Singular limit of an integrodifferential system related to the entropy balance

A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.

Equation (1.1) may be interpreted as an entropy balance equation. Note in particular that that the equation is singular with respect to the temperature, mainly for the presence of the logarithm, forcing the temperature to assume only positive values (which is in accordance with physical consistency). Similar systems have been studied in the literature from the point of view of the existence and regularity of solutions (see, among the others, [3,4,5,6,7,12]).
The well-posedness of a proper variational formulation of (1.1)-(1.4) has been proved in [5]. Here, our main goal is the following. By assuming that k τ converges to κ ′ 0 δ at τ ց 0 in a suitable sense, where δ is the Dirac mass at the origine of the real line and κ ′ 0 is a real constant satisfying κ := κ 0 + κ ′ 0 > 0, we prove that the solution (ϑ τ , χ τ ) to (1.1)-(1.4) converges in a proper topology to the solution (ϑ, χ ) of the problem stated below (1.6) ϑ| Γ = ϑ Γ and ∂ ν χ | Γ = 0 (1.7) ln ϑ(0) = ln ϑ 0 and χ (0) = χ 0 . (1.8) This convergence result is obtained by the use of an a priori estimates technique and passage to the limit arguments, based on monotonicity and compactness. Moreover, an error estimate, i.e. an estimate of suitable norms or quantities involving the difference of solutions, is shown.
Our paper is organized as follows. In the next section, we discuss a derivation of the system (1.5)-(1.6) from the basic laws of thermomechanics. Section 3 is devoted to the statement of our assumptions and of our results on the mathematical problem. In Section 4, we present some auxiliary material that is needed for the proof of our convergence Theorem 3.3, mainly. The last section is devoted to the proofs of the above theorem and of the error estimate stated in Theorem 3.4.

The model
In this section, we briefly introduce the modeling derivation of the equations (1.1)- (1.2) and discuss the convergence to (1.5)-(1.6), as the parameter τ (in the memory kernel) tends to 0. Here, the argument is mainly developed from a physical point of view, while we refer to subsequent sections for a more precise setting of analytical assumptions and comments. In particular, we aim to focus on the fact that (1.1) accounts for thermal evolution involving memory effects, on the basis of the memory kernel k τ .
Materials with thermal memory have been deeply studied in the literature, both from a modeling and analytical point of view. We refer, in particular, to the approach by Gurtin and Pipkin (see [15]) for thermal memory materials. Several authors have investigated phase transitions in special materials with thermal memory, both concerning modeling and analysis. For a fairly complete and detailed presentation of this kind of problems, let us mention the very recent monograph [1]. Now, we combine thermal memory with a new theory for phase transitions models, based on a generalization of the principle of virtual powers (see [11]). The idea is that micro-forces, which are responsible for the phase transition, have to be included in the whole energy balance of the system. Consequently, the phase (evolution) equation is derived as a micro-forces balance equation and it is coupled with an entropy evolution equation. This approach has been recently investigated in the literature by several authors (among the others, we mainly refer to the papers [3] and [7], in which the derivation of the model is detailed in the case when possible thermal memory effects are included, as in equations (1.1)-(1.2)).
Indeed, let us recall that in [3] the theory by Gurtin-Pipkin is considered, allowing the free energy functional to depend on the past history of the temperature gradient. The resulting functional accounts for non-dissipative contributions in the heat flux, which may be combined with additional dissipative instantaneous contributions coming from a pseudo-potential of dissipation. The use of an entropy balance has been recovered, in this approach, from a rescaling (with respect to the absolute temperature) of the energy balance, under the small perturbations assumption (see also [4,5]). In [7] a fairly general theory is introduced. The model is derived by a dual approach (mainly in the sense of convex analysis) in which the entropy and the history of the entropy flux are chosen as state variables (together with the phase parameter and possibly its gradient). Then, the dissipative functional is written in terms of a dissipative contribution in the entropy flux and for the time derivative of the phase parameter.
Let us point out that the above mentioned approach is not far from the theory proposed by Green-Naghdi [13] and Podio-Guidugli [16], in which some thermal displacement is introduced as state variable (it is a primitive of the temperature) and the equations come from a generalization of the principle of virtual powers, in which thermal forces are included. As a consequence, in this framework, the entropy equation is formally obtained as a momentum balance (i.e., a balance of thermal forces acting in the system). The reader may also examine [9,10], where some asymptotic analyses are carried out to find the interconnections among different Green and Naghdi types.
We aim to observe that the model we are investigating actually may be obtained by combining the above two theories, i.e. generalizing the principle of virtual powers accounting for microforces as well as thermal stresses. Let us present our position. First, we specify the expression of the power of internal forces. The power of interior forces is written for any virtual micro-velocity γ and thermal velocity v, as follows where B and H are interior forces responsible for the phase transition (as introduced in [11]), and Q stands for a thermal stress (corresponding to the entropy flux by [16]). Hence, the resulting balance equations are written as momentum balance equations. It is assumed that an external (density of) entropy source f is applied. A thermal momentum is introduced to measure reluctance to the order of the system (in analogy with the mechanical momentum measuring reluctance to quiet). We prescribe that it is given by the entropy s. It results that (see (2.1)) As far as the microscopic momentum balance, we assume that no acceleration and no external forces are contributing, so that we have Henceforth, (2.2)-(2.3) are combined with suitable boundary conditions. As usual, we assume that the flux through the boundary H · n is null, while (mainly for analytical reasons) we prescribe a known temperature on the boundary.
The entropy s, the entropy flux Q, and the new interior forces B and H are recovered by suitable energy and dissipation functionals, that we are going to make precise, in terms of state variables. The state variables are related to the equilibrium of the thermodynamical system: they are the absolute temperature ϑ, the phase parameter χ , the gradient ∇ χ (actually accounting for local interactions), and the history variable ∇ϑ t , which is defined as As in [3], we assume that the free energy of the system (depending on (ϑ, ∇ϑ t , χ , ∇ χ )) is split into two contributions: the first is related to present variables at time t (Ψ P ), the second accounts for some history in the system (Ψ H ), measured through a memory kernel (related to k τ in the equations). In particular, the history contribution of the free energy is given by where S τ is the space of the past histories (as it is introduced in the theory of thermal memory materials by Gurtin and Pipkin), defined by  Here, h τ : (0, +∞) → (0, +∞) (possibly depending on a parameter τ ) is a continuous, decreasing function such that The space S τ is endowed with the natural norm and the related scalar product is (v, u) Sτ = +∞ 0 h τ (s)v(s) · u(s)ds. Let us comment that in our system, to derive (1.1), we have introduced a kernel k τ such that −k ′ τ = h τ . More precisely, let k τ : (0, +∞) → R and require that Hence, by virtue of the assumptions on h τ we also have k ′ τ ≤ 0 and k ′′ τ ≥ 0 a.e. in (0, +∞). (2.10) Note that k ′ τ (t) vanishes for t going to +∞ and that k τ is a non-increasing function with k τ (0) ≥ 0, and in the case k τ (0) = 0 one has k τ ≡ 0. These assumptions on k τ actually ensure that the model is thermodynamically consistent, as it is detailed in [3].
Then, the free energy functional Ψ P (written at the present time t) is addressed where c V > 0 (in the sequel let us take c V = 1) is the specific heat, σ and λ are sufficiently smooth functions (with λ ′ ( χ ) denoting the latent heat), β is a proper convex and lower semicontinuous function, possibly accounting for internal constraints on the phase variable χ . For instance, a fairly classical choice is β( χ ) = I [0,1] ( χ ), which is equal to 0 if χ ∈ [0, 1] and takes value +∞ elsewhere (thus forcing χ ∈ [0, 1]).
Dissipation is rendered in terms of the time derivative χ t and of the dissipative variable ∇ϑ. It is derived by a pseudo-potential of dissipation (in the sense by Moreau, i.e. a convex, non-negative function assuming its minimum 0 for null dissipation): Note that, in order to ensure the validity of the second principle of thermodynamics, it is required that κ 0 ≥ 0. Now, we are in the position of recovering our system, after specifying constitutive relations for the involved physical quantities. We have that and β being the subdifferential (in the sense of convex analysis) of β, and Hence, the entropy flux vector Q is specified by where −Q nd results to be defined in S τ . It is obtained taking the derivative in S τ of the history functional with respect to the history variable. Integrating by parts in time, using the Fréchet derivative, and exploiting the hypotheses on k τ (see [3] for any further detail) lead to We have now to make precise the dissipative part of the entropy flux Hence, equations (1.1)-(1.2) are obtained by (2.2) and (2.3) exploiting the above introduced constitutive relations. We point out that in (1.1), the past history contribution of (2.17) (actually its divergence), i.e. 0 −∞ k τ (t − s)∇ϑ(s)ds, is assumed to be known and included in the external entropy source f (we have used the same notation of (2.2) for the sake of simplicity).
As we have already pointed out in the Introduction, the main aim of this paper is to investigate the asymptotic behavior of system (1.1)-(1.2) as the thermal memory kernel converges to κ ′ 0 δ, δ being the Dirac mass at the origin of the real line and More precisely, we are interested in proving that solutions to the system (1.1)-(1.2) converge to solutions to (1.5)-(1.6) (at least in some weak topology). Let us briefly comment that the system (1.5)-(1.6), obtained in our proof as a suitable limit of (1.1)-(1.2), can be actually derived by an analogous procedure as the one we have performed to formally derive (1.1)-(1.2). Indeed, (1.5)-(1.6) follow from (2.2) and (2.3) when exploiting (2.13)-(2.16). Here, the new energy and dissipative functionals are Ψ = Ψ P (i.e., no history contribution of type (2.5) in the free energy is given) and (cf.

Statement of the mathematical problem
In this section, we make our assumptions precise and state our results. First of all, we assume Ω to be a bounded connected open set in R 3 (lower-dimensional cases could be considered with minor changes) whose boundary Γ is supposed to be smooth. Next, we fix a final time T ∈ (0, +∞) and set: ∂ ν denoting the normal derivative operator on the boundary. We endow the spaces (3.2)-(3.3) with their standard norms, for which we use a self-explanatory notation like · V . Moreover, for p ∈ [1, +∞], we write · p for the usual norm in L p (Ω); as no confusion can arise, the symbol · p is used for the norm in L p (Q) as well. In the sequel, the same symbols are used for powers of the above spaces and the corresponding natural induced norms. It is understood that H ⊂ V * 0 as usual, i.e., any element u ∈ H is identified with the functional V 0 ∋ v → Ω uv which actually belongs to the dual space coincides with the dual space of L 2 (0, T ; V 0 ) and use the symbol · , · for the corresponding duality pairing.
As far as the structure of the system is concerned (see (1.1), (1.5) and (1.2), (1.6)), we are given the three functions β , λ, σ, the constant κ 0 and the memory kernel k τ depending on the parameter τ > 0 and we assume that the conditions listed below are satisfied.
and note that β is maximal monotone and that β(0) ∋ 0. In the sequel, we write D( β ) and D(β) for the effective domains of β and β, respectively, and we use the same symbol β for the maximal monotone operators induced on L 2 spaces.
As far as the data of our problem are concerned, we assume that the functions f , ϑ Γ , ϑ 0 , χ 0 and the constants ϑ * and ϑ * are given such that The function ϑ Γ is the boundary datum for the temperature and we would like to consider a function u := ϑ−ϑ H vanishing on the boundary as associated unknown function. Hence, a natural choice of ϑ H is the harmonic extension of ϑ Γ , so that ∆u = ∆ϑ. Therefore, we define ϑ H : Q → R as follows For the regularity of ϑ H (induced by (3.9)) see the subsequent Proposition 4.1.
Next, we list the a priori regularity conditions we require for any solution (ϑ, χ , ξ) of either (1.1)-(1.4) or (1.5)-(1.8). We ask that At this point, we are ready to state the problems we are dealing with in a precise form. For fixed τ > 0, we look for a triplet (ϑ τ , χ τ , ξ τ ) satisfying (3.15)-(3.18) and the following system We note that the boundary conditions (1.3) are contained in (3.15) and (3.17) (see the definitions (3.14) and (3.2)-(3.3)). We also remark that (3.16) implies that ln ϑ is a continuous V * 0 -valued function (while no continuity of ϑ is known), so that the Cauchy condition for ln ϑ τ contained in (3.21) makes sense. Similar remarks hold for the limit problem we are going to state (i.e., (1.5)-(1.8) in a precise form).
The following well-posedness result deals with a fixed τ > 0 and essentially follows from [5]. Just the notation is different, indeed. Our aim is to study the limit of the solution (ϑ τ , χ τ , ξ τ ) as τ tends to zero, under suitable assumptions on the behavior of the memory kernel k τ . Namely, we assume that for some real constant κ ′ 0 and set In (3.22) and later on, we use the same symbol for any real constant (like 1 and κ ′ 0 ) and for the corresponding constant function. We advice the reader that κ > 0 in the sequel, either by assumption or as a consequence of some condition we require, so that the limit problem we are going to state is parabolic with respect to ϑ. Such a problem consists in looking for a triplet (ϑ, χ , ξ) satisfying (3.15)-(3.18) and the following system By just taking k τ = 0 and replacing κ 0 by κ in Theorem 3.1, we obtain However, we can prove a convergence result under further assumptions, namely for some constants κ * , κ * > 0 and every v ∈ L 2 (0, T ) and τ > 0. By taking v = 0 on (T ′ , T ), we clearly see that the time T can be replaced by any T ′ ∈ (0, T ) in the first inequality of (3.27). Moreover, we observe that (3.22) and (3.27) imply that κ ≥ κ * , so that κ > 0 as a consequence. Here is our first result. The topology mentioned in Theorem 3.3 will be clear from the proof we give in Section 3.3 and is rather strong. Provided that a much weaker topology is considered, an error estimate can be proved. We have indeed for τ small enough, where M depends on the structure and the data, only.
Remark 3.6. Assumption (3.22) is a well-defined reinforcement of the condition roughly mentioned in the Introduction as k τ → κ ′ 0 δ, where δ is the Dirac mass at the origin. Indeed, if we introduce the Heaviside function H on (−∞, T ), i.e., H(t) = 0 for t < 0 and H(t) = 1 for t ∈ (0, T ), and the trivial extensionk τ of k τ , (3.22) reeds with an obvious new meaning of the convolution. By differentiating and observing that (H * k τ ) ′ = δ * k τ =k τ , we deduce that where δ is the actually well-defined Dirac mass at 0 in the open set (−∞, T ).
Remark 3.7. By checking the proofs in the next sections, the reader will be able to realize that our results can be suitably extended to the case of coefficients κ 0τ possibly depending on τ , with boundedness and convergence properties as τ ց 0.
We recall that Ω is bounded and smooth. So, throughout the paper, we owe to some well-known embeddings of Sobolev type, namely V ⊂ L p (Ω) for p ∈ [1,6], together with the related Sobolev inequality v p ≤ C v V for every v ∈ V and 1 ≤ p ≤ 6 (3.30) and W 1,p (Ω) ⊂ C 0 (Ω) for p > 3, together with v ∞ ≤ C p v W 1,p (Ω) for every v ∈ W 1,p (Ω) and p > 3. (3.31) In (3.30), C depends only on Ω, while C p in (3.31) depends also on p. In particular, the continuous embedding W ⊂ W 1,6 (Ω) ⊂ C 0 (Ω) holds. Some of the previous embeddings are in fact compact. This is the case for V ⊂ L 4 (Ω) and W ⊂ C 0 (Ω). We note that also the embeddings W ⊂ V , V ⊂ H, V 0 ⊂ H, and H ⊂ V * 0 are compact. Moreover, we often account for the well-known Poincaré inequalities where C depends only on Ω. Furthermore, we repeatedly make use of the notation and of well-known inequalities, namely, the Hölder inequality and the elementary Young inequality: ab ≤ δa 2 + 1 4δ b 2 for every a, b ≥ 0 and δ > 0. (3.35) As far as properties of the convolution are concerned, we take advantage of the elementary formulas (which hold whenever they make sense) and of the well-known Young theorem where X is a Banach space, 1 ≤ p, q, r ≤ ∞, and 1/r = (1/p) + (1/q) − 1 (cf., e.g., [14]). Finally, again throughout the paper, we use a small-case italic c for different constants, that may only depend on Ω, the final time T , the shape of the nonlinearities λ, β, σ, and the properties of the data involved in the statements at hand; a notation like c δ signals a constant that depends also on the parameter δ. The reader should keep in mind that the meaning of c and c δ might change from line to line and even in the same chain of inequalities, whereas those constants we need to refer to are always denoted by capital letters, just like C in (3.30).

Auxiliary material
This section contains a very short summary on the properties of the harmonic extension ϑ H of the boundary datum ϑ Γ (see (3.14)) and a preliminary result dealing with a generalized version of the limit problem (3.24)-(3.26). The properties listed in the following proposition will be extensively used in the sequel.
More precisely, owing to the theory of harmonic functions, in particular to the maximum principle, we have that where C is a constant depending on Ω, only.
Now, in order to help the reader, we sketch the outline of the proof of Theorem 3.3 we are going to develope in the next section. By accounting for a number of a priori estimates and using well-known compactness results, we derive that the family of solutions (ϑ τ , χ τ , ξ τ ) converges (for a subsequence) to a generalized solution to problem (3.24)-(3.26), in which ln ϑ is understood in a non standard sense. Next, in order to conclude that such a solution actually is the solution given by Corollary 3.2, we prove a preliminary well-posedness result for generalized solutions (Theorem 4.2). Therefore, we first have to introduce the ingredients that are needed to explain such a notion of solution.
Once uniqueness of the generalized solution is proved, we can easily conclude. Indeed, our assumptions allow us to apply Corollary 3.2. Hence, a solution exists in the strong sense, i.e., satisfying the regularity requirements (3.15)-(3.18). On the other hand, such a solution is also a generalized solution due to (4.4). Finally, it satisfies (4.11).

Proofs of Theorems 3.3 and 3.4
The argument we follow for our first proof uses compactness and monotonicity methods. So, we start estimating. However, we often proceed formally for the sake of simplicity. The correct procedure could be based on performing similar estimates on the solution of some approximating problem. One approximation is constructed in [5] and depends on the parameter ε: the solution is smoother than the solution to the problem we are dealing with (actually the limit as ε ց 0 keeps such estimates). Furthermore, in order to simplify the notation, we often avoid the subscript τ (on the solutions) during the calculation and restore it just at the end of each estimate.
First a priori estimate. We would like testing (3.19) by in the duality V * 0 -V 0 and integrate over (0, t), where t ∈ (0, T ) is arbitrary. However, we proceed formally, as just said. In particular, we behave as the logarithmic term were smoother. At the same time we multiply (3.20) by ∂ t χ τ and integrate over Q t (see (3.34)). Finally, we sum up and remark that the terms containing ∂ t χ τ partially cancel. Hence, by avoiding some subscripts in the notation for a while and adding the same integral Ω | χ (t)| 2 to both sides for convenience, we obtain Note that we have ϑ > 0 and can make use of the chain rule for subdifferentials. Now, we recall that β is nonnegative (cf. (3.4)) and treat each of the non-trivial terms, separately.
We integrate the second integral on the left-hand side by parts with respect to time and get Hence, by recalling Proposition 4.1 and observing that r − ϑ * ln + r ≥ (r/2) − c for every r > 0, we deduce that On the other hand, we notice that | ln r| ≤ c((r/2) + ϑ * ln − r) for r > 0, so that Qt | ln ϑ| |∂ t ϑ H | ≤ c Qt (ϑ/2) + ϑ * ln − ϑ |∂ t ϑ H | and the last integral can be treated on the right-hand side via the Gronwall lemma since ∂ t ϑ H ∈ L 1 (0, T ; L ∞ (Ω)). Next, thanks to (3.27) and to (3.32), we infer that where C is the constant of (3.32). Now, let us deal with the right-hand side. With the help of (3.27), the Young theorem (3.37), and (3.8), we immediately have Moreover, we observe that (3.5) and 0 ≤ ϑ H ≤ ϑ * by Proposition 4.1. Finally, we observe that (3.5) also yields |σ(r)| ≤ c(1 + r 2 ) for every r and deduce that By combining all the estimates we have derived with (5.2), applying the Gronwall lemma, and owing to the Poincaré inequality (3.33), we conclude that besides an estimate for β ( χ τ ) in L ∞ (0, T ; L 1 (Ω)).
Second a priori estimate. We write (3.20) in the form of a nonlinear monotone elliptic equation, namely and notice that each term on the right-hand side of (5.4) is bounded in L 2 (0, T ; H) by (3.5) and (5.3). Concerning the first term, notice that χ τ and ϑ τ are bounded in L ∞ (0, T ; L 4 (Ω)) and L 2 (0, T ; L 4 (Ω)), respectively, due to the Sobolev inequality (3.30). Then, a quite standard argument (formally test (5.4) by either −∆ χ τ or ξ τ in order to estimate both of them and then use the regularity theory for elliptic equations) yields Third a priori estimate. We want to estimate ∂ t ϑ τ in L 1 (0, T ; W −1,q (Ω)) for some q > 1 satisfying L 2 (Ω) ⊂ W −1,q (Ω), and the choice q = 4/3 will work. Therefore, by proceeding formally, we take any v ∈ W 1,4 0 (Ω) satisfying v W 1,4 (Ω) ≤ 1 and test (3.19) written at almost any time t by ϑ τ (t) v, which is a good test function belonging to V 0 , due to (3.30)-(3.31) and (5.3). We obtain (the first integral is intended as a duality pairing) We simplify the notation by dropping the time t (and the subscript τ , as usual) for a while and estimate each term on the right-hand side of (5.6), separately. We account for the Lipschitz continuity of λ ′ (see (3.5)) and for the Hölder and Sobolev inequality (3.30). Moreover, we observe that v ∞ ≤ c thanks to (3.31). We have We term C the maximum of the values of the above constant c's, for clarity. Then, we first collect the estimates just obtained and (5.6). Finally, we take the supremum with respect to v ∈ W 1,4 0 (Ω) under the constraint v W 1,4 (Ω) ≤ 1. We conclude that Therefore, the estimate ∂ t ϑ τ L 1 (0,T ;W −1,4/3 (Ω)) ≤ c (5.7) follows once we prove that φ τ is bounded in L 1 (0, T ). In view of the previous estimates (5.3), (5.5), and of the Sobolev inequality, we see that the only trouble could come from the term containing the convolution. By owing to the Young theorem (see (3.37)) and to the Sobolev inequality once more, we have Now, we recall (3.27) 2 and (5.3) and conclude that φ τ is bounded in L 1 (0, T ). Therefore, (5.7) is established.
Proof of Theorem 3.4. By arguing as in the last part of the previous proof, we consider the integrated version of the equations for temperature and test the difference by ϑ τ − ϑ. However, in the present situation we already know that ζ = ln ϑ and can integrate over Q t rather than Q. Thus, a quite similar calculation yields Qt (ln ϑ τ − ln ϑ)(ϑ τ − ϑ) + κ Qt ∇ 1 * (ϑ τ − ϑ) · ∇(ϑ τ − ϑ) At the same time, we multiply the difference between (3.20) and (3.25) by χ τ − χ and integrate over Q t . We obtain At this point, we sum (5.25) to (5.24), and it is clear that all the terms on the left-hand side are nonnegative. Thus, we estimate each term on the right-hand side. As far as the term containing the convolution kernel is concerned, we can repeat the argument that led to (5.23). Thus, we have The sum of all the terms involving λ and λ ′ can be first transformed and then estimated as follows (we use the Taylor formula and (3.5), besides standard inequalities, as usual) As the integral involving σ ′ can be treated in a trivial way due to (3.5), we can apply the Gronwall lemma (see, e.g., [8, p. 156]) and infer that the left-hand side of (3.28) is bounded by As the last integral is bounded by a constant thanks to (5.3), inequality (3.28) follows.