A singular cahn-hilliard-oono phase-field system with hereditary memory

We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.


1.
Introduction. In the Nineties, G. Caginalp introduced the following phase-field system ϕ t − ∆µ = 0, in order to describe phase transition phenomena such as melting/solidification processes, see [3] and [4]. Here, ϕ is the order parameter of the material undergoing the transition process, ϑ is its (relative) temperature, and Ψ is the derivative of a double-well potential of the form The model consists of the coupling of the Cahn-Hilliard equation, introduced in [5] and [6], with the heat equation, assuming the usual Fourier law for heat conduction (for simplicity, we have set all physical constants equal to 1 here). If we consider instead a linearized version of the Coleman-Gurtin law (see [7]) which accounts for (past) memory effects, we end up with the following equation for the relative temperature where k is a nonnegative and summable memory kernel and k d > 0 (we will take k d = 1 in what follows). Systems (1)- (2) and (1)- (4) are known as conserved phasefield models, in the sense that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity. They both have been much studied in the literature, see, e.g., [1,2,21], [12,14,16,17] and the references therein, mainly for regular potentials Ψ of the form (3). However, it is well known since the pioneering papers by J.W. Cahn and J.E. Hilliard that, in this context, logarithmic potentials of the form Ψ(ϕ) = Θ 2 (1+ϕ) ln(1+ϕ)+(1−ϕ) ln(1−ϕ) − Θ 0 2 ϕ 2 , where Θ 0 > Θ > 0, (5) are physically relevant, and provide a better description of transition processes. This motivates the present paper, where the goal is to investigate the conserved phase-field system with memory in presence of a general class of singular potentials (see below), including the logarithmic prototype. Actually, we also generalize (1) by considering the Cahn-Hilliard-Oono equation introduced in [24] (see also [28]) to account for long-ranged (i.e. nonlocal) interactions. This equation was studied in [20] for regular potentials and in [15] for logarithmic potentials.
In order to deal with (4) coupled with (6) as a dynamical system, it is convenient to exploit the past history formulation proposed in [10]. Namely, we introduce the auxiliary variable η t (s) = s 0 ϑ(t − y) dy, so that, setting g = −k , we (formally) arrive at the following problem: A SINGULAR CAHN-HILLIARD-OONO PHASE-FIELD SYSTEM WITH MEMORY 3035 in Ω × (0, ∞), where Ω is the domain occupied by the material, with boundary Γ. We couple the system with the Neumann boundary conditions and the initial conditions Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of attractors, of (7)-(9) with singular potentials. This is a quite challenging problem: indeed, as pointed out in [15] for the sole Cahn-Hilliard-Oono equation, the occurrence of singular potentials combined with the Oono term introduces essential difficulties. Furthermore, the memory effects yield a lack of regularization on the temperature, making the longterm analysis delicate.
We also study the (strict) separation of the order parameter from the pure phases. The latter is a very sensitive property in the context of the Cahn-Hilliard (see [23]) and Cahn-Hilliard-Oono (see [15] and [22]) equations and is only valid in one and two space dimensions in general. In our case, due to the aforementioned lack of regularization, we can prove this property for more regular initial temperatures only.

Preliminaries.
2.1. Assumptions. We assume that Ψ is a quadratic perturbation of a singular (strictly) convex function in [−1, 1], namely and there exists Θ > 0 such that We assume that Θ 0 − Θ > 0, so that Ψ has a double-well shape as in the prototype model (5). We also extend F by F (s) = +∞ for any s / ∈ [−1, 1]. Note that the above assumptions imply that there exists s 0 ∈ (−1, 1) such that F (s 0 ) = 0. Without loss of generality, we assume that s 0 = 0 and that F (s 0 ) = 0 as well. In particular, this entails that Concerning the memory kernel, we suppose that k is a nonnegative summable function of total mass equal to 1, having the explicit form where g ∈ L 1 (R + ) is a nonincreasing, nonnegative, absolutely continuous function satisfying, for some δ > 0, We agree to denote ∞ 0 g(s) ds = k(0) = κ.
2.2. Functional spaces. Let N ≤ 3 and Ω ⊂ R N be a smooth bounded domain. We denote by (H, ·, · , · ) the space L 2 (Ω) (or [L 2 (Ω)] N according to the context) endowed with the standard scalar product and norm. Besides, we denote by the standard Sobolev spaces, with scalar products ·, · σ and norms · H σ . We will use the notation V = H 1 , equipped with the norm u 2 V = ∇u 2 + u 2 , and we indicate by V the dual space of V , by · V its norm and again by ·, · the duality product ·, · V ,V . Denoting by the average of any measurable function u over Ω, let us recall that We introduce the space of zero-mean functions and its dual space We then consider the linear operator A : V → V defined by which is an isomorphism from V 0 onto V 0 . Its inverse map is an equivalent norm in V 0 , and We denote by In order to handle the convolution integral, we define as usual the so-called memory spaces where ·, · σ is the scalar product in H σ (we omit σ in the notation when σ = 0). Then, we introduce the operator T : (where η denotes the derivative of η with respect to the internal variable s), which is the infinitesimal generator of the strongly continuous semigroup of right translations on the memory space M 1 . We now recall some well-established facts, see e.g. [9].
It is well-known (see e.g. [13,Proposition 5.4]) that its closed balls are closed in M σ and the embedding K σ+1 M σ is compact.
We also recall the following result, useful to handle the norm in K σ+1 (see [9], Lemma 3.3 and Lemma 3.4).
Lemma 2.1. Assume that η satisfies the Cauchy problem on (0, T ), for some T > 0. Then, for every t ∈ (0, T ), we have η t (0) = 0 and Finally, we denote by the hierarchy of the extended spaces, endowed with their natural scalar product. Again, we omit the superscript σ whenever it equals zero. In particular, . By the above discussion, we learn that Notation. Throughout the paper c > 0 denotes a generic constant and Q(·) > 0 is a generic increasing function, only depending on the structural parameters of the problem. Moreover, given any measurable function u, we setū = u − u .
3. Weak solutions and their basic properties.
We associate to each weak solution z its (finite) energy

Continuity of the weak solutions.
It is apparent from the definition that, since ϕ ∈ H 1 (0, T ; V ) ∩ L 2 (0, T ; H 2 ), then ϕ ∈ C([0, T ], H). Indeed, it is possible to prove by a standard argument (see e.g. [8]) that ϕ ∈ C([0, T ], V ). On the other hand, if we consider the third equation of (7) written for H = ϕ + ϑ, namely 3.3. Conservation laws. Integrating (7) 1 over Ω, we deduce that d dt yielding Besides, we learn from the third equation of (7) that whence the conservation law The existence of a weak solution will be proved in Section 5 via an approximation procedure involving auxiliary problems in which the singular potential F is substituted by a family of globally Lipschitz potentials F λ converging (in a suitable sense) to F as λ → 0. This is done in the next two sections.
4. Approximating problems. Let us recall (see [11]) that, given F as in Section 2.1, for every λ > 0, there exists Accordingly, for any given λ > 0, we define and we consider the following approximated problems (P λ ) in the unknown z = (ϕ, ϑ, η): subject to the boundary and initial conditions (8)- (9), with the corresponding energy functionals Note that the solutions to the approximating problems satisfy the analogous conservation laws established in (14) and (15). Thanks to the Lipschitz regularity of F λ , exploiting a standard Galerkin method and a basic energy estimate (see below), it is a standard matter to prove the following existence result. In the next section we will prove several energy estimates for the solutions to (P λ ) which are uniform with respect to λ. This is the main step allowing to pass to the limit as λ → 0 in order to find a solution to the original problem (7).

Energy estimates and existence of a weak solution.
In what follows, α belongs to a given compact subset J ⊂ [0, ∞) and λ is fixed. Let be any global solution to (P λ ) with initial datum z 0 . Throughout the section, the generic constant Q > 0 depends on J, | ϕ 0 | and H 0 = ϕ 0 + ϑ 0 , but is independent of λ and of the specific initial datum. First, we prove some energy estimates in the less regular spacẽ involving the weaker energy functionals Lemma 5.1. There existω > 0 and λ > 0 such that, for any 0 < λ ≤ λ, for every t ≥ 0.
for every t ≥ 0.
We now provide estimates for ϕ t and ϑ t .
Proof. Notice first that, by comparison in (20), Then, since on account of (13) we have ϕ t = −α ϕ , we infer that ϕ t V ≤ c( ∇µ + ϕ V ), and the conclusion follows by (19). Analogously, we obtain the result for ϑ t by comparison in the third equation.
valid for every nonnegative locally summable function h and every ω > 0, we obtain from the Gronwall Lemma The desired conclusion for ϑ 2 V now follows recalling that ϑ = H − ϕ and (19). A final integration in time of (32) yields the integral control for ∆ϑ 2 .
Existence of a weak solution. Let z λ = (ϕ λ , ϑ λ , η λ ), λ ∈ (0, λ], be a family of solutions to (P λ ) departing from the same initial datum z 0 = (ϕ 0 , ϑ 0 , η 0 ) as in (18). Collecting Lemmata 5.1-5.4 we know that for every t ≥ 0 and for some C > 0 independent of λ. Hence, we can pass to the limit λ → 0, ending up with a triplet z = (ϕ, ϑ, η) such that (up to subsequences) Note that by the classical Aubin-Lions Theorem, we also have and the pointwise convergence Then, the uniform convergence of F λ to F on any compact set in (−1, 1) yields Note that the very same convergences hold true for ϑ λ → ϑ. Besides, we have The next step is to show that the limit triplet z is a weak solution to (7). This is proved by passing to the limit in the weak formulation of (P λ ) in a standard way (see e.g. [14] for the memory component), the only nontrivial parts being that |ϕ| < 1 a.e. in Ω × (0, T ). But this can be done reasoning as in [8,11] and we omit the details. Finally, we observe that all the energy estimates proved in the first part of this section for z λ pass to the limit λ → 0, whence they hold true for any limit triplet z. Summing up we have proved the following.
7. Partial regularization in finite time. In this section we show that the system exhibits a partial regularization effect in finite time. In particular, we prove that the order parameter is bounded in H 2 as soon as t > 0. To this aim, let z(t) = (ϕ(t), ϑ(t), η t ), t ≥ 0, be the solution departing from a given z 0 = (ϕ 0 , ϑ 0 , η 0 ) as in (18). In light of the estimates of Section 5, we have for any t ≥ 0, where, here and in what follows, the generic constant C > 0 may depend on E(z 0 ), | ϕ 0 | and H 0 = ϕ 0 + ϑ 0 . We start with the following lemma. and, for every t ≥ σ, Proof. Multiplying (7) 1 by µ t we have d dt Differentiating (7) 2 in time and multiplying the resulting equation by ϕ t yield where | ϕ 2 t | ≤ C on account of (13) and (14). We estimate the right-hand side as follows: and Exploiting the convexity of F , recalling that ϕ + ϑ ≤ C and F ≥ 0, we obtain We thus end up with where , for anyC ≥ 0. Note that, provided thatC is properly chosen (depending on E(z 0 ), in light of (39) and the control Λ ≤ ∇µ 2 + α ϕ µ + C ≤ C( µ 2 V + 1). As a consequence, the Uniform Gronwall Lemma allows to prove the desired bound for ∇µ(t) for t ≥ σ. A subsequent integration of (42) in time over [t, t + 1] yields On account of (13) the control holds true for the whole norm of ϕ t in V .

Corollary 1.
For every σ > 0, there exists C = C(σ) > 0 such that Proof. We multiply (7) 2 written as Thanks to (10) and the Young inequality, we have on account of Lemma 7.1. Then the elliptic regularity applied to (44) allows to conclude.
8. The dissipative semigroup and its attractor. Let m ∈ [0, 1) and ∈ R be arbitrarily fixed. On account of Theorem 5.5 and Theorem 6.1, we can consider the family of solution operators defined via the rule where z(t) = (ϕ(t), ϑ(t), η t ) is the unique solution at time t to the Cauchy problem (7)-(9) with initial datum Now consider the entering time of B 0 into itself, namely t 0 = t B0 , and define noticing that B 0 is again an absorbing set and is invariant (namely S(t)B 0 ⊂ B 0 for every t ≥ 0). Furthermore, according to Lemma 7.1 and Corollary 1, we have where, along the section, C > 0 is a generic constant depending on B 0 but independent of the specific initial datum. The main result of this section is the following. Proof. Let B 0 be the above absorbing set and let z = (ϕ 0 , ϑ 0 , η 0 ) ∈ B 0 be arbitrarily given. We decompose the solution departing from z as follows: We already know that ϕ(t) H 2 ≤ C.
Reasoning as in [8,Lemma 7.3] in order to estimate the term involving F in the right-hand side, we get 1 h [F (ϕ(t + h)) − F (ϕ(t))], ∆v

MONICA CONTI, STEFANIA GATTI AND ALAIN MIRANVILLE
Hence, since, v 2 L 6 (Ω) ≤ c ∇v 2 + c| v | 2 ≤ c v ∆v + c| v | 2 , we end up with the differential inequality 1 2 where Besides, recalling that ∂ h t u L 2 (t,t+1;H) ≤ u t L 2 (t,t+1;H) for every u, owing to (40) and (49) An application of the Uniform Gronwall Lemma and a final passage to the limit h → 0 complete the proof.
Final remarks. By a standard result in the theory of dynamical systems (see e.g. [19]), the global attractor of a semigroup S(t) : X → X has the form for any arbitrarily fixed t 0 ∈ R, where a complete bounded trajectory (cbt) of S(t) is a function z ∈ C(R, X), bounded on R and satisfying z(τ ) = S(t)z(τ − t), ∀t ≥ 0, ∀τ ∈ R.
Accordingly, as a consequence of Theorem 9.1 we deduce that the order parameter for all cbt's to our system is uniformly away from the pure phases for every time t ∈ R. Proof. Invoking Theorem 9.1, let δ * = δ(1). Then, for any cbtẑ of S(t) on H m, , we know that sup Indeed, for any τ > 0, we haveẑ(τ ) = S(τ )ẑ(0), whereẑ(0) ∈ A m, , owing to (52).