Long-time behaviour of a thermomechanical model for adhesive contact

This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Fremond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.


1.
Introduction. This paper is concerned with the large-time analysis of a PDE system describing adhesive contact between a thermo-viscoelastic body and a rigid support. The model has been recently introduced, and global-in-time existence results have been proved on finite-time intervals, both in isothermal cases (see [2] in the case of an irreversible damage evolution on the contact surface, and [3] for the reversible case), and for PDE systems including thermal effects (see [4]). The modelling approach for contact with adhesion which we apply refers to a damage theory described by phase transitions, and it is due to M. Frémond (see [13]). The idea consists in describing the adhesion between viscoelastic bodies in terms of a surface damage theory, in which the damage parameter is related to the active bonds which are responsible for the adhesion between the bodies. Hence, the equations of the evolutionary system are recovered from thermomechanical laws, and they are written in the domain of the viscoelastic body and on the contact surface.
It turns out to be interesting, both from a theoretical point of view and in view of applications, to investigate how the thermomechanical system (i.e., the body and the rigid support it is in contact with) behaves for large times. More precisely, we shall investigate if the trajectories of the solutions to the resulting PDE system present some cluster point, in the limit as time goes to +∞. Then, 274 ELENA BONETTI, GIOVANNA BONFANTI AND RICCARDA ROSSI we shall look for a relation between these limit states and the stationary system associated with our evolution problem. In particular, we aim to prove that, in the limit, solution trajectories reach a thermomechanical equilibrium state in which dissipation vanishes. This kind of large-time analysis was performed in [3] for the reversible model in the isothermal case.
Before introducing the long-time behaviour analysis of the problem, we shall briefly recall the model and make some comments on the existence of solutions on the whole time interval (0, +∞). Moreover, we shall point out that this paper also presents a novelty in the formulation of the model itself, as we generalize the convex potential usually ensuring internal constraints on the damage parameter.
The model. We mainly refer to the recent contribution [4], in which (a slightly different version of) the thermomechanical model has been introduced. The state variables, in terms of which the equilibrium of the system is established, are defined in the domain Ω ⊂ R 3 (where the body is located), and on the contact surface Γ c . Namely, we shall take Ω to be a sufficiently smooth bounded domain in R 3 , with boundary ∂Ω =Γ 1 ∪Γ 2 ∪Γ c . The sets Γ i are open subsets in the relative topology of ∂Ω, with smooth boundary and disjoint one from each other. In particular, Γ c is the contact surface. We suppose that Γ c and Γ 1 have strictly positive measures and, for the sake of simplicity, we identify Γ c with a subset of R 2 . Thus, we shall treat Γ c as a flat surface.
The state variables we shall consider in Ω are the absolute temperature ϑ of the body and the macroscopic deformations, given in terms of the linearized strain tensor ε(u) (u represents the vector of small displacements). On the contact surface, we introduce the surface absolute temperature (the reader may think of the temperature of the adhesive glue) ϑ s and a damage parameter χ, related to the active bonds in the glue ensuring adhesion. For the moment, we do not require any constraints on the values assumed by χ. Taking into account local interactions (in the glue and between the glue and the body), we include the gradient ∇χ and the displacement trace u |Γ c among the state variables on the contact surface. The free energy in Ω is written as follows where K is the elasticity tensor and the coefficient ϑ multiplying trε(u) accounts for the thermal expansion energy.
Remark 1. Notice that here we have taken the term ϑ(1 − ln(ϑ)) for the purely thermal contribution in the free energy Ψ Ω (and, similarly, for the free energy Ψ Γc below), while in [4] we have considered a more general concave function. The particular choice in this paper is very frequent in the literature, as it has some analytical and modelling advantages. From the latter viewpoint, the presence of the logarithm in (1) yields an internal constraint on the temperature: indeed, the domain of Ψ Ω is well-defined for ϑ > 0, which is in agreement with thermodynamical consistency. On the analytical level, this form of the thermal contribution shall allow us to simplify the procedure exploited in [4] to prove existence of solutions (see Remark 13).
Next, we specify the free energy in Γ c , which presents some novelty with respect to the model introduced in [4] (cf. also [2] and [3]). In fact, we shall consider Ψ Γc = ϑ s (1 − ln(ϑ s )) + λ(χ)(ϑ s − ϑ eq ) + β(χ) + σ(χ) where ϑ eq is a critical temperature, and β is a convex, lower-semicontinuous (proper) function. The indicator function I − forces the scalar product u |Γ c · n to be nonpositive, as it is defined on R by I − (y) = 0 if y ≤ 0 and I − (y) = +∞ for y > 0. This renders the impenetrability condition between the body and the support. In the same way, the term β may yield a constraint on the values assumed by χ. For example, a proper choice of β may enforce positivity of χ (see [2,3,4]). In particular, this occurs when, classically, β = I [0, 1] , forcing χ ∈ [0, 1]. Then, the coefficient of |u |Γ c | 2 remains non-negative, in accord with physical consistency. However, in the present paper we shall allow the potential β to be more general and we do not impose any a priori restriction on its domain. Hence, to ensure physical consistency, we take the deformation coefficient to be χ + (using the notation r + = max(r, 0) for every r ∈ R). Finally, the function λ is related to the latent heat, while σ takes into account possibly non-convex contributions in the free energy. In particular, we include in σ cohesive effects in the glue, which are represented by a non-increasing function in χ (a simple choice is σ(χ) = w(1 − χ), with a positive parameter w). Then, the evolution of the system is governed by two convex potentials (nonnegative and assuming their minimum 0 if there is no dissipation), namely the pseudo-potentials of dissipation written in Ω and in Γ c . We have (here K v is a viscosity matrix) and Notice that the dissipation in Ω depends on ε(u t ) and on ∇ϑ, while the dissipation in Γ c depends on ∇ϑ s , χ t , and on (ϑ |Γ c − ϑ s ). The function k, which accounts for the heat exchange between the body and the adhesive material, shall be taken nonnegative and smooth enough. Actually, to characterize the large-time behaviour of the system, we need to assume that k is bounded from below by some positive constant (see Remark 3).
The PDE system. Proceeding as in [4], we refer to thermomechanical laws and, after specifying the constitutive equations in terms of the above potentials, we arrive 276 ELENA BONETTI, GIOVANNA BONFANTI AND RICCARDA ROSSI at the following PDE system (T is a fixed final time) where h is an external entropy source, f a volume force, and g a traction. Moreover, β = ∂ β and H = ∂p, where p(r) = r + for all r ∈ R. Hence, the Heaviside maximal monotone operator H : R → 2 R is defined by H(r) = 0 if r < 0, H(0) = [0, 1], and H(r) = 1 if r > 0. We warn that, here and in what follows, we shall omit for simplicity the index v | Γc to denote the trace on Γ c of a function v, defined in Ω.

Remark 2.
Let us comment on the above equations, while referring the reader to [4] for their rigorous derivation. First of all, we point out that (5) and (7) are entropy equations. The possibility of describing thermal effects in phase transitions by the use of an entropy equation, in place of the more standard energy balance, has only recently been introduced. In particular, let us point out that the entropy in Ω is defined as ln(ϑ) − div(u), and in Γ c as ln(ϑ s ) − λ(χ). Using entropy equations brings to some advantages both for the analytical treatment and the modelling of the phenomenon. In particular, from (5) and (7) one directly recovers the positivity of the temperature, which is necessary for thermodynamical consistency, avoiding the application of any maximum principle argument. We do not enter the details of this theory and refer, among the others, to the papers [7] and [8]. Then, (9) is derived from the momentum balance, in which accelerations are not taken into account. Equation (12) is recovered as a balance equation for micro-movements related to the evolution of the phase parameter (see [13] for the theory of the generalized principle of virtual power including micro-movements and micro-forces responsible for the phase transition).
Remark 3. We emphasize that the structure of (12) is more complicated than the analogous equation in the model studied in [4]. Indeed, the maximal monotone operator β is more general than the one considered in [4], for we do not impose any restriction on its domain. Moreover, the presence of the operator H in (12) introduces a new nonlinearity in the equation. From the physical viewpoint, since H(0) = [0, 1], a residual influence of macroscopic displacements on the mechanical behaviour of the glue may persist also when the glue is damaged, e.g. if χ = 0 (the parameter χ representing here the proportion of active bonds). This corresponds to assuming that a local interaction between the body and the support is preserved even when the bonds in the glue are completely damaged. Notice that this is reasonable, if one takes into account distance forces. An analogous argument justifies the assumption that k in (7) is bounded from below by some positive constant, see (H5) later on. This ensures that the thermal local interaction between the body and the support is conserved even when the adhesion is not active.
Our first main result (see Theorem 2.1 later on) states that for every T > 0 the Cauchy problem for system (5)-(13) admits at least one solution. In this way, we parallel the global existence result of [4]: therein, as we mentioned before, we considered a slightly different free energy on the contact surface, which resulted in an equation governing the evolution of the parameter χ simpler than (12). Nonetheless, the proof of Theorem 2.1 (which shall be developed in Section 4), closely follows the argument developed in [4]. It hinges upon a double approximation procedure (depending on two approximating parameters), and a subsequent passage to the limit argument with respect to the mentioned parameters. One of them is used to regularize the nonlinearities in the equations by means of Yosida approximations. Furthermore, some viscosity terms in ϑ and ϑ s are added in (5) and (7), depending on the second parameter. The local existence of a solution for the approximate system (supplemented with suitable regularized initial data for ϑ and ϑ s , due to the presence of viscosity), is obtained with the Schauder theorem, while uniqueness for the approximate problem follows by contraction arguments. Hence, we conclude the existence of global-in-time solutions by proving suitable a priori estimates (which in fact directly hold in the time interval (0, +∞), as they do not depend on the final time horizon T ), independent of the approximating parameters. The very same estimates allow us to pass to the limit in the approximate problem, firstly as the viscosity parameter, and secondly as the parameter of the Yosida regularizations vanish. Finally, we point out that, due to the strongly nonlinear character of system (5)-(13), we do not expect uniqueness of solutions for the related Cauchy problem.
Large-time analysis. As previously mentioned, the ultimate aim of this paper is investigating the large-time behaviour of system (5)-(13) (supplemented with suitable initial conditions). More precisely, we are interested in finding cluster points of solution trajectories and characterizing a sort of thermomechanical equilibrium of the system in the limit, in which there is no dissipation. This corresponds to proving that solution trajectories converge to solutions of the stationary problem associated with our system, in which dissipation is zero. Now, some results in this direction have been obtained in the literature concerning the long-time behaviour of phase-field systems with non-convex potentials (see, for example, [11,12,14,16,17]). Typically, these results apply to binary systems (i.e., macroscopic deformations are not included), see among the others [8] dealing with a singular entropy equation.
The main difficulties related to our analysis are due to the singular character of the entropy equations, to the presence of general multivalued operators on the state variables, and to the nonlinear coupling between the equations written in the domain Ω and the ones set in Γ c .

Remark 4.
On the other hand, the analysis of the large-time behaviour in terms of the global attractor for the dynamical system generated by (5)-(13) might also be addressed. Indeed, the existence of the global attractor would signify that the system dissipation is controlled in the evolution. However, for the moment being, proving the existence of the attractor seems out of our reach. In fact, the strongly nonlinear character of the equations essentially prevents us from obtaining those estimates on the solutions which would guarantee the existence of a compact and absorbing set, for the dynamical system, in the phase space dictated by the choice of the initial data.
Prior to addressing the large-time analysis of system (5)-(13), we specify that we consider a quadruple (ϑ, ϑ s , u, χ) to be a solution of (5)- (13) in (0, +∞), if (ϑ, ϑ s , u, χ) fulfils (5)- (13) in the finite-time interval (0, T ), for every T > 0. Hence, to perform the asymptotic analysis on the solutions of (5)-(13) as time goes to +∞, we shall rely on some further estimates improving the solution regularity of the existence Theorem 2.1. Only in this enhanced setting, shall we obtain (see Proposition 1) the bounds on the solutions (in suitable functional spaces, on the whole half-line (0, +∞)), necessary to prove that, for every solution trajectory, its ω-limit set (i.e., the set of its cluster points) is non-empty. These further estimates shall be first formally derived in Section 3, and then made rigorous in Section 4 by performing all the related calculations on the approximate system used for proving Theorem 2.1. That is why, our asymptotic analysis solely applies to the solutions of (5)-(13) originating from the aforementioned double approximation procedure. Once proven that the ω-limit is non-empty, we shall show that its elements solve the stationary system associated with the evolutionary problem (5)-(13) (see Theorem 2.2). As a by-product of this procedure, we shall see that in the limit as t → ∞ the dissipation vanishes (cf., in particular, Remark 9). Plan of the paper. In Section 2 we enlist all of the assumptions on the problem data and state our results. A (partially formal) proof of our Theorem 2.2 on the long-time behaviour of the PDE system (5)-(13) is developed in Section 3 and rigorously justified in Section 4, which also contains the proof of the global existence Theorem 2.1.

Preliminaries.
Notation. Throughout the paper, given a Banach space X, we shall denote by X ·, · X the duality pairing between X and X itself, and by · X both the norm in X and in any power of X; C 0 w ([0, T ]; X) shall be the space of the weakly continuous X-valued functions on [0, T ].
Variational formulation of the elasticity equation. We introduce the standard bilinear forms which allow to give a variational formulation of (the boundary value problem for) equation (9). As usual in elasticity theory, we may assume that the material is isotropic and hence suppose that the rigidity matrix K in (9)-(11) can be represented as where λ, µ > 0 are the so-called Lamé constants and 1 is the identity matrix. Also, for the sake of simplicity but without loss of generality, we set K v = 1 in (9)- (11). Therefore, (9) may be formulated by means of the following bilinear symmetric forms a, b : Note that the forms a(·, ·) and b(·, ·) are continuous and, since Γ 1 has positive measure, by Korn's inequality they are W-elliptic as well, so that

2.2.
A global existence result. Statement of the assumptions. In equation (12) we consider a maximal monotone operator β : and denote by β : D(β) → (−∞, +∞] a proper, l.s.c. and convex function such that β = ∂ β. Instead of dealing with the pointwise operator ∂I − : R → 2 R in (11), we shall work with a suitable generalization, defined in the duality relation between H −1/2 (Γ c ) 3 and H 1/2 (Γ c ) 3 . To this aim, we introduce and set α : Remark 5. In order to render the impenetrability constraint mentioned in the Introduction by means of the operator α, we may proceed as follows. We consider j(u) = I − (u · n) and associate with j the following functional Since α is a proper, convex and lower semicontinuous functional on ( is a maximal monotone operator. Notice that, in this case, (19) implies that, if η ∈ α(v), then v belongs to the domain of j and thus fulfils v · n ≤ 0, which corresponds to the impenetrability condition.
We assume that the nonlinearities σ and λ comply with (and denote by L σ and L λ the Lipschitz constants of the functions σ : R → R and λ : R → R, respectively), and that (cf. Remark 3) (H5) Remark 6. We point out that (H3), (H4), and (H5) respectively entail that As far as the problem data are concerned, we suppose (23) Finally, we require that the initial data fulfil Note that the first of (24) and of (25) respectively yield Variational formulation and existence theorem. The variational formulation of the initial-boundary value problem for system (5)-(13) reads as follows. Problem (P). Under the standing assumptions (H1)-(H8), given a quadruple of initial data (ϑ 0 , ϑ 0 s , u 0 , χ 0 ) fulfilling (24)-(27), find functions (ϑ, ϑ s , u, χ, η, ξ, ζ), with the regularity satisfying the initial conditions and
then there exists a solution (ϑ, ϑ s , u, χ, η, ξ, ζ) having for all δ > 0 the further regularity (50) We refer to Remark 16 for some further comments concerning the above statement.
Remark 7. The regularity of ϑ 0 and ϑ 0 s required in (24)-(25) turns out to be necessary in the proof of our existence result Theorem 2.1 for (the Cauchy problem for) system (5)- (13). Indeed, since we are going to prove existence of solutions by passing to the limit in a viscosity approximation of equations (5) and (7), we shall need to dispose of more regular approximate initial data. Our construction of such data (see Lemma 4.2) apparently hinges upon the regularity (24)-(25) of ϑ 0 and ϑ 0 s . However, we are not able to recover for ϑ and ϑ s the regularity corresponding to assumptions (24) and (25), namely ϑ ∈ C 0 w ([0, T ]; Lp(Ω)) and ϑ s ∈ C 0 w ([0, T ]; Lq(Γ c )), withp andq as in (24)-(25). This is mainly due to the highly nonlinear character of PDE system and, in some sense, to the fact that the natural initial conditions for (40) and (41) are written for ln(ϑ) and ln(ϑ s ). Nonetheless, notice that the regularity required for the initial data is preserved (see (50)) for t ≥ δ > 0, for every δ > 0, in the more regular framework of (48).
The proof of the above result is based on a double approximation procedure which we shall detail in Section 4. The related passages to the limit rely on suitable a priori estimates on the approximate solutions, which we shall formally perform on the (un-approximated) system (40)-(47), and directly on the time-interval (0, +∞), within the (formal) proof of Proposition 1. Such estimates shall be rendered rigorous in Sec

ANALYSIS OF A THERMOMECHANICAL MODEL FOR ADHESIVE CONTACT 283
In view of the long-time analysis of the solutions to Problem (P), we shall hereafter suppose that Notice that (H9) is trivially fulfilled in the case dom( β) is a bounded interval, whereas, if dom( β) is unbounded, it is implied by a super-quadratic growth of β at infinity. We shall also require some summability on (0, +∞) for the problem data: The above assumptions yield in particular Furthermore, using (51), it is not difficult to prove (see [3,Remark 2.3] for all details) that Hence, Theorem 2.1 ensures that for every quadruple of initial data (ϑ 0 , ϑ 0 s , u 0 , χ 0 ) fulfilling (24)-(27) there exists (at least) one solution trajectory The ensuing Proposition 1 contains some suitable large-time a priori estimates for such trajectories. Such bounds shall enable us to conclude that the associated ω-limit set (57) is non-empty, and that its elements solve the stationary system associated with Problem (P) (see Theorem 2.2). As we shall see, these results in fact hold for a class of solutions of Problem (P), namely approximable solutions which, in order not to overburden the paper, we shall precisely define in Section 4 only (cf. Definition 4.4). Here, we may just mention that the notion of approximable solution is tightly linked to the approximation procedure developed in Sec. 4 to prove the global existence of solutions to Problem (P) (see Theorem 2.1). Such a solution notion allows us to perform rigourously on system (40)-(47) some of the a priori estimates on which our large-time analysis relies (cf. Remark 8).
No uniqueness result is available on the stationary system (58a)-(58c). Hence, one cannot deduce directly from Theorem 2.2 that ω(ϑ, ϑ s , u, χ) is a singleton and, thus, that the whole solution trajectory (ϑ, ϑ s , u, χ) converges, as t → +∞, to a unique equilibrium. However, the next result (whose proof is postponed to Section 3.2) shows that, under more specific assumptions on the operator β and on the nonlinearities λ and σ, it is possible to uniquely determine the χ-component of the elements in ω(ϑ, ϑ s , u, χ). Then, for any quadruple (ϑ ∞ , ϑ s,∞ , u ∞ , χ ∞ ) ∈ ω(ϑ, ϑ s , u, χ) there holds and we have, as t → +∞, that Remark 10. Corollary 1 ensures that, in the case when the latent heat is positive, β has a bounded domain (which is the interesting case from a physical point of view), and cohesion in the material (which is included in the decreasing part of σ, see (2)) is not too large with respect to the remaining part of the potential σ, the glue tends to be completely damaged in the large-time limit. We remark that this is the result one would expect from experience.

Proofs.
Notation. Henceforth, for the sake of notational simplicity, we shall write ·, · for all the duality pairings W ·, · W , V ·, · V , and H 1 (Γc) ·, · H 1 (Γc) , and, further, denote by the symbols c, c C, C most of the (positive) constants occurring in calculations and estimates. Notice that the above constants shall not depend on the final time T .
3.1. Proof of Proposition 1. The following is a formal proof, based on two main a priori estimates on system (40)-(47), which shall be revisited in Section 4.3. Therein, we shall also rigorously prove the further solution regularity (49a)-(49e).
First (formal) estimate. We test (40) by ϑ, (41) by ϑ s , (42) by u t and (44) by χ t , add the resulting relations and integrate them on the interval (0, t), with t ∈ (0, +∞). Now, we take into account the formal identities In the same way, an integration by parts and the chain rule for p(·) = (·) + lead to t 0 Γc where we recall that ζ ∈ H(χ) = ∂p(χ). Finally, we observe that, by the properties (17) and (18) of the forms a and b, respectively, we have Collecting (61)-(64) and observing that some terms cancel out, we get

Second (formal) estimate.
In what follows, we shall formally treat the maximal monotone operators α, H, and β as single-valued (in particular, H and β as nondecreasing functions). Indeed, the following estimates can be rigorously justified as in Section 4 regularizing these nonlinearities by their Yosida approximations. Further, we shall work with a solution (ϑ, ϑ s , u, χ) enjoying the further regularity (49) (see also (50)). Formally proceeding, for all δ > 0 we let a constant K δ , only depending on λ, k, σ, on the quantity M (54), and possibly on δ, such that This formal assumption shall be discarded once we put forth the rigorous arguments of Section 4, see Remark 11 for further comments. Hence, we are in the position of performing the following calculations. We test (40) by ϑ t and (41) by ∂ t ϑ s , differentiate (42) w.r.t. time and test it by u t , and differentiate (44) w.r.t. time and multiply it by χ t . We add the resulting relations and integrate them on the interval (δ, t), with t ∈ (δ, +∞), also adding 1/2( ϑ(t) 2 L 1 (Ω) + ϑ s (t) 2 L 1 (Γc) ) to both sides. Indeed, the Poincaré inequality for the zero mean value functions yields for some positive constant C P independent of t ∈ (0, +∞). We also notice that, for some other constant c also depending on the embeddings (15)- (16), there holds for all t ∈ (0, +∞). (71) Further, we remark that t δ Γc Taking into account the cancellation of some terms and the coercivity and continuity of the forms a and b (17)-(18), using the formal identities as well as (70)-(72), we end up with in which we have controlled the term 1/2( ϑ(t) 2 L 1 (Ω) + ϑ s (t) 2 L 1 (Γc) ) on the righthand side by (55b). Integrating by parts, we have
This concludes the proof.
Proof of Corollary 1. For the sake of completeness, here we repeat the same argument developed in the proof of [3,Prop. 2.5]. We test the first of (58c) by (χ ∞ − m * ) and integrate on Γ c . We obtain the latter inequality due to (59b)-(59c) and the fact that m * ≤ χ ∞ ≤ m * a.e. in Γ c . On the other hand, the second term on the left-hand side of the above inequality is non-negative by monotonicity, so that we deduce that ∇(χ ∞ − m * ) ≡ 0 a.e. in Γ c . Thus, there exists some constant ≥ m * such that χ ∞ ≡ a.e. in Γ c . Now, integrating (58c) and again recalling (59b)-(59c), we find that Γc ξ ∞ < 0 .

Rigorous estimates.
Outlook: This section is devoted to the rigorous proof of Theorem 2.1 and of Proposition 1. Thus, in Sec. 4.2 we shall specify the variational problem (depending on two parameters ε > 0 and µ > 0) approximating Problem (P), and outline the main steps of the proof of its global well-posedness. Hence, in Sec. 4.3 we shall prove some estimates on the approximate solutions, which are independent of the parameters ε and µ and in fact hold on (0, +∞). This shall enable us to pass to the limit in the approximate problem first as ε 0, and secondly as µ 0, and to conclude the proof of Theorem 2.1 in Sec. 4.4, and of Proposition 1 in Sec. 4.5.
Since the (double) approximation procedure for Problem (P) strongly relies on the usage of Yosida regularizations of the nonlinear operators ln, α , β, and of the Heaviside operator H, in the following section we recapitulate some related preparatory results.

Recaps on Yosida regularizations.
Regularization of ln: For fixed µ > 0, we denote by the resolvent operator associated with the logarithm ln (where Id : R → R is the identity function), and recall that r µ : R → (0, +∞) is a contraction.
The Yosida regularization of ln is then defined by It follows from [9, Prop. 2.6] that for all µ > 0 the function ln µ : R → R is non-decreasing and Lipschitz continuous, with Lipschitz constant 1/µ. For later convenience, as in [5] we also introduce the following function We point out that, since I µ is decreasing on (−∞, 0) and increasing on (0, +∞), there holds for all µ > 0 The following result collects some properties of ln µ and I µ which shall play a crucial role in the proof of the forthcoming Proposition 2.
Lemma 4.1. The following inequalities hold: As a consequence, I µ satisfies Proof. Ad (119). Using the definitions (114) and (116) of r µ and ln µ and repeating the calculations in the proof of [5,Lemma 4.2], it is possible to show that which, combined with (115), yields (119b). For the proof of (119a), which follows the very same lines, we directly refer to [5,Lemma 4.2]. Ad (120). Estimate (120a) is an immediate consequence of (119a) and of the definition of I µ . We shall now prove (120b) for x ≥ 0 (the inequality in the case x < 0 being completely analogous). Indeed, from the inequality for all s ≥ 0 (which is an immediate consequence of (119b)), we deduce that for some suitable positive constants C 1 and C 2 .
In particular, it can be checked that Using the definition of H µ and p µ , it is straightforward to verify that

Approximation of Problem (P).
A double approximation procedure. Let ε, µ > 0 be two strictly positive parameters. We consider the approximation of Problem (P) obtained in the following way: 1. we add to (40) the regularizing viscosity term εR(ϑ t ) and to (41) the viscosity term εR Γc (∂ t ϑ s ) (R and R Γc being the Riesz operators introduced in Notation 2.1); 2. we replace the operators α in (42), β and H in (44) with their Yosida regularization α µ : accordingly, we replace the term χ + u in equation (42) by p µ (χ)u; 3. both in (40) and in (41) we replace the logarithm ln with its Yosida regularization ln µ . Approximate initial data. In order to properly state our approximate problem, depending on the parameters ε > 0 and µ > 0, we shall need some enhanced regularity on the initial data for ϑ and ϑ s . The following result concerns the construction of sequences of suitable approximate initial data {ϑ 0 ε } and {ϑ 0 s,ε }, which in fact depend on the parameter ε > 0 only.
Variational formulation of the approximate problem. We thus obtain the following boundary value problem, which we directly state on the half-line (0, +∞) in view of the long-time a priori estimates of Proposition 2.
Problem (P µ ε ). Given a quadruple of initial data (ϑ 0 ε , ϑ 0 s,ε , u 0 , χ 0 ), u 0 and χ 0 being as in (26)-(27), and ϑ 0 ε and ϑ 0 s,ε fulfilling (126) and (127) respectively, find functions (ϑ, ϑ s , u, χ), with the regularity for all T > 0, fulfilling the initial conditions Remark 13. Note that the approximate system (132)-(135) presents fewer technical difficulties than the analogous approximate version introduced in [4]. This is mainly due to the fact that, as we deal with the specific choice of ln(ϑ) and ln(ϑ s ) in the entropy (cf. Remark 1), we are allowed to directly introduce the Yosida regularization of the logarithm, instead of the more intricate approximating procedure exploited in [4].

4.2.1.
Outline of the proof of global well-posedness for Problem (P µ ε ). Since Problem (P µ ε ) only slightly differs from the approximate problem considered in [4], the global well-posedness for Problem (P µ ε ), on any interval (0, T ), can be obtained arguing in the very same way as in [4,Sec. 3], to which we refer the reader for all details. Here, we shall just sketch the main steps of the proof.
Step Step 2.: Next, to extend the local solution to the whole interval (0, T ), one needs global (in time) a priori estimates. The latter substantially coincide with the ones formally performed in the proof of Proposition 1 and shall be repeated on the approximate system (132)-(135) within the proof of Proposition 2.
Step 3.: Finally, uniqueness of solutions to Problem (P µ ε ) follows from the very same contraction estimates performed in [4,Sec. 3.5].
The viscosity terms εR(ϑ t ) and εR | Γc (∂ t ϑ s ) have been inserted in (132) and (133), respectively, for technical reasons, related to the fixed point construction of a local solution for Problem (P µ ε ). In particular, the contributions εR(ϑ t ) and εR | Γc (∂ t ϑ s ) are essential to prove uniqueness of solutions to (the Cauchy problems for) approximate equations (132) and (133). Furthermore, they also play a crucial role to make the estimate leading to the further regularity (56a)-(56b) rigorous, see the ensuing proof of Proposition 2. For the same reason, we have replaced the maximal monotone operators ln, α and β by their Yosida regularizations, and correspondingly substituted the coupling terms χ + u and −1/2ζ|u| 2 in equations (42) and (44), with p µ (χ)u and −1/2H µ (χ)|u| 2 , respectively. Now, as in the approximation of the system considered in [4], we shall keep the viscosity parameter ε distinct from the Yosida parameter µ in both approximate equations (132) and (133). Thus, we shall prove the existence of solutions to Problem (P) by passing to the limit in Problem (P µ ε ) first as ε 0 for µ > 0 fixed, and then as µ 0. This procedure shall enable us to recover on any interval (0, T ) the L ∞ (0, T ; H)-regularity for ln(ϑ) (the L ∞ (0, T ; L 2 (Γ c ))-regularity for ln(ϑ s ), respectively) by testing the µ-approximation of (40) (of (41), respectively), by the term ln µ (ϑ) (ln µ (ϑ s ), resp.), and obtaining some bound independent of the approximation parameter µ. In fact, such an estimate may be performed on equation (132) (on (133), resp.) only when ε = 0. For, if one kept ε > 0, one would not obtain estimates on ln µ (ϑ) independent of the parameters ε and µ, essentially because the term εR(ϑ t ), ln µ (ϑ) ( εR | Γc (∂ t ϑ s ), ln µ (ϑ s ) , resp.) cannot be dealt with by monotonicity arguments.

4.4.1.
Passage to the limit in Problem (P µ ε ) as ε 0. First, we introduce the boundary value problem obtained by taking ε = 0 in Problem (P µ ε ), which we shall supplement with the initial data (24)-(27).
Notice that no uniqueness result is available for (the Cauchy problem for) Problem (P µ ).
The following result is a straightforward consequence of Proposition 2.