Analysis of a model coupling volume and surface processes in thermoviscoelasticity

We focus on a highly nonlinear evolutionary abstract PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of the adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction. We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-time solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation) of the problem. Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.


Introduction
This paper tackles the analysis of a PDE system describing a class of models where volume and surface processes are coupled. Our main example, and the motivation for our study, stems from a specific PDE system modeling adhesive contact, with frictional effects, in thermoviscoelasticity. A PDE system for contact with adhesion, friction, and thermal effects. Contact and delamination arise in many fields in solid mechanics: among others, we may mention here the application to (structural) adhesive materials in civil engineering, the investigation of earthquakes, and the study of layered composite structures within machine design and manufacturing. Indeed, the degradation of the adhesive substance between the various laminates leads to material failure. That is why, there is a rich literature on this kind of problems, both from the engineering and from the mathematical community: we refer to the monographs [15] and [21], and to the references in [4]- [9], for some survey.
In this paper, following up on the recent [9], we focus on a PDE system for frictional adhesion between a viscoelastic body, subject to thermal fluctuations, in contact with a rigid support. In [9] this system was derived, according to the laws of Thermomechanics, on the basis of the modeling approach proposed by M. Frémond (cf. [16,17,18]). Such approach has already been applied in the previous [4,5,6,7] to the investigation of adhesive contact, both for an isothermal and for a temperature-dependent system, as well as in [8], for isothermal adhesive contact with frictional effects. The model we consider here, encompassing friction, adhesion, and thermal effects, was analyzed in [9] in the case where no irreversibility of the degradation of the adhesive substance is enforced. In the present contribution, we broaden our investigation by encompassing in the model this unidirectionality feature. As we are going to demonstrate in what follows, this extension of the model from [9] brings about substantial analytical difficulties.
Let us now get a closer look at the PDE system under investigation. In accord with Frémond's modeling ansatz, adhesion is described in terms of an internal variable χ which can be interpreted as a surface damage paramater : it accounts for the state of the bonds between the body and the support, on which some adhesive substance is present. The other state variables are the strain tensor (in a small-strain framework), related to Date: January 31st, 2014. the displacement vector u, and the possibly different (absolute) temperatures ϑ and ϑ s in the body and on the contact surface. We allow for two different temperatures in the bulk domain and on the contact surface because we are modeling a physical situation in which some adhesive substance may be present on the contact surface, with different thermomechanical properties in comparison to the material in the domain. The evolution of (ϑ, ϑ s , u, χ ) is described by a rather complex PDE system, consisting of the quasi-static momentum balance (where the inertial contributions are neglected), a parabolic-type evolution equations for the temperature ϑ and ϑ s , and a doubly nonlinear differential inclusion for χ . Denoting by Ω ⊂ R 3 the (sufficiently regular) domain occupied by the body and by Γ C the part of the boundary ∂Ω on which the body may be in contact with the support, the system reads as follows ∂ t ln(ϑ) − div(∂ t u) − div(∇g(ϑ)) = h in Ω × (0, T ), (1.1a) ∇(g(ϑ)) · n = 0 in (∂Ω \ Γ C ) × (0, T ), (1.1b) (1.1k) In system (1.1), we have used the following notation: n is the outward unit normal to ∂Ω, for which we suppose ∂Ω = Γ D ∪ Γ N ∪ Γ C , with Γ D (Γ N , resp.) the Dirichlet (the Neumann) part of the boundary where zero displacement (a fixed traction) is prescribed. As for Γ C , we require that it is also sufficiently smooth (and denote by n s the outward unit normal to ∂Γ C ) and flat (cf. condition (2.1) below); for simplicity we shall write v, in place of v |Γ C , for the trace on Γ C of a function v defined in Ω. We also adopt the following convention: given a vector v ∈ R 3 , we denote by v N and v T its normal component and its tangential part, i.e. v N := v · n and v T := v − v N n. Analogously, the normal component and the tangential part of the Cauchy stress tensor σ (while ε(u) is the small-strain tensor), are denoted by σ N and σ T , with σ N := σn · n and σ T := σn − σ N n. Finally, the multivalued operator ∂I C : R ⇒ R (with C the interval [0, 1] or the half-line (−∞, 0]) is the subdifferential (in the sense of convex analysis) of the indicator function of the convex set C.
While referring the reader to [9,Sec. 2] for the rigorous derivation of system (1.1), let us now briefly comment on the features, and the meaning, of the single equations. The momentum balance (1.1f), where the elasticity and viscosity tensors K e and K v are positive-definite and satisfy suitable symmetry conditions and f is a given volume force, is supplemented by the boundary conditions (1.1g) on Γ D and Γ N (with g given), and by (1.1h)-(1.1i) on the contact surface Γ C . Observe that (1.1h) can be recast in complementarity form as (
(1. 5) In (1.4) and (1.5), R is a regularization operator with suitable properties, cf. Hypothesis 2.4. The regularized friction law resulting from the replacement (1.4) in (1.1i) can be interpreted as taking into account nonlocal interactions on the contact surface. We now dwell on the difficulties attached to the doubly nonlinear character of (1.1j), which is due to the inclusion of unidirectionality in the model. In order to prove the existence of a solution χ to a reasonably strong formulation of (1.1j), featuring two selections ζ ∈ ∂I (∞,0] (∂ t χ ) and ξ ∈ ∂I [0,1] ( χ ), it is essential to estimate the terms ∂I (∞,0] (∂ t χ ) and ∂I [0,1] ( χ ) (formally written as single-valued) separately in some suitable function space, in fact L 2 (0, T ; L 2 (Γ C )). Starting from the paper [3] on the analysis of a model for irreversible phase transition, it has been well known that such an estimate can be achieved by testing (1.1j) by the (formally written) term ∂ t (−∆ χ + ∂I [0,1] ( χ )) and employing monotonicity arguments. The related calculations also involve an integration by parts in time on the right-hand side of (1.1j), which in turn requires suitable time-regularity for ϑ s .
That is why, in order to carry out the crucial estimate for handling of the terms ∂I (∞,0] (∂ t χ ) and ∂I [0,1] ( χ ) in (1.1j), we will need to enhance the time-regularity of ϑ s . This can be done through a series of estimates, which partially rely on choosing in (1.1d) a function f tailored to the logarithmic nonlinearity therein.
It seems to us that the structure of these estimates, and the technical reasons underlying our hypotheses on the various nonlinearities of the system, can be highlighted upon developing our analysis on an abstract version of system (1.1). A generalization of system (1.1). Hereafter, we shall address the following PDE system coupling a volume process in a domain Ω ⊂ R 3 , with a surface process occurring on a portion Γ C of the boundary of Ω, which fulfills ∂Ω = Γ D ∪Γ N ∪Γ C . The surface process is described by a suitable internal variable χ , and thermal effects, in the bulk and on the surface, are accounted for through the temperature variables ϑ and ϑ s . Accordingly, we have ∂ t (L(ϑ)) − div(∂ t u) − div(∇g(ϑ)) = h in Ω × (0, T ), (1.6a) Observe that the temperature equations (1.6a) and (1.6c) are a generalization of the "concrete" equations (1.1a) and (1.1d). The logarithms therein have been replaced by two (possibly different) maximal monotone (singlevalued) operators L and ℓ fulfilling suitable conditions. The function g is a strictly increasing, bi-Lipschitz, and otherwise general. Instead, f depends on the choice of ℓ, as it is defined by Admissible choices are, for instance Nonetheless, in the second case the strict positivity of the temperature is no longer directly ensured by the form of ℓ like in the first case. Note moreover that the terms ∂I (−∞,0] (u N ) and d(∂ t u) in system (1.1) have been replaced by the subdifferentials ∂Φ(u) and ∂Ψ(∂ t u) in (1.6g), and accordingly in (1.6b) and (1.6c) (with R the regularization operator used in the analysis of frictional problems). Here, Φ and Ψ are (possibly nonsmooth) positive, lower semicontinuous, and convex functionals, and in addition Ψ is positively homogeneous of degree 1, i.e. Ψ(lv) = lΨ(v) for all l ≥ 0 and v ∈ R 3 . It turns out that the crucial requirement for tackling the simultaneous presence of the two terms ∂Φ(u) and ∂Ψ(∂ t u) in (1.6g) is that ∂Φ(u) and ∂Ψ(v) are orthogonal for all u, v ∈ R 3 , cf. Hypothesis 2.3, which is obviously fulfilled in the case of system (1.1).
Finally, in (1.6h) ρ and β are two convex and lower semicontinuous functionals, such that dom( β) ⊂ [0, +∞) in such a way as to ensure the positivity of the internal variable χ , which is also crucial for the analysis of system (1.6) and guarantees the physical consistency of the phase variable. An existence result for system (1.6). The main result of this paper, Theorem 2.7, states the existence of solutions to the (Cauchy problem for a) variational formulation of (1.6), in an appropriate functional framework which reflects the energy estimates for this system. This variational formulation is given in Section 2, where all the hypotheses on the various nonlinearities of the system and on the problem data are collected.
The proof follows by passing to the limit in a carefully devised approximate system, where some of the multivalued subdifferential terms featured in (1.6) are replaced by their Yosida regularizations. In particular, the possibly singular functions L and ℓ are regularized, and in addition the viscous terms ε∂ t ϑ and ε∂ t ϑ s are included into the left-hand sides of (1.6a) and (1.6c), thereby enhancing the time-regularity of the ϑand ϑ s -components of the approximate solutions. That is why, this procedure requires a technically delicate construction of approximate initial data for ϑ and ϑ s .
In Section 3 we develop the set-up of the approximate problem, we state its variational formulation and prove a local-in-time existence result for the approximate system via a Schauder fixed point argument. Then, we derive in Section 4 a series of a priori estimates on the approximate solutions. Relying on them, in Section 5 we perform the passage to the limit with the approximate solutions via refined compactness and lower semicontinuity arguments. We then obtain in Theorem 5.1 the (still local-in-time) existence of solutions to (1.6). In this passage to the limit we have to deal with a significant difficulty stemming from the coupling between thermal and frictional effects in the model. This is the dependence of the friction coefficient c on the thermal gap ϑ − ϑ s . To tackle the passage to the limit in the approximation of the terms c ′ (ϑ − ϑ s )|R(∂Φ(u))|Ψ(∂ t u) in (1.6b) and (1.6c), and c(ϑ − ϑ s )|R(∂Φ(u))|∂Ψ(∂ t u) in (1.6g), it is essential to prove strong compactness for (the sequences approximating) ϑ and ϑ s in suitable spaces. The key step for ϑ is to derive an estimate in BV(0, T ; W 1,3+ǫ (Ω) ′ ) for all ǫ > 0, which enables us to apply a suitable version of the Lions-Aubin compactness theorem generalized to BV spaces. As for ϑ s , exploiting condition (1.7), we are indeed able to obtain an a priori bound for ϑ s in H 1 (0, T ; L 2 (Γ C )). As previously mentioned, this enhanced time-regularity estimate for ϑ s plays a crucial role in the derivation of estimates for the terms ∂ ρ(∂ t χ ) and ∂ β( χ ) in (1.6h). Remark 1.1. This mismatch in the time-regularity properties of ϑ and ϑ s , as well as the fact that (as we are dealing with possibly different thermal properties of the substances in the domain and on the contact surface) we can allow for different choices of the functions L and ℓ, highlights that the temperature equations (1.6a) and (1.6c) have a substantially different character. In fact, in (1.6a) the function g can be fairly general, whereas in (1.6c) the function f needs to be chosen of the form (1.7), in order to allow for the enhanced time-regularity estimate for ϑ s . Such an estimate could not be carried out on equation (1.6a), due to the mixed boundary conditions (1.6b), which are also responsible for the low spatial regularity of ϑ.
The last step in our existence proof consists in the extension of the local-in-time solution to the Cauchy problem for system (1.1). This prolongation procedure follows the lines of an extension argument from [4].
Indeed, for technical reasons that shall be expounded at the beginning of Section 6, it is necessary to extend the local-in-time solution, whose existence is guaranteed by Theorem 5.1, along with its approximability properties. This makes the prolongation argument rather complex, that is why we have devoted to it the whole Sec. 6. Finally, in the Appendix we collect a series of auxiliary results, among which some lemmas addressing the approximation of the initial data for ϑ and ϑ s and the properties of the functions regularizing the nonlinearities of the system.

Main result
Before stating the analytical problem we are solving and the corresponding existence result, we first set up the notation and the assumptions.
Throughout the paper we shall assume that Ω is a bounded Lipschitz domain in R 3 , with open disjoint subsets in the relative topology of ∂Ω, such that and Γ C ⊂ R 2 a sufficiently smooth flat surface. (2.1) More precisely, by flat we mean that Γ C is a subset of a hyperplane of R 3 and H 2 (Γ C ) = L 2 (Γ C ), L d and H d denoting the d-dimensional Lebesgue and Hausdorff measures, respectively. As for smoothness, we require that Γ C has a C 2 -boundary.
Notation 2.1. Given a Banach space X, we denote by ·, · X the duality pairing between its dual space X ′ and X itself, and by · X the norm in X. In particular, we shall use the following short-hand notation for function spaces , is analogously defined. We will also use the space H 1/2 00,Γ D (Γ N ; R 3 ). The space W is endowed with the natural norm induced by V. We will make use of the operator and of the notation Linear viscoelasticity. We are in the framework of linear viscoelasticity theory (see e.g. [9] for some more details). In particular, we prescribe that the fourth-order tensors K e and K v (denoting the elasticity and the viscosity tensor, respectively) are symmetric and positive definite. Moreover, we require that they are uniformly bounded, in such a way that the following bilinear symmetric forms a, b : are continuous. In particular, we have Moreover, since Γ D has positive measure, by Korn's inequality we deduce that a(·, ·) and b(·, ·) are W-elliptic, i.e., there exist C a , C b > 0 such that Assumptions. In order to tackle the analysis of the PDE system (1.6), we require the following. L ∈ C 1 (D(L)) and 1 L ′ ∈ C 0,1 (D(L)).
(2.6b) Moreover, denoting by J a primitive of L, we impose that the Fenchel-Moreau convex conjugate J * of J (recall that its derivative coincides with the inverse function L −1 ), fulfills the following coercivity condition Analogously, we assume for ℓ as well as, again, the coercivity condition where j * is the Fenchel-Moreau convex conjugate of j, j denoting a primitive of ℓ.
A straightforward consequence of (2.6) and of (2.7) is that hence L and ℓ are invertible.
Furthermore, it is not restrictive to suppose that 0 ∈ D(J) with J(0) = 0 (2.8) (the latter relation is trivially obtained with a translation argument), and the same for j. Since D(ℓ) = D(j), this in particular implies that 0 ∈ D(ℓ) , (2.9) which will be convenient for the definition of f and f ε later on.
2. An example for L in accord with conditions (2.6a)-(2.6c) is In this case, with, e.g. J(ϑ) = ϑ(ln(ϑ) − 1) for all ϑ ∈ (0, +∞) as primitive of L, we see that J * (w) = e w = L −1 (w) for all w ∈ R, and (2.6c) is satisfied. This choice of L is particularly meaningful from a modeling viewpoint, since it enforces that ϑ > 0, in accord with its interpretation as the absolute temperature. Clearly, ℓ(ϑ s ) = ln(ϑ s ) is also an admissible choice for ℓ. As already mentioned in the introduction, let us point out that, in fact, for our analysis we do not need the positivity of ϑ and ϑ s (namely, that D(L), D(ℓ) ⊂ (0, +∞)). Hence, other admissible choices for L and ℓ are L(ϑ) = ϑ, ℓ(ϑ s ) = ϑ s with D(L) = D(ℓ) = R. (2.11) Let us also stress that, in system (1.6) we can in principle combine two distinct choices for L and ℓ. Hypothesis 2.2. As far as the functions g and f are concerned, we impose that As for f , as previously mentioned we require that it is is the primitive of 1 ℓ ′ ; in view of (2.9), we set Clearly, the function f depends on the choice for ℓ. For example, s , Hypothesis 2.3 (The subdifferential operators in the momentum balance equation). We suppose that Ψ : R 3 → [0, +∞) is convex, non-degenerate, and positively 1-homogeneous (2.14) i.e. Ψ satisfies (in fact, under positive homogeneity of degree 1, sublinearity is equivalent to convexity). As for the function Φ, we assume that Φ : R 3 → [0, +∞] is proper, convex and lower semicontinuous, with Φ(0) = 0 (2.15) and effective domain dom(Φ). We impose the following "compatibility" condition between the respective subdifferential operators ∂Ψ : R 3 ⇒ R 3 and ∂Φ : R 3 ⇒ R 3 : In the variational formulation of system (1.6) (cf. (2.36) ahead), in fact we are going to make use of the abstract realization of Φ as a functional on Y Γ C , viz.
Since ϕ : Y Γ C → [0, +∞] is a proper, convex and lower semicontinuous functional on Y Γ C , its subdifferential ∂ϕ : Y Γ C ⇒ Y ′ Γ C is a maximal monotone operator. With a slight abuse of notation, we will use the symbol η not only for the elements of ∂Φ, but also for those of ∂ϕ.
Instead, in formulation (2.36) we are going to stay with the "concrete" subdifferential operator ∂Ψ : H Γ C ⇒ H Γ C , which with a slight abuse we denote in the same way as the operator ∂Ψ : R 3 ⇒ R 3 inducing it. It follows from (2.14) (observe that dom(Ψ) = R 3 ), that ∂Ψ : Clearly, the orthogonality condition (2.16) is fulfilled in this case. Nonetheless, let us highlight that (2.16) allows for much more generality: for example, ∂Φ(u) and ∂Ψ(v) might be of the form with w 1 (u) and w 2 (v) depending on u and v, respectively, and such that Hypothesis 2.4 (The regularizing operator R). Following [8,9] we require that there exists ν > 0 such that ) is weakly-strongly continuous, viz.
) is bounded. We refer to [9, Example 3.2] for the explicit construction of an operator R complying with (2.20).
Observe that, with a translation we can always confine ourselves to the case in which ρ(0) = 0 = min x∈R ρ(x), therefore we may also suppose that 0 ∈ ρ(0).
Assumptions on the problem and on the initial data. We require that For later convenience, we remark that, thanks to (2.28b)-(2.28c) the function F : (0, T ) → W ′ defined by for all v ∈ W and almost all t ∈ (0, T ), For the initial data we impose that J * (L(ϑ 0 )) ∈ L 1 (Ω) and L(ϑ 0 ) ∈ H , Concerning the initial data ϑ 0 and ϑ 0 s , we observe that the first of (2.30a) implies ϑ 0 ∈ L 1 (Ω), in view of (2.6c). Moreover, the enhanced regularity (2.30b) required of ϑ 0 s reflects that we shall obtain a higher temporal regularity for ϑ s than for ϑ, see Theorem 2.7 ahead.

2.3.
Variational formulation of the problem and main result. We are now in the position to detail the formulation for the initial-boundary value problem associated with system (1.6). Observe that, while the temperature equations (1.6a) and (1.6c) and the momentum equation (1.6e) need to be formulated in dual spaces, the equation (1.6h) for the internal variable χ can be given a.e. in Γ C × (0, T ), due to the H 2 (Γ C )regularity obtained for χ . Problem 2.5. Given a quadruple of data (ϑ 0 , ϑ 0 s , u 0 , χ 0 ) fulfilling (2.30), find (ϑ, ϑ s , u, χ , η, µ, ξ, ζ), with and satisfying Definition 2.6. We call a solution to Problem 2.5 energy solution if, in addition, it satisfies the energy inequality (2.37) for almost all 0 ≤ s ≤ t ≤ T , and and for s = 0.
With our main result, Theorem 2.7, we state the existence of an energy solution to Problem 2.5, with the additional properties (2.38) below. Namely, ϑ has bounded variation, as a function of time, with values in some dual space: this in particular ensures that t → ϑ(t) is continuous, with values in that space, at almost all points t 0 ∈ (0, T ). For ϑ s we gain a better regularity, cf. (2.38b) and (2.38c), as a result of a further regularity estimate on the equation for ϑ s . Such an estimate also implies (2.38d)-(2.38f).
Then, Problem 2.5 admits an energy solution (ϑ, ϑ s , u, χ , η, µ, ξ, ζ), which in addition satisfies Outline of the proof. We set up a suitable approximation of system (2.36) by regularizing some of the (maximal monotone) operators featured therein; we shall denote the regularization parameter with the symbol ε and accordingly refer to the approximate problem as (P ε ). In Section 3 we prove the existence of local-intime solutions to Problem (P ε ) (cf. Proposition 3.3 ahead). Then, we show that the approximate solutions fulfill an energy identity, which serves as the basis for deriving a series of a priori estimates, uniform w.r.t. ε. Relying on them, in Section 5 we prove that, along a suitable subsequence, the approximate solutions converge to a local-in-time solution to Problem 2.5. Its extension to a global-in-time solution by means of a careful prolongation argument is the focus of Section 6. Some useful technical lemma we will use in the proofs are stated and proved in the Appendix.

Approximation
First, in Sec. 3.1 we introduce and explain our approximation of system (2.36), leading to Problem (P ε ); in the end, we state its variational formulation. The existence of a local-in-time solution to (P ε ) is proved in Sec. 3.2 via the Schauder fixed point theorem. Most of the calculations for the (uniform w.r.t. ε) a priori estimates on the approximate solutions which we shall derive in Sec. 5 hinge on a series of technical results on the functions approximating the nonlinearities of the problem, which we have collected in the Appendix.
3.1. The approximate problem. To motivate the regularization procedures we are going to adopt, we discuss in advance some of the a priori estimates we shall perform on system (2.36) in Sec. 5. As we will see, the related calculations cannot be performed rigorously on system (2.36), and indeed necessitate of the Yosida-type regularizations by which we are going to replace some of the maximal monotone nonlinearities in system (2.36). Outlook to the approximate problem. The basic energy estimate for system (2.36) (cf. the First a priori estimate in Sec. 4) follows by testing (2.36a) by ϑ, (2.36b) by ϑ s , (2.36c) by ∂ t u, (2.36f) by ∂ t χ , adding the resulting relations, and integrating in time. The formal calculations where we have used the auxiliary variable w := L(ϑ), the formal identity and, finally, the coercivity condition (2.6c), lead to a bound for ϑ in L ∞ (0, T ; L 1 (Ω)). The corresponding calculations on the level of (2.36b) yield an estimate for ϑ s in L ∞ (0, T ; L 1 (Γ C )).
Therefore, L ε ( ℓ ε , respectively) is differentiable, strictly increasing and Lipschitz continuous, see also the upcoming Lemma A.4. Nonetheless, this procedure only partially serves to the purpose of rigorously justifying (3.1), as expounded in Remark 4.1 at the end of Section 4. In accord with (2.13) and (3.3), (2) we thus replace the function f in (2.36b) by Combining the aforementioned energy estimate and a comparison argument in the momentum equation (2.36c) leads to the following estimate This follows from the orthogonality condition (2.16) only on a formal level, since (2.16) is not, in general, inherited by the abstract operator ∂ϕ : In order to justify this argument, we need to suitably approximate ∂ϕ : Γ C in such a way as to replace η ∈ Y ′ Γ C in (2.36c) with a term η ε orthogonal to ∂Ψ(∂ t u). Along the lines of [8,9], we (3) approximate the function Φ 2.15, which defines the functional ϕ through (2.17), by its Yosida approximation Φ ε : R 3 → [0, +∞).
We recall that Φ ε is convex, differentiable, and such that DΦ ε is the Yosida regularization of the subdifferential ∂Φ : where r ε denotes the resolvent of the operator ∂Φ. Therefore, in view of (2.16), any approximate solution u ε satisfies which will be crucial in order to deduce (3.6), cf. the Third a priori estimate in Sec. 4. Finally, along on the lines of [3] we will also perform on the doubly nonlinear equation (2.36f) the test by (the formal quantity) ∂ t (A χ + β( χ )). Let us mention that such an estimate is by now classical for this kind of doubly nonlinear diffusive evolutionary differential inclusions. It allows one to estimate the terms A χ and ξ ∈ β( χ ), separately, in L ∞ (0, T ; L 2 (Γ C )). Let us stress that this estimate requires ad hoc time-regularity for the right-hand side terms. Once the computations have been carried out, an estimate for ζ ∈ ρ(∂ t χ ) in L 2 (0, T ; L 2 (Γ C )) then follows from a comparison in (2.36f). In order to perform all the calculations in a rigorous way (cf. the Seventh a priori estimate in Sec. 4), it is necessary to (4) replace ρ and β by their Yosida approximations ρ ε and β ε .
Furthermore, it will be convenient to use the functions (cf. in particular the derivation of the approximate energy identity (3.52) later on), and the function H ε : (cf. the derivation of Sixth and Seventh a priori estimate in Sec. 4). Finally, we will supplement our approximate Problem (P ε ) by the initial data (ϑ ε 0 , ϑ 0,ε s , u 0 , χ 0 ), where the family approximates the data (ϑ 0 , ϑ 0 s ) from (2.30a)-(2.30b) in the sense that Observe that (3.15) and (3.16) guarantee that Finally, we also require that the family (ϑ 0,ε s ) ε fulfills Observe that, since f ε is bi-Lipschitz (cf. Lemma A.5 later on), (3.20) in fact implies that ϑ 0,ε s is also in V Γ C . In the Appendix we state a series of Lemmas, in which we will construct sequences of initial data (ϑ ε 0 , ϑ 0,ε s ) ε complying with the properties (3.12)-(3.20) and detail how the constantsS i , i = 1, · · · , 4 may depend on the data ϑ 0 and ϑ 0 s . All in all, the variational formulation of the approximate problem reads: and such that (u, χ , µ) comply with (2.31f), (2.38d), and (2.31i), satisfying the initial conditions Let us only briefly comment on the enhanced regularity properties (3.21)-(3.25). Since, for ε > 0 fixed, Analogous arguments apply to (3.23)-(3.25). Observe that ϑ s also inherits the regularity of f ε (ϑ s ), since f ε is bi-Lipschitz as well, cf. Lemma A.5.

3.2.
Local existence of approximate solutions. The main result of this section is the forthcoming Proposition 3.3, stating the existence of a local-in-time solution to Problem (P ε ). The latter features a structure very similar to the approximate problem for the PDE system analyzed in [9], cf. Problem 4.4 therein. Indeed, some of the arguments from [9] may be easily adapted to the present setting. Therefore, we only sketch the fixed point procedure yielding local existence. In particular, we only hint to the most relevant steps in the construction of the fixed point operator and in the proof of its continuity and compactness, referring to [9,Sect. 4.2] for all details. Fixed point setup. In view of hypothesis (2.20) on the regularizing operator R, we may choose δ ∈ (0, 1) such that R : ) is weakly-strongly continuous (3.29) (and therefore bounded). For a fixed τ > 0 and a fixed constant M > 0, we consider the set We are going to construct an operator T mapping Y T into itself for a suitable time 0 ≤ T ≤ T , depending on M , in such a way that any fixed point of T yields a solution to Problem (P ε ) on the interval (0, T ).
Notation 3.2. In the following lines, we will denote by S i , i = 1, ..., 5, positive constants depending on the problem data and on M > 0 in (3.30), but independent of ε > 0, and by S 6 (ε) a constant depending on the above quantities and on ε > 0 as well. Furthermore, with the symbols π i (A), π i,j (A), . . . , we will denote the projection of a set A on its i-, or (i, j)-component.

(3.31)
A well-posedness result for such a problem can be obtained easily adapting the arguments of the proof of [9, Lemma 4.6], observing that (3.31) has the very same structure of the corresponding momentum equation tackled in [9] (cf. Hypothesis 2.3 on the subdifferential operators). Then, there exists a constant S 1 > 0 and a unique pair (u, µ) ∈ H 1 (0, τ ; W) × L ∞ (0, τ ; L 2+ν (Γ C ; R 3 )) fulfilling the initial condition (2.34), equation (3.31) and the estimate For later convenience, let us detail the proof of the estimate for u H 1 (0,τ ;W) ; the estimate for z simply derives from (2.18), while the bound for µ follows from the calculations for the forthcoming Third a priori estimate, cf. Sec. 4. We choose v = ∂ t u in (3.31) and integrate in time over (0, t). In particular, we exploit the ellipticity properties (2.5) and integrate by parts. It follows from the Hölder inequality Note that here we have exploited the fact that in view of (2.25). Hence, the right-hand side of (3.33) can be handled by Young's inequality combined with trace theorems and Sobolev embeddings, which give Exploiting the Gronwall lemma, recalling (2.29) for F and (3.30) (in particular that χ 2 we infer that u is bounded in H 1 (0, τ ; W) by some constant S 1 depending on the data of the problem (and on M ) but not on ε. As a consequence, we may define an operator which maps every quadruple ( ϑ, ϑ s , u, χ ) ∈ Y τ into the unique solution u of the Cauchy problem for (3.31) (with associated µ ∈ |R(DΦ ε ( u))|∂Ψ(∂ t u T )).
Step 2: As a second step, we consider (the Cauchy problem for) (3.28e), with ϑ s ∈ π 2 (Y τ ) and u = T 1 ( ϑ, ϑ s , u, χ ) in U τ on the right-hand side, that is Standard results in the theory of parabolic equations (recall the Lipschitz continuity of β ε and ρ ε , as ε is fixed) ensure that there exists a constant S 2 > 0 (depending on M via S 1 ), and a unique function It follows that we may define an operator mapping ( ϑ s , u) ∈ π 2 (Y τ ) × U τ into the unique solution χ of the Cauchy problem for (3.37).
The well-posedness for the Cauchy problem for the doubly nonlinear equation (3.42) follows from standard results (cf. e.g. [23,Chap. 3,Thm. 4,1]), taking into account that both L ε and g are bi-Lipschitz. We only sketch here the uniqueness proof for (3.42). We subtract the equation for ϑ 2 from the equation for ϑ 1 and integrate on (0, t), with 0 ≤ t ≤ τ . Hence, we choose v = g(ϑ 1 ) − g(ϑ 2 ) as test function and integrate again in time. Using that L ε (r) = εr + L ε (r) and that the functions L ε and g are strictly increasing (cf. also (2.6a)-(2.6b), (2.12)), we obtain so that also g(ϑ) ∈ L 2 (0, τ ; V ). Thus, recalling (3.36) as well, by comparison in (3.42), we get Analogously, we handle equation (3.43) taking into account the monotonicity, the bi-Lipschitz continuity of ℓ ε and f ε and the coercivity of j * ε (cf. Lemma A.2 and Lemma A.4). In particular, in order to conclude suitable estimates for ϑ s , we test (3.43) by ϑ s and argue as in the derivation of the forthcoming First a priori estimate. We obtain ϑ s L ∞ (0,τ ;L 1 (Γ C )) + ϑ s L ∞ (0,τ ;HΓ C ) ≤ S 4 (3.48) for some constant S 4 > 0. Moreover, we test (3.43) by ℓ ε (ϑ s ). Proceeding as in the upcoming Second a priori estimate, we deduce for some constant S 5 > 0. Observe that the latter estimate holds uniformly w.r.t ε > 0 and this property will be crucial to deduce that the local-existence time does not depend on T > 0. Finally, by comparison in (3.43), taking into account that f ′ ε is bounded by a constant depending on ε (cf. (3.4) and (A.18)), we find for some constant S 6 (ε) > 0 depending on ε > 0 as well.
Therefore we may define an operator mapping ( ϑ, ϑ s , u, χ ) ∈ π 1,2 (Y τ ) × U τ × X τ into the unique solution (ϑ, ϑ s ) of the Cauchy problem for system (3.42)-(3.43)). We are now in the position to prove the existence of local-in-time solutions to Problem (P ε ), defined on some interval [0, T ] with 0 < T ≤ T . We stress that T in fact will not depend on the parameter ε > 0, and such a property will be crucial in the forthcoming passage to the limit procedure. Proposition 3.3 (Local existence for Problem (P ε )). Assume (2.1), Hypotheses 2.1-2.6, and conditions (2.28) on the data h, f , g, (2.30c)-(2.30d) on u 0 , χ 0 , and (3.12)-(3.20) on ϑ ε 0 , ϑ 0,ε s . Then, there exists T ∈ (0, T ] such that for every ε > 0 Problem (P ε ) admits a solution (ϑ, ϑ s , u, χ , µ) on the interval (0, T ), fulfilling the (approximate) energy identity We will prove (3.54) and (3.55) following the lines of the proof of [9, Proposition 4.9]. We shall not repeat the arguments leading to (3.55), as they are completely analogous to those in [9]. We only observe that, in the proof of the continuity of the operator T 2 from (3.39), providing the solution of the Cauchy problem for (3.37), the limit passage in the term ρ ε (∂ t χ ) can be easily handled in the very same way as in the forthcoming Sec. 5, cf. (5.40) later on. Arguing as in [9,Proposition 4.9], compactness and continuity of the operator T (3.53) in the (ϑ, ϑ s )-component can be proved by first deriving compactness for ( L ε (ϑ), ℓ ε (ϑ s )) (exploiting also estimates (3.47) and (3.50)). Since, for ε > 0 fixed, L ε and ℓ ε are bi-Lipschitz (cf. Lemma A.4), we can then infer compactness in (ϑ, ϑ s ).
Integrating by parts in time t s Γ C χ u·u t dx dr we also have a cancellation with the term − 1 2 t s Γ C χ t |u| 2 dx dr from (3.28e). We also use the formal chain rule (cf. Remark 4.1) and the same for the term holds rigorously. Furthermore, we exploit that z · ∂ t u = Ψ(∂ t u) a.e. in Γ C × (0, T ) by the 1-homogeneity of the functional Ψ. This gives rise to the tenth term on the left-hand side of (3.52). Then, we conclude (3.52).

Uniform w.r.t. ε a priori estimates
In this section, we perform a series of priori estimates on the solutions to Problem 3.1 (i.e. Problem (P ε )). From them, we derive bounds on suitable norms of the local solutions, which hold uniformly w.r.t. the parameter ε ∈ (0, 1). Exploiting them, we will pass to the limit with ε in Problem 3.1 (P ε ) and conclude in Theorem 5.1 the existence of a local-in-time solution to Problem 2.5.
Let us mention in advance that the forthcoming a priori estimates are not, however, global in time. This local character manifests itself already with the First a priori estimate (i.e. the energy estimate), which derives from the approximate energy identity (3.52). More precisely, the problem is to estimate the left-hand side term therein. Indeed, since in Problem (P ε ) the maximal monotone operator β has been replaced by its Yosida regularization β ε , the approximate solution χ is no longer guaranteed to be positive a.e. in Γ C ×(0, T ). Therefore the term in (4.1) is not, a priori, estimated from below by a constant. On the other hand, it cannot be moved to the right-hand side of (3.52) and absorbed into the left-hand side by Hölder and Young inequalities, or by use of the Gronwall lemma, mainly due to a lack of regularity of the terms in the left-hand side.
That is why, in order to estimate (4.1) we will resort to the following argument. It follows from the fixed point procedure in Sec. 3.2 that the local solution (ϑ, ϑ s , u, χ ) whose existence we have proved in Prop. 3.3 belongs to Y T from (3.30), whence where M does not depend on ε (see (3.30)). In addition, as estimates (3.32) and (3.38) do not depend on ε, we infer Exploiting (4.3) as well as well-known embedding and trace results, we then have where c depends here on S 1 , S 2 , on T , but not on ε > 0. Clearly, since the First a priori estimate is not global in time, neither of the subsequent estimates is. Nonetheless, for notational simplicity in the following calculations we shall write T in place of the localexistence time T . Moreover, we shall omit to indicate the dependence on ε for the approximate solutions. We also mention that with the symbols c and C we will denote possibly different positive constants, depending on the data of the problem, but not on ε. First a priori estimate. We consider the approximate energy identity (3.52) on the interval (0, t), with t ∈ (0, T ) (i.e. in (0, T )). Thanks to Lemma A.2 (cf. (A.9b) and (A.9c)) we have while (3.17) and (3.18) ensure that Since g ′ is bounded from below by a strictly positive constant (see (2.12)), and so is f ′ ε (cf. Lemma A.4), we also have We observe that the term t 0 Γ C k( χ )(ϑ − ϑ s ) 2 dx ds is positive thanks to (2.24), and so are the ninth, the tenth and the eleventh terms on the left-hand side of (3.52) by (2.15) and (2.25), respectively , i.e.
In particular, we (formally) have Thus, taking into account the monotonicity of β ε and ρ ε , we obtain HΓ C ≤ C + I 4 + I 5 + I 6 + I 7 + I 8 , where we have used the place-holder ξ := β ε ( χ ). Now, systematically using estimate (A.21), we have (cf. also (2.24)-(2.27)), f ε (ϑ s ) VΓ C + 1 dr , for some small δ > 0, and We plug the above estimates for I 4 , . . . , I 8 into (4.32), exploit the previously obtained bounds and apply the Gronwall Lemma. In this way, we conclude that We deduce moreover χ L ∞ (0,T ;H 2 (Γ C )) + ξ L ∞ (0,T ;HΓ C ) ≤ c, (4.34) yielding (by a comparison in (3.28e)) Remark 4.1 (A fully rigorous derivation of the a priori estimates). As already mentioned, the First and Fifth estimates are not yet rigorously justified in the framework of the approximate Problem 3.1. This is due to a lack of regularity for the term ∂ t L ε (ϑ ε ), which is only in L 2 (0, T ; V ′ ).
In order fully justify them, we should add a further viscosity contribution to the equation for ϑ, modulated by a second parameter ν > 0, hence perform a double passage to the limit, first as ν ↓ 0 and secondly as ε ↓ 0.
In the present paper we have chosen not to explore this, to avoid overburdening the analysis. We refer the reader to [6], where this procedure has been carried out in detail.

Local existence for Problem 2.5
In this section, we pass to the limit as ε ↓ 0 in Problem 3.1 and deduce from the local existence result in Proposition 3.3 the following Theorem 5.1. Assume (2.1) and Hypotheses 2.1-2.6. Suppose that the data (h, f , g) and (ϑ 0 , ϑ 0 s , u 0 , χ 0 ) fulfill (2.28) and (2.30).
For the latter task, we proceed as in [9] and test (3.28c) by u ε . By the previous convergences, lower semicontinuity arguments, and the fact that the limiting sextuple (ϑ, ϑ s , u, χ , µ, η) fulfills (2.36c), it is straightforward to check that lim sup for all t ∈ (0, T ). We use (5.22) to deduce that for all v ∈ Y Γ C there holds where the third inequality is due to the fact that Φ ε Mosco converges to Φ. We have thus shown that Instead of directly proving that z ∈ ∂Ψ(u t ) a.e. in Γ C × (0, T ), we will show that for all w ∈ L 2 (0, T ; L 4 (Γ C ; R 3 )), where the functional J (ϑ,ϑs,η) : L 2 (0, T ; L 4 (Γ C ; R 3 )) → [0, +∞) is defined for all v ∈ L 2 (0, T ; L 4 (Γ C ; R 3 )) by Clearly, J (ϑ,ϑs,η) is a convex and lower semicontinuous functional on L 2 (0, T ; L 4 (Γ C ; R 3 )). It can be easily verified that the subdifferential ∂J (ϑ,ϑs,η) : for almost all (x, t) ∈ Γ C × (0, T ). Therefore, (5.24) will yield whence (2.36e) for µ = |R(η)|z, also in view of (2.25). In order to show (5.24), we first observe that lim sup which can be checked by testing (3.28c) by ∂ t u ε and passing to the limit via the above convergences and lower semicontinuity arguments, and again the Mosco convergence of Φ ε to Φ. Therefore, by (5.27), the previously obtained weak-strong convergence we have (5.28) Then, (5.24) ensues. Furthermore, arguing as in the derivation of (5.28), a.e. in Γ C × (0, T ), we deduce lim inf Ultimately, from (5.27) and (5.29), taking into account that z ε · ∂ t u ε = Ψ(∂ t u ε ) and the same for z and u t , we conclude We can develop a lim sup argument completely analogous to the one leading to (5.22) and conclude that lim sup This gives, by the W-ellipticity of b (cf. (2.5)), the following strong convergence ∂ t u ε → ∂ t u in L 2 (0, T ; W). (5.32) Passage to the limit in the equation for ϑ. To pass to the limit in (3.28a) we combine the bi-Lipschitz continuity (2.12) of g with convergence (5.20) to conclude that g ′ (ϑ ε ) → g ′ (ϑ) in L q (Ω × (0, T )) for all q ∈ (1, ∞).
Passage to the limit in the equation for ϑ s . It proceeds exactly along the same lines as the limit passage to (2.36a) (cf. also the proof of [9, Thm. 1]). Let us only comment on the proof of Indeed, in view of convergence (5.15) for ϑ s,ε and of Lemma (A.6), we easily conclude, e.g., that f ε (ϑ s,ε ) → f (ϑ s ) in L 1 (Γ C × (0, T )), and thus a.e.. This is enough to identify the weak limit of f ε (ϑ s,ε ) in (5.11) and conclude (5.38).
Passage to the limit in the equation for χ . Finally, we pass to the limit in (3.28e) by virtue of convergences (5.2), (5.5), (5.12)-(5.14), and (5.15), also taking into account the properties (2.26) of λ. By the strong-weak closedness of the graph of (the maximal monotone operator induced by β on H Γ C ), we can directly identify ξ as an element of β( χ ) a.e. in Γ C × (0, T ). It remains to show that ζ ∈ ρ( χ t ) a.e. in Γ C × (0, T ). (5.39) To this aim, we test (3.28e) by ∂ t χ ε and prove that lim sup Therefore, (5.39) follows. As a byproduct, with the same arguments as in the previous lines we have Proof of the energy inequality (2.37). We take the limit as ε → 0 in the approximate energy inequality (3.52): let us only justify the passage to the limit in some of the terms on the left-and on the right-hand side. First of all, combining convergence (5.20) for (ϑ ε ) with the upcoming Lemma A.3 we obtain that for almost all t ∈ (0, T ) lim ε→0 Ω I ε (ϑ ε (t)) dx = Ω J * (L(ϑ(t))) dx .
All the remaining terms on the left-and on the right-hand side of (3.52) can be dealt with exploiting the previously proved convergences. This concludes the proof (2.37) on the interval (s, t) for almost all s, t ∈ (0, T ) and for s = 0.
6. Extension to a global-in-time solution and proof of Theorem 2.7 In this Section we show that the local solution to Problem 2.5 found in Theorem 5.1 (hereafter, we shall denote it by ( ϑ, ϑ s , u, χ , η, µ, ξ, ζ)), actually extends from the interval (0, T ) to the whole (0, T ).
To this aim, we first of all observe that the "energy estimates" (cf. the First estimate) derived from the energy inequality (2.37) have a global-in-time character. Nonetheless, we cannot derive from such global bounds the other estimates (i.e. the Second -Seventh estimates), and therefore we cannot directly extend the local solution ( ϑ, ϑ s , u, χ ), along with ( η, µ, ξ, ζ), to the whole interval (0, T ). The reason for this is that, as expounded in Sec. 3.1 and shown in Sec. 4, these estimates involve calculations which are only formal on the level of the limit problem. Thus, we need to perform them on the regularized system from Problem (P ε ). However, the energy estimates have only a local character for the approximate problem, since the term Γ C χ (t)|u(t)| 2 dx is estimated locally in time, cf. the discussion at the beginning of Sec. 4 and (4.2). Therefore, along the lines of the prolongation argument from [4], we shall proceed in the following way. We will extend the local solution ( ϑ, ϑ s , u, χ ) together with its approximability properties (cf. the notion of approximable solution in Definition 6.1 below). In this way, the approximate solutions shall "inherit" the global-in-time energy estimates from ( ϑ, ϑ s , u, χ ) (cf. (6.11) and (6.12)). Building on this, we will be able to perform rigorously all the estimates necessary for the extension procedure on the approximate level, and use them to conclude that ( ϑ, ϑ s , u, χ , η, µ, ξ, ζ) is defined on the whole interval (0, T ). More precisely, we will consider the maximal extension of our (approximable) solution ( ϑ, ϑ s , u, χ , η, µ, ξ, ζ) and show that it is defined (0, T ) with a standard contradiction argument (cf. Step 4 below). In doing so, we will meet with the technical difficulty that the (ϑ, ϑ s )-components of our solution need not be continuous w.r.t. time and therefore we will not be in the position to extend them by continuity. Indeed, in accord with the notion of approximable solution, we will argue on the level of the approximate solutions and rely on their time-regularity to carry out this procedure rigorously.
In what follows, (ϑ ε , ϑ s,ε , u ε , χ ε ) ε (with associated µ ε ) will be the family of solutions to Problem (P ε ) which converge, along a not-relabeled subsequence (cf. (5.1)-(5.21)), to the local solution ( ϑ, ϑ s , u, χ ) from Theorem 5.1. For simplicity, hereafter we shall omit the functions ( η, µ, ξ, ζ) and refer to the quadruple ( ϑ, ϑ s , u, χ ) as "the" local solution to our problem. Accordingly, we will give the definition of approximable solution only in terms of the (ϑ, ϑ s , u, χ )-components. We are now in the position to introduce the notion of "approximable solution". Definition 6.1. Let τ ∈ (0, T ]. We say that a quadruple (ϑ τ , ϑ s,τ , u τ , χ τ ) is an approximable solution on (0, τ ) to Problem 2.5 if the following conditions are verified • it is an energy solution on (0, τ ); • there exists a subsequence ε n ↓ 0 such that the related solutions of problem (P εn ) on (0, τ ) (here the dependence on τ is omitted in the notation) fulfill as n → ∞ ϑ sε n − ϑ s L 2 (0,τ ;L 2 (Γ C )) → 0. It follows from the passage to the limit argument in Theorem 5.1 that T is not an empty set, as at least T ∈ T.
As a consequence, letting T * = sup T we have 0 < T * ≤ T . Hence, proving that the local solution ( ϑ, ϑ s , u, χ ) extends to the whole (0, T ) reduces to showing that it extends to an approximable solution on (0, T * ), and that T * = T . To this aim let us outline the sketch of the proof. First, we prove that the "energy estimates" for an approximable solution hold with a constant independent of τ (Step 1). Then, we deduce that an approximable solution extends to (0, T * ) (Steps 2 and 3). Finally, we show that T = T * by a contradiction argument (Step 4).
Step 1. Let us prove the following Lemma, stating global estimates on the energy solutions to Problem 2.5.
Observe that the constant C below does not depend on τ .
Step 2. Let (ϑ τ , ϑ s,τ , u τ , χ τ ) τ be a family of approximable solutions depending on τ , with τ ∈ T. In view of the regularity required of an approximable solution, we have that (u τ , Hence we can consider the extension to (0, T * ) by continuity of these functions. More precisely, we define Due to (6.6) (where the constant C does not depend on τ ) there holds (independently of τ ) Thus, after fixing a sequence τ m ↑ T * , by (weak, weak * , and strong) compactness results we can conclude that there exists a pair such that, at least along some not relabeled subsequence, Hence, by construction of u τ and χ τ , we can infer that (u * (t), χ * (t)) = ( u(t), χ (t)) for all t ∈ [0, T ] (see (6.5)).
In particular, we get that We now extend the (ϑ, ϑ s )-components of the solution (along with (η, µ, ξ, ζ)), on the interval (0, T * ), together with their approximability properties. To this aim, we proceed with a diagonalization argument, which we sketch here for the sake of completeness, referring to [4] for all details. Let us take T k := T * (2 k − 1)/2 k , k ≥ 1. For k = 1 and for m sufficiently large m ≥m 1 we have [0, T 1 ] ⊂ [0, τ m ]. As a consequence we can apply compactness arguments analogous to the ones in the previous sections. In particular, we show that for some suitable subsequence (m 1 j ) j with m 1 j → ∞ and m 1 j ≥m 1 for all j ∈ N, the analogues of (5.1)-(5.17) hold on [0, T 1 ]. In particular, we deduce (as well as the existence of a limit quadruple (η * , µ * , ξ * , ζ * )). As a consequence it is straightforward to observe that ϑ * , ϑ * s can be identified (a.e.) with ϑ, ϑ s on (0, T ) (see the second of (6.5)). The aforementioned convergences allow us to apply a similar passage to the limit procedure in the approximate problem, and conclude that (ϑ * , ϑ * s , u * , χ * ) is a solution on (0, T 1 ) to Problem 2.5 (we omit details as they follow the already detailed argument in the proof Theorem 5.1). We can now proceed repeating the argument for T k with k = 2, and extending the above convergences to the interval [0, T 2 ] along a subsequence (m 2 j ) j larger than somē m 2 ≥m 1 . Iterating this construction for any k ∈ N (cf. [4, pag. 1061] for details), we get that the limit functions (u * , χ * , ϑ * , ϑ * s ) solve the limit Problem 2.5 on the interval (0, T * ) (along with some (η * , µ * , ξ * , ζ * )), and conclude indeed that (u * , χ * , ϑ * , ϑ * s ) is an approximable solution on (0, T * ).
Appendix A.
We develop here a series of technical results, collecting useful properties and estimates for the functions L ε , ℓ ε , f ε , and related quantities, which play a crucial role in deriving a priori estimates for Problem (P ε ). We also detail the construction of a family of approximate initial data complying with the properties (3.12)- (3.20).
In what follows, we shall rely on some definitions and results from the theory of maximal monotone operators, for which we refer to the classical monographs [2,11]. Preliminarily, we fix some notation.
Notation A.1. Hereafter, for fixed ε > 0 we will denote by the resolvent operators associated with L and ℓ, respectively. We recall that R ε and r ε are contractions, and that the Yosida regularizations of L and ℓ are defined, respectively, by and fulfill L ε (x) = L(R ε (x)), for a constant independent of ε: this follows from the fact that the convex function J is bounded from below by a linear function, and from R ε (0) → 0 as ε → 0, since 0 ∈ D(L).
Our next result is crucial for the passage to the limit as ε → 0, in particular to obtain the energy inequality (2.37). Then, Under conditions (2.7a) and (2.7c) on ℓ, the analogue of (A.14) holds for (i ε ) ε .
Finally, we address the properties of the functions f ε (3.4) and H ε (x) (3.11).
In order to check the last assertion, it is sufficient to observe that and use that 1 ℓ ′ ε is Lipschitz (cf. (A.19)) to estimate the first term, and (A.29) for the second summand.
The forthcoming Lemmas address the construction of a family of initial data (ϑ ε 0 , ϑ 0,ε s ) ε fulfilling properties (3.12)- (3.20). In Lemma A.7 we exhibit an example of sequence (ϑ ε 0 , ϑ 0,ε s ) ε complying with the first set of properties, i.e. (3.12)- (3.18). Since the construction developed in the proof of Lemma A.7 does not guarantee the other requirements, we tackle them in two different results. Namely, in Lemma A.8 we detail how the family (ϑ 0,ε s ) ε chosen in the Lemma A.7 satisfies the additional properties (3.19)- (3.20) in the case of a special class of functions ℓ (and function f ) in (2.36b). Finally, in Lemma A.9, a family of data (ϑ 0,ε s ) ε complying with the whole set of properties (3.12)-(3.20) is exhibited in the case of the (physically relevant) choice ℓ(ϑ s ) = ln(ϑ s ) (and f (ϑ s ) = (ϑ s ) 2 ).