Nonlocal phase-field systems with general potentials

We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.


Introduction
A well-known approach to study two-phase Stefan-like problems in more than one spatial dimension is the so-called phase-field (or diffuse interface) method. Roughly speaking, it consists in introducing an order parameter χ whose zero level set substitutes for the sharp interface, while χ = ±1 in the higher/lower energy phases. The classical problem is thus replaced by an order parameter dynamics, originated from the study of critical phenomena, coupled with the energy balance equation governing the temperature field. An important issue is to recover the original interface conditions and this is usually done by (formal) asymptotic expansions. Moreover, diffuse interface models are quite effective from the numerical viewpoint since there is no need for interface tracking. A significant and basic example of phase-field system is due to G. Caginalp (see [9], cf. also [8]), namely, on a given bounded domain Ω ⊂ R d , d ∈ {1, 2, 3}, for some time interval (0, T ). Here ϑ is a rescaled temperature so that ϑ = 0 is the equilibrium melting temperature, while ℓ, k, τ and ξ are given positive constants which represent the latent heat of fusion, the diffusivity, a relaxation time and a correlation length, respectively. The function W is (the density of) potential energy associated with the phase configuration. Such potential is usually a smooth double well function (typically W (r) = 1 8 (r 2 − 1) 2 ). However, this is just a convenient approximation of the physically more relevant logarithmic potential generally taking the form W (r) = (1 + r) log(1 + r) + (1 − r) log(1 − r) − γr 2 , γ ≥ 0. (1. 3) The mathematical literature on (1.1)-(1.2) is rather vast and we confine ourselves to quote the pioneering paper [14] and the more recent ones [17,18] (see also references therein).
In order to analyze the microscopic influences of anisotropy on the interface, in [10] a phasefield system has been derived from microscopic considerations based on Statistical Mechanics. This system is similar to (1.1)-(1.2) but for the diffusion term ξ∆ χ which is replaced by A : D 2 χ , where A ∈ R d×d is positive definite and D 2 χ is the Hessian of χ . However, this derivation is performed by truncating the expansion of the interaction function (see [10,Prop. 2.4], cf. also [11] for higherorder approximations). Then the author, by using formal asymptotics, deduces a modified Gibbs-Thompson relation in 2D. More recently, by using the same procedure, a related phase-field model has been obtained without approximating the interaction function (see [12,13]). Actually, working in a bounded domain and choosing λ = 0 in [13,Sec. 2], the system obtained there takes the following form: Here K ς (x) = ς −d K(ς −1 x) where ς > 0 is an atomic length scale and K : R d → R is a sufficiently smooth interaction kernel satisfying K(x) = K(−x) and such that κ(x) := Ω K(x − y) dy is bounded and nonnegative. The asymptotic limit ς ց 0 has been analyzed in [13] and a new anisotropic interface condition has been obtained. On the other hand, this class of systems was already considered in some previous papers (cf., e.g., [2,4,5,6,15] and their references). In particular, rigorous mathematical results were proven for smooth potentials. Well-posedness for Ω = R and d = 1 was established in [5] through semigroup theory. These results were then extended to bounded d-dimensional domains with either homogeneous Neumann or Dirichlet boundary conditions for ϑ (see [6,15]). Regarding the longtime behavior, the convergence of a solution to a single stationary state was shown in [15] by means of a suitable nonsmooth version of the Lojasiewicz-Simon inequality. Existence of an absorbing set was proven in [6] as well as an analysis of the ω-limit sets. More recently, the results of [15] have been extended to to a class of singular unbounded potentials which does not include the logarithmic ones (cf. [19]). Actually, in [19] equation (1.5) was modified by adding an inertial term of the form α χ tt , α > 0. However, the results proved there also hold for α = 0. Then, by exploiting [19], the existence of a finite-dimensional global attractor has been established in [16]. Here we want to generalize such results to both smooth potentials and more general singular potentials (e.g., of the logarithmic type (1.3)). This goal is connected with the property of the solutions of getting bounded in finite time uniformly with respect to sufficiently general initial data. In addition, in the case of singular potentials, a (uniform) separation property is also needed. Here we prove all these properties for weak solutions originating from initial data in the energy space. In particular, in the case of smooth potentials, our results generalize the corresponding ones in [15]. Moreover, for singular potentials, the separation property holds instantaneously, namely for t > 0, even though the initial datum is a pure state (see Remark 4.2 below). As a consequence, we also demonstrate the existence of a finite dimensional global attractor using the approach devised in [16]. This approach exploits the only source of compactness for χ , namely K ς * χ . Note that we cannot expect smoothing effects on χ .
For the sake of simplicity, we choose the constants in such a way that system (1.4)-(1.5) can be rewritten in the form in Ω × (0, T ). Here J is a linear operator which is a suitable generalization of the nonlocal convolution operator introduced above (see Section 2 below). Following [15], we endow the system with the following boundary and initial conditions It is worth mentioning that there are also (mainly) existence and uniqueness results for more refined nonlocal phase-field systems, formulated with respect to the absolute temperature, which are thermodynamically consistent also far from the equilibrium temperature (cf. [22,23,24,25,30]). It would be interesting and challenging to extend some of the present results to such systems. The plan of this paper goes as follows. The main results about well-posedness and regularization properties of the solutions are stated in Section 2. Then, the corresponding theorems for smooth potentials and singular potentials are proven in Section 3 and Section 4, respectively. The final Section 5 is devoted to the existence of the global attractor.

Well posedness and regularization results
In the sequel we will assume that |Ω| = 1, for simplicity. We set V := H 1 0 (Ω), H := L 2 (Ω) and note by · the norm in H, by (·, ·) the scalar product of H, and by · p the norm in L p (Ω) for p ∈ [1, ∞]. We also let A be the Laplace operator with homogeneous Dirichlet b.c., seen either as an unbounded linear operator on H with domain V ∩H 2 (Ω) or as a bounded linear operator from V to its topological dual V ′ . Then (1.6)-(1.8) can be rewritten as follows: where λ ∈ L ∞ (Ω) is a given function, and where I, the domain of f 0 , is an open and, possibly, bounded interval of R containing 0. We also set We assume J be a linear operator such that for some L > 0 independent of p and all p ∈ [1, +∞]. Moreover we assume that and, finally, J is a compact self-adjoint operator from H to H. (2.8) Observe that the concrete form of the nonlocal operator J (see (1.5)) satisfies assumptions (2.6)-(2.8), provided that K is smooth enough. We can then introduce the energy functional It is immediate to realize that under the above assumptions E could be unbounded from below. Thus, we need some condition implying that E has some coercivity. In particular, we will consider two different situations. The first one deals with what we will call a smooth potential: Assumption 2.1. We assume (2.4) with I = R. Moreover, we ask that and for some ǫ > 0, We will speak, instead, of singular potentials in the following case: Assumption 2.2. We assume (2.4) with I an open and bounded interval of R containing 0. Moreover, we ask that lim r→∂I f 0 (r) sign r = +∞. (2.11) Our first result deals with the case when F is smooth. In this situation, we define the energy space (i.e., the set of all (ϑ, χ )'s such that E(ϑ, χ ) is finite), as the Banach space solving, a.e. in (0,T), system (2.1)-(2.2), and enjoying the initial conditions Moreover, there exist a time T 0 ≥ 0 depending on the "initial energy" E 0 := E(ϑ 0 , χ 0 ) and a constant C 0 independent of E 0 , such that In the case when F is singular, the energy space is given by Actually, since the domain I of F 0 is bounded, it is clear that X ⊂ H × L ∞ (Ω). Due to the constraint term F 0 , X is not a linear space in this case. Nevertheless, it is easy to prove that has a complete metric structure with respect to a natural distance function (see, e.g., [28,Sec. 3] for details).

20)
and solving (2.1)-(2.2) together with the initial conditions (2.15). Moreover, there exist a time T 0 ≥ 0 depending on the "initial energy" E 0 := E(ϑ 0 , χ 0 ) and a constant C 0 independent of E 0 , such that (2.16) holds, together with the separation property A couple (ϑ, χ ) in the condition either of Theorem 2.3 or of Theorem 2.4 will be called an energy solution in what follows.
Remark 2.5. An autonomous heat source term in equation (2.1) could be easily handled (see, e.g., [16]). Some care is required in the non-autonomous case (cf. [18] for local systems).

Proof of Theorem 2.3
This existence proof is a slight generalization of [15, Thm. 1.1]. Thus we will proceed formally with the a priori estimates. Uniqueness goes exactly as in [15], while we will give all the details about (2.16). In the sequel we will note with the letter c a generic positive constant, allowed to vary on occurrence, depending only on f 0 , λ and L (cf. (2.6)). In particular, we will always assume c to be independent of the initial data and of time. The letter κ will note the positive constants, depending on the same quantities as c, appearing in estimates from below.
Energy estimate. We test (2.1) by ϑ, (2.2) by χ t and take the sum. This gives (recall that J is Next, we test (2.2) by χ , to obtain By (2.6), Assumption 2.1, and λ ∈ L ∞ (Ω), it is clear that Moreover, by Poincaré's and Young's inequalities, for small σ > 0 and c σ depending on σ. Then, summing (3.1) with (3.2) and taking (3.3), (3.4) into account, we arrive at Integrating (3.5), we then obtain for some new value of c, independent of T . In particular, by (2.10), this implies Moreover, integrating (3.1) over the generic interval (t, T ), and using (3.6), we infer Being c independent of T , the above bound can be rewritten also for T = +∞.
Regularization estimates for ϑ. From (3.8) and the Poincaré inequality, it is clear that for any Then, taking s ≥ 0 and correspondingly choosing τ ∈ [s, s + 1] such that (3.9) holds, testing (2.1) by −∆ϑ and integrating over (τ, T ) for a generic T ≥ τ , recalling (3.5), and using Hölder's and Young's inequalities, we obtain In particular, noting that the above holds at least for some τ ∈ (0, 1) (corresponding to the choice s = 0), we have still with c independent of T .
Regularization estimates for χ . We will now work on a generic time interval (S, S + 2) for S ≥ 1. Then, as a consequence of estimate (3.10) and interpolation, we have that Here and in what follows, C will always denote a quantity of the form where Q is a computable nonnegative-valued monotone function, whose expression is allowed to vary on occurrence, depending only on the fixed parameters of the system. That said, we choose a sequence of small time steps τ n , n ∈ N, defined by and proceed by induction. Namely, we set t 0 := S and assume that, for n ≥ 1, given t n−1 ≥ S, there exists t n ∈ (t n−1 , t n−1 + τ n ) such that Notice that this is surely true as n = 1 once one sets t 0 = S, thanks to (3.7). Then, we test (2.2) by | χ | nǫ χ . In principle this would be not an admissible test function (actually at this level (2.2) makes sense just as a relation in L 2 ((0, T )× Ω) and | χ | nǫ χ needs not have the L 2 -summability). However, the procedure could be easily justified by using a truncation of | χ | nǫ χ as a test function, and then passing to the limit w.r.t. the truncation parameter via the monotone convergence theorem (the details are left to the reader). Moreover, since we only need a finite number of iterations, we will not take care of the dependence of the various constants on n and on ǫ. We infer Then, by (2.10), Finally, by (2.6) and λ ∈ L ∞ (Ω), Then, integrating (3.16) over (t n , S+2), recalling (3.12), and using (3.15) and the induction hypothesis, we arrive at where the second term on the left hand side ensures that condition (3.15) will be fulfilled at the level n + 1. In particular, the procedure can be iterated at least until n satisfies the constraint 2 + (n + 1)ǫ 1 + ǫ ≤ 10, i.e., n ≤ 8 + 9ǫ ǫ . (3.21) More precisely, since (8 + 9ǫ)/ǫ may not be an integer, our method works at least until we reach some n max ≥ 8(1 + ǫ −1 ). Thus, we obtain (at least) the bound where C additionally depends on the chosen sequence τ n . The upper bound in hypothesis (2.10) then gives also f (·, χ ) L 9 ((S+1/4,S+2)×Ω) ≤ C, (3.23) whence, comparing terms in (2.2) and using (3.12), assumption (2.6) and (3.22)-(3.23), we also obtain χ t L 9 ((S+1/4,S+2)×Ω) ≤ C.
At this point, we can proceed with the iterations exactly as before and notice that, still in a finite number of steps, we arrive at χ L ∞ (S+3/4,S+2;L p * (Ω)) ≤ C. Consequently, using property (2.7), we also obtain for some constant M > 0 having the same dependence of C.
Remark 3.1. Unlike the case of parabolic type equations, it seems that, for equation (2.2), which has essentially an ODE structure, the Moser iteration method cannot be used to get directly an L ∞ -bound for χ . Actually, if we try to iterate the above procedure infinitely many times, we readily notice that the constants appearing on the right hand side's of the estimates (cf., e.g., (3.18) and (3.19)) would explode. This is due to the fact that, while in true parabolic equations the Moser exponents grow exponentially with respect to n, in the present case, the growth is just linear (and, hence, too slow to take the constants under control). This fact also forces us to assume that the kernel J has at least some regularizing effect (i.e., assumption (2.7)).
Once we have (3.25) and (3.28) at our disposal, we can apply a comparison principle to get the L ∞ -bound for χ . Namely, we have that, for t ∈ (S + 3/4, S + 2) and a.e. x ∈ Ω, there holds for a.e. x ∈ Ω and all t ∈ Λ + (x). Hence, dividing by χ 1+ǫ (t, x), we obtain It is then clear that Λ can be taken large enough (in a way only depending on the L ∞ -norm of λ and on the known constants Θ, M , c f and κ f ) so that, for t ∈ Λ + (x), In particular, for such times t, the function t → χ (t, x) is (strictly) decreasing. This implies that, if t ∈ (S + 3/4, S + 2) and t ∈ Λ + (x), then s ∈ Λ + (x) for all s ∈ [t, S + 2). In other words, if χ (t, x) is smaller than Λ, it can never become larger than it. Thus, assuming that x is such that χ S (x) := χ (S + 3/4, x) ≥ Λ (otherwise there is nothing to prove in view of the preceding discussion), integrating inequality (3.33) over (S + 3/4, t), we obtain at least for all t ≥ S such that χ (t, x) ≥ Λ. Equivalently, we can write Consequently, it is clear that there exists Λ ′ > 0, independent of the value of χ S (x), such that For instance, one can take Λ ′ = (8κ −1 f ǫ −1 ) 1/ǫ . Of course, a similar bound from below (of the form χ (t, x) ≥ − max{Λ, Λ ′ }) can be proved in the same way. Thanks to the arbitrariness of the starting time S ∈ [1, +∞) and recalling once more (3.25), we have obtained the bounds for ϑ and χ in (2.16), for instance with the choice of T 0 = 2 (however, see Remark 4.2 below). Then, the remaining bound for χ t follows by Assumption 2.1, (2.6), and a comparison of terms in (2.2). Theorem 2.3 is proved.

Proof of Theorem 2.4
This proof is presented in full detail (also for what concerns existence) since, to the best of our knowledge, this singular potential case has never been analyzed in the literature.
A priori, these regularity properties could depend on the approximation parameter δ. However, we shall see in a while that, in fact, δ-independent estimates are satisfied.

Uniform estimates and passage to the limit
In what follows we will assume that all constants κ, c are independent of δ. As we repeat the estimates performed in the proof of Theorem 2.3, it is easy to realize that, defining the approximate energy as the function E δ satisfies (3.5) with κ, c independent of δ. Then, noting that thanks to F δ ≤ F 0 , it is easy to check that the "Energy estimate" of the previous section can be repeated to obtain relations analogue to (3.6), (3.7), (3.8), that hold now uniformly w.r.t. δ. Moreover, we can test (4.9) by f δ ( χ δ ) and use (3.7), (3.8), and the properties of J to infer with Q independent of δ. We now show that the estimates detailed above suffice to take the limit δ ց 0. Actually, (3.6)-(3.8) and (4.12) guarantee that, for any T > 0, Here and below, we adopt the convention of overlining unidentified weak limits. Thanks to linearity and continuity of operator J , it is then easy to show that, at the limit δ ց 0, ϑ t + χ t + Aϑ = 0, (4.16) 4.17) and the initial conditions (2.15) are satisfied as well. Then, to conclude the proof, we have to show the identification f 0 ( χ ) = f 0 ( χ ) almost everywhere in (0, T ) × Ω. To do this, we follow with some variations the argument given in [15,Sec. 2.3], which we report for the reader's convenience.
First of all, letting using (4.14) it is a standard check to verify that Hence, we can assume that (here and below, all convergence relations are intended up to extraction of non-relabelled subsequences) (4.20) where ω is continuous and nonnegative. Now, as a further consequence of (4.14), we have that In particular, for all t ∈ [0, T ], χ δ (t) converges to χ (t) weakly in H. Next, we compute the difference between (4.9) and (4.17), test it χ δ − χ , and integrate with respect to the space variables. This gives Now, we notice that, by (4.12), (4.14) and the first inequality in (4.2), Consequently, By definition of subdifferential, we have, almost everywhere in (0, T ) × Ω, Let us now test (4.25) by a nonnegative test function φ ∈ D((0, T ) × Ω) and integrate. Then, by convexity and lower semicontinuity of the functional using (4.15) and (4.24), we obtain that To deduce the last inequality we have used the fact that the family of functionals {F δ }, being monotone increasing with respect to δ going to 0, converges to F 0 in the sense of Mosco (see, e.g., [3]) in L 2 ((0, T ) × Ω). In particular, we used here the lim inf-property of Mosco-convergence: Thus, we have, almost everywhere in (0, T ) × Ω, Moreover, we notice that, thanks to (4.13), (4.14), the Aubin-Lions Lemma, and assumption (2.8), at least in the sense of distributions over (0, T ). Then, we can take the limit, as δ ց 0, of (4.22). Using (4.24) and (4.29), and noting that the time-derivative operator is linear and continuous with respect to distributional convergence, we then obtain 1 2 or, equivalently, Since ω is nonnegative and ω(0) = 0, we then obtain ω(t) = 0 for all t ∈ [0, T ]. In other words, This fact, combined with (4.14) gives which entails in particular f 0 ( χ ) = f 0 ( χ ). Thus, (4.17) reduces to (2.2), as desired.

Regularization estimates and separation property
The above procedure is sufficient to get existence of an energy solution to our system under Assumption 2.2. Uniqueness of this solution is proved as in the other cases.
To prove rigorously the separation property (2.21), we go back to the δ-system (4.8)-(4.9) and start by noticing that the regularization estimates of Section 3 hold uniformly in δ. Actually, concerning the "Regularization estimates for ϑ" it is easy to see that nothing changes and the analogue of (3.9)-(3.11) hold uniformly in δ. Concerning the "Regularization estimates for χ ", we can proceed as before (where we now have ǫ = 1, of course), until we reach estimate (3.22). Indeed, in this part of the Moser iteration, we only use the estimates (3.7)-(3.8) and (3.9)-(3.11), which are uniform in δ, and the estimate from below (i.e., the first inequality) in (4.2), which is also independent of δ. Consequently, we now have the analogue of (3.22), which, for ǫ = 1 and in the current notation, becomes χ δ L ∞ (S+1/4,S+2;L 18 (Ω)) + χ δ L 19 ((S+1/4,S+2)×Ω) ≤ C. Here and in what follows, all constants c, κ and C have the same meaning as in the previous section and, in addition, are assumed to be independent of δ. However, at this point we can no longer deduce the analogue of (3.23) directly, since this requires use of the upper bound in (4.2), where the constants do depend on δ.
With this relation at disposal, we can repeat the ODE argument of Section 3, with essentially no variation. Actually, it is sufficient to use the lower bound in (4.2), which is uniform in δ. Summarizing, we have obtained with constants Θ and M independent of δ.
Remark 4.1. It is worth noting that, at the limit step, we have for free that −1 ≤ χ ≤ 1 almost everywhere (and starting from t = 0), since F 0 is singular. However, (4.45) says something more, i.e., that we have, for t ≥ 1, a uniform L ∞ -bound independent of the approximation parameter for both components of the approximate solution. This is a nontrivial information especially as far as ϑ δ is concerned (indeed, we just know that the initial datum ϑ 0 lies in H).
Remark 4.2. Looking at the statement (and at the proof) of (2.21) one could think that the separation property occurs only after some waiting time. However, the property was given in that form just for the sake of simplicity. Indeed, refining a bit the arguments in the proof, it is easy to demonstrate that (2.21) is in fact an instantaneous property. Namely, there holds × Ω and all τ > 0, (4.51) where ε(τ ) goes to 0 as τ ց 0. The details are left to the reader.

Global attractors
Here we establish the existence of a finite-dimensional global attractor for both smooth and singular potentials. The technique is the same as the one used in [16, Proof of Thm. 4.1] for singular unbounded potentials. However, here we start from very general initial data in the energy space and we exploit the previous regularization results to define a semigroup acting on a convenient invariant set. Let us consider the case of smooth potentials first. From Theorem 2.3 it is clear that we can define a semigroup S(t) : X → X (cf. (2.12)) by setting (ϑ(t), χ (t)) := S(t)(ϑ 0 , χ 0 ), where (ϑ, χ ) is the unique (energy) solution to (2.1)-(2.2), (2.15). Note that ϑ ∈ C 0 ([0, +∞); H) while χ ∈ C 0 ([0, +∞); H)∩C 0 w ([0, +∞); L 2+ǫ (Ω)). Moreover, on account of the Lipschitz continuity estimate [15, (1.11)], this semigroup is also closed in the sense of [27]. Thanks to (2.16) and (3.11), S(t) has an absorbing set B which is bounded in (V ∩ L ∞ (Ω)) × L ∞ (Ω). Hence we can find t 0 ≥ 0 such that S(t)B ⊂ B for all t ≥ t 0 . Then, without loss of generality, we can suppose that t 0 = 0 and assume that B is an invariant set for S(t). Moreover, we can endow B with the V × H-metric and obtain a complete metric space X.
We can then prove the following Then the dynamical system (X, S(t)) has a finite-dimensional connected global attractor.
For readers' convenience, we report here below the argument of [16].
Hence S(t) has a (connected) global attractor (see [ .17) as phase space, the semigroup S(t) defined as above takes X to itself for all t ≥ 0. Using again (2.16) and on account of the separation property (2.21), it is not difficult to realize that there exists an absorbing set of the following form (note that (3.11) still holds): for a suitable pair of constants (R, β) ∈ (0, +∞) × (0, 1). Then, reasoning as above, we can suppose that B(R, β) is invariant for S(t) and we can endow it with the V × H-metric. The resulting complete metric space X is now our phase space and we have Theorem 5.2. Let the assumptions of Theorem 2.4 and (5.1) hold. Then, the dynamical system (X, S(t)) has a finite dimensional connected global attractor.