A conserved Caginalp phase-field system with two temperatures and a nonlinear coupling term based on heat conduction

1.   Introduction

The Caginalp phase-field system

has been proposed in ^[1] to model phase transition phenomena, e.g. melting-solidification phenomena, in certain classes of materials. In this context, $u$ is the order parameter and $\theta$ the relative temperature (relative to the equilibrium melting temperature), $f$ is a given function (precisely, the derivative of a doublewell potential $F$). This system has been studied, e.g., ^{[2,6,7,8,9,10,11,13,15]} and ^[17].

Equations (1.1) and (1.2) are based on the total free energy

where $\Omega$ is the domain occupied by the material (we assume that it is a bounded and smooth domain of $\mathbb{R}^{n}$, $n = 2$ or 3 with boundary $\Gamma$).

We then introduce the enthalpy $H$ defined by

where ${\partial}$ denotes a variational derivative, so that

The gouverning equations for $u$ and $\theta$ are finally given by

where $\partial u$ stands for the variational derivative with respect to $u$, which yields (1.1). Then, we have the energy equation

where $q$ is the thermal flux vector. Assuming the classical Fourier law

we obtain $(1.1)$ and $(1.2)$.

Now, one drawback of the Fourier law is that it predicts that thermal signals propagate with an infinite speed, which violates causality (the so-called "paradox of heat conduction", see, e.g. $[5]$). Therefore, several modifications of $(1.8)$ have been proposed in the literature to correct this unrealistic feature, leading to a second order in time equation for the temperature.

A different approach to heat conduction was proposed in the Sixties (see, ^[14,16]), where it was observed that two temperatures are involved in the definition of the entropy: the conductive temperature $\theta$, influencing the heat conduction contribution, and the thermodynamic temperature, appearing in the heat supply part. For time-independent models, it appears that these two temperatures coincide in absence of heat supply. Actually, they are generally different in time for example, ^[8] and references therein for more discussion on the subject. In particular, this happens for non-simple materials. In that case, the two temperatures are related as follows (see ^[4,5]).

Our aim in this paper is to study a generalization of the Caginalp phase-field system based on these two temperatures theory and the usual Fourier law with a nonlinear coupling. In particular, we obtain the existence and the uniqueness of the solutions and we prove the existence of the exponential attractors and, thus, of finite-dimensional global attractors.

2.   Setting of the problem

We consider the following initial and boundary value problem:

where $\Gamma$ is the boundary of the spatial domain $\Omega$.

We make the following assumptions on nonlinearities $f$ and $g$:

where $k$ is an integer, $G(s) = \int^{s}_{0}g(\tau)d\tau\; , {\rm{and}}\; , F(s) = \int^{s}_{0}f(\tau)d(\tau)$.

We denote by $\|.\|$ the usual $L^{2}$-norm (with associated scalar product ((., .))) and set $\|.\|_{-1} = \|(-\Delta)^{-\frac{1}{2}}.\|$, where $-\Delta$ denotes the minus Laplace operator with Dirichlet boundary conditions. More generally, $\|.\|_X$ denotes the norm in the Banach space X. Throughout this paper, the same letters $c$, $c'$ and $c''$ denotes (generally positive) constants which may change from line to line, or even in a same line. Similarly, the same letter $Q$ denotes monotone increasing (with respect to each argument) functions which may change from line to line, or even in a same line.

Remark 2.1. In our case, to obtain equations (2.1) and (2.2), the total free energy reads in terms of the conductive temperature $\theta$

where $f = F'$ and $g = G'$, and (1.6) yields, in view of (1.9), the evolution equation for the order parameter (2.1). Furthermore, the enthalpy now reads

which yields thanks to (1.7), the energy equation,

Considering the usual Fourier law ($q = -\nabla \theta$), we have (2.2).

We can note that we still have an infinite speed of propagation here.

3.   A priori estimates

The estimates derived in this section are formal, but they can easily be justified within a Galerkin scheme. In what follows, the Poincaré, Hölder and Young inequalities are extensively used, Without further referring to them.

We rewrite (2.1) in the equivalent form:

We multiply (3.1) by $\partial_{t}u$ and integrate over $\Omega$, we have

which gives

We multiply (2.2) by $(\alpha-\Delta\alpha)$ and integrate over $\Omega$, we have

which gives,

(note that $\Vert \alpha-\Delta\alpha\Vert^{2} = \Vert\alpha\Vert^{2}+2\Vert\nabla\alpha\Vert^{2}+\Vert\Delta\alpha\Vert^{2}$).

Summing (3.2) and (3.3), we find

which yields,

where

Owing to (2.8), we obtain

We multiply (2.1) by $u$ and integrate over $\Omega$, we have

Summing (3.5) and $\delta$(3.8), where $\delta > 0$ is small enough, we have

where

satifies

In particular, we deduce from (3.9) and Gronwall's lemma the dissipative estimate

We multiply (2.1) by $\partial_{t}u$ and integrate over $\Omega$, we obtain

We multiply (2.2) by $-\Delta(\alpha-\Delta\alpha)$ and integrate over $\Omega$, we obtain

(note that $\Vert \nabla(\alpha-\Delta\alpha)\Vert^{2} = \Vert\nabla\alpha \Vert^{2}+2\Vert\Delta\alpha\Vert^{2}+\Vert\nabla\Delta\alpha\Vert^{2}$)

Summing (3.11) and (3.12), we find

This, let find estimates of (3.13) right terms, using H$\ddot{o}$lder inequality, owing to $\alpha \in H^{2}(\Omega)$ with continuous injection the $H^{2}(\Omega)\subset L^{\infty}(\Omega)$, we have

Furthermore,

and,

Inserting (3.14), (3.15) and (3.16) into (3.13), we find

We recall that $H^{2}(\Omega)\subset C(\overline{\Omega})$ and owing to (2.5), we obtain

and inserting (3.18) into (3.17), we find

In particular, we deduce

We set

we deduce from (3.20) an inequation of the form

Let z be the solution to the ordinary differential equation

It follows from the comparison principle, that there exists a time

$T_{0} = T_{0}(\Vert u_{0}\Vert_{H^{2}(\Omega)}, \Vert\alpha_{0}\Vert_{H^{3}(\Omega)}) > 0$ belonging to, say $(0, \frac{1}{2})$ such that

hence

We now differentiate (3.1) with respect to time, and have

Owing to (2.2), we have

We multiply (3.27) by $t\partial_{t}u$ and integrate over $\Omega$, we find for $t \leq T_{0}$

Owing to (2.9), (3.18) and (3.25), we obtain $t \leq T_{0}$

Using to the interpolation inequality note that, $\Vert\partial_{t}u\Vert^{2}\leq c\Vert\partial_{t}u\Vert_{-1}\Vert\nabla\partial_{t}u\Vert$.

Owing to (3.25) and that $-\Delta\alpha \in L^{2}(\Omega) \subset H^{-1}(\Omega)$, we have

Using the estimates (2.9) and owing to (3.25), we find

Owing to (3.25), we have

In inserting (3.29), (3.30), (3.31), (3.32) into (3.28) and owing to (3.25), we find

In particular, owing to (3.5), (3.10), (3.25) and (3.33), Gronwall's lemma and, we find

We multiply (3.27) by $\partial_{t}u$ and integrate over $\Omega$, we have

Owing to (3.10), (3.25) and Granwall's lemma, then the estimates (3.35) becomes

hence, owing to (3.34), we have

We now rewrite (3.1), for $t\geq T_{0}$ fixed, in the form

where

We multiply (3.39) by $h_{u}(t)$ and integrate over $\Omega$, we have

Owing to (3.34)-(3.37), we obtain

We multiply (3.38) by $u$, owing to (2.8) and integrate over $\Omega$, we find

We multiply (3.38) by $-\Delta u$, owing to (3.9)-(3.25) and integrate over $\Omega$, we have

we deduce the (3.41)-(3.43), we obtain

Owing to (3.25), we find

Then the estimate (3.3) becomes

Owing to (3.25) and (3.36), we have

Owing to (3.35)-(3.37) and we integrate over $T_{0}$ to $t$, we deduce that

which implies

Combining (3.44) and (3.49), we have

Finally, we deduce (3.35) and (3.50) that

Integrating (3.51) between $0$ to $1$, we obtain

We multiply (2.1) by $u$ and integrate over $\Omega$, we have

Owing to (3.18) and (3.25), we find

We deduce the (3.54), we have

The estimates (3.52) and (3.55) conclude that there exists $T \in (0, 1)$ such that

which implies

Owing to (3.10), (3.51) and (3.57), we have the estimate dissipative following

We multiply (2.2) by $\Delta\partial_{t}\alpha$ and integrate over $\Omega$, we have

Owing to (3.35)-(3.37) and integrate between $T_{0}$ to $t$, we deduce that

Setting $y = \Vert\nabla\Delta\alpha\Vert^{2}, g = 0$ and $h = \Vert\partial_{t}u\Vert^{2}_{-1}$, we deduce from (3.60) that

where, owing to the above estimates, y, g and h satisfy the assumptions of the uniform Gronwall's lemme (for $t\geq t_{0}$), and for $t\geq t_{0}+r$,

which implies

We deduce owing to (3.58) and (3.63) that

4.   The dissipative semigroup

Based on the a priori estimates, we have the

Theorem 4.1. We assume that $(u_{0}, \alpha_{0})\in (H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\cap L^{K+2}(\Omega))\times(H^{3}(\Omega)\cap H^{1}_{0}(\Omega))$. Then, the system (2.1)-(2.4) possesses at least solution $(u, \alpha)$ such that $u \in L^{\infty}(0, T; H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\cap L^{k+2}(\Omega))$, $\alpha \in L^{\infty}(0, T; H^{3}(\Omega)\cap H^{1}_{0}(\Omega))$ and $\partial_{t}u \in L^{2}(0, T; H^{-1}(\Omega)), \; \forall\; T > 0$.

Proof. The proof is based on the estimate (3.64) and, e.g., a standard Galerkin scheme.

We have, concerning the uniqueness, the following.

Theorem 4.2. We assume that the assumptions of Theorem 4.1 hold. Then, the solution obtained in Theorem 4.1 is unique.

Proof. Let now $(u_{1}, \alpha_{1})$ and $(u_{2}, \alpha_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}, \alpha_{1, 0})$ et $(u_{2, 0}, \alpha_{2, 0})$ $\in (H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\cap L^{K+2}(\Omega))\times(H^{3}(\Omega)\cap H^{1}_{0}(\Omega))$ respectively. We set $(u, \alpha) = (u_{1}, \alpha_{1})-(u_{2}, \alpha_{2})$ and $(u_{0}, \alpha_{0}) = (u_{1, 0}, \alpha_{1, 0})-(u_{2, 0}, \alpha_{2, 0})$. Then $(u, \alpha)$ verifies the following problem.

We multiply (4.1) by $\partial_{t}u$ and integrate over $\Omega$, we have

We multiply (4.2) by $(\alpha-\Delta\alpha)$ and integrate $\Omega$, we obtain

Summing (4.5) and (4.6) and integrate over $\Omega$, we find

which implies

where

satisfies

Find the estimates for (4.8),

Besides

Inserting (4.11) into the estimates (4.10), we find

Furthermore

and

Inserting (4.12), (4.13) and (4.14) into (4.8), owing (3.37), we find

which gives

Applying Granwall's lemme, into (4.16), we find

hence the uniqueness, as well as continuous depending with respect to the initial data.

We set $\Psi = (H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\cap L^{K+2}(\Omega))\times (H^{3}(\Omega)\cap H^{1}_{0}(\Omega))$. It follows from Theorem 4.2, that we have the continuous (with respect to the $H^{1}(\Omega)\times H^{2}(\Omega)$-norm) of the following semigroup

$(i.e, S(0) = I, S(t)oS(s) = S(t+s), \; t, s\geq 0)$. We then deduce from (3.47) the following theorem.

Theorem 4.3. The semigroup S(t) is dissipative in $\Psi$, i.e., there exists a bounded set $B\in \Psi$(called absorbing set) such that, for every bounded $B \in \Psi$, there exists $t_{0} = t_{0}(B)\geq 0$ such that $t\geq t_{0}$ implies $S(t)B\subset {B}_{0}$.

Remark 4.1. It is easy to see that we can assume, without loss of generality, that $B_{0}$ is positively invariant by $S(t), i.e., S(t){B}_{0} \subset {B}_{0}, \forall t \geq 0$. Furthermore, it follows from (3.64) that $S(t)$ is dissipative in $H^{2}(\Omega)\times H^{3}(\Omega)$ and it follows from (3.63) that we can take ${B}_{0}$ in $H^{2}(\Omega)\times H^{3}(\Omega)$.

Corollary 4.1. The semigroup $S(t)$ possesses the global attractor $\mathcal{A}$ who is bounded in $H^{2}(\Omega)\times H^{3}(\Omega)$ and compact in $\Psi$.

5.   Existence of exponential attractors

The aim of this section is to prove the existence of exponential attractors for the semigroup $S(t), t\geq 0$, associated to the problem (2.1)-(2.4). To do so, we need the semigroup that has to be Lipschitz continuous, satisfying the smoothing property and checking a Hölder continuous with respect to time. This is enough to conclude on the existence of exponential attractors.

Lemma 5.1. Let $(u_{1}, \alpha_{1})$ and $(u_{2}, \alpha_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}, \alpha_{1, 0})$ and $(u_{2, 0}, \alpha_{2, 0})$, respectively, belonging to ${B}_{0}$. Then, the corresponding solutions of the problem (2.1)-(2.4) satisfy the following estimate

where the constants only depend on ${B}_{0}.$

Proof. We set $(u, \alpha) = (u_{1}, \alpha_{1})-(u_{2}, \alpha_{2})$ and $(u_{0}, \alpha_{0}) = (u_{1, 0}, \alpha_{1, 0})-(u_{2, 0}, \alpha_{2, 0})$, then $(u, \alpha)$ satisfies

We first deduce from (4.16) that

and

where the constants only depend on $B_{0}$.

We differentiate (5.2) with respect to time and have, owing to (5.3), we obtain

where $\theta = \partial_{t}u$ and $u_{1} = u+u_{2}$.

We multiply (5.8) by $(t-T_{0})\theta$ and integrate over $\Omega$, where $T_{0}$ is same as in one of previous section, we have

We have,

Noting that $u_{1}, u_{2}\in H^{2}(\Omega)\subset L^{\infty}(\Omega)$, we have

Furthermore,

Using (2.9) and (4.17), we find

noting that $u_{1}\in H^{2}(\Omega)$ and $\alpha \in H^{3}(\Omega)$, then

after Green, we have

we have that $\alpha_{2}\in H^{2}(\Omega)$, then

moreover

and

where the constants only depend on $B_{0}$.

By substituting (5.10), (5.11), (5.12), (5.13), (5.14), (5.15), (5.16), (5.17) and (5.18) into (5.9), we have, owing to the interpolation inequality,

We now multiply (5.3) by $-(t-T_{0})\alpha$ and intégrate over $\Omega$, we obtain

which implies

For that, let find the estimates of (5.20) right terms, using Hölder inequality, we have

and

Inserting (5.21) and (5.22) into (5.20), we find

Noting that $(u, \alpha) = (u_{2}, \alpha_{2}) = (u_{1}, \alpha_{1})$, then

where the constants only depend on $B_{0}$.

Combining (5.19) and (5.23), we find

where

Applying Gronwall's lemma to (5.25) over $[T_{0}, t]$, we have

which implies

finally, we obtain

where the constants only depend on $B_{0}$.

We rewrite (5.8) in the form

for $t \geq 1$ fixed, where

We multiply (5.31) by $\tilde{h}_{u}(t)$ and integrate over $\Omega$, we have

which implies

Here

furthermore $u_{1}, u_{2} \in H^{2}(\Omega) \subset L^{\infty}(\Omega)$, then

if $n = 2$ where $n = 3$, for $k\leq 1$(in particular $k = 1$), preceding estimate give

if $n = 2$ where $n = 3$ with $k > 1$, owing we have

on the one hand,

on the other hand,

combining (5.33), (5.36), (5.37) and (5.38), we find

Using (5.29) and (5.6), we obtain

where the constants only depend on ${B}_{0}$

We multiply (5.30) by and intergrate over $\Omega$, we find

hence, owing to (5.40), we have

we finally deduce from (5.29) and (5.42), the estimate (5.1) which concludes the proof

Lemma 5.2. Let $(u_{1}, \alpha_{1})$ and $(u_{2}, \alpha_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}, \alpha_{1, 0})$ and $(u_{2, 0}, \alpha_{2, 0})$, respectively, belonging to ${B}_{0}$. Then, the semigroup $\lbrace S(t))\rbrace_{t\geq 0}$ is Lipschitz continuity with respect to space, i.e, there exists the constant $c > 0$ such that

where the constants only depend on ${B}_{0}$.

Proof. The proof of the Lemma 5.2 is a direct consequence of the estimate (5.6).

It just remains to prove the Hölder continuity with respect to time.

Lemma 5.3. Let $(u, \alpha)$ be the solution of (5.2)-(5.5) with intial data $(u_{0}, \alpha_{0})$\; in ${B}_{0}$. Then, the semigroup $\lbrace S(t))\rbrace_{t\geq 0}$ is Hölder continuous with respect to time, i.e, there exists the constant $c > 0$ such that $\forall t_{1}, t_{2}\in [0, T]$

where the constants only depend on ${B}_{0}$ and $\Gamma$.

Proof.

Noting that, thanks to (3.5) and (3.37), we have

where the constant c depends only on ${B}_{0}$ and $T\geq T_{0}$ such that $t_{1}, t_{2}\in [0, T]$.

Furthermore, multiplying (5.3) by $(-\Delta)^{-1}\partial_{t}\alpha$ and integrate over $\Omega$, we obtain

and it follows from (3.60), (5.24), (5.46) and (5.47) that

where c only depends on ${B}_{0}$ and T such that $t_{1}, t_{2}\in [0, T]$.

Finally, we obtain thanks to (5.46) and (5.48), the estimate (5.44). Thus, the Lemma is proved. We finally deduce from Lemma 5.1, Lemma 5.2 and Lemma 5.3 the following result (see, e.g. ^[12]).

Theorem 5.1. The semigroup $S(t)$ possesses an exponential attractor $M \subset {B}_{0}$, i.e.,

$(i)$ M is compact in $H^{1}(\Omega)\times H^{2}(\Omega)$;

$(ii)$ M is positively invariant, $S(t)M \subset M, t\geq 0$;

$(iii)$ M has finite fractal dimension in $H^{1}(\Omega)\times H^{2}(\Omega)$;

$(iv)$ M attracts exponentially fast the bounded subsets of $\Psi$

where the constant c is independent of B and $dist_{H^{1}(\Omega)\times H^{2}(\Omega)}$ denotes the Hausdorff semidistance between sets defined by

Remark 5.1. Setting $\tilde{M} = S(1)M$, we can prove that $\tilde{M}$ is an exponential attractor for $S(t)$, but now in the topology of $\Psi$.

Since M (or $\tilde{M}$) is a compact attracting set, we deduce from Theorem 5.1 and standard results (see, e.g, ^[3,12]) the

Corollary 5.1. The semigroup $S(t)$ possesses the finite-dimensional global attractor ${A}\subset {B}_{0}$.

Remark 5.2. We note that the global attractor ${A}$ is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e. $S(t){A} = {A}, \; \forall t\geq 0)$ and attractors all bounded sets of initial data as time goes to infinity; thus, it appears as a suitable object in view of the study of asymptotic behaviour of the system. Furthermore, the finite dimensionality means, roughly speaking, that, even though the initial phase space is infinite dimensional, the reduced dynamics is, in some proper sense, finite dimensional and can be described by a finite number of parameters.

The existence of the global attractor being established, one question is to know whether this attractor has a finite dimension in terms of the fractal or Hausdorff dimension. This is the aim of the final section.

Remark 5.3. Comparing to the global attractor, an exponentiel attractor is expected to be more robust under perturbations. Indeed, the rate of attraction of trajectories to the global attractor may be slow and it is very difficult, if not impossible, to estimate this rate of attraction with respect to the physical parameters of the problem in general. As a consequence, global attractors may change drastically under small perturbations.

6.   Conclusions

This manuscript explains in a clear way, the context of dynamic system with two temperatures, when the relative solution exists. The existence of exponential attractor, associated to the problem (2.1)-(2.4) that we have proved, allow to assert that the existing solution of the problem (2.1)-(2.4) that we have shown in this work, belongs to the finite-dimensional subset called global attractor, from a certain time.

Acknowledgments

The authors thank the referees for their careful reading of the paper and useful comments.

Conflict of interest

The authors declare that there is no conflict of interests in this paper.