ON TWO CLASSES OF GENERALIZED VISCOUS CAHN-HILLIARD EQUATIONS

This paper investigates two classes of generalized viscous CahnHilliard equations, featuring two different laws for the mobility, which is assumed to depend on the chemical potential. Both equations can be obtained with the new derivation of equations of Cahn-Hilliard type proposed by M.E. Gurtin [14]. Approximation and compactness tools allow to prove well-posedness and, in one case, regularity results for the equations supplemented with initial and suitable boundary conditions.


1.
Introduction. This paper is concerned with the analysis of the following fourthorder parabolic equation: where δ > 0 is a positive coefficient and Ω is a bounded, connnected domain in R N , N = 1, 2, 3, with smooth boundary Γ, occupied by a two-phase material (for instance, a binary alloy), subject to a phase separation process in the time interval (0, T ). The evolution of this phenomenon is described in terms of the order parameter χ, representing the local concentration of one of the two components. Furthermore, α : D(α) ⊂ R → R is a (strictly) increasing function, f may stand for a source term, while W is the derivative of a non-convex free energy potential W. Actually, throughout this paper we will assume W to have the double-well potential form We will refer to (1.1) as the generalized viscous Cahn-Hilliard equation, which includes as a special case (with the obvious choice α(r) := κr for every r ∈ R) both the standard viscous Cahn-Hilliard equation (where we have set δ = 1 for simplicity), and the Cahn-Hilliard equation The latter equation was originally proposed for modelling phase separation phenomena in a binary mixture, quickly quenched from a uniform mixed state into a w = −∆χ + W (χ).
Let us refer the interested reader to [20], and the references therein, for a rich survey of the mathematical analysis so far developed for (1.4) and for its viscous variant (1.3), which was first proposed in [19] (in the case f ≡ 0), to account for viscosity effects in the phase separation of polymer-polymer systems. We will focus instead on (1.1), which pertains to the class of the generalized Cahn-Hilliard equations derived by M.E. Gurtin. As a matter of fact, in the seminal paper [14] a different approach to (1.3) and (1.4) is proposed, based on the consideration that the work of the internal microforces associated with the changes of χ should be taken into account in the model. Then, a new derivation of (1. 3) and (1.4) is developed, in which the macroscopic mass balance (1.5) is coupled with a microforce balance, and both laws are combined with constitutive equations rigorously deduced from the second law of thermodynamics. In this connection, besides quoting the original paper [14], we also refer the reader to [17] and [18] for a detailed account of the derivation the generalized (viscous) Cahn-Hilliard equation χ t − div(M(Z)∇(δχ t − ∆χ + W (χ))) = 0 in Ω × (0, T ), (1.6) where δ ≥ 0 (δ > 0 in the viscous case), and Z denotes the set of the independent constitutive variables (χ, ∇χ, χ t , w, ∇w) the mobility tensor M (in general, a positive definite N ×N matrix), is allowed to depend on. Then, our equation (1.1) turns out to be a particular case of (1.6), of course with the choice M = M(w) := α (w)I, α being the (almost everywhere defined) derivative of the function α in (1.1). Namely, we assume M to depend on the chemical potential w only. Let us note that a concentration-dependent mobility tensor M has already been considered for the classical Cahn-Hilliard equation (see [13,6]), possibly supplemented with a source term nonlinearly depending on χ [16]. We may also mention [2], in which a viscous Cahn-Hilliard equation modelling phase separation in a tin-lead solder subject to internal and external mechanical stresses is investigated. Instead, the case of a mobility depending on the chemical potential, which naturally arises in the framework of Gurtin's derivation, has never been tackled so far to the author's knowledge. For the sake of convenience, hereafter we will rephrase (1.1) (for δ = 1), by introducing the new variable u := α(w), so that (1.1) may be split into the system χ t − ∆u = f in Ω × (0, T ), where ρ := α −1 . In the sequel, we will examine two different choices of α in (1.1), and we will analyse the initial-boundary value problem obtained by supplementing the system above with suitable initial and boundary conditions for χ and u, in accordance with the different properties of α.
Firstly, we will tackle the case α is a bi-Lipschitz, strictly increasing function α 1 : R → R, leading to the system (1.7) 9) and the Neumann boundary conditions and we will refer to the initial-boundary value problem given by (1.7)-(1.10) as Problem P 1 . Let us point out that, as soon as f ≡ 0 in (1.7) and g ≡ 0 in (1.10), then the no mass flux boundary condition (1.10) for u easily yields Namely, χ is a conserved parameter (i.e., its (spatial) mean value is constant on (0, T )), in compliance with the principle of matter conservation (recall that χ stands in fact for a concentration). Then, if non-zero data f and g are to be included in Problem P 1 , we have to require them to fulfil this compatibility condition (cf. Remark 2.2 later on) (1.11) in order to retain the mass conservation property. Secondly, we will consider the fourth-order system given by (1.7), coupled with where and α 2 is a stricly increasing function defined on the halfline (a, +∞), with lim r↓a α 2 (r) = −∞, and such that ρ 2 is a Lipschitz (but not bi-Lipschitz ) function. To fix ideas, we can think of Indeed, this is a particularly meaningful choice: it is straightforward to check that, in this case, the system ((1.7),(1.12)) can be obtained by setting ε = 0 in the system 14) 15) and formally putting u := ϑ − 1 ϑ = α 2 (− 1 ϑ ), (so that − 1 ϑ = ρ 2 (u)). Note that (1.14)-(1.15) is indeed a variant of the well-known phase field model proposed by O. Penrose and P.C. Fife (see the pioneering paper [21]), for the modelling of the kinetics of phase transitions. In this framework, ϑ is the absolute temperature of a physical system undergoing a phase transition. Moreover, the evolution of the phenomenon is described in terms of the order parameter χ, depending on the specific phase transition considered (see also [4,Chap. IV] for several examples in this connection). Let us point out that the energy balance equation (1.14) features the heat flux law (ϑ) := ϑ − 1 ϑ for ϑ > 0, which was first proposed in [8], on the grounds of physical feasibility, for modelling large temperatures.
Although we have so far highlighted just a formal link between the two systems ((1.7),(1.12)) and ((1.14),(1.15)), these considerations still have some interest. As a matter of fact, they fit well in the framework of a series of papers, investigating the (standard) viscous Cahn-Hilliard and Cahn-Hilliard equations as limits of the phase field system proposed by G. Caginalp [5] (see [11,12,22,25]) and as limits of (a variant of) the Penrose-Fife system (see [23]). While referring to [23], and the references therein, for a detailed account of the asymptotic analyses of the Caginalp and Penrose-Fife phase field systems, we just recall here that, for instance, in [25] it is shown that the solutions to the Caginalp phase field system converge as two suitable time relaxation parameters vanish to the solution of the Cahn-Hilliard equation; this kind of analysis is extended to the viscous Cahn-Hilliard equation in [22]. Actually, here we have not been able to prove rigorously that solutions to ((1.14),(1.15)) converge to solutions of ((1.7),(1.12)) (with α 2 given by (1.13)), as ε ↓ 0. Thus, we have restricted our attention to the limiting problem ((1.7),(1.12)), which is still interesting from the analytical point of view, as it features a singular law α 2 .
We will supplement the system ((1.7),(1.12)) with the initial condition (1.9) and the boundary conditions where γ is a positive constant and h : Γ×(0, T ) → R, and we will refer to the initialboundary value problem ((1.7),(1.12), (1.9), (1.16)) as Problem P 2 . Note by the way that in the case of the standard viscous Cahn-Hilliard equation (corresponding to α 2 (r) := r for every r ∈ R), (1.16) in fact prescribes Robin (or third type) boundary conditions on the chemical potential, which are not very usual for this kind of problems. Indeed, in this case the mean value of χ is no longer conserved, and the order parameter χ necessarily has a different physical interpretation. For instance, when Dirichlet boundary conditions, which could also be handled in our framework, are imposed on the chemical potential, the (viscous) Cahn-Hilliard equation models the propagation of a solidification front in a medium at rest with respect to the front. In turn, if we take into account our choice (1.13) of α 2 , then the conditions (1.16) appear somehow more natural from an analytical point of view, as we will detail in Remark 4.1 later on. Roughly speaking, a third type boundary condition on u allows us to recover some "coercivity" (namely, an H 1 (Ω)-a priori bound), for u from the first equation (1.7): such an estimate proves indeed to be crucial for obtaining a well-posedness result for P 2 . Note that this analytical difficulty is instead easily bypassed in the framework of Problem P 1 . Indeed, in that case, α 1 (and thus ρ 1 ) is bi-Lipschitz, which, basically, enables us to estimate the mean value of u from the second equation (1.8). Combining now this estimate with the bound on the L 2 (Ω)-norm of ∇u which we infer from (1.7), by Poincaré's inequality we still recover a H 1 (Ω)-a priori bound on u as needed. We will approximate both Problem P 1 and Problem P 2 by inserting the time derivative of u in the first equation of each system; focusing e.g. on P 1 , the related approximate system will be given by ν being a positive constant, coupled with (1.8). Note that ((1.17),(1.8)) has itself the structure of a phase field system in the variables u and χ. Then, we will show that the sequence of the solutions {(χ ν , u ν )} ν to the initial-boundary value problem for ((1.17),(1.8)) converges to the unique solution of Problem P 1 , thus establishing our first well-posedness result Theorem 2.1. In the same way, we will prove that Problem P 2 admits a unique solution (cf. Theorem 2.2), and we will investigate its further regularity under some additional assumptions on the data of P 2 , thus obtaining Theorem 2.3.
Plan of the paper. Our main results, together with some preliminary material, are presented in Section 2. Section 3 is devoted to the proof of our existence and continuous dependence results for P 1 , while the well-posedness and regularity for P 2 are tackled in Section 4. Acknowledgment. I would like to thank Professor Pierluigi Colli for introducing me to this subject, as well as for many valuable related discussions.
2. Preliminaries and Statement of the Main Results. we will identify H with its dual space H , so that W ⊂ V ⊂ H ⊂ V ⊂ W , with dense and compact embeddings. We will also deal with the Sobolev space H 1/2 (Γ), with dual H −1/2 (Γ); while referring the reader to [15,Chap. 1] for the definition and properties of H 1/2 (Γ), we just recall that for every v ∈ V , the trace v| Γ is in H 1/2 (Γ). We will denote by ( · , · ) H the inner product in H and in H N , by · , · ( ·, · Γ , resp.), the duality pairing between V and V (between H −1/2 (Γ) and H 1/2 (Γ), resp.), and by | · | H , · V , · V the norms in H, in V and in V ; furthermore, let V, H, and V be the subspaces of the elements v with zero mean value m Ω (v) = 1 |Ω| v, 1 in V , H, and V , respectively. We also consider the operator A : V → V given by and note that Au ∈ V for every u ∈ V . Indeed, the restriction of A to V is an isomorphism, and we can introduce its inverse operator N : V → V, defined by Throughout the following Section 3, we will make a systematic use of the relations

RICCARDA ROSSI
Moreover, we will consider on the spaces V and V the following norms which are equivalent to the standard ones on behalf of Poincaré's inequality for the zero mean value functions. It follows from the above formulae that In addition, we shall make use of the two following elementary inequalities for the functional W , (W being the double well potential in (1.2)), Variational formulation and main results for Problem P 1 . Let us detail our assumptions on Problem P 1 ((1.7)-(1.10)): α 1 : R → R is strictly increasing and bi-Lipschitz, namely As for the other data of Problem P 1 , we will suppose that We can then define the function G ∈ L 2 (0, T ; V ) by Remark 2.1. Let ρ 1 : R → R be the inverse function of α 1 : it follows from (2.6) that ρ 1 is strictly increasing and fulfils with of course d 1 = 1/M 1 , D 1 = 1/m 1 . Finally, let ρ o 1 := ρ 1 (0), and let us introduce the primitive Clearly, ρ 1 is strictly convex and (2.14) We can now give a rigorous variational formulation for the initial boundary value problem P 1 . 16) and the initial condition (1.9) holds for χ.
Of course, this is in accordance with the requirement χ 0 ∈ V . Let us also point out that the boundary conditions (1.10) for χ and u are contained in the above variational formulation, on account of (2.10) as well.
ii). Let us further require the function G (2.10) to fulfil (i.e., (1.11)). Then, any solution χ to Problem P 1 enjoys the conservation property To see this, it suffices to test (2.15) by 1, so that d Under the assumptions (2.6)-(2.9), Problem P 1 admits a unique solution (χ, u). The uniqueness statement of the above Theorem is of course a straightforward consequence of Proposition 2.1.

2.3.
Variational formulation and main results for Problem P 2 . Functional setting. Let us consider the operator J : V → V given by Of course, J is linear and bounded on V ; moreover, a standard version of Poincaré's inequality ensures that there exists a constant K * , depending on γ and on the geometry of the domain Ω only, such that Throughout this subsection and the following Section 4, we will consider, on the space V (V , respectively), the inner product ((v 1 , v 2 )) := Jv 1 , v 2 for every v 1 , v 2 ∈ V ( ((w 1 , w 2 )) * := w 1 , J −1 (w 2 ) for every w 1 , w 2 ∈ V , resp.). Accordingly, we will endow V and V with the norms which are equivalent to the usual norms on V and V . We will also renorm H so that the aforementioned constant K * fulfils K * = 1. In this setting, (V, H, V ) is still of course a Hilbert triplet, and J : V → V turns out to be the duality mapping: therefore, we have Formulation of Problem P 2 . We can now state our main assumptions on the data of P 2 : besides (2.7) and (2.8), we will suppose that As in (2.10), we also introduce the function F ∈ L 2 (0, T ; V ) given by Furthermore, let us consider the function ρ 2 (u) : . Then, ρ 2 is strictly convex and ρ 2 (u) ≥ ρ 2 (0) = 0; besides, ρ 2 fulfils the analogue of (2.14) (replacing of course D 1 by D 2 ).
In this setting, the variational formulation for Problem P 2 reads as follows.
Then there exists a positive constant S * , depending on M * , T , |Ω| , and m 2 , such that Our well-posedness result for Problem P 2 reads as follows. Lastly, we will give a regularity result for Problem P 2 (cf. Remark 4.2 later on), under the following additional regularity assumptions: . This is in agreement with our additional regularity requirement (2.33) on the initial datum χ 0 , as we will see later on in Section 4.3, cf. (4.30).
In the sequel, we will denote by the same symbol C several constants depending only on the quantities specified by the statement of each theorem, and possibly on the initial data; we will point out the few occurring exceptions.
Proof. Our argument follows the same outline of the proof of [8,Prop. 3.6], to which we refer the reader for further details. Thus, we first examine equations (3.2) and (3.3) separately.
As for the former, it follows from the well-known result [15,Thm.4.1,p.238] that for any j ∈ L 2 (0, T ; H) there exists a unique function u =: ∈ (0, T ), (3.5) and the first initial condition in (3.4). Moreover, there exists a constant C, depending on the data of the problem only, and not on ν, such that for any pair (j 1 , j 2 ) there holds Then, (3.6) follows, noting that the second summand on the left-hand side of the above inequality equals 3) (with k instead of ρ 1 (u ν ) on the right-hand side), and the second initial condition in (3.4). Moreover, the continuous dependence estimate holds for any pair k 1 , k 2 ∈ L 2 (0, T ; H), with χ i = S ν (k i ), i = 1, 2. While referring to [8] for further details, let us only say that, as in the previous case, it suffices to take the difference of (3.3) with data k 1 and k 2 , test the resulting equation by the difference (χ 1 − χ 2 ) of the corresponding solutions and exploit the monotonicity of the map χ → χ 3 , so that a straightforward application of Gronwall's Lemma leads to the estimate (3.7).

3.2.
Existence for Problem P 1 .  14) and the pair (χ, u) solves Problem P 1 . The existence statement of Theorem 2.1 is of course a direct consequence of Proposition 3.2.
Proof. As a first step towards the proof of this result, we shall provide some a priori estimates on the approximate solutions {(χ ν , u ν )} ν , which will enable us to pass to the limit in (3.2)-(3.4).
First estimate. We test (3.2) by ρ 1 (u ν ) − ρ o 1 (cf. Remark 2.1), (3.2) by ∂ t χ ν , we add the resulting equations and integrate on (0, t), for t ∈ (0, T ) (note that a regularization procedure is needed on the test function ∂ t χ ν , since it is not in 416 RICCARDA ROSSI V : for further details, we refer the interested reader to the proof of [22,Thm. 3], where such a regularization -originally proposed in [7, Lemma 2.9]-is applied to an analogous estimate). Two terms cancel out, and, taking into account (2.12), we easily obtain whereas we can deal with the last two terms on the left-hand side of (3.15) recalling (2.4), which yields Turning to the right-hand side of (3.15), we can estimate the first term therein: indeed, on behalf of (2.14) and (3.10). The latter hypothesis further enables us to control all the terms depending on the initial data χ 0ν . Finally, as for the last two summands in (3.15), we note that where the constant C in both inequalities depends on ρ o 1 , |Ω|, and T , only. The first term on the right-hand side of (3.20) is estimated by by virtue of Poincaré's inequality for the zero mean value functions, whereas we infer by comparison in (3.3) that Let us denote by S i , i = 1, 2, the two summands on the right-hand side of (3.22): using (2.5) as well, we see that where the constants c 1 and c 2 only depend on |Ω|, T , and on the initial datum χ 0 . Collecting (3.16)-(3.24), we infer from (3.15) (3.25) where the constant C 0 only depends on the initial data u 0ν , χ 0ν , cf. (3.10), and on ρ o 1 , |Ω|, T . Note that the first term in the left-hand side of the above inequality is positive thanks to (2.13). Then, an easy application of Gronwall's Lemma (see, e.g., [3,Lemma A.4]) to χ 4 ν (t) L 1 (Ω) allows us to conclude that for a constant C ≥ 0 independent of ν. Arguing by comparison in (3.3), we note that m Ω (ρ 1 (u ν )) L 2 (0,T ) ≤ C, so that we also infer that ρ 1 (u ν ) L 2 (0,T ;V ) ≤ C.
(3.27) Second estimate. Let us preliminarily note that (3.28) where we have used the crucial Lipschitz continuity assumption (2.6). Therefore, estimate (3.27) yields that Thus, we can test (3.2) by u ν and integrate over (0, t), getting We can control the latter summand with the second term on the left-hand side of (3.30), once again by Poincaré's inequality, while the other two terms are bounded in view of the estimate (3.26) for χ ν , and of (3.29) for m Ω (u ν ). We argue in the same way for the last summand on the right-hand side of (3.30), so that, using (3.10) again, we finally infer Note that the last estimate in (3.31) follows from a simple comparison argument in (3.2).
In the end, taking into account the previous estimates (3.26) and (3.27), and arguing by comparison in (3.3), we deduce that {Aχ ν } is bounded in L 2 (0, T ; H), whence by standard elliptic regularity results for a constant C independent of ν.
To conclude the proof, it remains to show that We observe that ρ 1 induces a maximal monotone graph on L 2 (Ω×(0, T ) Note that in the above chain of inequalities, we have substituted (3.2) for ∂ t χ ν ; then we have exploited (3.10) and combined the strong convergences (3.12) for χ ν and the convergences (3.13)-(3.14) for u ν in order to pass to the limit, deducing in the end the last equality.
We have so far obtained that the pair (χ, u) solves Problem P 1 , which on the other hand has a unique solution by Proposition 2.1. Then the limit pair (χ, u) does not depend on the subsequence that we have extracted, and we conclude that the convergences (3.11)-(3.14) hold indeed for the whole families {χ ν } and {u ν } as ν ↓ 0.
Proof of Proposition 2.1. Referring to the notation of the statement of Proposition 2.1, let us set The pair (χ, u) obviously satisfies m Ω (∂ t χ(t)) = m Ω (G(t)) = 0 for a.e. t ∈ (0, T ). (3.40) We can thus test (3.38) by N (ρ 1 (u 1 (t)) − ρ 1 (u 2 (t)) − m Ω (ρ 1 (u 1 (t)) − ρ 1 (u 2 (t)))), (3.39) by N (∂ t χ(t)) for a.e. t ∈ (0, T ), add the resulting equations and integrate over (0, t), t ∈ (0, T ). Note that two terms cancel out, since, in view of (2.2) and (3.40), we have for a.e. t ∈ (0, T ), and, in the same way, On the other hand, (2.1) and (2.6) yield that Finally, in view of (2.1), (2.2), and (2.3), we have for a.e. t ∈ (0, T ) Collecting (3.41)-(3.42), we may conclude (u(s), m Ω (ρ 1 (u 1 (s)) − ρ 1 (u 2 (s)))) H ds Let us first note that, by (3.39), Then, the second term on the right-hand side of (3.43) can be obviously estimated by Now, the first summand in (3.44) can be dealt with by using the elementary inequality where we have used Young's inequality for some σ 1 > 0 to be specified later. Note also that the last inequality above follows from the Lipschitz continuity (2.6) of α 1 and from (3.37), which yields an a priori bound on χ i L ∞ (0,T ;L 4 (Ω)) in terms of the data χ i 0 , f i , g i , i = 1, 2. Concerning the second summand in (3.44), once again by Young's inequality and (2.6), we have The fourth summand on the right-hand side of (3.43) can be coped with arguing in the same way as in ( Again, the last summand in the line above can be further estimated by means of (3.37). Finally, we note that, also on account of (2.3), for some σ 3 > 0 to be chosen later. Hence, collecting (3.43)-(3.50), we obtain where the constant K depends on σ 1 , σ 2 and the related constants C σi , i = 1, 2, as well as on m * and the data χ 0i , f i and g i , i = 1, 2, through the a priori estimate (3.37). Then, we now choose the constants σ i , i = 1, 2, 3, in such a way that σ 1 M 2 1 + σ 2 M 2 1 + σ 3 < m 1 : a direct application of Gronwall's Lemma allows us to conclude that where the constant s * depends on χ i 0 , f i and g i , i = 1, 2 through (3.37), and on the data m 1 , M 1 , T , and Ω. Finally, let us test (3.39) by χ and integrate over (0, t), t ∈ (0, T ): we easily obtain Note that the third summand on the left-hand side of the above inequality is positive thanks to the monotonicity of the map r → r 3 : then, taking into account (3.52), we easily deduce the continuous dependence estimate (2.18).

4.
Well-posedness and Regularity for Problem P 2 . We recall that throughout this section we will refer to the analytical setting specified in Section 2.3.

4.1.
Existence for Problem P 2 via an approximate problem. As the previous section, we will approximate Problem P 2 with the initial-boundary value problem for the system ((1.12),(1.17)), supplemented with (1.16) and suitable initial conditions. Problem P 2µ . Let χ 0µ and u 0µ fulfil (3.1) for every µ > 0, and let F be defined by (2.10).
Sketch of the proof. Our proof follows the same outline as the argument developed for Proposition 3.1: namely, a fixed point technique is applied, which leads to the analysis of the single equations (4.1) and (4.2). Then, we directly refer the reader to the proof of Proposition 3.1 for the analysis of the latter equation, as well as the definition of the related solution mapping S µ . As far as (4.1) is concerned, we can easily show that for any ∈ L 2 (0, T ; H) there exists a unique function Indeed, arguing as in [8,Lemma 3.4]), we may tackle the Cauchy problem above by means of the abstract theory of nonlinear semigroups generated by maximal monotone operators, so that its well-posedness follows from the well-known results [1,  6) and (3.7) may be obtained in this case as well, carrying out the same computations as in the proof of Proposition 3.1. Therefore, we can introduce the solution operator T µ : (L 2 (0, T ; H)) 2 → (L 2 (0, T ; H)) 2 by obviously adapting the definition (3.8) of the solution operator T ν for P 1ν . Note that a contraction estimate analogous to (3.9) holds in this setting too, in view of the analogues of (3.6) and (3.7), as well as of the Lipschitz continuity (2.26) of ρ 2 . As in the proof of Proposition 3.1, the contraction mapping principle then ensures the well-posedness of Problem P 2µ .
and the pair (χ, u) solves Problem P 2 .
Proof. A priori estimates. In order to derive some a priori estimates on the sequences of approximate solutions {χ µ }, {u µ }, we test (4.1) by u µ + ρ 2 (u µ ) − ρ o 2 , (4.2) by ∂ t χ µ , add the relations thus obtained and integrate over (0, t), for t ∈ (0, T ). By formal computations (which can be made rigorous by performing the regularization that we mentioned in the proof of Proposition 3.2), we get where we have cancelled out two terms and used (2.21) for the third summand on the left-hand side of (4.8). On the other hand, note that t 0 (∇(u µ (s)), ∇(ρ 2 (u µ (s)))) H ds + γ in view of the definition of ρ o 2 (cf. Remark 2.3), of our assumption (2.23) on α 2 , and of (2.21) once again to conclude the last equality. Furthermore, we can deal with the last two terms on the left-hand side of (4.8) as in (3.17). Turning to the right-hand side, all the terms depending on the initial data χ 0µ and u 0µ are easily controlled thanks to (3.10) and (2.14) (cf. (3.18)). Concerning the remaining summands, we note while the last summand in (4.8) can be estimated exactly as in (3.19). Collecting (4.9)-(4.12), we deduce from (4.8) where the constant C 0 only depends on the approximate data {u 0µ } and {χ 0µ } through (3.10), as well as on m 2 , ρ o 2 , |Ω|, and T . Therefore, there exists a positive constant C such that µ ρ 2 (u µ ) L ∞ (0,T ;L 1 (Ω)) +µ 1/2 u µ L ∞ (0,T ;H) + u µ L 2 (0,T ;V ) + ρ 2 (u µ ) L 2 (0,T ;V ) +µ ∂ t u µ L 2 (0,T ;V ) + χ µ H 1 (0,T ;H)∩L ∞ (0,T ;V )∩L 2 (0,T ;W ) ≤ C ∀µ > 0, (4.13) where the estimate for ρ 2 (u µ ) is an obvious consequence of the estimate for ρ 2 (u µ )− ρ o 2 , while the bound for µ ∂ t u µ L 2 (0,T ;V ) (for χ µ L 2 (0,T ;W ) , respectively), follows from a comparison in (4.1) (in (4.2), resp., yielding that {Aχ µ } is bounded in L 2 (0, T ; H), whence the bound for {χ µ } in L 2 (0, T ; W ) by elliptic regularity results). Passage to the limit. We can now refer the reader to the final part of the proof of Proposition 3.2. Arguing in the same way, we indeed deduce the convergences (4.4)-(4.7) from the a priori estimates (4.13). Then, the passage to the limit in the nonlinear coupling term ρ 2 (u µ ) in (4.2) may be performed by monotonicity exactly as in (3.36). Thus, we can pass to the limit in both (4.1) and (4.2), and conclude that the limit pair (χ, u) is the unique solution to Problem P 2 . Indeed, the main technical difficulty that one encounters in passing to the limit in Problems P 1ν and P 2µ is related to the deduction of suitable a priori estimates for the sequences {u ν } ν and {u µ } µ , respectively. Focusing on Problem P 1 , the (physically preferable) Neumann boundary conditions for u ν yield poor estimates on the sequence {u ν } ν , due to a lack of coercivity in (3.2). Nonetheless, it is possible to recover the H 1 -a priori bound that we need for u ν , by estimating m Ω (u ν ) through Note that our regularity requirements on u 0µ and χ 0µ are in agreement with (4.25) (cf. Remark 2.4 and (4.30) below), and with (4.26). Indeed, it follows from the latter that ∂ t u µ ∈ C 0 w ([0, T ]; H), whence we should have µ∂ t u µ (0) = −∂ t χ µ (0) + F (0) − Ju 0µ ∈ H, (4.27) which is in fact ensured by (2.33) and (4.24). We will not develop the proof of the above result, since it relies on the same estimates we shall perform later on for proving our main regularity result Theorem 2.3. Such estimates could indeed be made rigorous by a suitable approximation scheme for Problem P 2µ (possibly based on an implicit time discretization procedure). Nonetheless, we choose not to detail such a scheme here, in order to avoid too many technical details; besides, these approximation techniques are by now standard in the framework of phase field models.