Long time behaviour of a singular phase transition model

A phase-field system, non-local in space and non-smooth in time, with heat flux proportional to the gradient of the inverse temperature, is shown to admit a unique strong thermodynamically consistent solution on the whole time axis. The temperature remains globally bounded both from above and from below, and its space gradient as well as the time derivative of the order parameter asymptotically vanish in L2 -norm as time tends to infinity.


Introduction
We follow here the classical scheme for models of temperature-induced phase transitions in a physical body Ω as e. g. in [2,14], and derive equations for the temperature θ (we will consider the absolute temperature θ > 0 here) and the order parameter χ characterizing the physical state of the material. For example, in a simple meltingsolidification process, χ takes values in the interval [0, 1], where χ = 0 corresponds to the solid, χ = 1 to the liquid, and 0 < χ < 1 is the liquid fraction in a mixture of both phases. The mathematical model we describe below may or may not contain a restriction on the domain of admissible values of χ .
We consider the free energy density F in the form where c V > 0 is the specific heat, σ and λ are smooth functions describing the local dependence on χ of entropy and of latent heat, respectively, ϕ is a general proper, convex, and lower semicontinuous function (in the above example of solid-liquid phase transition, ϕ can be chosen for instance as the indicator function of the interval [0, 1]), β > 0 is a constant parameter, and B is a non-local operator of the form with a given sufficiently regular symmetric kernel k : Ω × Ω → R, and an even smooth function G which is bounded on the domain of ϕ together with its first two derivatives. In comparison with [8,13], we thus remove all restrictions on the convex potential ϕ. An interested reader will find a more detailed discussion on non-local phase-field models e. g. in [3,4,7].
The corresponding densities of internal energy E and entropy S have the form The evolution process is driven by the energy conservation principle and by the order parameter evolution equation The free energy contains a component which is Fréchet differentiable with respect to χ , and another component which is convex, but not necessarily differentiable. The symbol δ χ thus represents alternatively the Fréchet derivative and the subdifferential which may be multivalued. This also explains the inclusion sign in (1.6). Physically, the relation (1.6) expresses the tendency of the system to move towards local minima of the total free energy with speed proportional to 1/µ(θ).
Assuming that time differentiation and space integration can be interchanged in the energy conservation law (1.5), we obtain, using the symmetries in the operator B , that where we have set Formally, by (1.7), there exists a vector function q (the heat flux) such that q · n = 0 on ∂Ω (n is the unit outward normal) and Assuming now the Penrose-Fife law q := κ∇(1/θ), where κ > 0 is a constant parameter characterizing the heat conduction properties of the material, we obtain from (1.6) and (1.9) the following system of balance equations for the unknowns θ and χ : It is complemented with the boundary and initial conditions ∂ ∂n (1.13) where ∂/∂n denotes the outward normal derivative, and θ 0 , χ 0 are given functions.
It is easy to see that the system is thermodynamically consistent. The positivity of temperature and the smoothness that will be established in Theorem 2.2 below imply, by virtue of (1.4) and (1.10)-(1.11), that which is the Second Principle of Thermodynamics in Clausius-Duhem form.
As main results, we will prove that System (1.10)-(1.13) admits, under suitable assumptions on the data, a unique strong solution with positive temperature θ . If moreover the space dimension is at most three and the internal energy is a priori bounded from below, then the temperature remains globally bounded from above and from below by a positive constant. Moreover, as t → ∞, the functions χ t and ∇θ tend to 0 in the norm of L 2 (Ω).
The situation here differs from the problem treated e. g. in [12] in several respects. On the one hand, our heat flux does not contain the regularizing linear term in θ used there. On the other hand, the Ginzburg-Landau contribution |∇ χ | 2 in the free energy, which accounts for non-local interactions, is replaced here by the integral functional B . The regularizing effect in our setting is due to the positive constant β and to a specific growth of the coefficient µ as function of θ in Eq. (1.11).
The paper is organized as follows. The main results are stated in Section 2. Section 3 is devoted to a detailed study of a class of differential inclusions. The existence and uniqueness result is proved in Section 4, global bounds are derived in Section 5, and the asymptotic behaviour of solutions is discussed in Section 6.

Main results
Throughout the paper, the following assumptions on the data are supposed to hold. (ii) σ, λ ∈ W 2,∞ (D(ϕ)); is an absolutely continuous function, µ(0) > 0, and for a. e.
Let us first introduce some notation. For any C > 0 we denote Then there exists C > 0 and a unique solution (θ, χ ) to (1.10)-(1.13) such that Moreover, there exist positive constants c 1 , c 2 , independent of t, such that We are able to prove the global boundedness and stabilization results only in domains Ω of dimension N ≤ 3. We state them in the following form.
and let N ≤ 3. Then there exist constants 0 < θ * < θ * such that for a. e. (x, t) ∈ Ω × (0, ∞) we have It would also be interesting to describe the ω -limit set of the solution trajectory. This question seems to be still open and will be a subject of further research.
The proofs of the above results are postponed to the forthcoming sections. Theorem 2.2 is proved in Section 4, while Sections 5 and 6 are devoted to the proofs of Theorems 2.3 and 2.4, respectively. Before that, we investigate in detail a certain class of differential inclusions related to Eq. (1.11).

Solution operators to differential inclusions
In this section we derive some properties of solution operators to differential inclusions which slightly generalize (1.11). In addition to results established in [7], we prove the inequality (3.4) which is related to the "second-order energy inequality" for the underlying relaxation-hysteresis operator, see [5], and will play a crucial role in the proof of Theorem 2.4.
Consider a function ϕ as in Hypothesis 2.1 (i) and fix a final time T > 0. For a given initial condition χ 0 , and a given function θ ∈ L 1 (Q T ), we solve the following differential inclusion where α : R → R is a given function and f : is a given operator satisfying the following hypothesis.
Hypothesis 3.1. There exist positive constants α 0 , L, C such that Note that in all formulas throughout this section, we keep fixed the value of C from Hypothesis 3.1 (iii).
Proposition 3.2. Let Hypothesis 3.1 hold, and let D C (ϕ) be as in (2.1). Then for every θ ∈ L 1 (Q T ) and for every In addition, there exists a positive constant M such that the solutions Remark 3.3. The L 1 -Lipschitz continuity estimate (3.3) in Proposition 3.2 cannot be extended to L p (Q T ) for p > 1, see [6,Example 3], except in the case when ϕ is a C 1 -function with locally Lipschitz continuous derivative. Strong continuity L 1 (Q T ) → L p (Q T ) of the solution mapping for p < ∞ follows however from the uniform L ∞bound (3.2). Indeed, testing (3.1) by χ t , we obtain the identity e. in Q T , which implies in particular, by virtue of (3.2) and Hypothesis 3.1, that Let now θ (n) , θ be such that θ (n) → θ strongly in L 1 (Q T ) as n → ∞, and let χ (n) , χ be the corresponding solutions to Eq. (3.1). Using Proposition 3.2 and taking into account the L ∞ -bound (3.2), we see that Before proving Proposition (3.2), let us start with a space-independent problem. For a given initial condition χ 0 ∈ D(ϕ) and a given function θ ∈ L 1 (0, T ), we consider the differential inclusion where α : R → R is as in Hypothesis 3.1 and g ∈ L ∞ (0, T ) is such that We prove the following result.  .7), and we have In addition, there exists a positive constant R depending only on C , α 0 , and L, such that the solutions χ 1 , If moreover both θ and g belong to W 1,1 (0, T ), then for every non-negative function Proof of Proposition 3.4. We first prove the existence of solutions. We fix θ ∈ L 1 (0, T ), χ 0 ∈ D C (ϕ) and, for n ∈ N and k = 1, . . . , n, define the sequences corresponding to the partition t 0 = 0, t k = T k/n, where I(u) = u is the identity mapping. Assume that for some k ≥ 1 we have By hypothesis, we have |g k | ≤ C , hence max D C (ϕ) ≤ χ k < χ k−1 by the monotonicity of ∂ϕ. This yields, if k − 1 > 0, that and by induction we obtain max D C (ϕ) ≤ χ k < χ k−1 < · · · < χ 0 which is a contradiction. We obtain a similar contradiction by assuming that g k − (nα k /T )( χ k − χ k−1 ) < −C . Using the fact that α k ≥ α 0 , we thus have for all k = 1, . . . , n that We now define the interpolates , continuously extended to t = T . The functions χ (n) are bounded in W 1,∞ (0, T ) uniformly with respect to n ∈ N. Passing to a subsequence, if necessary, we find χ ∈ W 1,∞ (0, T ) such that χ (0) = χ 0 ,χ (n) →χ in L ∞ (0, T ) weakly-star, and χ (n) → χ uniformly in [0, T ]. Using the inequalities we also see thatχ (n) → χ , χ (n) → χ uniformly. Using the Mean Continuity Theorem for functions in L 1 (0, T ), we conclude that α (n) converge to α(θ(·)) strongly in L 1 (0, T ), and g (n) converge to g strongly in any L p (0, T ) for 1 ≤ p < ∞ and weakly-star in L ∞ (0, T ). Let now z ∈ L ∞ (0, T ) be a test function, z(t) ≥ 0 a. e. in (0, T ), and let w ∈ D(ϕ), ξ ∈ ∂ϕ(w) be arbitrary. By construction, we have Passing to the limit as n ∞ in (3.21) we obtain Since ∂ϕ is maximal monotone, the function χ satisfies Eq. (3.7). Estimate (3.9) follows from (3.16).
To prove inequality (3.11), we fix η with the required properties and h 0 > 0 such that suppη ⊂ (0, T − h 0 ). For 0 < h < h 0 we use the monotonicity of ∂ϕ which yields that for a. e. t ∈ (0, T − h). Testing the above inequality by η(t), dividing by h 2 , and integrating by parts, we obtain By the Mean Continuity Theorem for the Lebesgue integral, the difference quotients converge as h 0+ to the derivatives strongly in L 1 (0, T ), hence pointwise almost everywhere. Thanks to the L ∞ bound forχ, we can pass to the limit in (3.31) as h 0+ and obtain (3.11). The proof of Proposition 3.4 is complete.
We now use this result to prove Proposition 3.2.
Proof of Proposition 3.2. For given θ ∈ L 1 (Q T ) and χ 0 ∈ L ∞ (Ω), χ 0 (x) ∈ D C (ϕ) a. e., we prove the existence of a unique solution to (3.1) by the Banach contraction argument. We define the set a. e. in Ω .
For almost every x ∈ Ω, this inclusion is of the form (3.7) with right-hand side satisfying (3.8). By Proposition 3.4, the function χ belongs to S , and we may define the solution mapping T : S → S :χ → χ . We check that T is a contraction with respect to the norm (3.33). Indeed, we integrate the estimate (3.10) with θ 1 = θ 2 , g i (t) = f [χ i , θ i ](t) for i = 1, 2 from 0 to t. This, thanks to Hypothesis 3.1 (iv), leads to (3.35) Now, multiplying both sides of this inequality by e −2RLt , and integrating over [0, T ], we infer that Hence T is a contraction on S , and the Banach fixed point theorem yields the existence and uniqueness of a solution χ ∈ S of the differential inclusion (3.1). Estimate (3.2) follows directly from (3.9). Finally, in order to prove (3.3), take χ 01 , χ 02 ∈ D C (ϕ), θ 1 , θ 2 ∈ L 1 (Q T ), and let χ 1 , χ 2 be the corresponding solutions of Eq. (3.1). For almost all x we use (3.10) with , t), i = 1, 2. Integrating over Ω and over (0, t) for t ∈ (0, T ], and using Hypothesis 3.1, we obtain that Gronwall's argument then yields and (3.3) follows. Inequality (3.4) is a direct consequence of (3.11).

Existence and uniqueness
This section is devoted to the proof of Theorem 2.2. We rewrite Eq. (1.11) as We see that it is of the form (3.1), and we may use Proposition 3.2 for any T > 0 with some suitable C > 0 independent of T as an immediate consequence of Hypothesis 2.1. This enables us to define a mapping A : (0, ∞)) associates the solution χ of (4.1) satisfying the initial condition χ (x, 0) = χ 0 (x) a. e. in Ω. Problem (1.10)-(1.13) is thus of the form

Uniqueness
Let θ,θ be two positive solutions of (4.2)-(4.5) with the prescribed regularity, and set . We test the difference of equations (4.2) written for θ andθ by the sign of θ −θ (which is equal to the sign of (1/θ) − (1/θ)) and obtain Integrating this relation from 0 to t and using (3.3) and (3.5) leads to the inequality with some constant C 1 independent of t. As θ is locally bounded, we may use Gronwall's argument and conclude that θ =θ a. e.

A cut-off system
y =f ε (u) Figure 1: The cut-off diagram.
We fix some T > 0 and choose a parameter ε ∈ (0, 1) which will be specified later on.

Global bounds
We use here a variant of the Alikakos-Moser iteration scheme, see e. g. [10,14], to derive the bounds for θ(x, t) stated in Theorem 2.3. To simplify the formulas, we denote by | · | p the usual norm in L p (Ω) for 1 ≤ p ≤ ∞. We make repeated use of the well-known interpolation inequality which holds for every v ∈ W 1,2 (Ω) and every δ ∈ (0, 1) with α = 1−N ((1/2)−(1/q)), γ = N (1 − (1/q)), and with a constant K > 0 independent of v and δ . In fact, we deal only with values of q between 2 and 4, so that within this range, the constant K in (5.1) can be taken independent of q . On the other hand, the hypothesis N ≤ 3 is motivated by the fact that inequality (5.1) has to hold for all q ∈ [2,4].
The following elementary estimate will also be of interest.
Then for all t ≥ 0 we have Indeed, V is absolutely continuous, and ifV (t) = 0 for some t, then a w (t)−h(t) > 0, for all t and (5.3) follows.
Using (3.4) we find a constant C 28 such that for every T > 0 and every non-negative function η ∈ W 1,∞ (0, T ) with compact support in (0, T ) we have Put C * = C 27 + C 28 . We now claim that (6.5) the function q(t) := C * t − E(t) is non-decreasing.
Indeed, for every T > 0 and every non-negative function η ∈ W 1,∞ (0, T ) with compact support in (0, T ) we have and (6.5) follows. This implies, in particular, that E has bounded variation on every bounded time interval, and E(t+) ≤ E(t−) for every t > 0.