Multi-component conserved Allen-Cahn equations

We consider a multi-component version of the conserved Allen-Cahn equation proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for phase separation. In our case, the free energy is characterized by a mixing entropy density which belongs to a large class of physically relevant entropies like, e.g., the Boltzmann-Gibbs entropy. We establish the well-posedness of the Cauchy-Neumann problem with respect to a natural notion of (finite) energy solution which is more regular under appropriate assumptions and is strictly separated from pure phases if the initial datum is. We then prove that the energy solution becomes more regular and strictly separated instantaneously. Also, we show that any finite energy solution converges to a unique equilibrium. The validity of a dissipative inequality (identity for strong solutions) allows us to analyze the problem within the theory of infinite-dimensional dissipative dynamical systems. On account of the obtained results, we can associate to our problem a dissipative dynamical system and we can prove that it has a global attractor as well as an exponential attractor.


INTRODUCTION
Phase separation, namely the creation of two (or more) distinct phases from a single homogeneous mixture, is an important phenomenon which characterizes many important processes.In particular, it has recently become a paradigm in Cell Biology (see, for instance, [12,13] and references therein).A well-known mathematical model of phase separation for binary alloys was proposed by J.W. Cahn and J.E. Hilliard [8,9].This model leads to the so-called Cahn-Hilliard equation (see, for instance, [32] and references therein).More precisely, indicating by ϕ the concentration of one species, phase separation can be modeled as a competition between the Boltzmann-Gibbs mixing entropy S(ϕ) = −ϕ ln ϕ − (1 − ϕ) ln ϕ and the demixing effects due to the reciprocal attraction of the molecules of the same species which can be described, for instance, as follows D(ϕ) = −ϕ(1 − ϕ).
Thus the free energy density is given by the so-called Flory-Huggins potential (see, for instance, [6] and references therein) where Θ > 0 is the absolute temperature of the mixture and Θ 0 > 0 is its critical temperature (other constants are set equal to the unity).If Θ < Θ c then W has a double well shape and phase separation takes place.Assuming that the mixture occupies a bounded domain Ω ⊂ R d , d = 2, 3, the previous considerations lead to the following free energy functional where the penalization term allows the creation of diffuse interfaces between the two species and also allows a convenient mathematical treatment of the phenomenon (see [14]).Here γ > 0 is related to the thickness of the diffuse interface.The Cahn-Hilliard equation can be introduced as a conserved gradient flow generated by the gradient of the chemical potential µ defined by µ = δE δϕ = −γ∆ϕ + W ′ (ϕ), namely, taking constant mobility equal to a constant m > 0, This equation subject to no-flux (or periodic) boundary conditions entails the conservation of the total mass Ω ϕ(t)dx.An alternative model has been proposed by J. Rubinstein and P. Sternberg [39] by modifying another well-known equation proposed by S.M. Allen and J.W. Cahn [3] in order to ensure the mass conservation.
The equation has the form 2) where α > 0 and f is defined by for any integrable f .Here |Ω| d stands for the d-dimensional Lebesgue measure of Ω. Equation (1.2) equipped with homogeneous Neumann boundary condition, preserves the total mass.In [39] a (formal) asymptotic analysis was performed with respect to a specific scaling in order to understand the motion of the separating interfaces (see also [34] for an important application).More rigorous results can be found in [7] where the authors show that, in a radially symmetric setting, the sharp interface problem of a suitable scaling of (1.2) is a nonlocal motion by mean curvature.Moreover, they also prove that both (1.2) and the Cahn-Hilliard equation can be seen as degenerate limits of the viscous Cahn-Hilliard equation introduced in [36].
The corresponding motion by mean curvature is also analyzed in [10] under more general assumptions on the evolving surface In the quoted contributions, the mixing entropy is approximated with a smooth function defined all over R. In this case, the potential is called regular.In particular, the double-well potential W is usually represented by a fourth-order polynomial.However, in presence of the nonlocal constraint, one cannot ensure that ϕ takes its values in the physical range [0, 1] (see, however, [25] for an alternative model).Instead, if the mixing entropy is not approximated by a smooth function, then the image of ϕ is always contained in [0, 1].Well-posedness issues in the case of a smooth W are standard.However, if W given by (1.1) then proving the existence of sufficiently regular global solutions is less trivial because S ′ is singular at the endpoints and cannot be controlled by S like a polynomial.In this case, it would be nice to show that ϕ stays uniformly away from 0 and 1, that is, if the strict separation property holds, then S ′ would be globally Lipschitz and the analysis would simplify a lot (see, for instance, [18] and references therein for the Cahn-Hilliard equation in two dimensions, see also [37] for the case of three dimensions).In the non-conserved case, the strict separation is trivial for regular potentials and a bit less straightforward for logarithmic type potentials like (1.1) (see [26,Thm.2.3]).Concerning (1.2), it has been proven its instantaneous validity in dimension two (see [24]), while in dimension three the proof was given assuming that the initial datum is strictly separated (see [20]).Observe that the strict separation property combined with the uniqueness of a solution ϕ, allows to view the solution itself as the solution to a similar problem where S is replaced by a smooth approximation, defined on the whole real line, which coincides with S on the interval [δ, 1 − δ] and δ ∈ (0, 1) is such that ϕ ∈ [δ, 1 − δ].In other words, the validity of the strict separation can be interpreted as a rigorous justification of the regular approximation of a singular (e.g.Flory-Huggins) potential.
In this paper we want to reconsider these issues and say more for a multi-component version of (1.2).In many applications, it is important to account for the presence of multiple interacting species (see, for instance, [4,5,23,27,28,42,45] and references therein, see also [35] and its references for the motion by mean curvature in the non-conserved case and [38] for the importance of the Flory-Huggins potential).Nevertheless, to our knowledge, a comprehensive theoretical analysis of multi-component conserved Allen-Cahn equations is missing.Nonetheless, it is worth recalling [17,40] and their references for non-conserved stationary problems with a regular potential.Moreover, we mention that a rigorous solution to the so-called Keller-Rubinstein-Sternberg problem on the motion by curvature has recently been given in [16] (see also its references).On the contrary, multi-component Cahn-Hilliard equations have been analyzed long ago in the pioneering paper [15] (see also [21] and its references for further results and recent developments).As we shall see, one of the advantages (and our main result) is the fact that any weak solution becomes instantaneously strong and strictly separated also in dimension three, while this property is known only in dimension two for the corresponding multi-component Cahn-Hilliard equation.This regularization allows us to investigate the longtime behavior of solutions in some details, that is, we prove the existence of a global and an exponential attractor.Also, we can show that any weak solution converges to a single stationary state.The present analysis can also be viewed as a first step towards the analysis of multi-component Navier-Stokes-Allen-Cahn systems (see, for instance, [2,43,44], see also [24,31] and references therein for binary fluids).We also believe that this contribution is a significant addition to [29,Sec.9].
In the multi-component case, we denote by u : Ω × (0, T ) → R N the vector-valued function of concentration species whose components must satisfy the constraint (1. 3) The free energy density takes the form where A is a constant symmetric N × N matrix with the largest eigenvalue λ A > 0. Concerning ψ, here we are mainly interested in the Boltzmann-Gibbs mixing entropy, namely, where θ > 0 is the absolute temperature of the mixture.However, our framework also includes many other (physically relevant) entropy functions Ψ 1 : [0, 1] → R + (see [18,21]).The free energy E is thus defined as where Setting the vector µ 0 is the chemical potential without capillarity and is the chemical potential.
Summing up, arguing as in [15] for the Cahn-Hilliard case, the goal of this work is to study the following initial and boundary value problem (1.6) The (constant) mobility matrix α is a symmetric, positive semidefinite N × N matrix such that its kernel is given by span{ζ} (where ζ i = 1, for i = 1, . . ., N ).Here P is defined as follows (see also the next section) Then it is easy to check that, formally, a solution to the above problem with P∆u = ∆Pu in place of ∆u satisfies (1.3) if the initial datum does, using in (1.6) 1 the property and the fact that α ij = 0 for any j = 1, . . ., N (recall also that α is symmetric).Therefore P∆u = ∆u.
The plan of the paper goes as follows.In the next section we introduce the notation, the functional setup, and some basic assumptions on the mobility matrix α.Also, we discuss the basic assumptions on the potential (more general than (1.4)-(1.5))and its regularization.The main results are stated in Section 3 and the last subsection contains the proof of the convergence to a single equilibrium.The proofs of the well-posedness and regularity results, including the strict separation property, can be found in Section 4. The existence of the global attractor and of an exponential attractor are proven in Sections 5 and 6, respectively.

THE MATHEMATICAL FRAMEWORK
The (real) Sobolev spaces are denoted as usual by W k,p (Ω), where k ∈ N and 1 ≤ p ≤ ∞, with norm . Moreover, given a space X, we denote by X the space of vectors of three components, each one belonging to X.We then denote by (•, •) the inner product in L 2 (Ω) and by • the induced norm.We indicate by (•, •) X and • X the canonical inner product and its induced norm in a generic (real) Hilbert space X, respectively.Further, we introduce the affine hyperplane the Gibbs simplex and the tangent space to Σ We introduce the following notation: Notice that the spaces above are still Hilbert spaces with the same inner products given in L 2 (Ω) for the first two, and H 1 (Ω), for the others.We also have (see [21]) the Hilbert triplets Recalling (1.7), we now define rigorously the Euclidean projection P of R N onto T Σ, which is, for l = 1, . . ., N , where ζ := (1, 1, . . ., 1).Notice that the projector P is also an orthogonal L 2 (Ω)-projector, being symmetric and idempotent.We now assume that α is positive definite over T Σ.This will constitute the main assumption on the mobility matrix in this contribution, since it is enough to prove the existence of weak (and strong) solutions.Nevertheless it is not enough to show the validity of a continuous dependence estimate.Thus we need a second assumption (see (M1)).More precisely, we assume that: where Remark 2.1.Note that assumption (M1) can be also rewritten as follows: there exists ξ > 0 such that A matrix of this kind is the natural extension to the case N > 2 of the admissible matrix α when N = 2, which has necessarily the form (2.6), as one can easily verify.Observe that when ξ = 1 the matrix α is simply the representative matrix of the projector P, i.e., the identity operator over the space T Σ.We also point out that α is for sure positive semidefinite and satisfies (2.4), since it has a zero simple eigenvalue corresponding to the eigenspace T Σ ⊥ , whereas on T Σ we see by Lemma 4.1 below (with C equal to the N × N identity matrix) that α is positive definite.In particular, one could show that the eigenvalues of α are λ 1 = 0 (corresponding to the eigenvector (1, 1, . . ., 1)), and λ i = ξN , for i = 2, . . ., N , whose eigenspace is clearly T Σ.
Next, we define the set For the sake of simplicity we will adopt the compact notation v ≥ k, with v ∈ R N and k ∈ R to indicate the relations v i ≥ k, i = 1, . . ., N .Recalling (1.5), we now set In order to include a large admissible class of entropy functionals in (1.4), we suppose that has the following properties: As in [21], we also extend ψ(s) = +∞, for any s ∈ (−∞, 0), and extend ψ for all s ∈ [1, ∞) so that ψ is a C 2 function on (0, +∞) and (E0) holds for any s > 0. In particular we define with We refer the reader to [18,Section 6.3] for some other important classes of mixing potentials that are singular at 0. Furthermore, following the general scheme developed in [19,Section 3.1], by (E0)-(E1) we can define an approximation of the potential ψ by means of a sequence {ψ ε } ε>0 of everywhere defined non-negative functions.More precisely, let where J ε = (I + εA) −1 : R → (0, +∞) is the resolvent operator and According to the general theory of maximal monotone operators, as already developed in [21, Section 2], the following properties hold: (i) ψ ε is convex and ψ ε (s) ր ψ(s), for all s ∈ R, as ε goes to 0; , as ε goes to 0; (iv) for any ε ∈ (0, 1], there holds , for all s ∈ R; (v) for any compact subset M ⊂ (0, 1], ψ ′ ε converges uniformly to ψ ′ on M ; (vi) for any ε 0 > 0 there exists The latter property directly follows from a straightforward adaptation of [19,Lemma 3.11], which entails that for any ε 0 > 0 there exists C = C(ε 0 ) > 0 such that ψ ε (s) ≥ 1 4ε 0 s 2 − C, for any s ∈ R and any 0 < ε < ε 0 (see also [21,Section 2]).Let us now introduce where, as presented in the Introduction, A is a symmetric N × N matrix with λ A > 0 as the largest eigenvalue.We thus have that for any ε 0 > 0 sufficiently small there exist In particular, this comes from the fact that − and ε 0 has to be small enough so that, e.g., Remark 2.2.We point out that, differently from the standard assumptions on ψ (see, e.g., [18,24]) here we do not need the assumption since to deduce the validity of the instantaneous strict separation property we will make use only of assumptions (E0)-(E2).Clearly the logarithmic potential (1.4)-(1.5)satisfies assumptions (E0)-(E2) and is then included in our analysis.Indeed also assumption (E2) certainly holds for the logarithmic potential since ψ ′ (s) = θ(ln(s) + 1) and thus ψ ′ (s − 2s 2 ) − ψ ′ (2s 2 ) = θ(ln(s − 2s 2 ) − ln(2s 2 )) = θ ln 1 2s − 1 → +∞ as s → 0 + .Moreover, it seems that if we consider potentials exploding at infinity more slowly than the logarithm then (E2) is not satisfied.Indeed, if, for instance, we consider as s → 0.

MAIN RESULTS
This section is divided into several subsections according to the nature of the results.
Remark 3.8.As it will be clear from the proof (see also Remark 4.4), in the case N = 2 assumption (E2) is not needed to prove (3.14).This agrees with the result obtained in [20] for binary mixtures.
On account of the dissipative nature of the system, we have the following uniform control of the energy E. Theorem 3.9.Let the assumptions of Theorem 3.1, point (1), hold.Then the energy of solution u satisfies the following inequality ) where C 1 , C 2 > 0 depend on Ω, α, Ψ, and u 0 , while ω > 0 is a universal constant.
We can prove that any weak solution given by Theorem 3.1 instantaneously regularizes.Thanks to this, we can show the instantaneous strict separation property in both dimensions two and three.This means that, for any τ > 0, there exists , each component is strictly separated from both the pure phases 0 and 1.More precisely, the following result holds Theorem 3.10.Let the assumptions of Theorem 3.9 hold, together with (M1) and (E2).Then the energy solution (u, w), defined for all t ≥ 0, is such that, for any τ > 0, Moreover, u and w are uniformly bounded in the above spaces by positive constants only depending on Ω, α, Ψ, u 0 , and E(0).In particular, the energy identity (3.13) holds for almost any t ≥ τ .Moreover, there exists ) i.e., the instantaneous strict separation property holds.
In the proof of the strict separation property (see Section 4.4) we obtain Remark 3.12.We point out that, as observed in Remark 3.8, assumption (E2) is not needed to prove (3.21) when N = 2, i.e., for binary mixtures.

3.2.
Existence of the regular global attractor.We now define a complete metric space which will be the phase space of the dissipative dynamical system (see, for instance, [41]) associated with (1.6).For a given M ∈ Σ, such that M i ∈ (0, 1), for any i = 1, . . ., N , we set endowed with the H 1 -topology.In particular we consider the one induced by the equivalent norm u V M = ∇u + |u|.This is a complete metric space.Thus, under the same assumptions of Theorem 3.10, we can define a dynamical system (V M , S(t)) where Observe that S(t) satisfies the following properties: comes from the instantaneous regularization, so that, for any τ > 0, u ∈ C([τ, ∞); V M ), whereas the last property can be proved as follows.From (3.9) together with the H 2 -regularity (for any t > 0) and the interpolation estimate we deduce that u 0 → S(t)u 0 ∈ C(V M ; V M ), for any t ∈ (0, ∞).This is indeed a consequence of (3.17), since u ∈ L ∞ (τ, ∞; H 2 (Ω)) for any τ > 0 entails that, given two intial data u 0,1 , u 0,2 ∈ V M , for any t > 0, where in the last step we also used (3.9).The case t = 0 is trivial.
Furthermore, we recall that the global attractor is the unique compact set A ⊂ V M such that • A is fully invariant, i.e., S(t)A = A for every t ≥ 0; • A is attracting for the semigroup, i.e., for every bounded set B ⊂ V M .The dissipative inequality (3.16) and the instantaneous regularization of the energy solution allow us to prove Theorem 3.13.Let the assumptions of Theorem 3.10 hold.Then the dynamical system (V M , S(t)) admits a (unique) connected global attractor A ⊂ V M which is bounded in H 2 (Ω).
Remark 3.14.The proof of this result is based on showing that the dynamical system (V M , S(t)) admits a compact absorbing set B 0 (see Section 5 below).

3.3.
Existence of an exponential attractor.Thanks to the validity of the strict separation property in dimensions two and three, we can prove the existence of an exponential attractor in dimensions two and three.We first recall (see, e.g., [33]) that a compact set M ⊂ V M is an exponential attractor for (V M , S(t)) if • M is positively invariant, i.e., S(t)M ⊂ M for every t ≥ 0; • M is exponentially attracting, i.e, there exists ω > 0 such that for every bounded B ⊂ V M , where Q(•) denotes a generic increasing positive function; • M has finite fractal dimension in V M , where the fractal dimension is defined as and N (ǫ) is the minimum number of ǫ-balls of V M necessary to cover M. Observe that the exponential attractor is not unique, and that, by definition, A ⊂ M, so that from the existence result of an exponential attractor we deduce that the global attractor A is of finite fractal dimension.We thus have the following Theorem 3.15.Let the assumptions of Theorem 3.10 hold.Moreover, assume that ψ ∈ C 3 (0, 1].Then the dynamical system (V M , S(t)) possesses an exponential attractor M which is bounded in H 2 (Ω).Besides, A ⊂ M has finite fractal dimension in V M .
3.4.Convergence to equilibrium.In this section we discuss the convergence of any weak solution to a single equilibrium.We have all the ingredients to state and prove the result.
We consider the phase space V M as in the previous section.Under the assumptions of Theorem 3.10, we define the ω-limit set ω(u 0 ) of a given 1).In particular, we fix r ∈ ( d 4 , 1).We thus have Theorem 3.16.Let the assumptions of Theorem 3.10 hold and suppose, in addition, that ψ is (real) analytic in (0, 1).Then, for any and the (unique) weak solution u(t) is such that Proof.The proof of this Theorem is exactly the same as the one of [21,Thm. 3.16].Indeed, the only difference is in the energy estimate given by the application of Łojasiewicz-Simon inequality (see [21,Sec.7.3]), in which we need to substitute ∇w with w − w (basically we do not need to apply Poincaré's inequality, but we keep w − w in the inequality for E ′ ).
Theorem 3.16 is still valid without assumption (E2).Indeed, in the proof we do not need the instantaneous strict separation property, for which that assumption is essential.It is also worth noticing that, without assuming (E2), by the same proof of [21,Thm.3.13],we can show that the asymptotic strict separation property holds, i.e., Theorem 3.17.Let the assumptions of Theorem 3.10 hold except for (E2).Then, for any M ∈ (0, 1), M ∈ Σ, and for any initial datum u 0 ∈ V M , there exists δ > 0 and t * = t * (u 0 ) such that the corresponding (unique) solution u satisfies: 4. PROOFS OF SUBSECTION 3.1 Here we collect the proofs of Theorems 3.1, 3.9, and 3.10.
Remark 4.3.What is needed to prove (3.9) is actually (4.1).Nevertheless, the matrix structure (2.5) is the only example case we know that implies (4.1).
Continuous dependence estimate.We can now prove (3.9).Let us consider two solutions u 1 and u 2 and take the difference between the equations they solve.Taking u = u 1 − u 2 as a test function in the resulting equation and recalling, by mass conservation, that u ≡ 0, we deduce (note that αPΨ 1  ,u Notice that w 1 − w 2 does not appear in (4.2), since we have recalling in the last equality that u ≡ 0. Lemma 4.1 then entails γ (∇u, α∇u) ≥ 0.
Existence of a solution.Here we give the details of the Galerkin scheme since in previous related contributions (see, e.g., [15]) they are not given.We consider the approximation (2.10).In particular, for each ε > 0 sufficiently small, we set We then fix 0 < ε < ε 0 and consider the complete system of N -dimensional eigenfunctions {e i } i of the problem −∆e i = λ i e i , with homogeneous Neumann boundary conditions ∂ n e i = 0 on ∂Ω (λ i is the eigenvalue corresponding to e i ), subject to the constraints e i = 0 and N j=1 (e i ) j ≡ 0. The family {e i } i can be tuned to form an orthogonal basis in V 0 , orthonormal in H 0 (see also [21,Appendix 8.1]).We then set m := u 0 and introduce the finite-dimensional spaces and for w n,ε ∈ V 0 such that solving the equations for any v ∈ V n and for any t ∈ [0, T ] where u n,0 is the Let us first notice that the quantity w n,ε is necessary to be specified since any test function v ∈ V n has zero integral mean.Moreover, by construction, In the sequel we will denote by C a generic positive constant independent of n.Any other dependence is explicitly pointed out if necessary.
Recalling that ψ ′ ε is at least C 1 (R), we can locally solve the above Cauchy problem (4.3)-(4.4),(4.6) in the unknowns {α i } i and find a unique maximal solution α(n) ∈ C 1 ([0, t n,ε ]; R n ), from which we also obtain by comparison a unique δ (n) ∈ C 1 ([0, t n,ε ]; R n ).Then by substitution in (4.5) we immediately obtain the complete quantity w n,ε .It is now standard to test (4.3) by v = w n,ε − w n,ε ∈ V n and obtain the energy identity where Let us observe that, being ψ ′ ε Lipschitz (see (2.10)), and recalling that Ψ ε (u 0 ) ≤ Ψ(u 0 ), we obtain Therefore, since clearly u n,0 +m − u 0 → 0 as n → ∞, for any ε > 0 there exists n = n(ε) such that An application of Gronwall's Lemma then gives, thanks to (4.9) and ∇u n,0 ≤ ∇u 0 , Now, recalling property (vi) of ψ ε , it is immediate to see that, for any ε < ε 0 , for some K > 0, so that we can conclude, for any ε < ε 0 , where we also exploited (2.4).Clearly C does not depend on ε.From this we can easily deduce that local maximal time t n,ε is +∞.Moreover, from these estimates it clearly derives, by comparison, that These estimates, together with the fact that ψ ′ ε is Lipschitz, give from (4.5) Here C ε could depend on ε.The obtained bounds are enough to pass to the limit as n → ∞ by standard compactness arguments.However, since we also need to prove the existence of strong solutions, we now assume u 0 ∈ H 2 (Ω) such that ∂ n u 0 = 0 almost everywhere on ∂Ω, together with φ(u 0,i ) ∈ L 2 (Ω), for any i = 1, . . ., N , and find a higher order estimate, before passing to the limit.In particular, we test (4.
Recalling that P is selfadjoint and ∂ t u n,ε ≡ 0 by construction, we obtain Using (4.4), since ∂ t u n,ε ∈ V n , we find Being φ ′ ε ≥ 0 by property (iv) of ψ ε , we have only to treat the term related to the matrix A. This is readily done by comparison from (4.3): indeed, being v = ∂ t u n,ε ∈ V n , we get, by the Cauchy-Schwarz, Young, and Poincaré inequalities, Putting everything together in (4.13) and recalling (2.4), we end up with Observe now that, from (4.4), On the other hand, by the properties of the eigenfunctions, we have Thus, recalling properties (ii)-(iii) of ψ ε , we get Therefore, since u n,0 +m − u 0 → 0 as n → ∞ and by the stronger assumptions on the initial data, we deduce that for any ε < ε 0 there exists n = n(ε) > 0 such that We can thus conclude that, for any n > n 0 (ε) = max{n, n}, owing to Gronwall's Lemma and (2.4), it holds where C(T ) does not depend on ε.Furthermore, by comparison (choosing v = ∂ t u n,ε in (4.3)) it also holds We can now pass to the limit in n for both the situations (according to the regularity of the initial data), to deduce, by standard compactness arguments, that, for any ε < ε 0 , there exists a pair (u ε , w ε ), defined on [0, +∞), with w ε (t) ∈ V 0 for almost any t ≥ 0, such that (in the case of less regularity on u 0 ), for each T > 0, and for some C(T ) > 0 independent of ε, whereas there exists C ε > 0 such that If the stronger assumptions hold (see point (2) of Theorem 3.1), then there exists a constant C > 0, depending on the initial datum and on T , but independent of ε, such that where the L 2 (0, T ; H 1 (Ω)) control on the chemical potential differences is obtained by comparison in (4.19) below.
Notice that, to be precise, we find that u n,ε converges in suitable norms to a function u ε (t) ∈ V 0 (for almost any t ≥ 0) as n → ∞.We then define u ε := u ε + m to obtain the results above.Then, by elliptic regularity, being φ ε Lipschitz, from (4.20) we deduce its strong version, namely u ε ∈ L 2 (0, T ; H 2 (Ω)) and By standard computations (see also [15] for similar results), we then have • Conservation of mass: for a.a.x ∈ Ω and for all t ∈ [0, ∞).(4.24) • Conservation of chemical potential differences for any t ≥ 0, where At this point, we can argue as in the proof of [21, Thm 3.1] (which is based on [22]), in order to control w ε (t) which then allows us to control w ε (t) .Following the proof of [22, Lemma 3.3], we define where, on account of the boundary conditions, Taking advantage of (4.20), we have, Exploiting the convexity of Ψ 1 ε , for any k ∈ G, G being the Gibbs simplex, because k − u ε ∈ T Σ almost everywhere in Ω × (0, T ), we find where we used (see property (i) of ψ ε ) Here and in the sequel C > 0 stands for a generic constant independent of ε.Recalling that Ψ 1 ε,u (u ε ) = {φ ε (u ε,i )} i=1,...,N and choosing η = k − u ε in (4.27), on account of (4.28), we deduce that for almost all t ∈ (0, T ).On the other hand, we have (k ∈ G and thus 0 ≤ k ≤ 1) Then, using Cauchy-Schwarz's and Young's inequalities and recalling recalling property (vi) of ψ ε we obtain, where in the last estimate we have exploited (4.17).By the conservation of mass and Remark 3.3 we also deduce that, for all i = 1, . . ., N and all t ∈ [0, T ], Therefore, for any fixed k, l = 1, . . ., N , we choose 29), where ζ ζ ζ j := (0, . . ., 1 j , . . ., 0).Thus, from (4.29) we get that Integrating |(λ ε,k − λ ε,l )(t)| 2 over (0, T ) and using the identity , we find, owing to (4.17), This, using again (4.17), gives w ε L 2 (0,T ;L 2 (Ω)) ≤ C. (4.31)As a consequence, we deduce from (4.30) that for almost any t ∈ (0, T ).Therefore, in the case of a more regular initial datum (see (4.18)), we have We are now left with some estimates related to φ ε (u ε,i ).We follow again the proof of [21,Thm.3.1].Being φ ′ ε bounded for a fixed ε ∈ (0, ε 0 ), we have that Observe that and Thanks to (4.24), we have so that, being φ ε monotone, we infer Notice that C is independent of ε provided that we choose ε sufficiently small.Indeed, since we have the pointwise convergence φ ε ( 1 N ) → φ( 1 N ) as ε → 0, then there exists C > 0, independent of ε, such that |φ ε ( 1 N )| ≤ C for any ε ∈ (0, ε 0 ), with ε 0 > 0 sufficiently small.Then we get and (see (4.17)) Therefore, on account of the above inequalities and recalling that φ ′ ε ≥ 0, we deduce from (4.33) that as well as We have obtained all the bounds we need to pass to the limit as ε → 0. Being this step standard (see, e.g., [22]), we only present a sketch of the argument.By compactness we immediately deduce that, up to subsequences, u ε → u a.e. in Ω × (0, T ), Then, arguing as in [22,Section 6] and exploiting (4.36), we infer that for any k = 1, . . ., N .Thus the pair (u, w) satisfies (3.2)-(3.7).The energy inequality (3.13) is then retrieved by standard lower semicontinuity arguments.If the initial datum is more regular, then, up to subsequences,we also have the convergences and weakly in L 2 (0, T ; H 1 (Ω)), which ensure the regularity of Theorem 3.1, point (2).The energy identity (3.13) can be recovered since t → ∇u(t) 2 is absolutely continuous in [0, T ] and because of Ψ 1 (u) ∈ H 1 (0, T ; L 1 (Ω)) entailing that the function Step 1. Case N − 1.Let us then start from s = N − 1, having fixed σ.We consider the vector e N −1 σ as, for i = 1, . . ., N , Then we take η = ηe N −1 σ , for η ∈ H 1 (Ω), in (3.4).This gives for almost any t ∈ [0, T ].We now fix δ > 0 (to be chosen later on) and consider σ,δ and integrating by parts, we find where we used the property that, given any vector ζ ∈ R N , αPζ = αζ.Now notice that, being α ii = A > 0 for any i = 1, . . ., N , we have Since α ij = B < 0 for any i = j (clearly we have A + (N − 1)B = 0), we see that the second summand becomes Notice also that, recalling where we used the fact that, when u N −1 σ,δ ≤ δ, it holds Therefore, in the end we get Concerning the terms related to ψ ′ (u j ), we first observe that, on account of (2.5), we have Then we write where Thus, for δ ≤ 1 2 , we deduce being ψ ′ monotone increasing.Moreover, in E N −1 (t) it also holds, being Concerning the other terms in (4.43), we have, clearly, being 0 ≤ u k ≤ 1 for k = 1, . . ., N , that and observing that (see (3.11)) w ∈ L ∞ (0, T ), we have, similarly, Coming back to (4.43) and collecting all these results we end up with so that, assuming δ sufficiently small to satisfy (see assumption (E1)) Hence, having assumed the initial datum strictly separated, i.e., there exists 0 we can choose δ ≤ δ 0 in such a way that u N −1 σ,δ (0) ≡ 0 and Gronwall's Lemma yields Notice now that the choice of the set P N −1 σ is completely arbitrary, thus we infer that there exists δ N −1 , such that δ 0 ≥ δ N −1 > 0 and, for any possible Remark 4.4.We point out that in the case N = 2 the proof is ended.This means that (E2) is not necessary in this case, consistently with [20,Thm.3.5].
Step 2. Case N − 2. If N = 2 we are done.Otherwise we need to consider the sets P N −2 σ , σ = 1, . . ., N (N − 1) 2 .Let us fix σ and 0 < δ ≤ δ N −1 (to be chosen later on).Then, we set and define the vector e N −2 σ as We make a crucial observation: in the set Recall that δ < 1 2 and 0 < 2δ 2 < δ ≤ δ N −1 < 1.Then we take in (3.4), as in Step 1, the test function Choosing in the equation above η = −u N −2 σ,δ and integrating by parts, we find Recalling once more that α ii = A > 0 for any i = 1, . . ., N , and arguing exactly as in Step 1, we find Since α ij = B < 0 for any i = j, the second summand becomes Recall now that This entails The terms ψ ′ (u j ) can be handled as above.Indeed, observing that Thanks to (4.48) we know that in E N −2 (t), for δ sufficiently small, it holds being ψ ′ monotone increasing.This entails that and thus Concerning the other terms in (4.43), we have (recall that 0 and, arguing similarly (see (3.11)), we find Combining (4.49) with the obtained estimates, we end up with 1 2 Therefore, on account of (E2), for Then, thanks to (4.46) and to the choice δ ≤ δ N −1 (entailing also 2δ 2 ≤ δ N −1 ), we get u N −2 σ,δ (0) ≡ 0. Therefore, by Gronwall's Lemma, we get Again the choice of the set P N −2 σ is completely arbitrary, meaning that there exists a 0 < δ N −2 ≤ δ N −1 such that, for any possible The essential observation is again the following: in the set x ∈ Ω :

.51)
This implies that in E N −3 (t), for δ > 0 sufficiently small, it holds We can now argue as in Step 2 and conclude that there exists a δ N −3 ∈ (0, δ N −2 ] such that, for any possible Repeating iteratively these arguments, we reach a generic step m and we find δ N −m ∈ (0, δ N −m+1 ] such that, for any Therefore, we can continue the procedure until N − m = 1, which entails in the end that there exists a 0 < δ ≤ δ 0 ≤ 1 N such that, for any i = 1, . . ., N , i.e., the strict separation property holds.This concludes the proof of Theorem 3.1.

4.3.
Proof of Theorem 3.9.Let us take η = u(t) − u(t) in equation (3.5).This gives Moreover, by convexity of Ψ 1 (recall that u − u ∈ T Σ), we have where C > 0 depends on u 0 .Applying standard inequalities, from (4.55) we infer that and using (2.4) we get where in the last step we applied property (vi) of the potential ψ ε (recall that these estimates must be obtained in an approximating scheme, so for ε sufficiently small, see above).Therefore, we obtain Concerning the global bounds, for the sake of brevity, here we simply show the formal estimates.A rigorous argument can be performed within an approximation scheme like the previous one.First, we observe that (3.8) Notice that the constant C > 0 only depends on the initial energy E(0).Then, arguing as in (4.14), we obtain 1 2 Due to (4.57), we can apply the uniform Gronwall's Lemma (see, e.g., [41], by choosing, e.g., r = τ 2 ) to deduce, for any given τ > 0, From now on we can argue as in the proof of Theorem 3.1, to get where C > 0, now and in the sequel, stands for a generic constant depending on Ω, α, Ψ, u 0 , and E(0).This allows us to deduce Also, by comparison in (3.7), we find The proof is finished.
Instantaneous strict separation.We are in the case u 0 L ∞ (Ω) ≤ 1, that is, u 0 is not necessarily strictly separated like in Section 4.1.Therefore we need to adapt the proof we performed in Section 4.1.In order to do that, we perform a De Giorgi-type iterative scheme at each step.The basic tool is the following Lemma 4.5.Let {y n } n∈N∪{0} ⊂ R + satisfy the recursive inequalities for some C > 0, b > 1 and ε > 0. If then and consequently y n → 0 for n → ∞.
Lemma (4.5) can be found, e.g., in [11, Ch.I, Lemma 4.1] (see also [30, Ch.2, Lemma 5.6]) and can be easily proven by induction (see, e.g., [37,Lemma 3.8]).Since the iterative argument in which we sum up some components of u (in decreasing number at each step) is exactly the same as in the case treated in Section 4.1, we directly assume to be at Step m > 1 and show the differences with respect to estimate (4.43) (Step 1 is even easier, as we have seen in Section 4.1, thanks to the relation (4.45), thus it can be easily adapted following the analysis of the other steps).We assume to know, for an arbitrary τ > 0, that there exists 0 < δ N −m+1 ≤ 1 N such that, for any σ, with the same notation as in Section 4.1.Notice that the upper bound δ N −m+1 ≤ 1 N is set since clearly in the end the necessary condition for the separation will be that δ ≤ 1 N .We now consider the set of indices P N −m σ for a certain σ.Then, for i = 1, . . ., N , we set We can now perform De Giorgi's scheme.Let us set δ sufficiently small such that δ ≤ δ N −m+1 and fix τ such that where and the sequence of times Then, introduce a cutoff function on account of the above definition of {t n } n , and set

.70)
This means that, on A n (t), for δ > 0 sufficiently small, we have and where we used As in Section 4.1, recalling that α ii = A > 0 for any i = 1, . . ., N , we obtain Since α ij = B < 0 for any i = j, the second summand becomes Then, for d = 2, 3, we find so we get ds.
On the other hand, by (4.79), we obtain Similarly, using (4.79) once more, we have Therefore, we infer from (4.80) that Thus we can apply Lemma 4.5.In particular, we have b = 2 , to get that y n → 0, as long as
In the end, passing to the limit in y n as n → ∞, we have obtained that = 0, by uniqueness of the limit, since, as n → ∞, and, on the other hand, y n → 0. Notice that, due to the choice of T , we have (see (4.64)) T − τ = τ 2 + (m+1)τ

2N
≤ τ , therefore we can repeat the same procedure on the interval (T, T + τ ) (the new starting time will be t −1 = T −2 τ ≥ , +∞ .We can thus repeat the procedure increasing m, for a finite number of times, until each set P σ is a singleton (as in the case discussed in Section 4.1).This entails that there exists 0 < δ ≤ 1 N such that, for any i = 1 . . ., N , u i ≥ δ > 0 a.e. in Ω × [τ, +∞), (4.87) concluding the proof.Notice that the quantity δ depends on the initial datum only through the initial energy E(0) and u 0 since all the estimates involved in this proof are the ones mentioned in Theorem 3.10.

PROOF OF THEOREM 3.13
By Remark 3.14, we only need to show the existence of a compact absorbing set.From Theorem 3.9, we deduce that, for any u 0 ∈ V M , there exist constants C 3 , C 4 > 0 such that S(t)u 0 2 Indeed, being Ψ bounded on [0, 1] and 0 ≤ u 0 ≤ 1, it holds for some C > 0 independent of the initial datum u 0 .This means that the set is an absorbing set, i.e., for any bounded set B ⊂ V M there exists t e > 0 depending on sup for some C 0 = C 0 (R 0 ) > 0, and a time t R 0 , depending only on R 0 , such that S(t) B 0 ⊂ B 0 for any t ≥ t R 0 .Note that we can state for any t ≥ t R 0 instead of for almost any t (see Remark 3.11).This clearly implies that B 0 is a compact absorbing set and ends the proof.

PROOF OF THEOREM 3.15
We need some preliminary lemmas.First, recalling (5.2), we know that there exists t = t(R 0 , M) > 0 (with M fixed) such that S(t)B 0 ⊂ B 0 , for any t ≥ t.We then introduce the set B := Proof.The following computations are formal, but they can be performed within a suitable approximating scheme as the one used in the proof of Theorem 3.1.In particular, leaning on the strict separation property, which holds uniformly (depending only R 0 and M, this last one being fixed, see Remark 3.7) if the initial data belong to B (see Theorem 3.1), then we are able to interpret, by uniqueness, the solutions to problem (1.6) as the solutions to a similar problem where ψ is replaced by a suitable regular potential (i.e.obtained by extending ψ outside [δ, 1 − (N − 1)δ] in a smooth way).We start by observing that there exists δ > 0 (possibly smaller than the one in the definition of B) such that (see (3.14)  Set now u i = S(t)u 0,i , with u 0,i ∈ B, i = 1, 2. Then, taking the difference between the equations satisfied by u 1 and u 2 , multiplying it by ∂ t u, where u = u 1 − u 2 , and integrating over Ω, after an integration by parts, we get (α ij (Au) j , ∂ t u i ) + ∂ t u 2 = 0, where we exploited the following facts: ∂ t u ≡ 0, P(∂ t u) = ∂ t u, and the property α(Pξ) = αξ for any ξ ∈ R N .Thanks to (6.3), we have ψ ′′ (su 1 j + (1 − s)u 2 j ) L ∞ (Ω) ≤ C, for any j = 1, . . ., N , so that, by standard inequalities, N i,j=1 ).Thus (6.1) follows from (6.5) owing to Gronwall's Lemma and Poincaré's inequality.Notice that the constant C, thanks to (6.3), does not depend on the specific u 0,i ∈ B. Concerning (6.2), we write (3.4) for the difference (defined as u) between u 1 and u 2 and we differentiate the resulting equation with respect to time.Then, we multiply it by ∂ t u and integrate over Ω.This gives, after an integration by parts, the identity (α ij (A∂ t u) j , ∂ t u i ) + (α∇∂ t u, ∇∂ t u) = 0, where we exploited ∂ t u ≡ 0, P∂ t u = ∂ t u, and the properties of α.Using now (6.3) once more, standard inequalities, and on account of assumption ψ ∈ C 3 (0, 1], we get N i,j=1 where we exploited the embedding H 1 (Ω) ֒→ L 4 (Ω), the bound ∂ t u 1 L ∞ (0,T ;L 2 (Ω)) ≤ C with C depending only on R 0 (see point (2) of Theorem 3.1, Theorem 3.10, and (5.2)), Poincaré's inequality and the fact that (α∇∂ t u, ∇∂ t u) ≥ C ∇∂ t u 2 .This last estimate comes from (2.4), since we have )w = P(−Au + φ(u)) − γ∆u a.e. in Ω × (0, T ), (3.5) ∂ n u = 0 a.e. on ∂Ω × (0, T ),(3.6)u(0)= u 0 a.e. in Ω.(3.7)

•
i = 0, for a.a.x ∈ Ω and a.a.t ∈ (0, ∞).(4.25)It holds from (4.7)-(4.8),by standard arguments, the energy inequality from the regularity above.This concludes the proof of the existence part of Theorem 3.1.Strict separation property of strong solutions.We recall that (M1) is in force.Let us now introduce the following notation: we define P s σ , with s = 1, . . ., N − 1 and σ ∈ N, as any possible subset of s (non repeated) indices from 1, . . ., N .Note that σ indicates the choice of the subset, and σ = 1, . . ., N s .In case s = N − 1 we define the only index not belonging to P N −1 σ by j σ .
.69) Also, for any n ≥ 0, let us introduce the interval I n = [t n−1 , T ] and the set