Solvability and asymptotic analysis of a generalization of the Caginalp phase field system

We study a diffusion model of phase field type, which consists of a system of two partial differential equations involving as variables the thermal displacement, that is basically the time integration of temperature, and the order parameter. Our analysis covers the case of a non-smooth (maximal monotone) graph along with a smooth anti-monotone function in the phase equation. Thus, the system turns out a generalization of the well-known Caginalp phase field model for phase transitions when including a diffusive term for the thermal displacement in the balance equation. Systems of this kind have been extensively studied by Miranville and Quintanilla. We prove existence and uniqueness of a weak solution to the initial-boundary value problem, as well as various regularity results ensuring that the solution is strong and with bounded components. Then we investigate the asymptotic behaviour of the solutions as the coefficient of the diffusive term for the thermal displacement tends to 0 and prove convergence to the Caginalp phase field system as well as error estimates for the difference of the solutions.


Introduction
This paper is concerned with the initial and boundary value problem: w(·, 0) = w 0 , w t (·, 0) = v 0 , u(·, 0) = u 0 in Ω (1.4) where Ω ⊂ R 3 is a bounded domain with smooth boundary Γ, T > 0 represents some finite time, and ∂ n denotes the outward normal derivative on Γ. Moreover, α and β are two positive parameters, γ : R → 2 R is a maximal monotone graph (one can see [2, in particular pp. [43][44][45] or [1]), g : R → R is a Lipschitz-continuous function, f is a given source term in equation (1.1) and w 0 , v 0 , u 0 stand for initial data. The inclusion (in place of the equality) in (1.2) is due to the presence of the possibly multivalued graph γ.
Equations (1.1)-(1.2) yield a system of phase field type. Such systems have been introduced (cf. [3]) in order to include phase dissipation effects in the dynamics of moving interfaces arising in thermally induced phase transitions. In our case, we move from the following expression for the total free energy where the variables θ and u denote the (relative) temperature and order parameter, respectively. Let us notice from the beginning that our w represents the thermal displacement variable, related to θ by w(·, t) = w 0 + (1 * θ)(·, t) = w 0 + t 0 θ(·, s) ds, t ∈ [0, T ].
(1. 6) In (1.5), φ : [0, +∞] → R is the convex and lower semicontinuous function such that φ(0) = 0 = min φ and its subdifferential ∂φ coincides with γ, while G stands for a smooth, in general concave, function such that G ′ = g. A typical example for φ and G is the double obstacle case so that the two wells of the sum φ(u) + G(u) are located in −1 and +1, and one of the two is preferred as minimum of the potential in (1.5) according to whether the temperature θ is negative or positive. Indeed, note the presence of the term −θu besides φ(u) + G(u) in the expression of Ψ.
The example given in (1.7) is inspired by the systematic approach of Michel Frémond to non-smooth thermomechanics: we refer to the monography [7] which also deals with the phase change models. In the case of (1.7) the subdifferential of the indicator function of the interval [−1, +1] reads ξ ∈ ∂I [−1,+1] (u) if and only if ξ Let us point out that, with a different terminology motivated by earlier studies on the Stefan problem [6], some authors (cf. [7]) prefer to name "freezing index" the variable w defined by (1.6), having also in mind applications to frost propagation in porous media.
Another meaningful variable of the Stefan problem is the enthalpy e, which in our case is defined by e = −d θ Ψ (− the variational derivative of Ψ with respect to θ), whence e = θ + u = w t + u. Then, the governing balance and phase equations are given by e t + div q = f (1.8) u t + d u Ψ = 0 (1.9) where q denotes the thermal flux vector and d u Ψ stands for the variational derivative of Ψ with respect to u. Hence, (1.9) reduces exactly to (1.2) along with the Neumann homogeneous boundary condition for u. If we assume the classical Fourier law q = −∇θ (for the moment let us take the heat conductivity coefficient just equal to 1), then (1.8) is nothing but the usual energy balance equation as in the Caginalp model [3]. This is also as in the weak formulation of the Stefan problem, in which the mere pointwise inclusion Another approach, which is by now well established, consists in adopting the so-called Cattaneo-Maxwell law (see, e.g., [4,14] and references therein): such a law reads q + εq t = −∇θ, for ε > 0 small, (1.10) and leads to the following equation which has been investigated in [14]. On the other hand, if we solve (1.10) with respect to q we find q 0 (x, t) is known and can be incorporated in the source term, k(t) is a given kernel (depending on ε of course): from (1.8) we obtain the balance equation for the standard phase field model with memory which has a hyperbolic character and has been extensively studied in [4,5].
In [8,9,10,11] Green and Naghdi presented an alternative approach based on a thermomechanical theory of deformable media. This theory takes advantage of an entropy balance rather than the usual entropy inequality. If we restrict our attention to the heat conduction, these authors proposed three different theories, labeled as type I, type II and type III, respectively. In particular, when type I is linearized, we recover the classical theory based on the Fourier law q = −α∇w t , α > 0 (type I).
( We are interested in the study of existence, uniqueness, regularity of the solution to the initial-boundary value problem (1.1)-(1.4) when γ is an arbitrary maximal monotone graph, possibly multivalued, singular and with bounded domain. Of course, the case of Ψ shaped by a multiwell potential u → −w t u + φ(u) + G(u) is recovered as a sample. Then we study the asymptotic behaviour of the problem as β ց 0, obtaining convergence of solutions to the problem with β = 0, which corresponds to (1.11), the (type I) case of Green and Naghdi. We also prove two error estimates of the difference of solutions in suitable norms, showing a linear rate of convergence in both estimates. In a subsequent study we would like to address the investigation of the analogous limit α ց 0 to obtain the (type II) case in (1.12).
The paper is organized as follows. In Section 2 we state the main results related to the problem (1.1)-(1.4): existence and uniqueness of a weak solution, regularity results yielding a strong solution, further regularity results ensuring the boundedness of u, w t and of the appropriate selection of γ(u). Section 3 contains the related statements. Then we investigate the asymptotic limit as β ց 0: precisely, the convergence result and the error estimates under different assumptions on the data. In Section 4 we introduce some notation and present the uniqueness proof. The approximation of the problem (1.1)-(1.4) via a Faedo-Galerkin scheme and the derivation of the uniform a priori estimates are carried out in Section 5. Regularity and boundedness properties for the solutions are proved in Sections 6-8. Finally, the details of the asymptotic analysis as β ց 0 are developed in Section 9.
2 Well-posedness and regularity for α, β > 0 We point out the assumptions on the data and state clearly the formulation of the problem and the main results we achieve. Let Ω ⊆ R 3 be a bounded smooth domain with boundary Γ = ∂Ω and let T > 0. Set Q := Ω × (0, T ). We assume that The effective domain of γ will be denoted by D(γ). We consider Problem (P α,β ). Find (w, u, ξ) satisfying for all v ∈ H 1 (Ω) and a.a. t ∈ (0, T ) (2.13) (2.14) We can prove the well-posedness of this problem.
Next, in addition to (2.1)-(2.7), we suppose (2.16) in this case, we are able to prove a regularity result, which allows us to solve a strong formulation of Problem (P α,β ). In particular, (w, u, ξ) solves Problem (P α,β ) in a strong sense, that is, w and u satisfy The aim of the subsequent results is to provide L ∞ estimates. We will need to strengthen again the hypotheses on the initial data. For s ∈ D(γ) let us denote by γ 0 (s) the element of γ(s) having minimal modulus. Then, we require that The above results still hold if the dimension N of the domain Ω is arbitary. On the other hand, since (2.22) implies in particular that u is continuous from [0, T ] to the space H s (Ω) for all s < 2, then, if we let N ≤ 3 and s sufficiently large, it turns out that H s (Ω) ⊂ C 0 (Ω) and consequently u ∈ C 0 (Q) .
Finally, we assume for the data enough regularity to get L ∞ estimates for w t and ξ. The hypothesis N ≤ 3 is essential in the proof of the following result.
Remark 2.5. All the statements contained in this paper still hold if Ω ⊆ R 3 is, for instance, a convex polyhedron, for which standard results on Sobolev embeddings and regularity for elliptic problems apply.
3 Asymptotic behaviour as β ց 0 Let us fix the parameter α once and for all. We shall concentrate on the asymptotic behaviour of the solution as β ց 0, so we let β vary in a bounded subset of (0, +∞). We allow the source term and the initial data in Problem (P α,β ) to vary with β, by replacing f , w 0 , v 0 and u 0 in (2.12) and (2.14) with f β , w 0,β , v 0, β and u 0,β respectively. We will denote by (w β , u β , ξ β ) the solution to Problem (P α,β ).
If we set β = 0 in the statement of Problem (P α,β ), we get a first-order system of differential equations, with respect to time, in the variable w t , which is of physical relevance (recall that w t = θ). Anyway, we avoid this change of variable, in order to preserve the formalism. We introduce the formulation of Problem (P α ), in which β is set to be zero.
Problem (P α ). Find (w, u, ξ) satisfying (2.8)-(2.11) as well as We state at first the well-posedness of Problem (P α ) and a convergence result.
With slightly strengthened hypotheses, we are able to prove the strong convergence for the solution and even to give an estimate for the convergence rate.  .5), we assume for some constant c which is independent of β. Then one has the estimate where c does not depend on β.
If γ is a (single-valued) smooth function, and if enough regularity on the data is assumed, it is possible to obtain much stronger estimates. The assumption N ≤ 3 on the spatial dimension is essential for the proof of the following result.
where r > 4/3. Then the estimate holds for a suitable constant c, which may depend on α but not on β.

Notation and uniqueness proof
Before facing the proof of all the results, for the sake of convenience we fix some notation: We embed H in V ′ , by means of the formula Furthermore, the same symbol · H will denote both the norm in L 2 (Ω) and in L 2 (Ω) N ; we behave similarly with · V . If a, b are functions of space and time variables, we introduce the convolution product with respect to time We also point out that the symbols c, c i -even in the same formula -stand for different constants, depending on Ω, T and the data, but not on the parameters α, β. However, as we will be interested in the study of convergence as β ց 0, if a constant c depends on α, β in such a way that c is bounded whenever α, β lie bounded, then we will accept the notation c. A constant depending on the data and on α, but not on β, may be denoted by c α or c α,i or simply c, as it will happen in Section 9.
In our computations, we will often exploit the Hölder and Young inequalities to infer where a, b ∈ L 2 (Q) and σ > 0 is arbitrary. We point out another inequality which will turn out to be useful: if ϕ ∈ H 1 (0, T ; H), then the fundamental theorem of calculus and the Hölder inequality entail for all 0 ≤ t ≤ T . Now, let us concentrate on the uniqueness proof.
Let (w 1 , u 1 , ξ 1 ) and (w 2 , u 2 , ξ 2 ) be solutions to the Problem (P α,β ); we claim that they coincide. Setting w = w 1 − w 2 , u = u 1 − u 2 and ξ = ξ 1 − ξ 2 , we easily get for all v ∈ V and a.a. 0 ≤ t ≤ T , along with the initial conditions We choose v = u(t) in equation (4.3) and integrate over (0, t); thus, we obtain Accounting for the Lipschitz-continuity of g, the Hölder inequality and the monotonicity of γ, frow the above equality we easily derive Integrating in time the equation (4.2) (this is possible thanks to (2.8)) and taking the initial data (4.4) into account, we have we choose v = w t (t) in (4.6) and integrate over (0, t). Noticing that the equality holds, we get (4.8) The Hölder inequality and (4.1) allow us to deal with the right-hand side of this formula: Collecting now (4.5), (4.8) and (4.9), it follows that then, by applying the Gronwall lemma and recalling (4.4), we obtain u = w = 0 almost everywhere in Q. A comparison in (2.13) and the density of H 1 (Q) as a subspace of L 2 (Q) entail ξ = 0 almost everywhere in Q; thus, the proof of uniqueness is complete.

Approximation and a priori estimates
We are going to prove the existence of a solution to Problem (P α,β ) via a Faedo-Galerkin method. First, we approximate the graph γ with its Yosida regularization: for all ε ∈ (0, 1] say, we let where I denotes the identity on R. We recall that φ ε is a nonnegative, convex and differentiable function, γ ε is Lipschitz-continuous, monotone and We look for a solution of the approximating problem in a finite-dimensional subspace V n ⊆ V , chosing a sequence {V n } filling up V ; then we get a priori estimates and use compactness arguments to take the limit as n −→ +∞. In a second step we let ε ց 0.
A special choice of the approximating subspaces will be useful. Let {v i } i∈N be an orthonormal basis for V satisfing where {λ i } i∈N are the eigenvalues of the Laplace operator; also, let V n be the subspace of V spanned by v 1 , . . . , v n , for all n ∈ N. Thus, we have defined an increasing sequence of subspaces, whose union is dense in V , and hence in H; furthermore, we notice that the regularity of Ω implies V n ⊆ W , for all n ∈ N.
As approximations of the data w 0 , v 0 , u 0 we choose the projections on V n : let w 0,n be the projection of w 0 , with respect to V , and let v 0,n , u 0,n be the projections of v 0 , u 0 , with respect to H. We notice that We also need to regularize the source term f : so, we first write then we assume f we also set f n = f (1) n + f (2) n . Now we are ready to state the approximated problem. For the sake of simplicity, we do not specify explicitly the dependency on ε in the solution.
Problem (P α,β ) n, ε . Find T n ∈ (0, T ] and (w n , u n ) satisfying w n (0) = w 0,n , ∂ t w n (0) = v 0,n , u n (0) = u 0,n . (5.8) Writing w n and u n as linear combinations of v 1 , . . . , v n with time-dependent coefficients, and testing equations (5.6) and (5.7) by v = v 1 , . . . , v n , we obtain a system of ordinary differential equations, for whose local existence and uniqueness standard results apply. Thus, Problem (P α,β ) n, ε admits a solution, defined on some interval [0, T n ]. The following estimates imply that these solutions can be extended over the whole interval [0, T ].
First a priori estimate. We choose v = u n (t) in equation (5.7) and integrate over (0, t): The last term in the left-hand side is non negative, because γ ε is increasing and γ ε (0) = 0; it will be ignored in the following estimates. Meanwhile, the right-hand side can be easily estimated using the Lipschitz-continuity of g and (5.3); so we get Following the same computation as in the uniqueness proof, we integrate equation (5.6) with respect to time: for all v ∈ V n and 0 ≤ t ≤ T n . We take v = ∂ t w n (t) in the previous equation and integrate over (0, t). Recalling the identity (4.7), we have where we have set (v 0,n + u 0,n , ∂ t w n (s)) H ds , T 7 (t) = α t 0 (∇w 0,n , ∇∂ t w n (s)) H ds .
We do not need any estimate on terms T 1 and T 3 . With simple applications of the Hölder inequality, we estimate T 2 , T 5 and T 6 : We deal with T 7 by direct integration and the use of the Hölder inequality: Now we pay attention to T 4 and integrate by parts in time: where σ > 0 is arbitrary, to be set later. According to the definition of the norm in V and the inequality (4.1), we have We collect all the terms containing ∂ t w n L 2 (0,t; H) and ∇w n (t) H in the left-hand side of (5.11); their coefficients turn out to be, respectively, We choose σ ≤ min {α/4, 1/4T }, so that k 1 ≥ 1/4, k 2 ≥ α/8. We also remark that the assumptions (5.5) and (5.3) enable us to get a bound for terms involving f (1) n , f (2) n and the initial data. Finally, adding (5.9) and (5.11) and taking into account all the previous inequalities, we obtain The Gronwall lemma entails Second a priori estimate. Since φ ε is at most of quadratic growth, by definition, and γ ε is Lipschitz-continuous, from the estimate (5.12) we directly derive where the symbols c ′ α,i denote positive constants, possibly depending on ε and α, but not on n and β.
By (5.2), we can easily check that (y, z) H = (P n y, z) H for all y ∈ V , z ∈ V n where P n y is the projection of y in V n , with respect to V . Then, as we have a uniform estimate for u n in L 2 (0, T ; V ), it is not difficult to extract from (5.7) the property Third a priori estimate. We take v = ∂ t w n (t) as a test function in equation (5.6) and integrate over (0, t); thanks to the Hölder inequality, we get We consider the term involving f Because of the estimate (5.15) and the properties (5.5) and (5.3), from (5.16) we deduce where c ′ depends on ε, α. Hence, by a generalized version of the Gronwall lemma (see, e.g., [2, pp. 156-157]), we infer that Passage to the limit as n −→ +∞. From the estimates (5.12), (5.13)-(5.15), (5.17), with standard arguments of weak or weak* compactness we can find functions (w ε , u ε ) such that, possibly taking a subsequence as n −→ +∞, thus, since g and γ ε are Lipschitz-continuous, we easily check that where ξ ε = γ ε (u ε ). We then take the limit as n −→ +∞ in (5.6)-(5.8) and see that (w ε , u ε , ξ ε ) fulfills equations (2.11)-(2.14), where γ is replaced by γ ε . Indeed, by (5.22)-(5.23) and (5.3), it is obvious that w ε (0) = w 0 , u ε (0) = u 0 . To deal with the last initial condition properly, we fix a test function v ∈ V m , where m ≥ 1 is arbitrary, and we integrate in time equation (5.6); we get equation (5.10), for 0 ≤ t ≤ T and n ≥ m. Arguing as in [13, pp. 12-13], we can take the limit in (5.10), (5.7) and check that (w ε , u ε , ξ ε ) fulfills for a.a. t ∈ (0, T ), m ≥ 1 and v ∈ V m ; by a density argument, the same equalities hold when v ∈ V . Since the right-hand side in (5.24) is a continuous function in [0, T ], taking t = 0 we find that whence the second of (2.14) follows.
Fifth a priori estimate. As a consequence of the weak lower semi-continuity of the norm in a Banach space, (w ε , u ε , ξ ε ) satisfy the estimate (5.12); we now need to improve estimates (5.13)-(5.15), (5.17).
We first notice that, because of the Lipschitz-continuity of γ ε , ξ ε (t) ∈ V for all t; thus, we can choose v = ξ ε (t) in equation (5.25) and integrate over (0, t), to get In view of (5.1), we have on the other hand, because of the Lipschitz continuity of g, From these estimates and (5.26), we derive We notice that the second term in the lef-hand side is nonnegative, because of the monotonicity of γ ε . Secondly, accounting for (5.12), (5.1) and (2.7), we infer that and consequently we can also establish the estimate (5.17), now for a constant which is independent of ε.
Passage to the limit as ε ց 0. We are able to repeat the compactness argument as above and find (w, u, ξ), a candidate for the solution to Problem (P α,β ), as a limit of a subsequence of (w ε , u ε , ξ ε ). The proof will be easily completed by the passage to the limit as ε ց 0, provided that we deduce (2.11).
By construction, we can assume that

Regularity and strong solutions
This section is devoted to the derivation of further a priori estimates on the approximating solutions (w n , u n , ξ n ), which are independent of n and ε, under stronger assumptions. The same compactness -passage to the limit arguments then apply, and this will prove Theorem 2.2. We first notice that the hypothesis (2.16) and V n ⊆ W make it possible to assume w 0,n −→ w 0 in W , v 0,n −→ v 0 and u 0,n −→ u 0 in V ; (6.1) on the other hand, owing to (2.15), we can require f n ∈ L 1 (0, T ; V ) for all n ∈ N and Sixth a priori estimate. We choose v = ∂ t w n (t) in the equation (5.6) and integrate over (0, t); an application of the Hölder inequality yields for all ε > 0 and n large enough, depending on ε; these requests on parameters are not restrictive, as we first take the limit for n −→ +∞, then for ε ց 0. From (7.2) and (7.3) we deduce that u n W 1,∞ (0,T ;H)∩H 1 (0,T ;V ) ≤ c α,6 . (7.4) Finally, we consider equation (5.7) and we rewrite it in the form for all v ∈ V n and a.a. t ∈ (0, T ), where F n = ∂ t w n − ∂ t u n − g(u n ). Testing with v = −∆u n (t) the previous equation and integrating by parts in space, we obtain Since the estimates (6.5) and (7.4) entail

L ∞ estimates
The aim of this section is to obtain L ∞ estimates on w t and on ξ, under the hypotheses (2.23) and (2.24).
Since we already know that w ∈ L ∞ (Q) (as it is implied, for example, by (6.10)), we have w t ∈ L ∞ (Q) and We notice that, being α fixed and letting β vary in a bounded set, we can find an upper bound for the constant c α, 8 .
In order to prove a L ∞ estimate for ξ, we consider the solution (w ε , u ε ) to the approximating problem, in which the Yosida regularization appears; we then fix p ∈ (1, +∞) and get a bound for γ ε (u ε ) L p (Q) , which is independent of p, ε. From this, we will obtain a uniform bound for and, via a weak* compactness argument, ξ ∈ L ∞ (Q). For the sake of simplicity, we do not plug in the subscript ε in the solution any more.
We know that the equalities
Now, we come back to the equation (8.4); according to the previous estimates, we infer that 9 Well-posedness of (P α ) and convergence as β ց 0 Now we set the notation as in Section 3, since we are interested in the proof of Theorems 3.1-3.4. We assume that the hypotheses (2.1)-(2.7) are satisfied, and we start by studying the convergence as β ց 0, by a compactness argument.