ ∈ H 1 (Ω) and for a.e. t ∈ (0, T ): • for all ζ ∈ H 1 (Ω) and for a.e. t ∈ (0, T ): • for all ζ ∈ H 1 ΓD (Ω; R n ) and for a.e. t ∈ (0, T ): • for all ζ ∈ W 1,p − (Ω) and for a.e. t ∈ (0, T ): • for all ζ ∈ L ∞ + (Ω) and for a.e. t ∈ (0, T ): • total energy inequality for a.e. t ∈ (0, T ): with free energy: kinetic energy: The proof of the main result, see Theorem 3.3, is based on the existence of weak solutions for the regularized system. For the existence proof of the regularized system, we will use a semi-implicit Euler scheme solved by a recursive minimization procedure.
Let τ > 0 denote the discretization fineness and let M τ := ⌊T /τ ⌋ be the number of discrete time points. We fix a k ∈ 1, . . . , M τ and define the functional F k τ : We refer to [Gar00] for details. A minimizer of F k τ in the subspace obtained by the direct method in the calculus of variations is denoted by (c k τ , u k τ , z k τ ). More precisely, by a recursive minimization procedure starting from the initial values (c 0 , u 0 , z 0 ) and u −1 := u 0 − τ v 0 , we obtain functions (c k τ , u k τ , z k τ ) for k = 0, . . . , M τ . The velocity field v k τ is set to (u k τ − u k−1 τ )/τ and b k τ and l k τ are given by b(τ k) and l(τ k).
Proof. We split the proof into two steps. We first prove the a priori estimates (i), (ii) and (iv) and then we deduce estimate (iii). First a priori estimates. Testing (24) with τ µ τ , testing(25) with c τ − c − τ , testing (26) with , and adding everything, yield These terms are estimated in the following.
Second a priori estimates.
By applying Poincaré's inequality, standard weak and weakly-star compactness results to the above a priori estimates, we obtain the following convergence properties.
Strong convergence of a subsequence of {∇z τ } in L p (Ω T ; R n ) can be shown as in [HK13a] by a tricky approximation argument.
For a time discrete solution of the regularized system, we can prove the validity of an energy inequality of type (16) except the additional discretization error terms e 1 τ , . . . , e 4 τ which will turn out to converge to 0 in a certain sense as τ ց 0.
Lemma 4.5 (Discrete energy inequality) Let a time-discrete weak solution be given as in Lemma 4.1. Then the following energy estimate is satisfied for a.e. t ∈ (0, T ): with the discrete energies and the error terms Proof. We compute by using convexity of W with respect to e: We , apply (48), use further convexity arguments and end up with Using the convexity estimate Next we test equation (24) with τ µ τ and (25) with (c τ −c − τ ) and add the two derived equations. We obtain by means of the convexity property Adding the estimates (49)-(51), we end up with with the error terms e 1 τ (t), e 2 τ (t), e 3 τ (t) and e 4 τ (t). Summing over the discrete time points and taking into account the discrete integration by parts formula (28), we finally obtain the claim.
Proof of Theorem 3.5 We are going to establish the equalities and inequalities of the weak formulation (17)-(22).

• (Energy inequality)
To treat the energy inequality (47), we set Then, (47) is equivalent to Furthermore, by the a priori estimates, we observe that for all t ∈ [0, T ] and for all τ > 0 (along a subsequence τ k ). Next, we consider the lim inf τ ց0 of each term in (53) separately.
-By the already proven convergence properties and by lower semi-continuity arguments, we obtain where A is defined as A τ but c τ , u τ , z τ , v τ and b τ are substituted by their continuous limits. Note that this lim inf-estimate does not necessarily hold pointwise a.e. in t because, for instance, we do not know v τ (t) → v(t) weakly in L 2 (Ω; R n ) for a.e. t (see (41)).
-Let 0 ≤ t 1 ≤ t 2 ≤ T be arbitrary. By Fatou's lemma, by (44) and by a lower semi-continuity argument, we obtain lim inf Taking also (54) and the already known convergence properties into account, we obtain where B is defined as B τ but c τ , c τ , u τ , u τ , v − τ , z τ , z τ , µ τ and b τ are substituted by their continuous counterparts and ∂t bτ (t)−∂t bτ (t−τ ) τ by ∂ tt b(t).
-Due to the differentiability of C we have Hence, we obtain t 0 Ω 1 2 in Ω T as τ ց 0 we conclude by Lebesgue's generalized convergence theorem → 0 for every q ≥ 1.
The convergence can be shown as above.
-Noticing the linearity of e * , a short calculation yields Due to the already known convergence properties, we obtain and, consequently, E 2 τ (t) → 0 as τ ց 0. Together with the uniform boundedness (54), this implies can be shown by the following arguments: On the one hand, convexity of Ψ 1 yields On the other hand, by using the differentiability property of Ψ 2 , we obtain (cf. (60)) In the non-trivial case c − τ − c τ = 0, we can argue as follows. Since Indeed, it converges pointwise to 0 a.e. in Ω T and applying the mean value theorem . Therefore, the left hand side is bounded in L ∞ (0, T ; L 2 * (Ω)). Lebesgue's generalized convergence theorem yields (66). We end up with lim inf τ ց0 E 3 τ (t) ≥ 0 as τ ց 0. Fatou's lemma shows the claim.
Hence, we obtain existence of weak solutions in the sense of Definition 3.4.

Existence proof for the limit system
We now study the limit δ ց 0. For each δ > 0, we obtain a weak solution (c δ , u δ , z δ , µ δ , ξ δ ) in the sense of Definition 3.4.
Proof. From the energy inequality (22), we infer the second inequality of (i), the first two inequalities of (ii), (iii) and the second inequality of (iv). By considering (19), we get and, therefore, Due to Ω c δ (t) dx = const. and the boundedness of ∇c δ (t) L 2 (Ω) , we derive by Poincaré's inequality the first inequality of (i).