In this paper, we are interested in the study of the following three equations:

$\frac{\partial u}{\partial t}+{\text{Δ}}^{2}v-\text{Δ}f\left(u\right)=-\text{Δ}\theta ,$ (1.1)

$u=v-\epsilon \text{Δ}v,\epsilon >0,$ (1.2)

$\frac{\partial \theta }{\partial t}-\text{Δ}\theta =-\frac{\partial u}{\partial t},$ (1.3)

where $\epsilon$ is expected to be small (it is related to the inverse of the penalty modulus), $u$ is the order parameter, while $v$ is the microconcentration and $\theta$ the (relative) temperature. Furthermore, here and below, we set all physical parameters equal to one and we refer the interested reader to [1] - [12] . In particular, in the presence of the microconcentration, the total (Ginzburg-Landau type) free energy associated with the problem read (see [13] ):

$\Psi \left(u,v,\nabla v,\theta \right)={\displaystyle {\int }_{\Omega }\left(\frac{1}{2\epsilon }{\left(u-v\right)}^2+\frac{1}{2}{\left|\nabla v\right|}^2+F\left(u\right)-\theta u-\frac{1}{2}{\theta }^2\right)\text{d}x},$ (1.4)

where the potential $F$ is such that $F\left(s\right)={\displaystyle {\int }_0^{s}f\left(\tau \right)\text{d}\tau }$, $\Omega$ 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

$H=u+\theta .$ (1.5)

As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)

$\frac{\partial u}{\partial t}=\text{Δ}\frac{D\text{Ψ}}{Du},$ (1.6)

where $Du$ denotes a variational derivative with respect to u, which yields (1.1) and (1.2). Then, we have the energy equation

$\frac{\partial H}{\partial t}=-\text{div}q,$ (1.7)

where q is the heat flux. Assuming finally the usual Fourier law for heat conduction

$q=-\nabla \theta ,$ (1.8)

we obtain (1.3).

The microconcentration model was used in [14] in an application to lithium-ion batteries, coupled with finite deformation elastoplasticity. The computational advantage of the microconcentration approach, compared to the standard classical conserved phase-field model, is that less regularity of shape functions is required for the concentration variables in a finite element setting [15] .

Our aim in this paper is to prove the aforementioned convergence. We also prove the well-posedness to the conserved phase-field model based on microconcentrations and obtain error estimates on the difference of the solutions to this model and the classical conserved phase-field model, on finite time intervals. Finally, we prove the existence of global attractor and we give some numerical simulations.

We recall that we are interested in the following initial and boundary value problem:

$\frac{\partial u}{\partial t}+{\text{Δ}}^{2}v-\text{Δ}f\left(u\right)=-\text{Δ}\theta ,$ (2.1)

$u=v-\epsilon \text{Δ}v,$ (2.2)

$\frac{\partial \theta }{\partial t}-\text{Δ}\theta =-\frac{\partial u}{\partial t},$ (2.3)

$\frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial \text{Δ}v}{\partial \nu }=\frac{\partial \theta }{\partial \nu }=0\text{on}\Gamma ,$ (2.4)

${u|}_{t=0}=u_0,{\theta |}_{t=0}={\theta }_0,$ (2.5)

in a bounded and regular domain $\Omega \subset {ℝ}^n$, $n=1,2$ or 3, with boundary Γ; $\nu$ denotes the unit outer normal to Γ and $\frac{\partial \phi }{\partial \nu }=\nabla \phi \cdot \nu$ denotes the normal derivative on Γ. In particular, we assume, throughout this paper, that

$0<\epsilon \le {\epsilon }_0<1.$ (2.6)

As far as the nonlinear term is concerned, we make the following assumptions:

$f\in {\mathcal{C}}^2\left(ℝ\right),f\left(0\right)=0,$ (2.7)

$f^{\prime }\ge -1.$ (2.8)

$F\left(s\right)\ge \frac{1}{8}{s}^4-c_1,c_1\ge 0,$ (2.9)

$f\left(s\right)s\ge c_{2}F\left(s\right)-c_3,F\left(s\right)\ge -c_4,c_2>0,c_3,c_4\ge 0,s\in ℝ,$ (2.10)

where $F\left(s\right)={\displaystyle {\int }_0^{s}f\left(\tau \right)\text{d}\tau }$.

Remark 2.1. In particular, the usual cubic nonlinear term $f\left(s\right)=s^3-s$ satisfied these assumptions and $F\left(s\right)=\frac{1}{4}{s}^4-\frac{1}{2}{s}^2$.

We introduce here our main assumptions, together with several mathematical tools which are needed in order to give a precise analytical statement of our results.

We denote by $\langle \phi \rangle$ the spatial average of a function $\phi \in L^1\left(\Omega \right)$,

$\langle \phi \rangle =\frac{1}{\text{Vol}\left(\Omega \right)}{\displaystyle {\int }_{\Omega }\phi \left(x\right)\text{d}x}$

and, for $\phi \in H^{-1}\left(\Omega \right)=H^1{\left(\Omega \right)}^{\prime }$,

$\langle \phi \rangle =\frac{1}{\text{Vol}\left(\Omega \right)}{\langle \phi ,1\rangle }_{H^{-1}\left(\Omega \right),H^1\left(\Omega \right)},$

where $\langle .,.\rangle$ denotes the duality product. Furthermore, we set

$\bar{\phi }=\phi -\langle \phi \rangle ,$

where $\bar{\phi }$ denotes the conjugate of $\phi$.

We then set

${\stackrel{\dot{}}{H}}^1\left(\Omega \right)=\left\{\phi \in H^1\left(\Omega \right),\langle \phi \rangle =0\right\}$

and

${\stackrel{\dot{}}{L}}^2\left(\Omega \right)=\left\{\phi \in L^2\left(\Omega \right),\langle \phi \rangle =0\right\}.$

Integrating (2.1) over the spatial domain $\Omega$, we have, owing to (2.4),

$\frac{\text{d}}{\text{d}t}\langle u\rangle =0.$ (3.1)

Furthermore, integrating then (2.2) and (2.3) over $\Omega$, we obtain resp.

$\langle u\rangle =\langle v\rangle ,$ (3.2)

so that, also,

$\frac{\text{d}}{\text{d}t}\langle v\rangle =0$ (3.3)

and

$\frac{\text{d}}{\text{d}t}\langle H\rangle =0,$ (3.4)

which also yields, owing to (3.1),

$\frac{\text{d}}{\text{d}t}\langle \theta \rangle =0.$ (3.5)

We thus have the conservation of temperature, the conservation of mass, both for the order parameter $u$ and the microconcentration $v$.

Let $A$ be the operator defined by

${\langle Au,\phi \rangle }_{H^{-1}\left(\Omega \right),H^1\left(\Omega \right)}=\left(\left(\nabla u,\nabla \phi \right)\right),\forall \phi \in {\stackrel{\dot{}}{H}}^1\left(\Omega \right),$

where $\left(\left(.,.\right)\right)$ denotes the usual $L^2$-scalar product, with associated norm $\Vert .\Vert$, and the operator $A$ is an unbounded linear, selfadjoint and positive operator with compact inverse and is an isomorphism from ${\stackrel{\dot{}}{H}}^1\left(\text{Ω}\right)$ onto its dual. Furthermore,

$\mathcal{D}\left(A\right)=\left\{\phi \in H^2\left(\Omega \right)\cap {\stackrel{\dot{}}{H}}^1\left(\Omega \right),\frac{\partial \phi }{\partial \nu }=0\text{on}\Gamma \right\}$

and $Au=h$, $u\in \mathcal{D}\left(A\right)$, $h\in {\stackrel{\dot{}}{L}}^2\left(\Omega \right)$, is equivalent to

$-\Delta u=h,\frac{\partial u}{\partial \nu }=0\text{on}\Gamma .$

We will thus write $-\text{Δ}$ instead of $A$ in what follows, meaning that we consider that this operator acts on functions with null spatial average; of course, it can also be defined on functions with nonvanishing spatial average. We refer the interested reader to, e.g., [16] for more details on this. Having this, we can rewrite, equivalently, (2.1) in the (weaker) form:

${\left(-\text{Δ}\right)}^{-1}\frac{\partial u}{\partial t}-\text{Δ}v+\overline{f\left(u\right)}=\theta -\langle {\theta }_0\rangle ,$ (3.6)

noting that

$\langle \frac{\partial u}{\partial t}\rangle =0$

and

$\langle \theta \rangle =\langle {\theta }_0\rangle ,\forall t\ge 0.$

We only keep one boundary condition on $v$, namely

$\frac{\partial v}{\partial \nu }=0\text{on}\Gamma .$ (3.7)

Remark 3.1. In particular, it follows from (2.1) to (2.3) that

$\frac{\partial }{\partial t}\left(v-\epsilon \text{Δ}v\right)+{\text{Δ}}^{2}v-\text{Δ}f\left(v-\epsilon \text{Δ}v\right)=-\text{Δ}\theta ,$ (3.8)

$\frac{\partial \theta }{\partial t}-\text{Δ}\theta =-\frac{\partial }{\partial t}\left(v-\epsilon \text{Δ}v\right),$ (3.9)

which we can rewrite in the following (at least formally) equivalent form:

$\frac{\partial }{\partial t}\left({\left(-\text{Δ}\right)}^{-1}\bar{v}+\epsilon \bar{v}\right)-\text{Δ}\bar{v}+\overline{f\left(\langle u_0\rangle +\stackrel{¯}{v}-\epsilon \text{Δ}\stackrel{¯}{v}\right)}=\bar{\theta },$ (3.10)

$\frac{\partial \bar{\theta }}{\partial t}-\text{Δ}\bar{\theta }=\frac{\partial }{\partial t}\left(\bar{v}+\epsilon \bar{v}\right),$ (3.11)

where $-\text{Δ}$ denoting the minus Laplace operator with Neumann boundary conditions and acting on functions with null average. Also recall that

$\langle u\rangle =\langle v\rangle =\langle u_0\rangle ,\langle \theta \rangle =\langle {\theta }_0\rangle ,\forall t\ge 0.$ (3.12)

Alternatively, we can rewrite (2.2) in the equivalent form:

$\langle u\rangle =\langle u_0\rangle ,\bar{v}={\left(I-\epsilon \text{Δ}\right)}^{-1}\bar{u},$ (3.13)

allowing us to rewrite (3.10)-(3.11) in the equivalent form:

${\left(-\text{Δ}\right)}^{-1}\frac{\partial \bar{u}}{\partial t}-\text{Δ}{\left(I-\epsilon \text{Δ}\right)}^{-1}\bar{u}+\overline{f\left(\langle u_0\rangle +\stackrel{¯}{u}\right)}=\bar{\theta }.$ (3.14)

$\frac{\partial \bar{\theta }}{\partial t}-\text{Δ}\bar{\theta }=\frac{\partial \bar{u}}{\partial t},$ (3.15)

This shows that we can rewrite (2.1)-(2.4) as an equivalent problem for the sole unknown $\left(u,v,\theta \right)$.

We set ${\Vert .\Vert }_{-1}=\Vert {\left(-\text{Δ}\right)}^{-\frac{1}{2}}.\Vert$; ${\Vert .\Vert }_{-1}$ is a norm on $\left\{v\in H^{-1}\left(\Omega \right),\langle v\rangle =0\right\}$ which is equivalent to the usual $H^{-1}\left(\Omega \right)$-norm. More generally, we denote by ${\Vert .\Vert }_X$ the norm on the Banach space X.

Throughout this paper, the same letters $c$, $c^{\prime }$ and $c^{″}$ denote (nonnegative or positive) constants which may vary from line to line, or even in a same line, and which are independent of $\epsilon$ (but may depend on ${\epsilon }_0$).

In this section, we will establish a number of important inequalities that will be used later in the proof of the existence of the solution, the existence of global attractor and the convergence to the conserved phase-field model.

We assume that

$\left|\langle u_0\rangle \right|\le M_1,\left|\langle {\theta }_0\rangle \right|\le M_2$ (4.1)

for fixed positive constants $M_1$ and $M_2$, which yields, owing to (3.12),

$\left|\langle u\left(t\right)\rangle \right|=\left|\langle v\left(t\right)\rangle \right|\le M_1,\left|\langle \theta \left(t\right)\rangle \right|\le M_2,t\ge 0.$ (4.2)

We start with the following proposition.

Proposition 4.1 Any sufficiently regular solution to (2.1)-(2.5) satisfies the following estimates:

$\begin{array}{l}{\Vert u\left(t\right)\Vert }_{L^4\left(\Omega \right)}^2+{\Vert v\left(t\right)\Vert }_{H^2\left(\Omega \right)}^2+{\Vert \theta \left(t\right)\Vert }^2\\ \le c{\text{e}}^{-c^{\prime }t}\left({\Vert u_0\Vert }_{L^4\left(\Omega \right)}^2+{\Vert v_0\Vert }_{H^2\left(\Omega \right)}^2+{\Vert {\theta }_0\Vert }^2\right)+{c^{″}}_{M_1},c^{\prime }>0,t\ge 0\end{array}$ (4.3)

and

$\begin{array}{l}{\displaystyle {\int }_0^t\left({\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+{\Vert \theta \Vert }_{H^1\left(\Omega \right)}^2+{\Vert v\Vert }_{H^3\left(\Omega \right)}^2\right)\text{d}s}\\ \le c{\text{e}}^{-c^{\prime }t}\left({\Vert u_0\Vert }_{L^4\left(\Omega \right)}^2+{\Vert v_0\Vert }_{H^2\left(\Omega \right)}^2+{\Vert {\theta }_0\Vert }^2\right)+{c^{″}}_{M_1},c^{\prime }>0,t\ge 0.\end{array}$ (4.4)

Proof. The estimates below will be formal, but they can easily be justified within, e.g., a standard Galerkin scheme.

We multiply (3.6) by $\frac{\partial u}{\partial t}$ and have, integrating over $\Omega$ and by parts,

${\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2-\left(\left(\Delta v,\frac{\partial u}{\partial t}\right)\right)+\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }F\left(u\right)\text{d}x}=\left(\left(\theta ,\frac{\partial u}{\partial t}\right)\right)-\left|\Omega \right|\langle {\theta }_0\rangle \langle \frac{\partial u}{\partial t}\rangle .$

Noting that, owing to (2.2),

$\frac{\partial u}{\partial t}=\frac{\partial v}{\partial t}-\epsilon \text{Δ}\frac{\partial v}{\partial t}$ (4.5)

and

$\langle \frac{\partial u}{\partial t}\rangle =0,$

we thus deduce from the above that

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla v\Vert }^2+\epsilon {\Vert \Delta v\Vert }^2+2{\displaystyle {\int }_{\Omega }F\left(u\right)\text{d}x}\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2=2\left(\left(\theta ,\frac{\partial u}{\partial t}\right)\right).$ (4.6)

We then multiply (2.3) by $\theta$ to obtain

$\frac{\text{d}}{\text{d}t}{\Vert \theta \Vert }^2+2{\Vert \nabla \theta \Vert }^2=-2\left(\left(\theta ,\frac{\partial u}{\partial t}\right)\right).$ (4.7)

The sum of (4.6) and (4.7) gives, setting

$E_1={\Vert \nabla v\Vert }^2+\epsilon {\Vert \Delta v\Vert }^2+2{\displaystyle {\int }_{\Omega }F\left(u\right)\text{d}x}+{\Vert \theta \Vert }^2$

the differential equality

$\frac{\text{d}{E}_1}{\text{d}t}+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \nabla \theta \Vert }^2=0,$ (4.8)

i.e., the decay of the total free energy.

We now multiply (3.6) by $u$ and have, owing to (3.2), (2.10) and (4.2),

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{-1}^2+c\left({\Vert \nabla v\Vert }^2+\epsilon {\Vert \Delta v\Vert }^2+{\displaystyle {\int }_{\Omega }F\left(u\right)\text{d}x}\right)\le {\Vert \theta \Vert }^2+{c^{\prime }}_{M_1},c>0.$ (4.9)

We next multiply (3.6) by $-\text{Δ}u$ and find, owing to (2.8),

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert u\Vert }^2+\left(\left(\text{Δ}u,\text{Δ}v\right)\right)\le \frac{3}{2}{\Vert \nabla u\Vert }^2+\frac{1}{2}{\Vert \nabla \theta \Vert }^2.$ (4.10)

Noting that, owing to (2.2),

$-\text{Δ}v=\frac{1}{\epsilon }\left(u-v\right),$

we have

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\Vert u\Vert }^2+\left(\frac{1}{\epsilon }-3\right){\Vert \nabla u\Vert }^2\le {\Vert \nabla \theta \Vert }^2+\frac{2}{\epsilon }\left(\left(\nabla u,\nabla v\right)\right)\\ \le {\Vert \nabla \theta \Vert }^2+\left(\frac{1}{\epsilon }-1\right){\Vert \nabla u\Vert }^2+\frac{1}{\epsilon \left(1-\epsilon \right)}{\Vert \nabla v\Vert }^2\end{array}$

and we find

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }^2+\left(\frac{1}{\epsilon }-2\right){\Vert \nabla u\Vert }^2\le {\Vert \theta \Vert }^2+\frac{1}{\epsilon \left(1-\epsilon \right)}{\Vert \nabla v\Vert }^2,$ (4.11)

which gives

$\frac{\text{d}}{\text{d}t}\left(\epsilon {\Vert u\Vert }^2\right)+\left(1-2{\epsilon }_0\right){\Vert \nabla u\Vert }^2\le {\epsilon }_0{\Vert \theta \Vert }^2+\frac{1}{1-{\epsilon }_0}{\Vert \nabla v\Vert }^2.$ (4.12)

Writing now, in view of (2.2),

$\text{Δ}u=\text{Δ}v-\epsilon {\text{Δ}}^{2}v,$

we deduce from (4.10) that

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }^2+{\Vert \Delta v\Vert }^2+{\Vert \nabla \text{Δ}v\Vert }^2\le 3{\Vert \nabla u\Vert }^2+{\Vert \nabla \theta \Vert }^2.$ (4.13)

Summing (4.8), ${\delta }_1$ times (4.9) and (4.13), where ${\delta }_1>0$ is small enough, we obtain, setting

$E_2=E_1+{\delta }_1{\Vert u\Vert }_{-1}^2+{\Vert u\Vert }^2$

an inequality of the form

$\frac{\text{d}{E}_2}{\text{d}t}+c\left(E_2+{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+{\Vert \theta \Vert }_{H^1\left(\Omega \right)}^2+{\Vert v\Vert }_{H^3\left(\Omega \right)}^2\right)\le {c^{\prime }}_{M_1},$ (4.14)

where $E_2$ satisfies

$E_2\ge c\left({\Vert u\Vert }_{L^4\left(\Omega \right)}^2+{\Vert v\Vert }_{H^2\left(\Omega \right)}^2+{\Vert \theta \Vert }^2\right)-c^{\prime }.$ (4.15)

In particular, it follows from (4.14)-(4.15) and Gronwall’s lemma the dissipative estimate (4.3) and (4.4), where

$v_0={\left(I-\epsilon \text{Δ}\right)}^{-1}{u}_0.$

Remark 4.1. When $\epsilon =0$, then we have $v=u$ and (4.8) reads:

$\frac{\text{d}{E}_1}{\text{d}t}+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \nabla \theta \Vert }^2=0,$ (4.16)

where

$E_1={\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }F\left(u\right)\text{d}x}+{\Vert \theta \Vert }^2,$

which is precisely the energy decay for the classical conserved phase-field model (see [17] ).

In this section we consider that $\epsilon >0$ is fixed. We have the following.

Theorem 5.1. Let $T>0$ be given. We assume that (2.6) holds, $\left(u_0,{\theta }_0\right)\in H^1\left(\Omega \right)\times L^2\left(\Omega \right)$ and $\epsilon u_0\in H^2\left(\Omega \right)$. Then, (2.1)-(2.5) possesses a unique weak solution $\left(u,\theta \right)$ such that

$u\in L^{\infty }\left({ℝ}^{+};L^4\left(\Omega \right)\right)\cap L^2\left(0,T;H^1\left(\Omega \right)\right),$

$\frac{\partial u}{\partial t}\in L^2\left(0,T;H^{-1}\left(\Omega \right)\right),$

$v\in L^{\infty }\left({ℝ}^{+};H^2\left(\Omega \right)\right)\cap L^2\left(0,T;H^3\left(\Omega \right)\right),$

$\frac{\partial v}{\partial t}\in L^2\left(0,T;H^1\left(\Omega \right)\right)$

and

$\theta \in L^{\infty }\left({ℝ}^{+};L^2\left(\Omega \right)\right)\cap L^2\left(0,T;H^1\left(\Omega \right)\right).$

Furthermore, $u\in \mathcal{C}\left(\left[0,T\right];L^4{\left(\Omega \right)}_w\right)$, $v\in \mathcal{C}\left(\left[0,T\right];H^2\left(\Omega \right)\right)$ and $\theta \in \mathcal{C}\left(\left[0,T\right];L^2\left(\Omega \right)\right)$ where the index $w$ denotes the weak topology.

Proof. Existence: The proof of existence, as well as of further regularity, is based on the above a priori estimates and a proper Galerkin scheme. Furthermore, the continuity results follow from the Lions-Magenes theorem and the Strauss lemma (see, e.g., [18] for details).

We can note that (3.10)-(3.11) is associated with the following weak formulation:

Find $\left(u,v,\theta \right):\left[0,T\right]\to {\stackrel{\dot{}}{H}}^1{\left(\Omega \right)}^3$, such that

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\left({\left(-\Delta \right)}^{-1}\bar{v}+\epsilon \bar{v},\phi \right)\right)+\left(\left(\nabla \bar{v},\nabla \phi \right)\right)+\left(\left(f\left(\langle u_0\rangle +\bar{v}-\epsilon \Delta \bar{v}\right),\phi \right)\right)\\ =\left(\left(\bar{\theta },\phi \right)\right)\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),\end{array}$ (5.1)

$\frac{\text{d}}{\text{d}t}\left(\left(\bar{\theta },\phi \right)\right)+\left(\left(\nabla \bar{\theta },\nabla \phi \right)\right)=-\frac{\text{d}}{\text{d}t}\left(\left(\bar{v}-\epsilon \Delta \bar{v},\phi \right)\right)\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),$ (5.2)

$\forall \phi \in {\stackrel{\dot{}}{H}}^1\left(\Omega \right),$

${\bar{v}|}_{t=0}={\bar{v}}_0,{\bar{\theta }|}_{t=0}={\bar{\theta }}_0.$ (5.3)

Let ${\left(e_i\right)}_{i\in \mathbb{N}}$ be eigenvectors of the minus Laplace operator associated with Neumann boundary conditions; these eigenvectors are associated with the eigenvalues $0<{\lambda }_1\le {\lambda }_2\le \cdots$,

$-\text{Δ}{e}_i={\lambda }_i{e}_i$

and acting on functions with null spatial average. We assume that the $e_i\text{s}$ are orthonormal in ${\stackrel{\dot{}}{L}}^2\left(\Omega \right)$ and orthogonal in ${\stackrel{\dot{}}{H}}^1\left(\Omega \right)$.

We set for $m\in \mathbb{N}$,

$V_m=Span\left\{e_1,\cdots ,e_m\right\}.$

Actually, here, the only difficulty is to prove the existence of a local in time solution to an approximated problem. To do so, keeping the same notation as in the previous section, we consider the following approximated problem, for $m\in \mathbb{N}$ given:

Find $\left(u_m,v_m,{\theta }_m\right):\left[0,T\right]\to V_m^3$, such that

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\left({\left(-\Delta \right)}^{-1}{\bar{v}}_m+\epsilon {\bar{v}}_m,\phi \right)\right)+\left(\left(\nabla {\bar{v}}_m,\nabla \phi \right)\right)+\left(\left(f\left(\langle u_0\rangle +{\bar{v}}_m-\epsilon \Delta {\bar{v}}_m\right),\phi \right)\right)\\ =\left(\left({\bar{\theta }}_m,\phi \right)\right)\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),\end{array}$ (5.4)

$\frac{\text{d}}{\text{d}t}\left(\left({\bar{\theta }}_m,\phi \right)\right)+\left(\left(\nabla {\bar{\theta }}_m,\nabla \phi \right)\right)=-\frac{\text{d}}{\text{d}t}\left(\left({\bar{v}}_m-\epsilon \Delta {\bar{v}}_m,\phi \right)\right)\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),$ (5.5)

$\forall \phi \in V_m,$

${{\bar{v}}_m|}_{t=0}={\bar{v}}_{0,m},{{\bar{\theta }}_m|}_{t=0}={\bar{\theta }}_{0,m},$ (5.6)

where

${\bar{u}}_{0,m}={\bar{v}}_{0,m}-\epsilon {\bar{v}}_{0,m},$ (5.7)

${\bar{u}}_{0,m}=P_m{\bar{u}}_0,{\bar{\theta }}_{0,m}=P_m{\bar{\theta }}_0$ (5.8)

$P_m$ being the orthogonal projector onto $V_m$ (for the $L^2\left(\Omega \right)$-norm).

The existence of a local in time solution to (5.4)-(5.8) then follows from the Cauchy-Lipschitz theorem. Having this, we can pass to the limit in a standard way, owing to the above a priori estimates (which also hold at the approximated level) and standard Aubin-Lions compactness results, and deduce the existence of a solution.

Uniqueness: Let $\left(u_1,v_1,{\theta }_1\right)$ and $\left(u_2,v_2,{\theta }_2\right)$ be two solutions with initial data $\left(u_{0,1},{\theta }_{0,1}\right)$ and $\left(u_{0,2},{\theta }_{0,2}\right)$, respectively, such that $\left|\langle u_{0,i}\rangle \right|\le M_1$ and $\left|\langle {\theta }_{0,i}\rangle \right|\le M_2$, $i=1,2$. We set

$\left(u,v,\theta \right)=\left(u_1,v_1,{\theta }_1\right)-\left(u_2,v_2,{\theta }_2\right)$

and have

${\left(-\text{Δ}\right)}^{-1}\frac{\partial u}{\partial t}-\text{Δ}v+\overline{f\left(u_1\right)-f\left(u_2\right)}=\theta ,$ (5.9)

$u=v-\epsilon \text{Δ}v,$ (5.10)

$\frac{\partial \theta }{\partial t}-\text{Δ}\theta =-\frac{\partial u}{\partial t},$ (5.11)

$\langle u\rangle =\langle v\rangle =\langle \theta \rangle =0,t\ge 0,$ (5.12)

$\frac{\partial v}{\partial \nu }=\frac{\partial \theta }{\partial \nu }=0\text{on}\Gamma ,$ (5.13)

${u|}_{t=0}=u_0,{\theta |}_{t=0}={\theta }_0.$ (5.14)

We multiply (5.9) by $u$ and have, owing to (2.8),

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{-1}^2+\left(\left(-\Delta v,u\right)\right)\le \frac{3}{2}{\Vert u\Vert }^2+\frac{1}{2}{\Vert \theta \Vert }^2.$ (5.15)

Next, multiplying (5.11) by ${\left(-\text{Δ}\right)}^{-1}\left(u+\theta \right)$, we find

$\frac{\text{d}}{\text{d}t}{\Vert u+\theta \Vert }_{-1}^2+{\Vert \theta \Vert }^2\le {\Vert u\Vert }^2.$ (5.16)

Summing finally ${\delta }_2$ times (5.15) and (5.16), where ${\delta }_2>0$ is small enough, we find

$\frac{\text{d}}{\text{d}t}\left({\delta }_2{\Vert u\Vert }_{-1}^2+{\Vert u+\theta \Vert }_{-1}^2\right)+{\delta }_2\left(\left(-\text{Δ}v,u\right)\right)+\left(1-{\delta }_2\right){\Vert \theta \Vert }^2\le \left(3{\delta }_2+1\right){\Vert u\Vert }^2,$ (5.17)

where

$1-{\delta }_2>0.$

Note that it follows from (5.10) that

$\left(\left(-\text{Δ}v,u\right)\right)={\Vert \nabla v\Vert }^2+\epsilon {\Vert \text{Δ}v\Vert }^2$

and

${\Vert u\Vert }^2={\Vert v\Vert }^2+2\epsilon {\Vert \nabla v\Vert }^2+{\epsilon }^2{\Vert \text{Δ}v\Vert }^2.$

We thus deduce that

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\delta }_2{\Vert u\Vert }_{-1}^2+{\Vert u+\theta \Vert }_{-1}^2\right)+{\delta }_2\left({\Vert \nabla v\Vert }^2+\epsilon {\Vert \text{Δ}v\Vert }^2\right)+\left(1-{\delta }_2\right){\Vert \theta \Vert }^2\\ \le \left(3{\delta }_2+1\right)\left({\Vert v\Vert }^2+2\epsilon {\Vert \nabla v\Vert }^2+{\epsilon }^2{\Vert \text{Δ}v\Vert }^2\right)\end{array}$

and (2.6) yields, employing the Poincar-Wirtinger inequality,

$\frac{\text{d}}{\text{d}t}\left({\delta }_2{\Vert u\Vert }_{-1}^2+{\Vert u+\theta \Vert }_{-1}^2\right)+\left(1-{\epsilon }_0\right)\epsilon {\delta }_2{\Vert \text{Δ}v\Vert }^2+\left(1-{\delta }_2\right){\Vert \theta \Vert }^2\le c{\Vert \nabla v\Vert }^2.$ (5.18)

Writing next

${\Vert u\Vert }_{-1}^2={\Vert {\left(-\text{Δ}\right)}^{-\frac{1}{2}}\left(v-\epsilon \text{Δ}v\right)u\Vert }_{-1}^2={\Vert v\Vert }_{-1}^2+2\epsilon {\Vert v\Vert }^2+{\epsilon }^2{\Vert \nabla v\Vert }^2,$

it follows that, setting

$E_3={\delta }_2{\Vert u\Vert }_{-1}^2+{\Vert u+\theta \Vert }_{-1}^2$

an inequality of the form

$\frac{\text{d}{E}_3}{\text{d}t}\le \frac{c}{{\epsilon }^2}{E}_3,$ (5.19)

$E_3$ satisfies

$E_3\ge c^{\prime }\left({\Vert u\Vert }_{-1}^2+{\Vert \theta \Vert }_{-1}^2\right),c^{\prime }>0.$ (5.20)

It follows from (5.19), (5.20) and Gronwall’s lemma that

${\Vert u\left(t\right)\Vert }_{-1}^2+{\Vert \theta \left(t\right)\Vert }_{-1}^2\le c^{\prime }\exp \left(\frac{c}{{\epsilon }^2}t\right)\left({\Vert u_0\Vert }_{-1}^2+{\Vert {\theta }_0\Vert }_{-1}^2\right),t\ge 0,$ (5.21)

finally yields the continuous (with respect to the $H^{-1}{\left(\Omega \right)}^2$-norm) dependence on the initial data, as well as the uniqueness, for $\left(u,v,\theta \right)$.

This yields uniform in time estimates (i.e., on ${ℝ}^{+}$) on the solutions, as well as the dissipativity of the corresponding solution operators, we set

$\Phi =H^1\left(\Omega \right)\times L^2\left(\Omega \right).$

We have the continuous (with respect to the semigroup

$S_{\epsilon }\left(t\right):\Phi \to \Phi ,\left(u_0,{\theta }_0\right)\mapsto \left(u^{\epsilon }\left(t\right),{\theta }^{\epsilon }\left(t\right)\right),t\ge 0,\epsilon \in \left(0,{\epsilon }_0\right]$

(i.e., $S_{\epsilon }\left(0\right)=I$ (identity operator) and $S_{\epsilon }\left(t+s\right)=S_{\epsilon }\left(t\right)\circ S_{\epsilon }\left(s\right),t,s\ge 0$). We then deduce from (4.3) the following theorem.

Theorem 5.2. The semigroup $S_{\epsilon }\left(t\right)$ is dissipative in $\Phi$, i.e., there exists a bounded set ${ℬ}_0\subset \Phi$ (called absorbing set) such that, for every bounded set $ℬ\subset \Phi$, there exists $t_0=t_0\left(ℬ\right)\ge 0$ such that $t\ge t_0$ implies $S_{\epsilon }\left(t\right)ℬ\subset {ℬ}_0$.

We thus deduce from standard results the following theorem.

Theorem 5.3. The semigroup $S_{\epsilon }\left(t\right)$ possesses the connected global attractor ${\mathcal{A}}_{\epsilon }$ such that ${\mathcal{A}}_{\epsilon }$ is compact in $\Phi$, verifying:

1. ${\mathcal{A}}_{\epsilon }$ is invariant, i.e., $S_{\epsilon }\left(t\right){\mathcal{A}}_{\epsilon }={\mathcal{A}}_{\epsilon },\forall t\ge 0$;

2. ${\mathcal{A}}_{\epsilon }$ attracts all bounded sets of initial data in the following sense:

$\forall B\subset \Phi$ bounded, $dist\left(S_{\epsilon }\left(t\right)B,{\mathcal{A}}_{\epsilon }\right)\to 0$ as $t\to +\infty$,

where dist denotes the Hausdorff semi-distance between sets defined by

$dist\left(A,B\right)=\underset{a\in A}{\text{sup}}\underset{b\in B}{\text{inf}}{\Vert a-b\Vert }_{\Phi }$

(we refer the reader to, e.g., [18] for more details.)

The next step would be to study the existence of finite-dimensional attractors for , $t\ge 0$, $\epsilon \in \left(0,{\epsilon }_0\right]$, and their stability with respect to $\epsilon$, as well as their convergence to (proper $S_{\epsilon }\left(t\right)$ liftings of) those corresponding to $S_{\epsilon }\left(t\right)$ as $\epsilon \to 0^{+}$. In particular, one interesting and important problem would be to construct a robust (i.e., both upper and lower semicontinuous as $\epsilon \to 0^{+}$) family of exponential attractors, meaning that the dynamics of the original and limit problems are close in some proper sense. This will be addressed elsewhere. We also refer the interested reader to, e.g., [19] for discussions on such objects.

Our aim in this section is to pass to the limit in (2.1)-(2.5) as $\epsilon$ goes to $0^{+}$. Note that the limit problem for $\epsilon =0$ corresponds to the classical conserved phase-field model,

$\frac{\partial u^0}{\partial t}+{\text{Δ}}^2{v}^0-\text{Δ}f\left(u^0\right)=-\text{Δ}{\theta }^0,$ (6.1)

$u^0=v^0,$ (6.2)

$\frac{\partial {\theta }^0}{\partial t}-\text{Δ}{\theta }^0=-\frac{\partial u^0}{\partial t},$ (6.3)

$\frac{\partial u^0}{\partial \nu }=\frac{\partial v^0}{\partial \nu }=\frac{\partial \text{Δ}{v}^0}{\partial \nu }=\frac{\partial {\theta }^0}{\partial \nu }=0\text{on}\Gamma ,$ (6.4)

${u^0|}_{t=0}=u_0,{{\theta }^0|}_{t=0}={\theta }_0.$ (6.5)

To do so, we first need to derive estimates on the solutions to (2.1)-(2.5) which are independent of $\epsilon$ (we consider here strong solutions as given in Theorem 5.1). We thus consider the initial and boundary value problem

$\frac{\partial u^{\epsilon }}{\partial t}+{\text{Δ}}^2{v}^{\epsilon }-\text{Δ}f\left(u^{\epsilon }\right)=-\text{Δ}{\theta }^{\epsilon },$ (6.6)

$u^{\epsilon }=v^{\epsilon }-\epsilon \text{Δ}{v}^{\epsilon },$ (6.7)

$\frac{\partial {\theta }^{\epsilon }}{\partial t}-\text{Δ}{\theta }^{\epsilon }=-\frac{\partial u^{\epsilon }}{\partial t},$ (6.8)

$\frac{\partial u^{\epsilon }}{\partial \nu }=\frac{\partial v^{\epsilon }}{\partial \nu }=\frac{\partial \Delta v^{\epsilon }}{\partial \nu }=\frac{\partial {\theta }^{\epsilon }}{\partial \nu }=0\text{on}\Gamma ,$ (6.9)

${u^{\epsilon }|}_{t=0}=u_0,{{\theta }^{\epsilon }|}_{t=0}={\theta }_0,$ (6.10)

Repeating, for (6.6)-(6.10), the estimates leading to (4.14), we obtain

$\frac{\text{d}{E}_2^{\epsilon }}{\text{d}t}+c\left(E_2^{\epsilon }+{\Vert \frac{\partial u^{\epsilon }}{\partial t}\Vert }_{-1}^2+{\Vert {\theta }^{\epsilon }\Vert }_{H^1\left(\Omega \right)}^2+{\Vert v^{\epsilon }\Vert }_{H^3\left(\Omega \right)}^2\right)\le c^{\prime },$ (6.11)

where

$E_2^{\epsilon }=E_1^{\epsilon }+{\delta }_1{\Vert u^{\epsilon }\Vert }_{-1}^2+{\Vert u^{\epsilon }\Vert }^2$

and

$E_1^{\epsilon }={\Vert \nabla v^{\epsilon }\Vert }^2+\epsilon {\Vert \text{Δ}{v}^{\epsilon }\Vert }^2+2{\displaystyle {\int }_{\Omega }F\left(u^{\epsilon }\right)\text{d}x}+{\Vert {\theta }^{\epsilon }\Vert }^2.$ (6.12)

We can now prove the following.

Theorem 6.1. We assume that the assumptions of Theorem 5.1 hold. Then, the sequence of solutions $\left(u^{\epsilon },v^{\epsilon },{\theta }^{\epsilon }\right)$ to (6.6)-(6.10) converges to a solution to (6.1)-(6.5) on finite time intervals $\left[0,T\right]$, $T>0$, as $\epsilon \to 0^{+}$.

Proof. It follows from the uniform (with respect to $\epsilon$) a priori estimates derived and standard Aubin-Lions compactness results that, at least for a subsequence that we do not relabel, there exist $\left(u^0,v^0,{\theta }^0\right)$ and $\chi$ such that, in particular,

$u^{\epsilon }\to u^{0}in{L}^{\infty }\left(0,T;L^4\left(\Omega \right)\right)\text{weak-}\star ,\text{in}{L}^2\left(0,T;H^1\left(\Omega \right)\right)\text{weakly}\text{and}\text{a}\text{.e}.,$

$\frac{\partial u^{\epsilon }}{\partial t}\to \frac{\partial u^0}{\partial t}\text{in}{L}^2\left(0,T;H^{-1}\left(\Omega \right)\right)\text{weakly}$

$v^{\epsilon }\to v^0\text{in}{L}^{\infty }\left(0,T;H^1\left(\text{Ω}\right)\right)\text{weak-}\star \text{and}{L}^2\left(0,T;H^2\left(\text{Ω}\right)\right)\text{weakly},$

${\epsilon }^{\frac{1}{2}}{v}^{\epsilon }\to \chi \text{in}{L}^{\infty }\left(0,T;H^2\left(\Omega \right)\right)\text{weak-}\star \text{and}{L}^2\left(0,T;H^3\left(\Omega \right)\right)\text{weakly}$

and

${\theta }^{\epsilon }\to {\theta }^0\text{in}{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right)\text{weak-}\star \text{and}{L}^2\left(0,T;H^1\left(\Omega \right)\right)\text{weakly}.$

For a proper $u$, which implies that

$f\left(u^{\epsilon }\right)\to f\left(u^0\right)\text{a}\text{.e}\text{.}\text{and}f\left(u^{\epsilon }\right)\text{is}\text{bounded}\text{in}{L}^{\frac{4}{3}}\left(\left(\Omega \right)\times \left(0,T\right)\right).$

Therefore, $f\left(u^{\epsilon }\right)\to f\left(u^0\right)$ in $L^{\frac{4}{3}}\left(\left(\Omega \right)\times \left(0,T\right)\right)$ weakly, which is sufficient to pass to the limit in the weak formulation.

Having this, it is now standard to pass to the limit in (6.6)-(6.10) to find, at the limit,

$\frac{\text{d}}{\text{d}t}\left(\left(u^0,\phi \right)\right)+\left(\left(\Delta u^0,\Delta \phi \right)\right)+\left(\left(\nabla f\left(u^0\right),\nabla \phi \right)\right)=0\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),$ (6.13)

$\frac{\text{d}}{\text{d}t}\left(\left({\theta }^0,\phi \right)\right)+\left(\left(\nabla {\theta }^0,\nabla \phi \right)\right)=-\frac{\text{d}}{\text{d}t}\left(\left(u^0,\phi \right)\right)\text{in}{\mathcal{D}}^{\prime }\left(\Omega \right),$ (6.14)

${u^0|}_{t=0}=u_0,{{\theta }^0|}_{t=0}={\theta }_0$ (6.15)

i.e., $\left(u^0,{\theta }^0\right)$ is solution to the classical conserved phase-field model.

Noting finally that the solution to the classical conserved phase-field model (6.1)-(6.5) is unique, we see that the whole sequence $\left(u^{\epsilon },v^{\epsilon },{\theta }^{\epsilon }\right)$ converges.

We can also derive error estimates and prove the following.

Theorem 6.2. Under the assumptions of Theorem 6.1, then, $\forall T>0$,

${\Vert u^{\epsilon }-u^0\Vert }_{-1}^2+{\Vert {\theta }^{\epsilon }-{\theta }^0\Vert }_{-1}^2\le c\left(\epsilon ,T\right){\text{e}}^{c^{\prime }T}\left({\Vert u_0\Vert }_{L^4\left(\Omega \right)}^2+{\Vert v_0\Vert }_{H^2\left(\Omega \right)}^2+{\Vert {\theta }_0\Vert }^2\right).$

Proof. We set $\left(u,v,\theta \right)=\left(u^{\epsilon },v^{\epsilon },{\theta }^{\epsilon }\right)-\left(u_0,v_0,{\theta }_0\right)$. Note that

$\langle u\rangle =\langle v\rangle =\langle \theta \rangle =0,t\ge 0.$ (6.16)

Furthermore, $\left(u,v,\theta \right)$ solves

${\left(-\text{Δ}\right)}^{-1}\frac{\partial u}{\partial t}-\text{Δ}v+\overline{f\left(u^{\epsilon }\right)-f\left(u^0\right)}=\theta ,$ (6.17)

$u=v-\epsilon \text{Δ}v$ (6.18)

$\frac{\partial \theta }{\partial t}-\text{Δ}\theta =\frac{\partial u}{\partial t}$ (6.19)

$\frac{\partial v}{\partial \nu }=\frac{\partial \theta }{\partial \nu }=0\text{on}\Gamma ,$ (6.20)

${u|}_{t=0}=u_0,{\theta |}_{t=0}={\theta }_0.$ (6.21)

Multiplying (6.17) by $u$, we obtain, owing to (2.8),

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{-1}^2-2\left(\left(u,\text{Δ}v\right)\right)\le 3{\Vert u\Vert }^2+{\Vert \theta \Vert }^2.$ (6.22)

Note that, employing the interpolation inequality

${\Vert u\Vert }^2\le {\Vert u\Vert }_{-1}\Vert \nabla u\Vert$

we can write, owing to (6.18),

${\Vert u\Vert }^2\le {\Vert u\Vert }_{-1}\Vert \nabla u\Vert +{\Vert u\Vert }_{-1}\Vert \nabla \text{Δ}{v}^{\epsilon }\Vert .$ (6.23)

Moreover, employing again (6.18), we can see that

$-2\left(\left(u,\text{Δ}v\right)\right)=2\parallel \nabla v{\parallel }^2-2\epsilon \left(\left(\nabla v,\nabla \text{Δ}{v}^{\epsilon }\right)\right).$ (6.24)

It thus follows from (6.22) to (6.24) that

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{-1}^2+2{\Vert \nabla v\Vert }^2\le 3{\Vert u\Vert }_{-1}\Vert \nabla u\Vert +3{\Vert u\Vert }_{-1}\Vert \nabla \text{Δ}{v}^{\epsilon }\Vert +2\Vert \nabla v\Vert \Vert \nabla \text{Δ}{v}^{\epsilon }\Vert +{\Vert \theta \Vert }^2,$

which yields, employing Young’s inequality,

$\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{-1}^2+{\Vert \nabla v\Vert }^2\le c\left({\Vert u\Vert }_{-1}^2+{\epsilon }^2{\Vert \nabla \text{Δ}{v}^{\epsilon }\Vert }^2+{\Vert \theta \Vert }^2\right).$ (6.25)