An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter and the chemical potential; each equation includes a viscosity term and the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In the recent paper arXiv:1103.4585v1 [math.AP] we proved that this problem is well posed and investigated the long-time behavior of its solutions. Here we discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0. We prove convergence of solutions to the corresponding solutions for the limit problem, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.


Introduction
The system we study was proposed for mathematical investigation in [5]; as to modeling issues, its most directly relevant antecedents are two papers by Fried & Gurtin [8] and Gurtin [9], and a paper by one of us [11].
A nonstandard phase-field evolution problem. The initial/boundary-value problem we dealt with in [5] consists in finding two phase fields, the chemical potential µ and the order parameter ρ, such that in Ω × (0, +∞), in Ω × (0, +∞), (1.2) ∂ ν µ = ∂ ν ρ = 0 on Γ × (0, +∞), (1.3) µ( · , 0) = µ 0 and ρ( · , 0) = ρ 0 in Ω, (1.4) where Ω denotes a bounded domain of R 3 with (sufficiently) smooth boundary Γ, and f ′ stands for the derivative of a double-well potential f . This nonstandard phase-field model can be regarded as a variant of the classic Cahn-Hilliard system for diffusion-driven phase segregation by atom rearrangement: Note, in (1.1), the unpleasant nonlinear terms involving time derivatives, and the fact that we have taken the mobility coefficient κ > 0 equal to 1. Moreover, recall that equations (1.5) are customarily combined so as to obtain the well-known Cahn-Hilliard equation: ∂ t ρ = κ∆(−∆ρ + f ′ (ρ)). (1.6) Fried & Gurtin's generalization of Cahn-Hilliard equation. In [8,9] a broad generalization of (1.6) was arrived at, with three measures: (i) by regarding the second of (1.5) as a balance of microforces: where the distance microforce per unit volume is split into an internal part π and an external part γ, and the contact microforce per unit area of a surface oriented by its normal n is measured by ξ · n in terms of the microstress vector ξ; 1 (ii) by interpreting the first equation of (1.5) as a balance law for the order parameter : where the pair (h, σ) is the inflow of ρ; (iii) by requiring that the constitutive choices for π, ξ, h, and the free energy density ψ, be consistent in the sense of Coleman and Noll [2] with a postulated "dissipation inequality that accomodates diffusion": ∂ t ψ + (π − µ)∂ t ρ − ξ · ∇(∂ t ρ) + h · ∇µ ≤ 0 (1.9) (cf. equation (3.6) in [9]).
With the use of (1.7), (1.8), (1.10), and (1.11) 1 , a general equation for diffusive phase segregation processes is arrived at: ∂ t ρ = div M ∇ ∂ ρ ψ(ρ, ∇ρ) − div ∂ ∇ρ ψ(ρ, ∇ρ) − γ + σ; in particular, the Cahn-Hilliard equation (1.6) is obtained by taking the external distance microforce γ and the order-parameter source term σ identically null, and by choosing ψ(ρ, ∇ρ) = f (ρ) + 1 2 |∇ρ| 2 , M = κ1 . (1.12) An alternative generalization of Cahn-Hilliard equation. In [11], a major modification of Fried & Gurtin's approach to phase-segregation modeling was proposed. While the crucial step (i) was retained, both the order-parameter balance (1.8) and the dissipation inequality (1.9) were dropped and replaced, respectively, by the microenergy balance and the microentropy imbalance (1.14) A further new feature was that the microentropy inflow (h, σ) was deemed proportional to the microenergy inflow (h, σ) through the chemical potential µ, a positive field; consistently, the free energy was defined to be with the chemical potential playing the same role as the coldness in the deduction of the heat equation. 2 Combination of (1.13)-(1.15) gives: an inequality that replaces (1.9) in restrictingà la Coleman-Noll the possible constitutive choices. On taking all of the constitutive mappings delivering π, ξ, η, and h, dependent in principle on ρ, ∇ρ, µ, ∇µ, and on choosing compatibility with (1.16) implies that we must have: If we now choose for H the simplest expression H = κ1 , implying a constant and isotropic mobility, and if we once again assume that the external distance microforce γ and the source σ are null, then, with the use of (1.18) and (1.15), the microforce balance (1.7) and the energy balance (1.13) become, respectively, (R3) in front of both ∂ t µ and ∂ t ρ there are nonconstant factors that should remain nonnegative during the evolution. Thus, the system (1.19)-(1.20) deserves a careful analysis. We must confess that we boldly attacked this problem as is, prompted to optimism by the successful outcome of a previous joint research effort [3,4] devoted to tackling the system of Allen-Cahn type one arrives at via the approach in [11] for no-diffusion phase-segregation processes. Unfortunately, the evolution problem ruled by (1.19)-(1.20) turned out to be too difficult for us. Therefore, we decided to study its regularized version (1.1)-(1.4), which we obtained by introducing the extra terms ε ∂ t µ in (1.20) and δ ∂ t ρ in (1.19), for small positive coefficients ε and δ. Motivations for the introduction of such terms are proposed and discussed in [5]; interestingly, while the second can be interpreted as a dissipative part of the distance microforce, so far we have not been able to find a convincing physical interpretation for the first. But, our present study demonstrates -so we believe -that it can legitimately be regarded as an efficient mathematical device.
Limit as the first parameter tends to 0. In [5], by assuming (as we did in [3,4]) that f ′ is the sum of a strictly increasing C 1 function f ′ 1 with domain (0, 1) that is singular at the endpoints, and of a smooth bounded perturbation f ′ 2 (to allow for a double-or multi-well potential f ), we first proved existence of a strong solution (µ, ρ) to (1.1)-(1.4) satisfying µ ≥ 0 and 0 < ρ < 1 almost everywhere in Ω × (0, +∞) (of course, we stipulated that the initial data meet same requirements in Ω). Then, under some additional technical assumptions, we showed that the component µ is bounded, and so is f ′ (ρ); as a consequence, ρ stays away from the threshold values 0 and 1. These boundedness properties are very useful in proving uniqueness of solutions.
In some sense, passing to the limit as the regularizing parameters tend to zero is the challenging final aim of our research project. For the moment being, we are able to deal with ε and to deduce, by a rather delicate asymptotic analysis, an existence theorem for the limit problem. Precisely, we let ε tend to zero and show that any weak or weak star limit of any subsequence of solutions (µ ε , ρ ε ) to (1.1)-(1.4) yields a solution (µ, ρ) to the resulting limit problem, which is obtained by putting ε = 0 in (1.1)-(1.4) and rewriting the corresponding first equation in the form 2∂ t (µρ) − ∆µ = µ ∂ t ρ rather than as 2ρ ∂ t µ + µ ∂ t ρ − ∆µ = 0. (1.21) This we do because it is not clear to us from the structure of the system whether a suitably regular representation for ∂ t µ could be recovered in the limit, while we are able to show that the time derivative ∂ t (µρ) actually exists, at least in some generalized sense. 3 Here, as we did in [5], we also deal with the long-time behavior of the system. We prove that each element (µ ω , ρ ω ) of the ω-limit set for a certain trajectory is a steady state solution to (1.1)-(1.4); therefore, in particular, µ ω is a constant (cf. (1.21) and (1.3)). 4 An outline of our paper is the following: in Section 2, we carefully state assumptions and results; Section 3 is devoted to the proof of the convergence theorem, so as to deduce 3 In fact, (1.1) has the equivalent formulation: which singles out the time derivative of the auxiliary variable εµ + 2µρ for ε > 0. 4 Note that the steady state problem associated with both cases ε > 0 and ε = 0 is the same.
the existence of solutions to the limit problem; finally, in Section 4, we develop our argument for the characterization of the ω-limit.

Assumptions and main results
First of all, we assume Ω to be a bounded connected open set in R 3 with smooth boundary Γ and set, for convenience, We endow these spaces with their standard norms, for which we use self-explaining notation like · V . However, we write · H for the norm in any power of H as well. The symbol · , · denotes the duality product between V * , the dual space of V , and V itself.
Since Ω is bounded and smooth, the embeddings W ⊂ V ⊂ H are compact. Moreover, since V is dense in H, we can identify H with a subspace of V * in the usual way, i.e., in order that u, v = (u, v) H , where ( · , ·) H denotes the inner product in H, holds for every u ∈ H and v ∈ V . Then, also the embedding H ⊂ V * is compact.
As in [5], we assume that As to initial data, we start with the assumptions µ 0 ∈ V and µ 0 ≥ 0 a.e. in Ω; The reader is referred to the forthcoming Remark 2.5 for weaker conditions.
Since we aim to let ε tend to zero, we stress the dependence of the solution found in [5] on the parameter ε. In that paper, for any fixed T > 0, the following a priori regularity is required: where Q T := Ω×(0, T ). We note that homogeneous Neumann boundary conditions follow from (2.7)-(2.8), in view of the definition of W (see (2.1)). Then, the ε-problem is: a.e. in Ω. (2.13) We recall the existence result of [5].
As to uniqueness, the result in [5, Thm. 2.2] holds for solutions that, in addition to (2.7)-(2.10), have certain properties that, in turn, are guaranteed whenever the initial data fulfil the following conditions, additional to (2.5) and (2.6): Within such a framework, since T > 0 is arbitrary, the existence of a unique solution (µ ε , ρ ε ) defined for every positive time was ensured, and its long-time behavior was studied.
(2.22) Remark 2.3. The last integral in (2.20) makes sense because V ⊂ L 3 (Ω) and µ∂ t ρ belongs at least to L 1 (0, T ; L 3/2 (Ω)), as a consequence of (2.15)-(2.16). Note that (2.20) incorporates the homogeneous Neumann boundary condition for µ in a generalized sense. Moreover, note that (2.19) implies that µρ is a continuous V * -valued function, so that the first equality in (2.22) has a precise meaning. On the contrary, no continuity for µ is ensured at the moment.
Remark 2.5. The assumptions (2.5)-(2.6) are strong. In fact, while they are needed for Theorem 2.1 and Proposition 2.2, some of them will not play any role in the following, as it can be seen by looking at our a priori estimates. For instance, the last condition in (2.6) will not be important, since just f (ρ 0 ) ∈ L 1 (Ω) will be used. Accordingly, one can prove a result similar to Theorem 2.4, but involving ε-approximations of less regular initial data that, this notwithstanding, satisfy (2.5)-(2.6). Precisely, suppose we assume that Then, it would be possible to construct ε-approximations (µ 0ε , ρ 0ε ) of such initial data (µ 0 , ρ 0 ) that satisfy (2.5)-(2.6) and whose norms of type (2.23) remain bounded as ε ց 0 : e.g., as to ρ 0ε , one could take the solution to the elliptic equation Our final aim is to study the long-time behavior of any solution constructed according to Theorem 2.4. To this end, we introduce the ω-limit of the trajectory in a proper topology, and prove that every element of it coincides with a steady state. We set: weakly in H × V for some sequence t n ր +∞ . (2.24) The above definition has a precise meaning, because the pointwise values of the (H × V )valued function (µ, ρ) are well defined thanks to the continuity properties stated in our next result.  , (µ, ρ) is bounded, and its components µ and ρ are weakly and strongly continuous, respectively. In particular, the ω-limit (2.24) is nonempty. Moreover, every element of ω(µ, ρ) coincides with a pair (µ s , ρ s ) such that µ s is a nonnegative constant, in Ω, (2.25) i.e., it coincides with a steady state.
We stress that the above result does not necessarily hold for all possible solutions. Indeed, it only deals with solutions obtained as limits of solutions to the ε-problem as ε ց 0. We also observe that there is no reason for the function ρ s mentioned in the statement to be a constant, since f is not required to be convex.
The rest of the paper is organized as follows: in the next section, we prove both Proposition 2.2 and Theorem 2.4; the proof of Theorem 2.7 will be given in the last section.

Global solutions
We first prove the existence of a global solution to the ε-problem. The major part of the section is devoted to the proof of Theorem 2.4 and the subsequent existence of a global solution to the limit problem.
Proof of Proposition 2.2. We imitate, with minor changes, the proof of Thm 2.1 in [5], where the final time T was fixed once and for all. Let ε be fixed and, for notational conciseness, let the dependence on ε be omitted. The main tool used in [5] was an approximation procedure using a time delay τ = T /N, for N a positive integer. Approximating τ -problems were constructed and solved step by step in the time intervals I n := [0, nτ ], n = 1, . . . , N. It turned out that the resulting unique solution (µ τ , ρ τ ) coincided a posteriori with the one obtained by glueing together solutions on the time steps [(n − 1)τ, nτ ], n = 1, . . . , N.
Preliminaries to the proof of Theorem 2.4. We begin by listing some of the tools we shall use. First of all, the well-known continuous embedding, with the related Sobolev inequality, holds in our 3-dimensional case: with the constant C p in (3.1) depending only on Ω and p; moreover, where C depends only on Ω and the embedding V ⊂ L q (Ω) is compact if q < 6. Furthermore (see, e.g., where C T depends on Ω and T . In our proof, we shall make use also of the well-known Hölder inequality, mainly in the form Remark 3.1. To avoid a cumbersome notation, the lowercase letter c stands for different constants, each of which may depend on one or another of the data involved in our current statement and on the coefficient δ, but never depends either on ε or on the final time T ; consequently, the relative estimates continue to hold when we discuss both the system's asymptotic limit as ε tends to 0 and its long-time behavior. Moreover, a notation like c σ signals that that constant has an additional dependence on the parameter σ. Hence, the meaning of c and c σ may change from line to line, and even in the same chain of inequalities. On the contrary, we use the uppercase letter C for precise constants we are going to refer to after their introduction, such as C p in (3.1) or C T in (3.4). Finally, in order to lighten our notation, we do not write the subscript ε in performing our a priori estimates until each estimate is completely proved; the same we do for the auxiliary function u ε := εµ ε + 2µ ε ρ ε . (3.6) Next, we prove that ρ ε is bounded away from zero uniformly with respect to ε. Such a result is essentially known from the proof of [5, Thm 2.3], among other properties there established for a fixed ε. Nevertheless, we prefer to repeat the proof here, in order to make sure that the constructed lower bound is in fact independent of ε, and that just the additional assumption (2.14) is used.
Proof of Theorem 2.4. Our proof will proceed as follows. For a fixed finite final time T , we shall perform a number of a priori estimates and use well-known compactness results to prove that, as ε tends to 0, the solution (µ ε , ρ ε ) to the ε-problem (2.11)-(2.13) we are considering converges to a solution (µ, ρ) to problem (2.20)-(2.22), at least for a subsequence ε n ց 0; in particular, this holds for T = 1, 2, . . . . Having established this result, we shall be able to argue as in the proof of Proposition 2.2. Indeed, a diagonal procedure provides a subsequence ε n ց 0 such that (µ ε , ρ ε ) with ε = ε n converges to a global solution (µ, ρ) to problem (2.20)-(2.22) defined in the whole of [0, +∞). Therefore, just the case of a fixed final time T has to be considered.
In summary, so far we have proved (2.15)-(2.18), (2.21), and the second condition in (2.22). It remains for us to show (2.19), (2.20), and the first condition in (2.22). For this purpose, further arguments are needed.
Fourth a priori estimate. We recall (3.6), and we notice that the second (2.11) reads: (3.15) Moreover, µ ε satisfies homogeneous Neumann boundary conditions, since it is W -valued. Therefore, we have From (3.16), we derive an estimate for ∂ t u ε as a V * -valued function, in the framework of the Hilbert triplet (V, H, V * ). We treat the integrals on the right-hand side individually (for a while, we omit the subscript ε in order to simplify the notation).
In order to prove (2.20), one can try to let ε ց 0 in (3.16) first, then to get rid of time integration. But, a trouble arises in dealing with the first term on the right-hand side, since, for the moment being, both µ ε and ∂ t ρ ε are just weakly convergent. Hence, we have to prepare a new tool.
Fifth a priori estimate. We want to find a bound for ∇u ε , i.e., for the partial derivatives D i u ε , i = 1, 2, 3. As usual, we omit the subscript ε for a while. We have: Now, on taking (3.7) into account, we see that D i µ is bounded in L 2 (0, T ; H), while µ is bounded in L 2 (0, T ; L 6 (Ω)) thanks to the Sobolev inequality (3.4). On the other hand, (3.9) provides a bound for D i ρ in L ∞ (0, T ; H). Hence, using Hölder inequality, we see that the product µD i ρ is bounded in L 2 (0, T ; L 3/2 (Ω)). Therefore, we conclude that u ε L 2 (0,T ;W 1,3/2 (Ω)) ≤ c T . (3.21) Consequence. As (3.19) holds, from (3.21) we infer that u ε → 2µρ weakly in L 2 (0, T ; W 1,3/2 (Ω)). We stress that, in particular, u ε → 2µρ strongly in L 2 (0, T ; H). Proof. We set u := 2µρ and argue a.e. in Q T for a while. Thanks to Lemma 3.2, we have On the other hand, we have By combining these inequalities, we deduce that The first two terms on the right-hand side tend to zero as ε ց 0, by (3.23); as to the last term, it suffices to recall (2.15), (3.14), and (3.4) for q = 4.
Conclusion. The strong convergence guaranteed by Lemma 3.3, together with the weak convergence ∂ t ρ ε → ∂ t ρ in L 2 (0, T ; H) given by (3.12), imply that On the other hand, (3.20) and (3.11) hold. Hence, by letting ε ց 0 in (3.16), we easily obtain that for every z ∈ L 5 (0, T ; V ) ∩ L ∞ (Q T ). Now, take any v ∈ V ∩ L ∞ (Ω) and any ζ ∈ L ∞ (0, T ). Then the function z : t → ζ(t)v is admissible in (3.25), and a standard argument yields for every v ∈ V ∩ L ∞ (Ω) and for a.a. t ∈ (0, T ). Now, we note that, for a.a. t ∈ (0, T ), each term in the above equation defines an element of V * . This is clear as far the lefthand side is concerned, since ∂ t (µρ) is V * -valued and µ is V -valued. For the remaining term, we recall Remark 2.3. On the other hand, V ∩ L ∞ (Ω) is dense in V . Therefore, (2.20) follows, and the proof is complete.

Long-time behavior
This section is devoted to proving Theorem 2.7. We first derive some a priori estimates, then we prove the continuity property announced in the statement; finally, we characterize the ω-limit.
A priori estimates. We recall Lemma 3.2 and the a priori estimates (3.7) and (3.8), which involve constants that do not depend on the final time. Hence, we immediately obtain that ρ(t) ≥ r * and µ(t) H + ∇ρ(t) H ≤ c for a.a. t > 0. (4.1) Recalling that 0 < ρ < 1, we see, in particular, that (µ, ρ) is a bounded (H × V )-valued function, as stated. For the same reason, we also deduce that (µρ)(t) H ≤ c for a.a. t > 0. Strong and weak continuity. As far as ρ is concerned, we have ρ ∈ H 1 (0, T ; H) ∩ L 2 (0, T ; W ) for every T < +∞ by (3.12). Since the embedding holds, we immediately deduce that ρ is a strongly continuous V -valued function. The weak continuity of µ is less obvious: we prove it by using the following well-known tool, whose proof is left as an exercise to the reader.
Proposition 4.1. Let Z be a Hausdorff topological space, and let Z be a reflexive Banach space such that Z ⊂ Z, where the embedding is continuous with respect to the weak topology of Z. Assume that z : [0, T ] → Z is continuous and that z(t) ∈ Z and z(t) Z ≤ M for a.a. t ∈ (0, T ) for some constant M. Then z is Z-valued, i.e., z(t) belongs to Z for every t ∈ [0, T ], and is continuous with respect to the weak topology of Z. Moreover, In our case, we argue on any fixed finite time interval [0, T ] and apply Proposition 4.1 twice, first with Z = V * , with either the weak or the strong topology, then with Z = L 1 (Ω), endowed with the weak topology. We set: u := 2µρ, (4.4) in order to agree with (3.6), and we recall that u ∈ W 1,5/4 (0, T ; V * ), by (3.20); in particular, u ∈ C 0 ([0, T ]; V * ). On the other hand, we have proved (4.2). We conclude that u(t) ∈ H for every t ∈ [0, T ] and that u is continuous with respect to the weak topology of H. Besides, ρ is strongly continuous as an H-valued function, and the first condition in (4.1) holds. Hence, the same is true for 1/ρ, and we infer that µ = u/(2ρ) is a weakly continuous L 1 (Ω)-valued function. Now, we recall the estimate of µ given by (4.1) and conclude that µ is weakly continuous as an H-valued function.