ON THE NONCONSERVED CAGINALP PHASE-FIELD SYSTEM BASED ON THE MAXWELL-CATTANEO LAW WITH TWO TEMPERATURES AND LOGARITHMIC POTENTIALS

. Our aim in this article is to study generalizations of the nonconserved Caginalp phase-ﬁeld system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Fur-thermore, we give some numerical simulations, obtained with the FreeFem ++ software [24], comparing the nonconserved Caginalp phase-ﬁeld model with regular and logarithmic nonlinear terms. Mathematics Subject Classiﬁcation. 35B45, 35K55, 35L15.


Introduction. The nonconserved Caginalp phase field system
has been proposed in [5] to model phase transition phenomena, such as meltingsolidification phenomena. Here, u is the order parameter, T is the relative temperature (defined as T =T − T E , whereT is the absolute temperature and T E is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F (s) = 1 4 (s 2 − 1) 2 , hence the usual cubic nonlinear term f (s) = s 3 − s). Furthermore, here and below, we set all physical parameters equal to one. This system has been extensively studied; we refer the reader to, e.g., [1], [2], [3], [4], [10], [11], [13], [17], [18], [19], [20], [21], [22], [27], [35] and [37]. These equations can be derived as follows. One introduces the (total Ginzburg-Landau) free energy where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R n , n = 1, 2 or 3, with boundary Γ) and the enthalpy As far as the evolution equation for the order parameter is concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one) where D Du denotes a variational derivative with respect to u, which yields (1). Then, we have the energy equation where q is the heat flux. Assuming finally the usual Fourier law for heat conduction, we obtain (2). Now, one essential drawback of the Fourier law is that it predicts that thermal signals propagate at an infinite speed, which violates causality (the so-called paradox of heat conduction, see [14]). To overcome this drawback, or at least to account for more realistic features, several alternatives to the Fourier law, based, for example, on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed and studied in, e.g., [25], [26], [28], [29], [30], [31] and [32].
In the late 1960's, several authors proposed a heat conduction theory based on two temperatures (see [7], [8] and [9]). More precisely, one now considers the conductive temperature T and the thermodynamic temperature θ. For time-independent problems the difference between these temperatures is proportional to the heat supply; they thus coincide when there is no heat supply. However, for time-dependent problems, they are generally different even in the absence of heat supply: this is in particular the case for non-simple materials. In that case, the two temperatures are related as follows: The nonconserved Caginalp system was studied in [15] for the classical Fourier law with two temperatures and in [33] for the type III thermomechanics theory [23] with two temperatures recently proposed in [37] (see also [16]).
In this article, we consider the theory of two-temperature-generalized thermoelasticity proposed in [39] and based on the Maxwell-Cattaneo law.
In that case, the free energy reads, in terms of the (relative) thermodynamic temperature θ, and (5) yields, in view of (8), the following evolution equation for the order parameter: Furthermore, to obtain the corresponding generalized heat equation, one writes where the heat flux q satisfies the Maxwell-Cattaneo law [39], In particular, it follows from (11) that hence, in view of (13), We thus deduce from (12) and (14) the generalized heat equation Here, the presence of the second derivative ∂ 2 u ∂t 2 makes the mathematical analysis of the equation particularly difficult and, to overcome such a difficulty, we will rewrite the equation in a different way, keeping the enthalpy H as unknown. Indeed, it follows from (12) and (14) that Furthermore, owing again to (12), (10) can be written as In [34], the authors studied the well-posedness of the nonconserved Caginalp system (16)- (17), for regular nonlinear terms f and Dirichlet boundary conditions. It is however important to note that, in phase transition, regular nonlinear terms actually are approximations of thermodynamically relevant logarithmic ones of the form

AHMAD MAKKI, ALAIN MIRANVILLE AND GEORGES SADAKA
with s ∈ (−1, 1) and 0 < κ 1 < κ 0 , which follow from a mean-field model (see [6], [27]; in particular, the logarithmic terms correspond to the entropy of mixing). In order to compare the logarithmic potentials with the cubic ones in the numerical simulations that we will perform, we will choose a cubic polynomial which has the same extrema as the logarithmic potential. To this end, we will consider the following cubic nonlinear terms: 2.5(s 3 − .9315 2 s), when (κ 0 , κ 1 ) = (ln(6), 1).
The original Caginalp phase-field system, with the aforementioned logarithmic nonlinear terms, was studied in [27]; see also [19] for a more general Caginalp phasefield system, with a nonlinear coupling between u and T .
In this article, we consider the nonconserved phase-field model (16)- (17), with the logarithmic nonlinear terms (18). The article is organized as follows. In Section 2, we derive a priori estimates which are of fundamental significance for what follows. In Section 3, we prove that the solutions are separated from the singular points of f , which allows us to prove the existence of global (in time) solutions. In Section 4, we study the dissipativity of the associated dynamical system. In Section 5, we prove the existence of the global attractor. In Section 6, we write the spatial and time discretizations of (16)- (17), which allows us finally, in Section 7, to give a comparison for the nonconserved Caginalp model with regular and the logarithmic nonlinear terms, first, by comparing the convergence rate of our codes and then by computing the propagation of a cross function for u and a constant enthalpy H. In particular, we give an example for which both potentials are comparable (this is expected when the quench is shallow, i.e., when κ 1 is close to κ 0 ) and a second one for which the logarithmic potential gives much better results.
Notation. We denote by (( · , · )) the usual L 2 -scalar product, with associated norm · . More generally, · X denotes the norm in the Banach space X.
Throughout the article, the same letter c, c (and, sometimes, c , C) denotes (generally positive) constants which may vary from line to line. Similarly, the same letter Q denotes (positive) monotone increasing (with respect to each argument) functions which may vary from line to line.
Setting the problem. We consider the following initial and boundary value problem, in a bounded and regular domain Ω ⊂ R n , n = 1, 2 or 3, with boundary Γ: For simplicity, we set τ and ε equal to one in what follows. The nonlinear term f is defined as with s ∈] − 1, 1[ and 0 < κ 1 < κ 0 . We then have Lemma 1.1. The nonlinear term f in (25) is of class C ∞ and satisfies where F (s) = s 0 f (τ ) dτ , and Proof. We have, for s ∈ (−1, 1), Note that, for s ∈ (−1, 1), Therefore, we obtain −c 0 F (s) f (s)s + c 0 , with c 0 = κ 0 s 2 > 0. Finally, it easily follows from (26) that Remark 1.1. We can also endow the problem with periodic or Neumann boundary conditions. In these cases, we have, integrating (22) over Ω, where · denotes the spatial average, which yields d H dt and Taking (29)-(31) into account, we can adapt the proofs below and derive the same well-posedness results. Note however that, in order to study the existence of attractors, we need to assume that It thus follows from (31) that We can then define the family of solving operators This family of operators forms a semigroup which is continuous for the L 2 (Ω) × L 2 (Ω) × L 2 (Ω)-topology. We refer the interested reader to [33] for more details on the necessary modifications.
Remark 2.1. We will make formal calculations here, keeping in mind that all these calculations can be rigorously justified by approaching the singular function f by regular functions of class C 1 . We will assume a priori that u L ∞ ((0,T )×Ω) < 1 and that u 0 L ∞ < 1.
We finally multiply (21) by −∆u and obtain, owing to (28) and classical elliptic regularity results, Summing (45) and δ 4 times (48), where δ 4 > 0 is chosen small enough, we find a differential inequality of the form where In a second step, we differentiate (21) with respect to time to have the initial and boundary value problem Note that, if u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) and H 0 ∈ L 2 (Ω), then ∂u ∂t (0) ∈ L 2 (Ω) and Indeed, it follows from the continuity of f and the continuous embedding Multiplying (52) by ∂u ∂t , we obtain, in view of (28), Summing then (49) and δ 5 times (56), where δ 5 > 0 is chosen small enough, we find a differential inequality of the form where We finally rewrite (21) as an elliptic equation, for t > 0 fixed, Multiplying (60) by −∆u, we have, owing to (28), hence, owing to classical regularity results, so that u ∈ L ∞ (0, T, H 2 (Ω)).
In particular, it follows from (57) and Gronwall's lemma that which yields, owing to (59), the continuity of f and the continuous embedding It then follows from (61), (67) and (68) that and from (64), (66)-(68) and Gronwall's lemma that 3. Existence and uniqueness of solutions. One of the difficulties here is precisely to ensure that the order parameter u remains in the physical interval (−1, +1), in order to give a meaning to the equations. We should note that the values −1 and +1 correspond to the pure phases. To prove the well-posedness of our problem, it suffices to obtain an estimate of H in L ∞ ((0, T ) × Ω) (see [27]). We start with the following result.
Then, the order parameter u satisfies the strict separation property for some δ ∈ (0, 1) depending on T .
Proof. It follows from the previous section that H ∈ L ∞ (0, T, H 2 (Ω)) and, since Set now v + = max{0, v}. Multiplying (71) by v + , we find, integrating over Ω, We note that, by definition, v + = 0 on Γ, since v = 0 on Γ. Then, thanks to (28), By choosing δ such that we then deduce that In particular, Gronwall's lemma then yields, noting that v This means that v + (t) = 0, ∀t 0, and, as v v + , then Therefore, As f is an odd function, we set v = u − λ, with λ = −δ. We define the quantity v − = min{0, v}. Proceeding as above, replacing δ by λ, we obtain Consequently, v − (t) = 0, ∀t 0. Since This means that which is equivalent to Finally, Therefore, the order parameter u is strictly separated from the singular points of f . Furtheremore, there exists a constant δ = δ(T, u 0 ) ∈ (0, 1) such that Proof. a) Existence: We first regularize the function f by a C 1 function f δ defined by where δ is the constant defined above. We can choose δ sufficiently close to 1 so that f (δ) 0 and f (δ) 0, taking δ small enough if necessary. We then consider the problem (21)- (24) with f replaced by f δ and u replaced by u δ , that is, It is known that problem (P δ ) admits a unique solution (see [11]). We further have Lemma 3.2. We assume that The functions f δ and F δ satisfy the following properties: where c 0 and c 1 are the positive constants in (27) and (28) (taking δ smaller if necessary).
Proof. We consider, e.g., the case where s ∈]δ, +∞) and we have It is clear that Furthermore, As a consequence of Lemma 3.2, the a priori estimates established in Section 2 for the solutions to problem (21)-(24) still hold for the solutions to (P δ ). In particular, we deduce from Lemma 3.1 that Hence we have f δ (u δ ) = f (u δ ) and we conclude that (u δ , H δ , ∂H δ ∂t ) is also a solution to (21)- (24).
It follows from Theorem 3.1 that we can define the family of solving operators . Furthermore, this family of solving operators forms a semigroup, that is, S(0) = I and S(t + τ ) = S(t) • S(τ ), ∀t, τ 0, which is continuous with respect to the L 2 -topology. 4. Dissipativity. In this section, we study the dissipativity of our system. Moreover, one difficulty is that δ depends on the initial data and time T ; note that the constant δ that appears in the strict separation property satisfied by the order parameter u is such that u 0 L ∞ (Ω) δ < 1. Our aim is therefore to have an estimate that does not depend on the initial data nor on the time, at least for large times. To do this, we proceed as in [27]; see also, e.g., [11], [12] and [19].
Let R 0 > 0 be given and assume that We then have, owing to Theorem 3.1 and (69)-(70), the existence of t 0 = t 0 (R 0 ) 0 such that where C is independent of R 0 . Furthermore, there holds whereδ =δ(R 0 ). Here, we can assume without loss of generality that C δ . We now choose β ∈ (0, 1) independent of R 0 and t 1 t 0 such that and γ(= γ(R 0 )) = 1 − β t 1 small enough so that We finally set if t t 1 .
(106) Proceeding in a similar way to derive a lower bound, we finally deduce that there exists β ∈ (0, 1) independent of R 0 such that hence a dissipative L ∞ -estimate on u.
The dynamical system (S(t), Φ) is thus dissipative (i.e., it possesses a bounded absorbing set B 0 , that is, ∀B ∈ Φ bounded, ∃t 0 = t 0 (B) such that t t 0 implies S(t)B ⊂ B 0 ; it is understood here that B bounded means that ∃R 0 such that Theorem 4.1. The semigroup S(t), t 0, associated to our system is dissipative on Φ, i.e., it possesses a bounded absorbing set B 0 in Φ.

5.
Existence of the global attractor. Proof. According to the previous section, it is known that the semigroup possesses a bounded absorbing set B 0 in Φ. To prove the existence of the global attractor A, it suffices to prove that the semigroup is asymptotically compact in the sense of the Kuratowski measure of noncompactness. We consider the following decomposition: and w, b, ∂b ∂t is solution of with the initial data in the bounded absorbing set B 0 . We will now write a certain number of a priori estimates. First, repeating the same estimates leading to (69)-(70), but now taking f ≡ 0, we obtain We can see that S 1 (t)(u 0 , H 0 , H 1 ) = v(t), a(t), ∂a ∂t tends to zero as t tends to infinity.
Hence, the operator S 2 (t)(u 0 , H 0 , H 1 ) = w(t), b(t), ∂b ∂t (t) is asymptotically compact in the sense of the Kuratowski measure of noncompactness, which proves the existence of the global attractor A.
6. Discretization of the nonconserved Caginalp phase-field system. In this section, we present the spatial discretization using a finite element method with P 1 continuous piecewise linear functions and a first-order semi-implicit scheme for the time marching scheme.
6.1. Spatial discretization. We let Ω be a convex, planar domain and T h be a regular, quasi-uniform triangulation of Ω with triangles of maximum size h < 1.
where P 1 is the set of all polynomials of degree 1 with real coefficients, we consider the weak formulation of (21)- (22): 6.2. Time marching scheme. We will discretize system (126) in time using a first-order semi-implicit scheme. To this end, let us denote by (u n+1 h , H n+1 h ) and (u n h , H n h ) the approximate values at time t = t n+1 and t = t n respectively and by δt the time step. Then, owing to (126), the unknown fields at time t = t n+1 are defined as the solution of: in which (127) can be written equivalently in the following matrix form (AX = B): where Finally, the simplest method to solve (128)-(129) is to use Picard's iterate as follows:  [24], comparing the nonconserved Caginalp phase-field system (128)-(129) with the cubic nonlinear term f (s) satisfying (19) (respectively, (20)) and the logarithmic one f (s) = −2κ 0 s + κ 1 ln 1 + s 1 − s when (κ 0 , κ 1 ) = (ln(3), 1) (respectively, (κ 0 , κ 1 ) = (ln(6), 1)). We first start by considering the rate of convergence of the first-order in time semi-implicit scheme. We then compute the propagation of a cross function for u and a constant enthalpy H.
We also use here the adaptmesh of FreeFem ++ with uadapt=Hn+un,err=1.e-4 each 100 iteration, where, in that case, we obtain an error of order 10 −5 with the solution without using adaptmesh as shown in [38] and we gain a lot of computational time.
We further note that the solution u starts from [−.01, .01] and goes to one of the following values: [−α, α], α, −α (α ∈] − 1., 1.[); the solution can also explode depending on the value of ε and the initial datum for f H (x, y, 0). This explains why we use periodic boundary conditions and not Dirichlet ones.
More precisely, we display in Tables 7 −→ 9 the convergence of the solution u for different values of f H (x, y, 0) when f (s) satisfies (19), while in Tables 10 −→ 12, we take f (s) satisfying (20). We obtain opposite values of α when f H (x, y, 0) = −δ and f H (x, y, 0) = δ, with δ > 0. We can clearly see that, for (20), the logarithmic potential works much better than the polynomial one.   Table 9. Comparison of the convergence of the solution u with ε = .001.
We thus deduce that, the smaller ε is, the faster the solution converges to a constant (negative or positive) solution 0.5 < |α| < 1. We also note that, when   the solution lies between [−0.5, 0.5], we need more iterations and time in order to possibly obtain the convergence. In Figures 1 −→ 8, we consider the convergence of u and H with different values of ε (ε = 0.1 (left), ε = 0.01 (center) and ε = 0.001 (right)) with f H (x, y, 0) = .1 or f H (x, y, 0) = 1.5 when f (s) satisfies (19). We also note that we did not observe any influence of τ on the simulations.