A vanishing diffusion limit in a nonstandard system of phase field equations

We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011 and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

This system has been recently addressed in the paper [6]: the existence of solutions has been proved, thus complementing and extending the results of the papers [3,4,5] concerned with simpler or reduced versions of the problem.
Here, we are interested to investigate the asymptotic behavior of the above initialboundary value problem (1.1)-(1.4) as the positive diffusion coefficient σ appearing in (1.2) tends to 0.
Let us briefly explain the modelling background for (1.1)-(1.4). Such a system comes from a generalization of the phase-field model of viscous Cahn-Hilliard type originally proposed in [14], and it aims to describe the phase segregation of two species (atoms and vacancies, say) on a lattice in presence of diffusion. The state variables are the order parameter ρ, interpreted as the volume density of one of the two species, and the chemical potential µ. For physical reasons, µ is required to be nonnegative, while the phase parameter ρ must of course take values in the domain of f ′ .
We also recall the features of [3] and what has been generalized in [5,6]. Firstly, the nonlinearity f considered in [3] is a double-well potential defined in (0, 1), whose derivative f ′ diverges at the endpoints ρ = 0 and ρ = 1: e.g., for f = f 1 + f 2 with f 2 smooth, one can take f 1 (ρ) = c (ρ log(ρ) + (1 − ρ) log(1 − ρ)), (1.5) with c a positive constant. In this paper, we let f 1 : R → [0, +∞] be a convex, proper and lower semicontinuous function so that its subdifferential (and not the derivative) is a maximal monotone graph from R to R. Then, we rewrite equation (1.2) as a differential inclusion, in which the derivative of the convex part f 1 of f is replaced by the subdifferential β := ∂f 1 , i.e., ∂ t ρ − σ∆ρ + ξ + f ′ 2 (ρ) = µg ′ (ρ) with ξ ∈ β(ρ). (1.6) Note that f 1 need not be differentiable in its domain, so that its possibly nonsmooth and multivalued subdifferential β := ∂f 1 appears in (1.2) in place of f ′ 1 . In general, β is only a graph, not necessarily a function, and it may include vertical (and horizontal) lines, as for example when and β = ∂I [0,1] is specified by ξ ∈ β(ρ) if and only if ξ (1. 8) Secondly, while in [3] g was simply taken as the identity map g(ρ) = ρ, in [5,6] g is allowed be any nonnegative smooth function, defined (at least) in the domain where f 1 and its subdifferential live. The presence of such a function g allows for a more general behavior of (the related term in) the free energy, which reads (1.9) Indeed, in particular g(ρ) is not obliged, as it was instead for g(ρ) = ρ, to take its minimum value at ρ = 0, be increasing and with maximum value at ρ = 1 (when D(f 1 ) = [0, 1]), but we may have many other instances like, e.g., a specular behavior of g around the extremal points of the domain of f . Here, we have to impose an additional restriction on g, which however looks reasonable from the modelling point of view: we postulate that g is a (smooth) concave function, which in turn implies convexity with respect to ρ of the term −µ g(ρ) in the free energy (1.9). However, let us recall that f may stand for a multi-well potential in which the nonconvex perturbations are incorporated into f 2 , so that ψ in its entirety needs not be convex with respect to ρ.
An important generalization that is considered in this paper concerns the diffusivity κ. In [3], κ was just assumed to be a constant function, but it can be a positive-valued, continuous, bounded, and nonlinear function of µ (and this was the setting of [5]), or of µ and ρ as it is postulated in [6]. For simplicity, we confine ourselves to study of the convergence properties of the solution under an assumption that guarantees uniform parabolicity, i.e., κ ≥ κ * > 0. We point out that [5] treats the situation of κ depending only on µ and possibly degenerating somewhere. Therefore, the system turns out the initial and boundary value problem for a nonstandard and highly nonlinear phase field system in which however the role usually played by the temperature is here conducted by the chemical potential µ. In the study of phase field systems, it has been always considered rather important to analyze the behavior of the problem as the coefficient σ of the diffusion term in the phase parameter equation tends to 0. The limiting case σ = 0 corresponds indeed to a pointwise ordinary differential equation (or inclusion) 14) in place of (1.11), and to an expression for the free energy (1.9) in which the last quadratic term accounting for nonlocal interactions is removed.
In fact, especially for the choice (1.7)-(1.8), the limiting problem can be formulated in terms of hysteresis operators: in particular, the so-called stop and play operators are involved; the interested reader can find some discussion and various results on this class of problems in [7,8,9,10,11,12,13].
By collecting a number of estimates independent of σ for the solution (µ σ , ρ σ ) to the problem (1.10)-(1.13), by weak and weak star compactness we prove that any limit in a suitable topology of a convergent subsequence of {(µ σ , ρ σ )} yields a solution to the limiting problem in which (1.11) is replaced by (1.14). Furthermore, under natural compatibility conditions on the nonlinearities and the initial data, we show boundedness for all the components of any solution to the limit problem. Finally, in the special case of a constant mobility κ in (1.10), we prove that the solution is unique and more regular than required.
The paper is organized as follows. In the next section, we state precise assumptions along with our results. The basic a priori estimates independent of σ are proved in Section 3 and they allow us to pass to the limit by compactness and monotonicity techniques. Finally, the last section is devoted to the study of the limit problem and our boundedness, uniqueness, and further regularity properties are proved.

Assumptions and results
The aim of this section is to introduce precise assumptions on the data for the mathematical problem under investigation, and establish our main result. We assume Ω to be a bounded connected open set in R 3 with smooth boundary Γ (treating lower-dimensional cases would require only minor changes) and let T ∈ (0, +∞) stand for a final time. We introduce the spaces and endow them with their standard norms, for which we use a self-explanatory notation like · V . For powers of these spaces, norms are denoted by the same symbols. We remark that the embeddings W ⊂ V ⊂ H are compact, because Ω is bounded and smooth. The symbol · , · denotes the duality product between V * , the dual space of V , and V itself. Moreover, for p ∈ [1, +∞], we write · p for the usual norm in L p (Ω); as no confusion can arise, the symbol · p is used for the norm in L p (Q) as well, where Q := Ω × (0, T ). Now, we present the structural assumptions we make. It is useful to fix an upper bound for σ, that is, Then, for the diffusivity coefficient κ we assume that 3) the partial derivatives ∂ r κ and ∂ 2 r κ exist and are continuous, (2.4) κ * , κ * ∈ (0, +∞), and for other nonlinearities we require that (2.7) f 1 is convex, proper, l.s.c. and f 2 is a C 2 function, (2.8) g ∈ C 2 (R), g(r) ≥ 0 and g ′′ (r) ≤ 0 for r ∈ R, (2.9) f ′ 2 , g, and g ′ are Lipschitz continuous. We also note that the structural assumptions of [5] are fulfilled if κ only depends on m, and that, due to the presence of β(ρ), a strong singularity in equation (1.11) is allowed. On the other hand, equation (1.10) is uniformly parabolic, since g is nonnegative and κ is bounded away from zero. , whence the assumption f 1 ≥ 0 looks reasonable, as one can suitably modify the smooth perturbation f 2 . Moreover, we point out that the sign conditions g ≥ 0 and g ′′ ≤ 0 are just needed on the set D(β), for g can be extended outside of D(β) accordingly.
(2.24) div κ(µ σ , ρ σ )∇µ σ ∈ L 2 (Q) and κ(µ σ , ρ σ )∇µ · ν = 0 a.e. on Σ, (2.25) and solving the system of equations and conditions in the following strong form a.e. in Q, a.e. in Ω. (2.28) Let us point out that equation (2.26) can be rewritten as and the auxiliary variable u σ has been added. Now, we take advantage of a variational formulation of (2.29) which also accounts for the boundary condition in (2.25), that is, The main result of this paper reads as follows.
Now, we list a number of tools and notations we owe to throughout the paper. We repeatedly use the elementary Young inequalities as well as the Hölder and Sobolev inequalities. The precise form of the latter we use is the following with a constant C p,q in (2.48) depending only on Ω, p, and q, since Ω ⊂ R 3 . Moreover The particular case p = 2 of (2.48) becomes where C depends only on Ω. Moreover, the compactness inequality 6), and ε > 0 (2.51) holds for some constant C q,ε depending on Ω, q, and ε, only. We also recall the interpolation inequalities, which hold for any ϑ where p, q, r ∈ [1, +∞] and 1 thanks to the Young inequality, and a similar remark holds for (2.53). Thus, we have the continuous embeddings L p (Ω) ∩ L q (Ω) ⊂ L r (Ω) and L p 1 (0, T ; L p 2 (Ω)) ∩ L q 1 (0, T ; L q 2 (Ω)) ⊂ L r 1 (0, T ; L r 2 (Ω)).
Notice that we can take v ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ) in (2.54)-(2.55), since V ⊂ L 6 (Ω). Finally, we set and, again throughout the paper, we use a small-case italic c for different constants, that may only depend on Ω, the final time T , the shape of the nonlinearities f and g, and the properties of the data involved in the statements at hand; a notation like c ε signals a constant that depends also on the parameter ε. The reader should keep in mind that the meaning of c and c ε might change from line to line and even in the same chain of inequalities, whereas those constants we need to refer to are always denoted by capital letters, just like C in (2.50).

The asymptotic analysis
In this section, we prove Theorem 2.3, which ensures the existence of a solution to problem (2.35)-(2.38) along with the convergence properties stated in (2.31)-(2.34).
Then, for any σ ∈ (0, 1] we let (µ σ , ρ σ , ξ σ ) denote the triplet defined by Proposition 2.2 and set u σ := (1+2g(ρ σ ))µ σ . The existence of (µ σ , ρ σ , ξ σ ) has been proved in [6]: we follow in parts the arguments developed there in order to recover useful estimates independent of σ. Before that, let us remark that the property µ σ ≥ 0 can be verified by simply multiplying equation (2.26) by −µ − σ , the negative part of µ σ , and integrate over Q t . In principle, in this computation one has to define κ everywhere, e.g., by taking an even extensionκ with respect to the first variable. We observe that Hence, by using µ 0σ ≥ 0 and owing to the boundary condition in (2.25), we have As both g andκ are nonnegative, this implies µ − σ = 0, that is, µ σ ≥ 0 a.e. in Q.

Properties of the limit problem
In this section, we prove Theorem 2.6. In the whole section, it is understood that the assumptions of Theorem 2.6 are satisfied, and sometimes we do not remind the reader about that. As far as the first part of Theorem 2.6 is concerned, the true result regards ordinary variational inequalities and we present it in the form of a lemma. For convenience, we use the same notation ρ, etc., even though it is clear that everything is independent of x: the dot over the variable ρ denotes the (time) derivative, here.
Next, if (µ, ρ, ξ, u) is a solution to problem (2.35)-(2.38), it is clear that, for almost all x ∈ Ω, the functions µ(x, ·) and ρ(x, ·), and the constant ρ 0 (x) satisfy the assumptions of Lemma 4.1. Thus, the first part of Theorem 2.6 concerning bounds (2.44) is proved. We derive an interesting consequence. Proof. We already know that both ξ and π(ρ) are bounded. Moreover, µg ′ (ρ) belongs to L ∞ (0, T ; H) ∩ L 2 (0, T ; L 6 (Ω)) since µ does so and g ′ (ρ) is bounded. We see that also ∂ t ρ belongs to such a space, just by comparison in (2.37). It follows that ∂ t ρ ∈ L 7/3 (0, T ; L 14/3 (Ω)) by (2.55). From this and assumption (2.45), we derive the boundedness of µ. Indeed, we can reproduce the proof carried out in [6, Fifth a priori estimate], since that proof acts only on the equation for µ and works provided that an estimate of ∂ t ρ in L 7/3 (0, T ; L 14/3 (Ω)) is known. At this point, by comparing in (2.37) once more, we conclude that ∂ t ρ is bounded as well.
Let now µ 0 ∈ V ∩ L ∞ (Ω) be arbitrary, and consider a sequence {µ 0 k } ⊂ W bounded in L ∞ (Ω) and converging to µ 0 in V as k → ∞. Let (µ k , ρ k , ξ k , u k ) be the corresponding solutions to (4.13)-(4.16). Then, we can use equation (4.24) written with the index k and test it by ∂ t µ k . We obtain with an obvious choice of ψ k ∈ L 2 (Q) bounded in this space (even better) independently of k. By time integration, it is straightforward to obtain a bound for ∂ t µ k L 2 (Q) and for ∇µ k L ∞ (0,T ;H) independent of k. Then, by weak star compactness we infer that µ k →μ weakly star in H 1 (0, T ; H) ∩ L ∞ (0, T ; V ) at least for a subsequence, which implies (see, e.g., [15,Cor. 4,p. 85]) strong convergence in C 0 ([0, T ]; H). In particular,μ(0) = µ 0 . On the other hand, (µ k , ρ k , ξ k , u k ) satisfies the estimates stated in Lemma 4.1 and the boundedness properties for µ k and ∂ t ρ k given by Corollary 4.2, which are uniform with respect to k. This yields weak or weak star limits ρ andξ. Moreover, strong convergence in L 1 (Q) for {ρ k } and {∂ t ρ k } is ensured via a Cauchy sequence argument based on (4.3), integration over Ω, and Gronwall's lemma. Hence, {µ k }, {ρ k }, {∂ t ρ k } converge strongly in L p (Q) for every p ∈ [1, ∞). At this point, it is not difficult to verify that (μ,ρ,ξ,ũ), with the correspondingũ, actually solves problem (2.35)-(2.38) and thus coincides with the unique solution (µ, ρ, ξ, u). Therefore, the proof is complete.