A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

Abstract

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter $δ$ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as $δ$ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model.

Keywords

Elliptic–parabolic system asymptotic analysis inhomogeneous thermoelastic body thin inhomogeneity Galerkin method generalized solution

1. Introduction

Progress in the development of aerospace, automobile, medical, and other industries is associated with an increase in the use of composite materials. This is due to the fact that composite materials enhance the strength properties of structural elements. Commonly, by a composite material, we mean a heterogeneous medium consisting of several materials and bodies, the properties and behavior of which can differ significantly from each other [1, 2]. As an example, let us mention the composite materials with carbon nanotubes that are widely used nowadays, since they have high strength and rigidity (see, e.g., [3, 4]).

It is known (see, e.g., [5, 6]) that numerical calculations of models of composites with thin layers and inhomogeneities are fraught with large computational costs. To overcome this difficulty, approximating models are derived. In these models, inhomogeneities are replaced by some submanifolds of codimension one, on which the interface conditions inherited from the original full model are properly set.

This paper is devoted to a study of the quasi-static thermoelasticity problem for a body with a narrow inclusion. We suppose that the elastic properties of the inclusion are proportional to a small parameter $δ$ characterizing its width. The mixed boundary conditions are prescribed on the external boundary of the body. As a result, we deal with an elliptic–parabolic system. We seek for a weak solution of this system, and for this purpose, the system is formulated as a variational problem.

A brief note on the physical background of the basic quasi-static thermoelastic model when the accelerations are negligible is given further in section 2; a more detailed discussion of this topic can be found in Boley and Weiner [7] and Day [8]. Besides, we note that the differential equations of thermoelasticity are similar to those for the poroelasticity. Such problems were investigated in Ern and Meunier [9], Serpilli [10], and Showalter [11] within the linear approach and in Kovtunenko [12] and Kovtunenko and Lazarev [13] for variational inequalities.

Relying on the variational formulation, we investigate the behavior of solutions as $δ$ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). The main purpose of this paper is to prove existence, uniqueness, and stability of a weak solution to the limit model. There are many methods to prove existence of solutions to elliptic–parabolic systems, e.g., the theory of $m$ -accretive operators [11, 14], Shauder’s fixed point theorem (mainly for variational inequalities) [15, 16], and the Rothe semi-discretization method [12, 17]. In this paper, we use the Galerkin method and the classical techniques of derivation of energy estimates because, in our opinion, this is the simplest method, which allows to prove existence, uniqueness, and stability of a weak solution of the elliptic–parabolic system, as well as to derive some additional regularity in time of a weak solution.

Finalizing the introduction, we give a short review of the works related to thin inclusion (thin inhomogeneity) problems in the theory of elasticity. Asymptotic analysis was performed for different models of elasticity with a wide range of constitutive relations, such as elastic models [18 –22], multiphysics models [23, 24], and models of plates [25 –29]. In this line, let us mention also the recent articles [25, 29 –37]. At present, there are not so many works (see, e.g., [38 –41]) dealing with asymptotic analysis of dynamical problems in elasticity. In the most cases, authors investigated such problems using the method of formal asymptotic expansions without providing mathematically rigorous proofs of well-posedness of limit problems.

The further layout of this paper is as follows. In section 2, we discuss the quasi-static model of thermoelasticity and define the strong (differential) and the weak (variational) formulations of the problem under consideration. After refining the geometrical setting and properly adapting the formulations defined in section 2 and using the method of formal asymptotic expansions in section 3, we derive the model of thermoelastic body with a thin (of zero thickness) thermoelastic inclusion (inhomogeneity). After this, in section 4, we prove the well-posedness of the model with the thin inclusion. Moreover, we prove stability and additional regularity in time of the weak solution.

2. Preliminaries: the quasi-static model in thermoelasticity

This section provides the necessary information about the model under consideration, which describes quasi-static behavior of a thermoelastic medium.

By $u = (u_{1}, u_{2})$ and $σ_{y} = {σ_{yij}}$ denote the displacement vector and the strain tensor, respectively. Then, the equation of motion has the form:

- di v_{y} σ_{y} (u, θ) + ρ \partial_{tt} u = f,

(1)

where $f = (f_{1}, f_{2})$ is a given volumetric force, $ρ$ is the medium density, $θ$ is the temperature, $t$ is the time variable, and $y = (y_{1}, y_{2})$ is a coordinate in the space of physical positions in $R^{2}$ . Linear constitutive equation, i.e., Duhamel–Neumann’s law, has the form:

σ_{y} (u, θ) = 2 μ ε_{y} (u) + (λ di v_{y} u - α θ) I,

(2)

where $ε_{y} (u) = {ε_{yij} (u)}$ is the strain tensor:

ε_{yij} (u) = (1 / 2) (\partial_{y_{j}} u_{i} + \partial_{y_{i}} u_{j}),

(3)

$μ$ and $λ$ are the Lamé coefficients, such that $μ > 0, 3 λ + 2 μ > 0, α$ is the coefficient corresponding to thermal dilatation, and $I = diag {1, 1}$ is the unit 2×2-matrix.

The heat conduction is described by the balance of energy equation:

c \partial_{t} θ - di v_{y} q (θ) + α di v_{y} \partial_{t} u = g,

(4)

supplemented with Fourier’s law of heat conduction:

q (θ) = κ \nabla_{y} θ,

(5)

where $κ$ is the heat conductivity coefficient, $c$ is the specific heat coefficient, $g$ is a distributed heat source, and $q (θ)$ is the heat flux. We assume that the thermoelastic body occupies a domain $O \subset R_{y}^{2}$ with a Lipschitz boundary $Γ = \partial O$ , so that $O$ is a domain of all known and unknown functions in equations (1)–(5).

System (1)–(5) is the classical model of linear thermoelasticity [42, Chapter I]. In this paper, we assume that the accelerations in the system are negligible. Thus, we omit the acceleration term $ρ \partial_{tt} u$ in the equation of motion (1) and arrive at the coupled quasi-static approximate model. Following Day [6, sections 1.1, 2.1] and Boley and Weine [7, sections 2.1, 2.2, 2.5], we note that this approximation is justified for a wide class of applied problems in the case when the dimensionless coupling constant $a = \frac{α^{2}}{c (λ + 2 μ)}$ , which is a measure of the coupling between thermal and mechanical effects, is positive and rather large, while the dimensionless inertial constant $b = \frac{κ^{2} ρ}{c^{2} (λ + 2 μ) {(diam O)}^{2}}$ is sufficiently small.

Now, let $Γ$ be divided into two parts $Γ_{D}$ and $Γ_{N}$ with the nonzero one-dimensional Hausdorff measure. By $n$ denote the unit outward normal vector to $Γ$ .

With account of the above assumptions and notation, following Malqvist and Persson [43, section 2] and Showalter [11, sections 3 and 4], we set up the basic formulation of the problem under study in both differential and variational forms.

Problem A_diff. In the space-time cylinder $Q_{T} : = O \times [0, T]$ , where $T = const > 0$ is a given time moment, find a pair of functions $u : Q_{T} \mapsto R^{2}$ and $θ : Q_{T} \mapsto R$ satisfying the system of equations:

- {div}_{y} (2 μ ε_{y} (u) + (λ {div}_{y} u - α θ) I) = f, (y, t) \in Q_{T},

(6a)

c \partial_{t} θ - {div}_{y} (κ \nabla_{y} θ) + α {div}_{y} \partial_{t} u = g, (y, t) \in O \times (0, T),

(6b)

the boundary conditions:

u = 0, (y, t) \in Γ_{D} \times (0, T),

(6c)

(2 μ ε_{y} (u) + (λ {div}_{y} u - α θ) I) n = 0, (y, t) \in Γ_{N} \times (0, T),

(6d)

θ = 0, (y, t) \in Γ_{D} \times (0, T),

(6e)

κ \nabla_{y} θ \cdot n = 0, (y, t) \in Γ_{N} \times (0, T),

(6f)

and the initial condition:

θ |_{t = 0} = θ_{0}, y \in O .

(6g)

In equation (6f) and further, by the dot “·” we standardly denote the inner product in $R^{2}$ .

Introduce the functional spaces:

V_{u} = {u \in H^{1} (O)^{2} : u = 0 on Γ_{D}}, V_{θ} = {ξ \in H^{1} (O) : ξ = 0 on Γ_{D}} .

Here and further, by $H^{1} (O)$ for various open connected domains $O \subset R^{2}$ , we denote the Sobolev space equipped with the standard norm:

∥ w ∥_{H^{1} (O)} = {(∥ w ∥_{L^{2} (O)}^{2} + ∥ \nabla_{y} w ∥_{L^{2} {(O)}^{2}}^{2})}^{1 / 2} .

Suppose that $f, \partial_{t} f \in L^{2} {(Q_{T})}^{2}$ , $g \in L^{2} (Q_{T})$ , $θ_{0} \in V_{θ}$ and that $μ$ , $λ$ , $α$ , $c$ , and $κ$ depend on $y$ (in general) are measurable, bounded, and strictly positive, i.e., they are separated from zero from below by a positive constant.

Solution of Problem A_diff is understood in the weak sense. More certainly, we further consider the following variational formulation, which is equivalent to Problem A_diff in the sense of distributions.

Problem A_var. Find a pair of functions:

(u, θ) \in C ([0, T]; V_{u} \times V_{θ}) \cap C^{1} ((0, T]; V_{u} \times V_{θ}),

satisfying the integral equalities:

\begin{matrix} \int_{O} (2 μ ε_{y} (u (t)) : ε_{y} (v) + λ {div}_{y} u (t) {div}_{y} v - α θ (t) {div}_{y} v) dy = \int_{O} f (t) \cdot v dy, \\ \forall v \in V_{u}, \forall t \in [0, T], \end{matrix}

(7a)

\begin{matrix} \int_{O} (c \partial_{t} θ (t) + α {div}_{y} \partial_{t} u (t)) ξ dy + \int_{O} κ \nabla_{y} θ (t) \cdot \nabla_{y} ξ dy = \int_{O} g (t) ξ dy, \\ \forall ξ \in V_{θ}, \forall t \in (0, T], \end{matrix}

(7b)

and the initial condition (6g) in the $H^{1} - strong$ trace sense, i.e.,

∥ θ (t) - θ_{0} ∥_{H^{1} (O)} \underset{t \to 0 +}{\to} 0 .

(7c)

In equation (7a) and further, by the colon “:” we standardly denote the inner product in the space of 2×2-matrices, i.e., $A : B = \sum_{i, j = 1}^{2} a_{ij} b_{ij}$ for all $A = (a_{ij})_{i, j = 1, 2}$ and $B = (b_{ij})_{i, j = 1, 2}$ .

Problem A_var is well-posed. Namely, the following assertion holds true.

Proposition 1. Assume the given functions $μ$ , $λ$ , $α$ , $c$ , $κ$ , $f$ , $g$ , and $θ_{0}$ satisfy the above-formulated assumptions. Then, Problem A _var has a unique solution $(u, θ)$ .

Proof follows directly from Showalter [11, Theorems 3.1 and 4.1].

3. Derivation of the model with a thin inclusion

3.1. Refinement of the geometric structure and appropriate adaptation of the formulation of Problem A_var

We introduce a specific structure for the thermoelastic body in question, following Fankina et al. [32, section 2(a)] and Sazhenkov et al. [37, section 1.1].

In the two-dimensional space $R^{2}$ referred to the coordinate system $O y_{1} y_{2}$ , let $Ω$ be a bounded domain with a Lipschitz boundary $\partial Ω$ . Let $γ = (Ω \cap {y_{2} = 0})$ be the intersection of $Ω$ with axis $O y_{2}$ , which is supposed to have a positive one-dimensional measure:

γ = {(y_{1}, y_{2}) \in R^{2} : 0 < y_{1} < l^{*}, y_{2} = 0}, l^{*} = const > 0 .

The line segment $γ$ divides $Ω$ into subdomains $Ω_{\pm}$ with Lipschitz boundaries $\partial Ω_{\pm}$ (see Figure 1).

Let us introduce a small real parameter $δ > 0$ . We consider the assembly consisting of the three thermoelastic isotropic bodies. To describe this assembly, we introduce the following geometric objects:

\begin{matrix} Ω_{m}^{δ} = {0 < y_{1} < l^{*}} \times {- δ / 2 < y_{2} < δ / 2}, \\ Ω_{\pm}^{δ} = {(y_{1}, y_{2}) \in R^{2} : (y_{1}, y_{2} \mp δ / 2) \in Ω_{\pm}}, \\ γ_{\pm}^{δ} = γ \pm (0, δ / 2) = \\ = {(y_{1}, y_{2}) \in R^{2} : (y_{1}, y_{2}) \in ({0 < y_{1} < l^{*}} \times {y_{2} = \pm δ / 2})}, \end{matrix}

which depend on a fixed parameter $δ$ . Also, to the entire region occupied by the material, we relate the domain $Ω^{δ} = Ω_{+}^{δ} \cup Ω_{-}^{δ} \cup Ω_{m}^{δ} \cup γ_{+}^{δ} \cup γ_{-}^{δ}$ and introduce the sets $Γ_{\pm}^{δ} = \partial Ω^{δ} \cap \partial Ω_{\pm}^{δ}$ , $Γ_{D}^{δ} = Γ_{+}^{δ} \cup Γ_{-}^{δ}$ and $Γ_{N}^{δ} = \partial Ω^{δ} \ \bar{Γ_{D}^{δ}}$ , which are the parts of the outer boundary $Γ^{δ} = \partial Ω^{δ}$ . We call $Ω_{m}^{δ}$ and $Ω_{\pm}^{δ}$ the adhesive and the adherents, respectively. Notice that the sets $Ω_{\pm}$ and $γ$ presented before are exactly what the adhesive and the adherents tend to, as $δ$ tends to zero. Comparing the notation of the sets in $R_{y}^{2}$ in this section with the notation introduced in section 1, we have:

O = Ω^{δ}, Γ = Γ^{δ}, Γ_{D} = Γ_{D}^{δ}, Γ_{N} = Γ_{N}^{δ} .

(8)

From now on, we consider Problems A_diff and A_var for the geometry fixed above and depending on $δ$ , so that all sought and given functions depend on $δ$ , as well:

u = u^{δ}, θ = θ^{δ}, θ_{0} = θ_{0}^{δ}, f = f^{δ}, g = g^{δ}, μ = μ^{δ}, λ = λ^{δ}, α = α^{δ}, and so on .

It is convenient to introduce a geometrical setting of the problem that is independent of $δ$ . To this end, we employ an appropriate coordinate scaling and shift. In the adhesive, the change of variables $\hat{p} : (z_{1}, z_{2}) \to (y_{1}, y_{2})$ is implemented, which is such that:

y_{1} = z_{1}, y_{2} = δ z_{2},

giving:

\partial_{y_{1}} = \partial_{z_{1}}, \partial_{y_{2}} = δ^{- 1} \partial_{z_{2}} .

In the adherents, the following change of variables $\bar{p} : (x_{1}, x_{2}) \mapsto (y_{1}, y_{2})$ is used:

y_{1} = x_{1}; y_{2} = 0 for x_{2} = 0, y_{2} = x_{2} \pm δ / 2 for \pm x_{2} > 0 .

By this, alongside $O y_{1} y_{2}$ , we introduce the two more coordinate systems: $O x_{1} x_{2}$ and $O z_{1} z_{2}$ . We take:

{\hat{u}}^{δ} = u^{δ} ° \hat{p}, {\hat{θ}}^{δ} = θ^{δ} ° \hat{p}, {\bar{u}}^{δ} = u^{δ} ° \bar{p}, {\bar{θ}}^{δ} = θ^{δ} ° \bar{p},

to denote unknown fields in the scaled adhesive and adherents. In the same way:

{\hat{f}}^{δ} = f^{δ} ° \hat{p}, {\hat{g}}^{δ} = g^{δ} ° \hat{p}, {\hat{θ}}_{0}^{δ} = θ_{0}^{δ} ° \hat{p}, {\bar{f}}^{δ} = f^{δ} ° \bar{p}, {\bar{g}}^{δ} = g^{δ} ° \bar{p}, {\bar{θ}}_{0}^{δ} = θ_{0}^{δ} ° \bar{p} .

Here and further, by $ϕ ° ψ$ , we standardly denote the superposition of two admissible functions $ϕ$ and $ψ$ . We emphasize that $\bar{p}$ maps $Ω_{\pm}$ to $Ω_{\pm}^{δ}$ , meanwhile we denote the preimages of $Ω_{m}^{δ}$ and $γ_{\pm}^{δ}$ under $\hat{p}$ by $Ω_{m} = {0 < z_{1} < l^{*}} \times {- 1 / 2 < z_{2} < 1 / 2}$ and $γ_{\pm} = {0 < z_{1} < l^{*}} \times {z_{2} = \pm 1 / 2}$ .

For simplicity, we assume that the exterior force $f^{δ}$ and the source $g^{δ}$ act only on the adherents, more certainly, they have the forms:

\begin{matrix} f^{δ} (y, t) = f (y_{1}, y_{2} \mp δ / 2, t) = f (x, t) for (y, t) \in Ω_{\pm}^{δ} \times [0, T], \\ i . e ., for (x, t) \in Ω_{\pm} \times [0, T], \\ g^{δ} (y, t) = g (y_{1}, y_{2} \mp δ / 2, t) = g (x, t) for (y, t) \in Ω_{\pm}^{δ} \times [0, T], \\ i . e ., for (x, t) \in Ω_{\pm} \times [0, T], \\ f^{δ} (y, t) = 0, g^{δ} (y, t) = 0 for (y, t) \in Ω_{m}^{δ} \times [0, T], \\ i . e ., for (z, t) \in Ω_{m} \times [0, T], \end{matrix}

where $f \in L^{2} (Ω)^{2}$ and $g \in L^{2} (Ω)$ are the given functions and $\partial_{t} f$ belongs to $L^{2} (Ω)^{2}$ , and that the initial distribution of temperature $θ_{0}^{δ}$ has the form:

\begin{matrix} θ_{0}^{δ} (y) = θ_{0} (y_{1}, y_{2} \mp δ / 2) = θ_{0} (x) for y \in Ω_{\pm}^{δ}, i . e ., for x \in Ω_{\pm}, \\ θ_{0}^{δ} (y) = θ_{m 0} (y_{1}, δ^{- 1} y_{2}) = θ_{m 0} (z) for y \in Ω_{m}^{δ}, i . e ., for z \in Ω_{m}, \end{matrix}

(9)

where $θ_{0} \in H^{1} (Ω_{\pm})$ and $θ_{m 0} \in H^{1} (Ω_{m})$ are given functions such that:

θ_{0} (\cdot, 0 +) = θ_{m 0} (\cdot, 1 / 2 - 0) = θ_{m 0} (\cdot, 0 -) = θ_{0} (\cdot, - 1 / 2 + 0) \in H^{1 / 2} (0, l^{*}) .

(10)

Here recall that $H^{1 / 2} (0, l^{*})$ is the space where the traces of the functions from $H^{1} (Ω_{\pm})$ on $γ_{\pm}$ belong to.

Furthermore, for $ϕ^{δ} : = μ^{δ}, λ^{δ}, α^{δ}, c^{δ}, κ^{δ}$ , we assume that:

\begin{matrix} ϕ^{δ} (y) = ϕ (y_{1}, y_{2} \mp θ / 2) = ϕ (x) for y \in Ω_{\pm}^{δ}, i . e ., for x \in Ω_{\pm}, \\ ϕ^{δ} (y) = δ^{- 1} ϕ_{m} (y_{1}, δ^{- 1} y_{2}) = δ^{- 1} ϕ_{m} (z) for y \in Ω_{m}^{δ}, i . e ., for z \in Ω_{m}, \end{matrix}

(11)

where $ϕ : = μ, λ, α, c, κ$ and $ϕ_{m} : = μ_{m}, λ_{m}, α_{m}, c_{m}, κ_{m}$ are the given functions such that $ϕ_{*} \leq ϕ (x) \leq ϕ^{*}$ for a.e. $x \in Ω$ and $ϕ_{*} \leq ϕ_{m} (z) \leq ϕ^{*}$ for a.e. $z \in Ω_{m}$ with some positive constants $ϕ_{*}, ϕ^{*}$ . (Here, we mean that $ϕ_{*} : = μ_{*}, λ_{*}, \dots, ϕ^{*} : = μ^{*}, λ^{*}, \dots, and so on$ .)

The demand that $ϕ^{δ} = O (δ^{- 1})$ in $Ω_{m}^{δ}$ as $δ \to 0 +$ is related to the desire to obtain a thermoelastic thin inclusion as a limit on the place of the adhesive. In this paper, we do not make any further assumptions about the asymptotic behavior of the known and unknown functions.

Now, in equations (7a) and (7b) with account of equation (8), we fulfill the inverse changes of variables ${\hat{p}}^{- 1} : (y_{1}, y_{2}) \mapsto (z_{1}, z_{2})$ in $Ω_{m}^{δ}$ and ${\bar{p}}^{- 1} : (y_{1}, y_{2}) \mapsto (x_{1}, x_{2})$ in $Ω_{\pm}^{δ}$ . As before, for any function $ϕ : Ω^{δ} \mapsto R^{d} (d = 1, 2)$ , we denote ${\hat{ϕ}}^{δ} = ϕ^{δ} ° \hat{p}$ and ${\bar{ϕ}}^{δ} = ϕ^{δ} ° \bar{p}$ .

We note that:

\begin{matrix} {div}_{y} v = \partial_{z_{1}} {\hat{v}}_{1} + δ^{- 1} \partial_{z_{2}} {\hat{v}}_{2}, \\ ε_{y} (v) = (\begin{matrix} \partial_{z_{1}} {\hat{v}}_{1} & (1 / 2) (δ^{- 1} \partial_{z_{2}} {\hat{v}}_{1} + \partial_{z_{1}} {\hat{v}}_{2}) \\ (1 / 2) (δ^{- 1} \partial_{z_{2}} {\hat{v}}_{1} + \partial_{z_{1}} {\hat{v}}_{2}) & δ^{- 1} \partial_{z_{2}} {\hat{v}}_{2} \end{matrix}), \\ σ_{y} (v) = (\begin{matrix} \begin{matrix} δ^{- 1} (2 μ_{m} + λ_{m}) \partial_{z_{1}} {\hat{v}}_{1} \\ + δ^{- 2} λ_{m} \partial_{z_{2}} {\hat{v}}_{2} \end{matrix} & δ^{- 1} μ_{m} (δ^{- 1} \partial_{z_{2}} {\hat{v}}_{1} + \partial_{z_{1}} {\hat{v}}_{2}) \\ δ^{- 1} μ_{m} (δ^{- 1} \partial_{z_{2}} {\hat{v}}_{1} + \partial_{z_{1}} {\hat{v}}_{2}) & \begin{matrix} δ^{- 2} (2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{v}}_{2} \\ + δ^{- 1} λ_{m} \partial_{z_{1}} {\hat{v}}_{1} \end{matrix} \end{matrix}), \end{matrix}

for $y \in Ω_{m}^{δ}$ , i.e., for $z \in Ω_{m}$ , for any rather regular vector-function $v = (v_{1} (y), v_{2} (y))$ . For the sake of brevity, we introduce the integral expressions:

A^{u} (\bar{u} (t), \bar{θ} (t), \bar{v}) = \int_{Ω_{+} \cup Ω_{-}} (2 μ ε_{x} (\bar{u} (t)) : ε_{x} (\bar{v}) + (λ {div}_{x} \bar{u} (t) - α \bar{θ} (t)) {div}_{x} \bar{v}) dx,

(12a)

\begin{matrix} A^{θ} (\bar{u} (t), \bar{θ} (t), \bar{ξ}) = \int_{Ω_{+} \cup Ω_{-}} (c \partial_{t} \bar{θ} (t) + α {div}_{x} \partial_{t} \bar{u} (t)) \bar{ξ} dx \\ + \int_{Ω_{+} \cup Ω_{-}} κ \nabla_{x} \bar{θ} (t) \cdot \nabla_{x} \bar{ξ} dx . \end{matrix}

(12b)

In equations (7a) and (7b), we take the test functions $v$ and $ξ$ satisfying the structural demands:

v = v^{δ} (y) = {\begin{matrix} \bar{v} (y_{1}, y_{2} \mp δ / 2) = \bar{v} (x) for y \in Ω_{\pm}^{δ}, i . e ., for x \in Ω_{\pm}, \\ \hat{v} (y_{1}, δ^{- 1} y_{2}) = \hat{v} (z) for y \in Ω_{m}^{δ}, i . e ., for z \in Ω_{m}, \end{matrix}

(13a)

ξ = ξ^{δ} (y) = {\begin{matrix} \bar{ξ} (y_{1}, y_{2} \mp δ / 2) = \bar{ξ} (x) for y \in Ω_{\pm}^{δ}, i . e ., for x \in Ω_{\pm}, \\ \hat{ξ} (y_{1}, δ^{- 1} y_{2}) = \hat{ξ} (z) for y \in Ω_{m}^{δ}, i . e ., for z \in Ω_{m}, \end{matrix}

(13b)

where $\bar{v}$ , $\hat{v}$ , $\bar{ξ}$ , and $\hat{ξ}$ are the arbitrary functions that meet the matching conditions:

\bar{v} (\cdot, 0 +) = \hat{v} (\cdot, 1 / 2 - 0), \hat{v} (\cdot, - 1 / 2 + 0) = \bar{v} (\cdot, 0 -) on [0, l^{*}],

(13c)

\bar{ξ} (\cdot, 0 +) = \hat{ξ} (\cdot, 1 / 2 - 0), \hat{ξ} (\cdot, - 1 / 2 + 0) = \bar{ξ} (\cdot, 0 -) on [0, l^{*}] .

(13d)

Remark that equations (13c) and (13d) ensure that $v^{δ} \in H^{1} (Ω^{δ})^{2}$ and $ξ^{δ} \in H^{1} (Ω^{δ})$ . Thus, $v^{δ}$ and $ξ^{δ}$ are the admissible test functions.

Using the above-introduced assumptions and expressions, after the change of variables ${\hat{p}}^{- 1}$ and ${\bar{p}}^{- 1}$ , we write out equations (7a) and (7b) as follows:

\begin{matrix} A^{u} ({\bar{u}}^{δ} (t), {\bar{θ}}^{δ} (t), \bar{v}) \\ + \int_{Ω_{m}} {(2 μ_{m} + λ_{m}) \partial_{z_{1}} {\hat{u}}_{1}^{δ} (t) \partial_{z_{1}} {\hat{v}}_{1} + μ_{m} \partial_{z_{1}} {\hat{u}}_{2}^{δ} (t) \partial_{z_{1}} {\hat{v}}_{2} - α_{m} {\hat{θ}}^{δ} (t) \partial_{z_{1}} {\hat{v}}_{1} \\ + δ^{- 1} [λ_{m} \partial_{z_{2}} {\hat{u}}_{2}^{δ} (t) \partial_{z_{1}} {\hat{v}}_{1} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{δ} (t) \partial_{z_{1}} {\hat{v}}_{2} + μ_{m} \partial_{z_{1}} {\hat{u}}_{2}^{δ} (t) \partial_{z_{2}} {\hat{v}}_{1} \\ + λ_{m} \partial_{z_{1}} {\hat{u}}_{1}^{δ} (t) \partial_{z_{2}} {\hat{v}}_{2} - α_{m} {\hat{θ}}^{δ} (t) \partial_{z_{2}} {\hat{v}}_{2}] \\ + δ^{- 2} [(2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{u}}_{2}^{δ} (t) \partial_{z_{2}} {\hat{v}}_{2} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{δ} (t) \partial_{z_{2}} {\hat{v}}_{1}]} dz = \\ = \int_{Ω_{+} \cup Ω_{-}} f (t) \cdot \bar{v} dx, \forall t \in [0, T], \end{matrix}

(14a)

\begin{matrix} A^{θ} ({\bar{u}}^{δ} (t), {\bar{θ}}^{δ} (t), \bar{ξ}) \\ + \int_{Ω_{m}} {(c_{m} \partial_{t} {\hat{θ}}^{δ} (t) + α_{m} \partial_{z_{1}} \partial_{t} {\hat{u}}_{1}^{δ} (t)) \hat{ξ} + κ_{m} \partial_{z_{1}} {\hat{θ}}^{δ} (t) \partial_{z_{1}} \hat{ξ} \\ + δ^{- 1} α_{m} \partial_{z_{2}} \partial_{t} {\hat{u}}_{2}^{δ} (t) \hat{ξ} + δ^{- 2} κ_{m} \partial_{z_{2}} {\hat{θ}}^{δ} (t) \partial_{z_{2}} \hat{ξ}} dz = \\ = \int_{Ω_{+} \cup Ω_{-}} g (t) \bar{ξ} dx, \forall t \in (0, T] . \end{matrix}

(14b)

3.2. Formal asymptotic analysis

Next, we make use of the method of formal asymptotic expansions (see, e.g., [21, 23, 24]). We postulate the ansatz:

\begin{matrix} {\hat{u}}^{δ} = {\hat{u}}^{0} + δ {\hat{u}}^{1} + δ^{2} {\hat{u}}^{2} + \dots, \\ {\hat{θ}}^{δ} = {\hat{θ}}^{0} + δ {\hat{θ}}^{1} + δ^{2} {\hat{θ}}^{2} + \dots, \end{matrix} \begin{matrix} {\bar{u}}^{δ} = {\bar{u}}^{0} + δ {\bar{u}}^{1} + δ^{2} {\bar{u}}^{2} + \dots, \\ {\bar{θ}}^{δ} = {\bar{θ}}^{0} + δ {\bar{θ}}^{1} + δ^{2} {\bar{θ}}^{2} + \dots, \end{matrix}

(15)

and substitute these expansions into equations (14a) and (14b). As usual in formal asymptotic analysis, we equate the coefficients before $δ^{l} (l = - 2, - 1, 0, 1, \dots)$ to zero, thus obtaining the set of variational equations for finding ${\hat{u}}^{l}$ , ${\hat{θ}}^{l}$ , ${\bar{u}}^{l}$ , and ${\bar{θ}}^{l} (l = 0, 1, 2, \dots)$ :

δ^{- 2} : \int_{Ω_{m}} [(2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{u}}_{2}^{0} (t) \partial_{z_{2}} {\hat{v}}_{2} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{0} (t) \partial_{z_{2}} {\hat{v}}_{1}] dz = 0,

(16a)

\int_{Ω_{m}} κ_{m} \partial_{z_{2}} {\hat{θ}}^{0} (t) \partial_{z_{2}} \hat{ξ} dz = 0;

(16b)

\begin{matrix} δ^{- 1} : \int_{Ω_{m}} [λ_{m} \partial_{z_{2}} {\hat{u}}_{2}^{0} (t) \partial_{z_{1}} {\hat{v}}_{1} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{0} (t) \partial_{z_{1}} {\hat{v}}_{2} + μ_{m} \partial_{z_{1}} {\hat{u}}_{2}^{0} (t) \partial_{z_{2}} {\hat{v}}_{1} \\ + λ_{m} \partial_{z_{1}} {\hat{u}}_{1}^{0} (t) \partial_{z_{2}} {\hat{v}}_{2} - α_{m} {\hat{θ}}^{0} (t) \partial_{z_{2}} {\hat{v}}_{2} \\ + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{1} (t) \partial_{z_{2}} {\hat{v}}_{1} + (2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{u}}_{2}^{1} (t) \partial_{z_{2}} {\hat{v}}_{2}] dz = 0, \end{matrix}

(16c)

\int_{Ω_{m}} [α_{m} (\partial_{z_{2}} \partial_{t} {\hat{u}}_{2}^{0} (t) \hat{ξ} + κ_{m} \partial_{z_{2}} {\hat{θ}}^{1} (t) \partial_{z_{2}} \hat{ξ}] dz = 0;

(16d)

\begin{matrix} δ^{0} : A^{u} ({\bar{u}}^{0} (t), {\bar{θ}}^{0} (t), \bar{v}) \\ + \int_{Ω_{m}} [(2 μ_{m} + λ_{m}) \partial_{z_{1}} {\hat{u}}_{1}^{0} (t) \partial_{z_{1}} {\hat{v}}_{1} + μ_{m} \partial_{z_{1}} {\hat{u}}_{2}^{0} (t) \partial_{z_{1}} {\hat{v}}_{2} \\ + λ_{m} \partial_{z_{2}} {\hat{u}}_{2}^{1} (t) \partial_{z_{1}} {\hat{v}}_{1} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{1} (t) \partial_{z_{1}} {\hat{v}}_{2} + μ_{m} \partial_{z_{1}} {\hat{u}}_{2}^{1} (t) \partial_{z_{2}} {\hat{v}}_{1} \\ + λ_{m} \partial_{z_{1}} {\hat{u}}_{1}^{1} (t) \partial_{z_{2}} {\hat{v}}_{2} + μ_{m} \partial_{z_{2}} {\hat{u}}_{1}^{2} (t) \partial_{z_{2}} {\hat{v}}_{1} \\ + (2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{u}}_{2}^{2} (t) \partial_{z_{2}} {\hat{v}}_{2} \\ - α_{m} {\hat{θ}}^{0} (t) \partial_{z_{1}} {\hat{v}}_{1} - α_{m} {\hat{θ}}^{1} (t) \partial_{z_{2}} {\hat{v}}_{2}] dz = \int_{Ω_{+} \cup Ω_{-}} f (t) \cdot \bar{v} dx, \end{matrix}

(16e)

\begin{matrix} A^{θ} ({\bar{u}}^{0} (t), {\bar{θ}}^{0} (t), \bar{ξ}) \\ + \int_{Ω_{m}} [c_{m} \partial_{t} {\hat{θ}}^{0} (t) \hat{ξ} + κ_{m} \partial_{z_{1}} {\hat{θ}}^{0} (t) \partial_{z_{1}} \hat{ξ} + κ_{m} \partial_{z_{2}} {\hat{θ}}^{2} (t) \partial_{z_{2}} \hat{ξ} \\ + α_{m} \partial_{z_{1}} \partial_{t} {\hat{u}}_{1}^{0} (t) \hat{ξ} + α_{m} \partial_{z_{2}} \partial_{t} {\hat{u}}_{2}^{0} (t) \hat{ξ}] dz = \int_{Ω_{+} \cup Ω_{-}} g (t) \bar{ξ} dx, \end{matrix}

(16f)

and so on.

Due to sufficient arbitrariness of $\hat{v}$ and $\hat{ξ}$ , from equations (16a) and (16b), we derive that ${\hat{u}}^{0}$ and ${\hat{θ}}^{0}$ do not depend on $z_{2}$ a.e. in $Ω_{m}$ , i.e.,

{\hat{u}}^{0} = {\hat{u}}^{0} (z_{1}), {\hat{θ}}^{0} = {\hat{θ}}^{0} (z_{1}) a . e . in Ω_{m} .

(17)

Using equation (17), from equations (16c) and (16d), we deduce the simpler integral equalities:

\int_{Ω_{m}} (λ_{m} \partial_{z_{1}} {\hat{u}}_{1}^{0} (t) - α_{m} {\hat{θ}}^{0} (t) + (2 μ_{m} + λ_{m}) \partial_{z_{2}} {\hat{u}}_{2}^{1} (t)) \partial_{z_{2}} {\hat{v}}_{2} dz = 0,

(18a)

\int_{Ω_{m}} μ_{m} (\partial_{z_{1}} {\hat{u}}_{2}^{0} (t) + \partial_{z_{2}} {\hat{u}}_{1}^{1} (t)) \partial_{z_{2}} {\hat{v}}_{1} dz = 0,

(18b)

\int_{Ω_{m}} κ_{m} \partial_{z_{2}} {\hat{θ}}^{1} (t) \partial_{z_{2}} \hat{ξ} dz = 0 .

(18c)

Here, in order to obtain equation (18a), we take ${\hat{v}}_{1} = 0$ and ${\hat{v}}_{2} \neq 0$ in equation (16c), and, in order to obtain equation (18b), we take ${\hat{v}}_{1} \neq 0$ and ${\hat{v}}_{2} = 0$ in equation (16c). From equation (18), we readily deduce that:

\partial_{z_{2}} {\hat{u}}_{1}^{1} = - \partial_{z_{1}} {\hat{u}}_{2}^{0}, \partial_{z_{2}} {\hat{u}}_{2}^{1} = \frac{α_{m} {\hat{θ}}^{0} - λ_{m} \partial_{z_{1}} {\hat{u}}_{1}^{0}}{2 μ_{m} + λ_{m}}, {\hat{θ}}^{1} = {\hat{θ}}^{1} (z_{1}) a . e . in Ω_{m} .

(19)

In equations (16e) and (16f), we take test functions such that:

\partial_{z_{2}} \hat{v} = 0, \partial_{z_{2}} \hat{ξ} = 0 in Ω_{m},

and insert relations (17) and (19) to get:

\begin{matrix} A^{u} ({\bar{u}}^{0} (\cdot, t), {\bar{θ}}^{0} (\cdot, t), \bar{v} (\cdot)) \\ + \int_{Ω_{m}} [(2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})) \partial_{z_{1}} {\hat{u}}_{1}^{0} (z_{1}, t) \\ + \frac{λ_{m} (z_{1}, z_{2})}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})} (α_{m} (z_{1}, z_{2}) {\hat{θ}}^{0} (z_{1}, t) - λ_{m} (z_{1}, z_{2}) \partial_{z_{1}} {\hat{u}}_{1}^{0} (z_{1}, t)) \\ - α_{m} (z_{1}, z_{2}) {\hat{θ}}^{0} (z_{1}, t)] \partial_{z_{1}} {\hat{v}}_{1} (z_{1}) dz = \int_{Ω_{+} \cup Ω_{-}} f (x, t) \cdot \bar{v} (x) dx, \end{matrix}

(20a)

\begin{matrix} A^{θ} ({\bar{u}}^{0} (\cdot, t), {\bar{θ}}^{0} (\cdot, t), \bar{ξ} (\cdot)) \\ + \int_{Ω_{m}} [(c_{m} (z_{1}, z_{2}) \partial_{t} {\hat{θ}}^{0} (z_{1}, t) + α_{m} (z_{1}, z_{2}) \partial_{z_{1}} \partial_{t} {\hat{u}}_{1}^{0} (z_{1}, t)) \hat{ξ} (z_{1}) \\ + κ_{m} (z_{1}, z_{2}) \partial_{z_{1}} {\hat{θ}}^{0} (z_{1}, t) \partial_{z_{1}} \hat{ξ} (z_{1}) \\ + \frac{α_{m}^{2} (z_{1}, z_{2})}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})} \partial_{t} {\hat{θ}}^{0} (z_{1}, t) \hat{ξ} (z_{1}) \\ - \frac{α_{m} (z_{1}, z_{2}) λ_{m} (z_{1}, z_{2})}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})} \partial_{z_{1}} \partial_{t} {\hat{u}}_{1}^{0} (z_{1}, t) \hat{ξ} (z_{1})] dz = \\ = \int_{Ω_{+} \cup Ω_{-}} g (x, t) \bar{ξ} (x) dx . \end{matrix}

(20b)

Introduce the notation:

\begin{matrix} c_{in} = c_{in} (z_{1}) : = \int_{- \frac{1}{2}}^{\frac{1}{2}} (c_{m} (z_{1}, z_{2}) + \frac{α_{m}^{2} (z_{1}, z_{2})}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})}) d z_{2}, \\ κ_{in} = κ_{in} (z_{1}) : = \int_{- \frac{1}{2}}^{\frac{1}{2}} κ_{m} (z_{1}, z_{2}) d z_{2}, \\ α_{in} = α_{in} (z_{1}) : = \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 α_{m} (z_{1}, z_{2}) μ_{m} (z_{1}, z_{2})}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})} d z_{2}, \\ E_{in} = E_{in} (z_{1}) : = \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{4 μ_{m} (z_{1}, z_{2}) (μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2}))}{2 μ_{m} (z_{1}, z_{2}) + λ_{m} (z_{1}, z_{2})} d z_{2} . \end{matrix}

(21)

Using this notation, re-denoting $x_{1} : = z_{1}$ in the integrals over $Ω_{m}$ and, accordingly, noticing that $γ = {0 < x_{1} < l^{*}}$ , we reduce equation (20) to the form:

\begin{matrix} A^{u} ({\bar{u}}^{0} (\cdot, t), {\bar{θ}}^{0} (\cdot, t), \bar{v} (\cdot)) \\ + \int_{γ} (E_{in} (x_{1}) \partial_{x_{1}} {\hat{u}}_{1}^{0} (x_{1}, t) - α_{in} (x_{1}) {\hat{θ}}^{0} (x_{1}, t)) \partial_{x_{1}} {\hat{v}}_{1} (x_{1}) d x_{1} = \\ = \int_{Ω_{+} \cup Ω_{-}} f (x, t) \cdot \bar{v} (x) dx, \end{matrix}

(22a)

\begin{matrix} A^{θ} ({\bar{u}}^{0} (\cdot, t), {\bar{θ}}^{0} (\cdot, t), \bar{ξ} (\cdot)) \\ + \int_{γ} (c_{in} (x_{1}) (\partial_{t} {\hat{θ}}^{0} (x_{1}, t)) \hat{ξ} (x_{1}) + κ_{in} (x_{1}) \partial_{x_{1}} {\hat{θ}}^{0} (x_{1}, t) \partial_{x_{1}} \hat{ξ} (x_{1}) \\ + α_{in} (x_{1}) (\partial_{x_{1}} \partial_{t} {\hat{u}}_{1}^{0} (x_{1}, t)) \hat{ξ} (x_{1})) d x_{1} = \int_{Ω_{+} \cup Ω_{-}} g (x, t) \bar{ξ} (x) dx . \end{matrix}

(22b)

3.3. Setting the limit problem

Let us make several final assumptions, notations, and remarks regarding the integral equalities (22).

Suppose that the leading terms ${\hat{u}}^{0}$ , ${\hat{θ}}^{0}$ , ${\bar{u}}^{0}$ , and ${\bar{θ}}^{0}$ in the formal expansions (15) inherit the smoothness of the solution $(u^{δ}, θ^{δ})$ of Problem A_var. Then, in particular, due to equation (17) and notation $x_{1} : = z_{1}$ , the following matching conditions hold:

{\bar{u}}^{0} (x_{1}, 0 \pm) = {\hat{u}}^{0} (x_{1}), {\bar{θ}}^{0} (x_{1}, 0 \pm) = {\hat{θ}}^{0} (x_{1}) for a . e . x_{1} \in (0, l^{*}) .

(23)

By $H_{0}^{1} (γ) = H_{0}^{1} (0, l^{*})$ denote the standard Sobolev space of functions defined on the one-dimensional set (interval) ${0 < x_{1} < l^{*}}$ . This space is equipped with the norm:

∥ ϕ ∥_{H^{1} (γ)} = {(∥ ϕ ∥_{L^{2} (γ)}^{2} + ∥ \partial_{x_{1}} ϕ ∥_{L^{2} (γ)}^{2})}^{1 / 2}, ϕ \in H_{0}^{1} (γ) .

(24)

Simplify the notation and denote:

Ω : = (Ω_{+} \cup Ω_{-} \cup γ) \subset R_{x}^{2} .

(25)

Similarly to Sazhenkov et al. [37, section 1.4], we introduce the space:

V = {(v, ξ) \in H_{0}^{1} {(Ω)}^{2} \times H_{0}^{1} (Ω) : v_{1} |_{γ} \in H_{0}^{1} (γ), ξ |_{γ} \in H_{0}^{1} (γ)} .

(26)

Note that the limiting relation:

\begin{matrix} \int_{Ω^{δ}} θ_{0}^{δ} (y) ξ^{δ} (y) dy = \\ = \int_{Ω_{+} \cup Ω_{-}} θ_{0} (x) \bar{ξ} (x) dx + δ \int_{Ω_{m}} θ_{m 0} (z) \hat{ξ} (z) dz \underset{δ \to 0 +}{\to} \int_{Ω} θ_{0} (x) \bar{ξ} (x) dx, \end{matrix}

(27)

holds true due to equations (9) and (13b). This means that $θ_{0}^{δ} \underset{δ \to 0 +}{\to} θ_{0}$ , in a sense. Moreover, note that $θ_{0}^{δ} \in V_{θ}$ and $θ_{0} \in H_{0}^{1} (Ω)$ due to equation (10).

In line with equation (23), for the sake of brevity, we omit the upper caps ^{^} and ^¯ and the superscripts “zero” in notation for the leading terms:

u : = {\hat{u}}^{0}, θ : = {\hat{θ}}^{0} on γ \times [0, T], u : = {\bar{u}}^{0}, θ : = {\bar{θ}}^{0} on (Ω_{+} \cup Ω_{-}) \times [0, T] .

(28)

Using equations (12), (25), and (28), we rewrite equation (22) in a modified form, and, taking into account the rest of the above arguments, we finally set up the limit variational formulation:

Problem B_var. For $t \in [0, T]$ , find a pair of functions $(u (t), θ (t)) \in V$ such that:

\begin{matrix} \int_{Ω} (2 μ ε_{x} (u (t)) : ε_{x} (v) + (λ {div}_{x} u (t) - α θ (t)) {div}_{x} v) dx \\ + \int_{γ} (E_{in} \partial_{x_{1}} u_{1} (t) - α_{in} θ (t)) \partial_{x_{1}} v_{1} d x_{1} = \int_{Ω} f (t) \cdot v dx, \forall t \in [0, T], \end{matrix}

(29a)

\begin{matrix} \int_{Ω} ((c \partial_{t} θ (t) + α {div}_{x} \partial_{t} u (t)) ξ + κ \nabla_{x} θ (t) \cdot \nabla_{x} ξ) dx \\ + \int_{γ} ((c_{in} \partial_{t} θ (t) + α_{in} \partial_{x_{1}} \partial_{t} u_{1} (t)) ξ + κ_{in} \partial_{x_{1}} θ (t) \partial_{x_{1}} ξ) d x_{1} = \int_{Ω} g (t) ξ dx, \\ \forall t \in (0, T], \\ for all (v, ξ) \in V, \end{matrix}

(29b)

∥ θ (\cdot, t) - θ_{0} (\cdot) ∥_{L^{2} (Ω)} \underset{t \to 0 +}{\to} 0 .

(29c)

In the formulation of Problem B_var, the exterior volumetric force $f$ , the distributed heat source $g$ , the initial temperature field $θ_{0}$ , and the coefficients $μ$ , $λ$ , $α$ , $c$ , $κ$ , $E_{in}$ , $α_{in}$ , $c_{in}$ , and $κ_{in}$ are given. In equations (29a) and (29b), the coefficients $E_{in}$ , $α_{in}$ , $c_{in}$ , and $κ_{in}$ are calculated by the formulas (21).

Remark 1. Asymptotics (15) mean that the solution $(u, θ)$ of Problem B_var is a limit, to which the family of functions $(u^{δ}, θ^{δ})$ tends as $δ \to 0 +$ , at least, formally. In line with this, Problem B_var is the limit problem arising as $δ$ tends to zero.

Remark 2. In the sense of distributions, a solution of Problem B_var (if any) is a weak solution of the following initial-boundary value problem for the elliptic–parabolic system of partial differential equations.

Problem B_diff. In the space-time cylinder $Q_{T} : = Ω \times (0, T)$ , we find a pair of functions $u = (u_{1} (x, t), u_{2} (x, t))$ and $θ = θ (x, t)$ satisfying the following system of equations and boundary and initial conditions:

- {div}_{x} (2 μ ε_{x} (u) + (λ {div}_{x} u - α θ) I) = f, (x, t) \in (Ω \ γ) \times (0, T),

(30a)

\begin{matrix} - {div}_{x} (2 μ ε_{x} (u (0)) \\ + (λ {div}_{x} u (0) - α θ_{0}) I) = f (0), (x, t) \in (Ω \ γ) \times {t = 0}, \end{matrix}

(30b)

c \partial_{t} θ - {div}_{x} (κ \nabla_{x} θ) + α {div}_{x} \partial_{t} u = g, (x, t) \in (Ω \ γ) \times (0, T),

(30c)

u = 0, θ = 0 on Γ,

(30d)

\begin{matrix} [(2 μ ε_{x} (u) + (λ {div}_{x} u - α θ) I) ν] = \\ = \partial_{x_{1}} (- E_{in} \partial_{x_{1}} u_{1} + α_{in} θ), (x_{1}, t) \in γ \times (0, T), \end{matrix}

(30e)

\begin{matrix} [κ \nabla_{x} θ \cdot ν] = \\ = c_{in} \partial_{t} θ - \partial_{x_{1}} (κ_{in} \partial_{x_{1}} θ + α_{in} \partial_{t} \partial_{x_{1}} u_{1}), (x_{1}, t) \in γ \times (0, T), \end{matrix}

(30f)

θ (\cdot, 0) = θ_{0}, x \in Ω,

(30g)

where $Γ = \partial Ω$ , as usually, $ν = (0, 1)$ is a unit normal to $γ$ , and […] designates a jump on $γ$ . Here, equations (30a)–(30c) are the standard equations of a quasi-static behavior of the thermoelastic body. Conditions (30d) characterize immobility and isothermicity of the body edge. Equations (30e) and (30f) describe a quasi-static behavior of the thermoelastic rod $γ$ incorporated in the body $Ω$ , wherein the force $[(2 μ ε_{x} (u) + (λ {div}_{x} u - α θ) I) ν]$ and the distributed heat source $[κ \nabla_{x} θ \cdot ν]$ act on the rod $γ$ by the body $Ω$ . On the contrary, conditions (30e) and (30f) are the so-called imperfect conditions between two coupled bodies $Ω_{-}$ and $Ω_{+}$ (see, e.g., [5, 6, 18, 24]). It is notable that they are of the Ventcel type. In view of this, see the paper [44], in which the Ventcel-type equations for strong interfaces were obtained by means of $Γ$ -convergence, and see the work [23], in which the authors study similar interfaces related to thermoelasticity. It is worth to note that in Problem B_diff, the governing equations for the inclusion are set up on the subset of codimension one (with respect to the domain, occupied by the entire body). Relation (30g) is the initial condition for temperature $θ_{0}$ . In turn, the initial distribution of displacements $u (\cdot, 0)$ is not prescribed but should be determined as the solution of the Dirichlet problems (30b) and (30d)₁.

4. Well-posedness of the model with a thin inclusion

4.1. Existence of a solution to Problem B_var

The following theorem manifests existence of at least one solution to Problem B_var.

Theorem 1. Whenever $θ_{0} \in H_{0}^{1} (Ω)$ , there is at least one solution $(u (t), θ (t)) \in V$ to Problem B_var having the additional regularity properties:

\begin{matrix} \partial_{t} u \in L^{2} (0, T; H_{0}^{1} {(Ω)}^{2}), \partial_{t} u_{1} \in L^{2} (0, T; H_{0}^{1} (γ)), \\ \partial_{t} θ \in L^{2} (0, T; L^{2} (Ω) \cap L^{2} (γ)) . \end{matrix}

Proof is based on the application of the Galerkin method and consists of the several steps.

Step 1. The Galerkin system. Before constructing the Galerkin system, let us define a special basis in $V$ . Namely, in $V$ and, additionally, in:

L : = ((L^{2} (Ω) \cap L^{2} (γ)) \times L^{2} (Ω)) \times (L^{2} (Ω) \cap L^{2} (γ)),

we introduce the scalar products and corresponding norms in the form associated with the considered problem. That is, we set:

\begin{matrix} {((u, θ), (v, ξ))}_{V}^{*} : = \int_{Ω} (2 μ ε_{x} (u) : ε_{x} (v) + λ {div}_{x} u {div}_{x} v) dx \\ + \int_{γ} E_{in} \partial_{x_{1}} u_{1} \partial_{x_{1}} v_{1} d x_{1} + \int_{Ω} κ \nabla_{x} θ \cdot \nabla_{x} ξ dx + \int_{γ} κ_{in} \partial_{x_{1}} θ \partial_{x_{1}} ξ d x_{1}, \end{matrix}

(31)

∥ (u, θ) ∥_{V}^{*} : = \sqrt{{((u, θ), (u, θ))}_{V}^{*}};

(32)

{((u, θ), (v, ξ))}_{L}^{*} : = \int_{Ω} u \cdot vdx + \int_{γ} u_{1} v_{1} d x_{1} + \int_{Ω} c θ ξ dx + \int_{γ} c_{in} θ ξ d x_{1},

(33)

∥ (u, θ) ∥_{L}^{*} : = \sqrt{{((u, θ), (u, θ))}_{L}^{*}} .

(34)

On the strength of the Korn and the Poincaré–Friedrichs inequalities [45, Proposition 2.11], [46, Chapter I, section 6], we have that equations (32) and (34) are the norms equivalent, respectively, to the norm in $V$ that was introduced earlier and to the standard norm in $L$ .

By the Rellich theorem, $V$ is compactly embedded into $L$ :

V ↪ ↪ ℒ .

Let us recall the well-known result of the spectral theory of linear compact operators [47, Chapter 2, section 5].

Proposition 2. Assume $X$ and $Y$ are the Hilbert spaces such that $X ↪ ↪ Y$ . Then, there exist a sequence of real positive eigenvalues ${\hat{ψ} = {\hat{ψ}}_{n}}_{n = 1, 2, . . .} \subset X$ and the corresponding set of linearly independent eigenfunctions ${U = U_{n}}_{n = 1, 2, . . .} \subset X$ satisfying the spectral problem:

(U, ϕ)_{X} = \hat{ψ} (U, ϕ)_{Y}, \forall ϕ \in X,

(35)

such that ${U_{n}}_{n = 1, 2, . . .}$ is an orthogonal basis in both $X$ and $Y$ .

In line with Proposition 2 set $X : = V$ (supplemented with the scalar product defined by equation (31)) and $Y : = L$ (supplemented with the scalar product defined by equation (33)). According to Proposition 2, let ${U_{n} = ({\tilde{u}}_{n}, {\tilde{θ}}_{n} {)}}_{n = 1, 2, . . .}$ be an orthogonal basis in $V$ and $L$ . Furthermore, let ${U_{n}}_{n = 1, 2, . . .}$ be orthonormal in $V$ .

Now, using this definition of the basis, we introduce the Galerkin approximations:

u_{m} = \sum_{k = 1}^{m} a_{mk} (t) {\tilde{u}}_{k} (x),

(36)

θ_{m} = \sum_{k = 1}^{m} b_{mk} (t) {\tilde{θ}}_{k} (x),

(37)

and the Galerkin system for Problem B_var (equivalently, for Problem B_diff):

\begin{matrix} \int_{Ω} (2 μ ε_{x} (u_{m} (t)) : ε_{x} ({\tilde{u}}_{k}) + λ {div}_{x} u_{m} (t) {div}_{x} {\tilde{u}}_{k}) dx \\ - \int_{Ω} α θ_{m} (t) {div}_{x} {\tilde{u}}_{k} dx + \int_{γ} (E_{in} \partial_{x_{1}} u_{m 1} (t) - α_{in} θ_{m} (t)) \partial_{x_{1}} {\tilde{u}}_{k 1} d x_{1} = \\ = \int_{Ω} f (t) \cdot {\tilde{u}}_{k} dx, \forall t \in [0, T], \end{matrix}

(38a)

\begin{matrix} \int_{Ω} c (\partial_{t} θ_{m} (t)) {\tilde{θ}}_{k} dx + \int_{Ω} α ({div}_{x} \partial_{t} u_{m} (t)) {\tilde{θ}}_{k} dx + \int_{Ω} κ \nabla_{x} θ_{m} (t) \cdot \nabla_{x} {\tilde{θ}}_{k} dx \\ + \int_{γ} c_{in} (\partial_{t} θ_{m} (t)) {\tilde{θ}}_{k} d x_{1} + \int_{γ} α_{in} (\partial_{x_{1}} \partial_{t} u_{m 1} (t)) {\tilde{θ}}_{k} d x_{1} \\ + \int_{γ} κ_{in} \partial_{x_{1}} θ_{m} (t) \partial_{x_{1}} {\tilde{θ}}_{k} d x_{1} = \int_{Ω} g (t) {\tilde{θ}}_{k} dx, \forall t \in (0, T], \end{matrix}

(38b)

θ_{m} (\cdot, t) = θ_{m 0} (\cdot),

(38c)

where $k = 1, \dots, m$ , the function $θ_{m 0}$ is the $m th$ partial sum of the Fourier series of $θ_{0}$ , and the initial condition (38c) is understood in the strong trace sense, i.e., in the sense of the limiting relation:

∥ θ_{m} (\cdot, t) - θ_{m 0} (\cdot) ∥_{L^{2} (Ω)} \underset{t \to 0 +}{\to} 0 .

Let us show that the problems (36)–(38) for the unknown functions $a_{mk}$ and $b_{mk} (k = 1, \dots, m)$ is well-posed.

Insert equations (36) and (37) into equation (38a) and use notation (31) to get:

\begin{matrix} {((\sum_{l = 1}^{m} a_{ml} (t) {\tilde{u}}_{l}, 0), ({\tilde{u}}_{k}, 0))}_{V}^{*} - \int_{Ω} α \sum_{l = 1}^{m} b_{ml} (t) {\tilde{θ}}_{l} {div}_{x} {\tilde{u}}_{k} dx \\ - \int_{γ} α_{in} \sum_{l = 1}^{m} b_{ml} (t) {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{u}}_{k 1} d x_{1} = \int_{Ω} f (t) \cdot {\tilde{u}}_{k} dx, k = 1, \dots, m, \end{matrix}

(39)

which yields the expression for $a_{mk} (t)$ :

\begin{matrix} a_{mk} (t) = \int_{Ω} α \sum_{l = 1}^{m} b_{ml} (t) {\tilde{θ}}_{l} {div}_{x} {\tilde{u}}_{k} dx + \int_{γ} α_{in} \sum_{l = 1}^{m} b_{ml} (t) {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{u}}_{k 1} d x_{1} \\ + \int_{Ω} f (t) \cdot {\tilde{u}}_{k} dx, k = 1, \dots, m, \end{matrix}

(40)

Insert equations (36) and (37) into equation (38b) to get:

\begin{matrix} \sum_{l = 1}^{m} \frac{d b_{ml} (t)}{dt} \int_{Ω} c {\tilde{θ}}_{l} {\tilde{θ}}_{k} dx + \sum_{l = 1}^{m} b_{ml} (t) \int_{Ω} κ \nabla_{x} {\tilde{θ}}_{l} \cdot \nabla_{x} {\tilde{θ}}_{k} dx \\ + \sum_{l = 1}^{m} \frac{d a_{ml} (t)}{dt} \int_{Ω} α ({div}_{x} {\tilde{u}}_{l}) {\tilde{θ}}_{k} dx + \sum_{l = 1}^{m} \frac{d b_{ml} (t)}{dt} \int_{γ} c_{in} {\tilde{θ}}_{l} {\tilde{θ}}_{k} d x_{1} \\ + \sum_{l = 1}^{m} b_{ml} (t) \int_{γ} κ_{in} \partial_{x_{1}} {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{θ}}_{k} d x_{1} + \sum_{l = 1}^{m} \frac{d a_{ml} (t)}{dt} \int_{γ} α_{in} (\partial_{x_{1}} {\tilde{u}}_{l 1}) {\tilde{θ}}_{k} d x_{1} = \\ = \int_{Ω} g (t) \tilde{θ} dx . \end{matrix}

Joining this equality with equation (40) and using notation (33), we deduce the equality:

\begin{matrix} \sum_{l = 1}^{m} \frac{d b_{ml} (t)}{dt} {((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{L}^{*} + \sum_{l = 1}^{m} [\sum_{r = 1}^{m} \frac{d b_{mr} (t)}{dt} \int_{Ω} α {\tilde{θ}}_{r} {div}_{x} {\tilde{u}}_{l} dx \\ + \sum_{r = 1}^{m} \frac{d b_{mr} (t)}{dt} \int_{γ} α_{in} {\tilde{θ}}_{r} \partial_{x_{1}} {\tilde{u}}_{l 1} d x_{1} + \int_{Ω} \frac{\partial f (t)}{\partial t} \cdot {\tilde{u}}_{l} dx] \int_{Ω} α ({div}_{x} {\tilde{u}}_{l}) {\tilde{θ}}_{k} dx \\ + \sum_{l = 1}^{m} [\sum_{r = 1}^{m} \frac{d b_{mr} (t)}{dt} \int_{Ω} α {\tilde{θ}}_{r} {div}_{x} {\tilde{u}}_{l} dx \\ + \sum_{r = 1}^{m} \frac{d b_{mr} (t)}{dt} \int_{γ} α_{in} {\tilde{θ}}_{r} \partial_{x_{1}} {\tilde{u}}_{l 1} d x_{1} + \int_{Ω} \frac{\partial f (t)}{\partial t} \cdot {\tilde{u}}_{l} dx] \int_{γ} α_{in} (\partial_{x_{1}} {\tilde{u}}_{l 1}) {\tilde{θ}}_{k} d x_{1} \\ + \sum_{l = 1}^{m} b_{ml} (t) (\int_{Ω} κ \nabla_{x} {\tilde{θ}}_{l} \nabla_{x} {\tilde{θ}}_{k} dx + \int_{γ} κ_{in} \partial_{x_{1}} {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{θ}}_{k} d x_{1}) = \int_{Ω} g (t) \tilde{θ} dx, \\ k = 1, \dots, m . \end{matrix}

(41)

We rewrite equation (41) using the notion of the scalar product $(\cdot, \cdot)_{V}^{*}$ and the properties of the basis $({\tilde{u}}_{k}, {\tilde{θ}}_{k})$ . More precisely, arranging the terms properly and exchanging the indices $l$ and $r$ in the double sums, we represent equation (41) in the following form:

\begin{matrix} \sum_{l = 1}^{m} {{((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{L}^{*} + \sum_{r = 1}^{m} [\int_{Ω} α {\tilde{θ}}_{l} {div}_{x} {\tilde{u}}_{r} dx + \int_{γ} α_{in} {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{u}}_{r 1} d x_{1}] \times \\ \times [\int_{Ω} α ({div}_{x} {\tilde{u}}_{r}) {\tilde{θ}}_{k} dx + \int_{γ} α_{in} (\partial_{x_{1}} {\tilde{u}}_{r 1}) {\tilde{θ}}_{k} d x_{1}]} \frac{d b_{ml} (t)}{dt} \\ + \sum_{i = 1}^{m} b_{ml} (t) {((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{V}^{*} \\ + (\sum_{l = 1}^{m} \int_{Ω} \frac{\partial f (t)}{\partial t} \cdot {\tilde{u}}_{l} dx) (\int_{Ω} α ({div}_{x} {\tilde{u}}_{l}) {\tilde{θ}}_{k} dx + \int_{γ} α_{in} (\partial_{x_{1}} {\tilde{u}}_{l 1}) {\tilde{θ}}_{k} d x_{1}) = \\ = \int_{Ω} g (t) \tilde{θ} dx, k = 1, \dots, m . \end{matrix}

(42)

Here, ${((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{V}^{*} = δ_{lk}$ due to the orthonormality property.

Now, we clearly see that equation (42) is the system of the linear first-order ordinary differential equations of the form:

\hat{R} \frac{d b_{m} (t)}{dt} + b_{m} (t) = F (t),

(43)

where $\hat{R} = ({\hat{R}}_{kl})$ is an $m \times m - matrix$ with the constant components, each component ${\hat{R}}_{kl}$ is the coefficient {…} before $\frac{d b_{ml} (t)}{dt}$ in the $k th$ equation (42), $b_{m} (t) = {(b_{1 m} (t), \dots, b_{mm} (t))}^{t} = {(b_{ml} (t))}_{l = 1, \dots, m}$ is the vector of the sought functions, and $F (t) = (F_{k} (t))_{k = 1, \dots, m}$ is the given vector-function of $t$ . Here and further, the superscript $(\dots)^{t}$ denotes the transpose operation; hence, in particular, $b_{m} (t)$ is a vector-column.

Note that matrix $\hat{R}$ is symmetric. Furthermore, for any $ψ = (ψ_{1}, \dots, ψ_{m})^{t} \in R^{m}$ , we have:

\begin{matrix} \hat{R} ψ \cdot ψ = \sum_{k, l = 1}^{m} {\hat{R}}_{kl} ψ_{l} ψ_{k} \\ = \sum_{k, l = 1}^{m} {((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{L}^{*} ψ_{l} ψ_{k} \\ + \sum_{k, l = 1}^{m} \sum_{r = 1}^{m} [\int_{Ω} α ψ_{l} {\tilde{θ}}_{l} {div}_{x} {\tilde{u}}_{r} dx + \int_{γ} α_{in} ψ {\tilde{θ}}_{l} \partial_{x_{1}} {\tilde{u}}_{r 1} d x_{1}] \times \\ \times [\int_{Ω} α ψ_{k} {\tilde{θ}}_{k} {div}_{x} {\tilde{u}}_{r} dx + \int_{γ} α_{in} ψ_{k} {\tilde{θ}}_{k} \partial_{x_{1}} {\tilde{u}}_{r 1} d x_{1}] \\ = \sum_{k = 1}^{m} c_{k}^{θ} | ψ_{k} |^{2} + \sum_{r = 1}^{m} {(\sum_{k = 1}^{m} [\int_{Ω} α ψ_{k} {\tilde{θ}}_{k} {div}_{x} {\tilde{u}}_{r} dx + \int_{γ} α_{in} ψ_{k} {\tilde{θ}}_{k} \partial_{x_{1}} {\tilde{u}}_{r 1} d x_{1}])}^{2} \\ \geq \sum_{k = 1}^{m} c_{k}^{θ} | ψ_{k} |^{2} \geq c^{m θ} | ψ |^{2} . \end{matrix}

Here, due to the choice of the basis $(u_{k}, θ_{k})_{k = 1, 2, \dots}$ , we have that:

{((0, {\tilde{θ}}_{l}), (0, {\tilde{θ}}_{k}))}_{L}^{*} = {\begin{matrix} c_{k}^{θ} = {const}_{k} > 0 if k = l, \\ 0 if k \neq l, \end{matrix}

where $c_{k}^{θ} = ψ_{k}^{θ} (k = 1, \dots, m)$ are the eigenvalues of the spectral problem (35) (with $X = V$ and $Y = L$ ), and we denote $c^{m θ} = min_{k = 1, \dots, m} c_{k}^{θ} > 0$ .

Hence, there exists the inverse matrix ${\hat{R}}^{- 1}$ , and therefore, we can bring equation (43) to the normal form:

\frac{d b_{m} (t)}{dt} + {\hat{R}}^{- 1} b_{m} (t) = {\hat{R}}^{- 1} F (t)

(44a)

supplemented with the initial data:

b_{m} (0) = b^{0} = (b_{k}^{0})_{k = 1, \dots, m},

(44b)

where

b_{k}^{0} = {((0, θ_{0}), (0, {\tilde{θ}}_{k}))}_{V}^{*} = \int_{Ω} κ \nabla_{x} θ_{0} \cdot \nabla_{x} {\tilde{θ}}_{k} dx + \int_{γ} κ_{in} \partial_{x_{1}} θ_{0} \partial_{x_{1}} {\tilde{θ}}_{k} dx .

By the Peano theorem, whenever $\partial_{t} f \in C ([0, T]; L^{2} {(Ω)}^{2})$ , there is a unique solution $b_{m} (t) = (b_{m 1} (t), \dots, b_{mm} (t))^{t}$ in $C^{1} [0, T]$ to the problem (44).

Furthermore, we substitute $b_{mk}$ into equation (40). Since $f, \partial_{t} f \in C ([0, T]; L^{2} (Ω)^{2})$ and $b_{mk} \in C^{1} [0, T]$ , from equation (40), it follows that $a_{mk} \in C^{1} [0, T]$ . Thus, the Galerkin system (38a)–(38c) has a unique solution.

Step 2. Uniform estimates for the Galerkin approximations. Now, we turn to the study of the limiting passage in equation (38), as $m \to + \infty$ . To this end, we derive the necessary estimates for the Galerkin approximate solutions. We multiply the $k th$ equation in equation (38a) by $\partial_{t} a_{mk} (t)$ and the $k th$ equation in equation (38b) by $b_{mk} (t)$ . After this, we integrate the resulting equations in $t$ over $(0, τ)$ and sum up. From the obtained equalities, using the Korn and the Poincaré–Friedrichs inequalities, the $ε - version$ of Young’s inequality, and the Cauchy–Bunyakovsky inequality, we derive the bounds:

\begin{matrix} ∥ u_{m} ∥_{L^{\infty} (0, T; H_{0}^{1} {(Ω)}^{2})}^{2} \leq c, \\ ∥ u_{m 1} ∥_{L^{\infty} (0, T; H_{0}^{1} (γ))}^{2} \leq c, \end{matrix} \begin{matrix} ∥ θ_{m} ∥_{L^{\infty} (0, T; L^{2} (Ω))}^{2} \leq c, \\ ∥ θ_{m} ∥_{L^{\infty} (0, T; L^{2} (γ))}^{2} \leq c, \end{matrix} \begin{matrix} ∥ θ_{m} ∥_{L^{2} (0, T; H_{0}^{1} (Ω))}^{2} \leq c, \\ ∥ θ_{m} ∥_{L^{2} (0, T; H_{0}^{1} (γ))}^{2} \leq c, \end{matrix}

(45)

where the positive constant $c$ does not depend on $m$ .

Step 3. Limiting passage as $m \to + \infty$ . On the strength of equation (45), there exist a subsequence of ${(u_{m}, θ_{m})}_{m = 1, 2, \dots}$ (still labeled by $m$ ) and a pair of limit functions $(u_{*}, θ_{*})$ such that:

\begin{matrix} u_{m} \underset{m \to + \infty}{\to} u_{*} weakl y^{★} in L^{\infty} (0, T; H_{0}^{1} {(Ω)}^{2}), \\ u_{m 1} \underset{m \to + \infty}{\to} u_{* 1} weakl y^{★} in L^{\infty} (0, T; H_{0}^{1} (γ)), \\ θ_{m} \underset{m \to + \infty}{\to} θ_{*} weakl y^{★} in L^{\infty} (0, T; L^{2} (Ω) \cap L^{2} (γ)), \\ weakly in L^{2} (0, T; H_{0}^{1} (Ω) \cap H_{0}^{1} (γ)) . \end{matrix}

(46)

Using equation (46), we pass to the limit in equations (38a) and (38b). We multiply equations (38a) and (38b) by the test functions $ϕ_{k} \in C^{1} [0, T]$ and $ψ_{k} \in C^{1} [0, T]$ and integrate in $t$ over $[0, τ]$ , where $τ \in (0, T]$ . Thus, we get:

\begin{matrix} \int_{0}^{τ} \int_{Ω} (2 μ ε_{x} (u_{m}) : ε_{x} (ϕ_{k} {\tilde{u}}_{k}) + λ {div}_{x} u_{m} {div}_{x} (ϕ_{k} {\tilde{u}}_{k})) dxdt \\ - \int_{0}^{τ} \int_{Ω} α θ_{m} {div}_{x} (ϕ_{k} {\tilde{u}}_{k}) dxdt \\ + \int_{0}^{τ} \int_{γ} (E_{in} \partial_{x_{1}} u_{m 1} - α_{in} θ_{m}) \partial_{x_{1}} (ϕ_{k} {\tilde{u}}_{k 1}) d x_{1} dt = \\ = \int_{0}^{τ} \int_{Ω} f \cdot (ϕ_{k} {\tilde{u}}_{k}) dxdt, \end{matrix}

(47a)

\begin{matrix} \int_{0}^{τ} \int_{Ω} c (\partial_{t} θ_{m}) (ψ_{k} {\tilde{θ}}_{k}) dxdt + \int_{0}^{τ} \int_{Ω} α ({div}_{x} \partial_{t} u_{m}) (ψ_{k} {\tilde{θ}}_{k}) dxdt \\ + \int_{0}^{τ} \int_{Ω} κ \nabla_{x} θ_{m} \cdot \nabla_{x} (ψ_{k} {\tilde{θ}}_{k}) dxdt + \int_{0}^{τ} \int_{γ} c_{in} (\partial_{t} θ_{m}) (ψ_{k} {\tilde{θ}}_{k}) d x_{1} dt \\ + \int_{0}^{τ} \int_{γ} α_{in} \partial_{x_{1}} (\partial_{t} u_{m 1}) (ψ_{k} {\tilde{θ}}_{k}) d x_{1} dt \\ + \int_{0}^{τ} \int_{γ} κ_{in} \partial_{x_{1}} θ_{m} \partial_{x_{1}} (ψ_{k} {\tilde{θ}}_{k}) d x_{1} dt = \int_{0}^{τ} \int_{Ω} g (ψ_{k} {\tilde{θ}}_{k}) dxdt, \\ k = 1, \dots, m, τ \in (0, T] . \end{matrix}

(47b)

Introduce the functions $Φ$ and $Ψ$ such that:

Φ \in C^{1} ([0, T]; (H_{0}^{1} (Ω) \cap H_{0}^{1} (γ)) \times H_{0}^{1} (Ω)), Ψ \in C^{1} ([0, T]; H_{0}^{1} (Ω) \cap H_{0}^{1} (γ)) .

Let us consider the partial sums of Fourier series:

Φ_{l} = \sum_{k = 1}^{l} c_{k}^{Φ} (t) {\tilde{u}}_{k},

(48)

where $c_{k}^{Φ} (t) = {((Φ (t), 0), ({\tilde{u}}_{k}, 0))}_{V}^{*}, l \leq m$ , and:

Ψ_{l} = \sum_{k = 1}^{l} c_{k}^{Ψ} (t) {\tilde{θ}}_{k},

(49)

where $c_{k}^{Φ} (t) = {((0, Ψ (t)), (0, {\tilde{θ}}_{k}))}_{V}^{*}, l \leq m$ . We have $c_{k}^{Φ}, c_{k}^{Ψ} \in C^{1} [0, T]$ .

Insert $ϕ_{k} : = c_{k}^{Φ}$ into equation (47a) and $ψ_{k} : = c_{k}^{Ψ}$ into equation (47b) and sum in the both resulting equalities from 1 till $l$ . Thus, we get the integral equalities:

\begin{matrix} \int_{0}^{τ} \int_{Ω} (2 μ ε_{x} (u_{m}) : ε_{x} (Φ_{l}) + λ {div}_{x} u_{m} {div}_{x} Φ_{l}) dx dt - \int_{0}^{τ} \int_{Ω} α θ_{m} {div}_{x} Φ_{l} dx dt \\ + \int_{0}^{τ} \int_{γ} (E_{in} \partial_{x_{1}} u_{m 1} - α_{in} θ_{m}) \partial_{x_{1}} Φ_{l 1} d x_{1} dt = \int_{0}^{τ} \int_{Ω} f \cdot Φ_{l} dx dt, \end{matrix}

(50a)

\begin{matrix} \int_{0}^{τ} \int_{Ω} c (\partial_{t} θ_{m}) Ψ_{l} dx dt + \int_{0}^{τ} \int_{Ω} α ({div}_{x} \partial_{t} u_{m}) Ψ_{l} dx dt \\ + \int_{0}^{τ} \int_{Ω} κ \nabla_{x} θ_{m} \cdot \nabla_{x} Ψ_{l} dx dt + \int_{0}^{τ} \int_{γ} c_{in} (\partial_{t} θ_{m}) Ψ_{l} d x_{1} dt \\ + \int_{0}^{τ} \int_{γ} α_{in} \partial_{x_{1}} (\partial_{t} u_{m 1}) Ψ_{l} d x_{1} dt + \int_{0}^{τ} \int_{γ} κ_{in} \partial_{x_{1}} θ_{m} \partial_{x_{1}} Ψ_{l} d x_{1} dt = \\ = \int_{0}^{τ} \int_{Ω} g Ψ_{l} dxdt, \\ k = 1, \dots, m, τ \in (0, T] . \end{matrix}

(50b)

Take $τ = T$ in equations (50a) and (50b). In equation (50b), integrate by parts in $t$ assuming $Ψ$ such that $Ψ (x, T) = 0$ , hence $Ψ_{l} (x, T) = 0$ . Furthermore, in equations (50a) and (50b), we pass to the limit as $m \to \infty$ on the strength of equation (46). After this, we pass to the limit, as $l \to \infty$ . Thus, from equation (50a), we derive:

\begin{matrix} \int_{0}^{T} \int_{Ω} (2 μ ε_{x} (u_{*}) : ε_{x} (Φ) + λ {div}_{x} u_{*} {div}_{x} Φ) dx dt - \int_{0}^{T} \int_{Ω} α θ_{*} {div}_{x} Φ dx dt \\ + \int_{0}^{T} \int_{γ} (E_{in} \partial_{x_{1}} u_{* 1} - α_{in} θ_{*}) \partial_{x_{1}} Φ_{1} d x_{1} dt = \int_{0}^{T} \int_{Ω} f \cdot Φ dx dt . \end{matrix}

(51)

Remark 3. Due to arbitrariness of $Φ$ , equation (51) yields equation (29a).

From equation (50b), we derive:

\begin{matrix} - \int_{0}^{T} \int_{Ω} c θ_{*} \partial_{t} Ψ dx dt - \int_{0}^{T} \int_{Ω} α ({div}_{x} u_{*}) \partial_{t} Ψ dx dt \\ + \int_{0}^{T} \int_{Ω} κ \nabla_{x} θ_{*} \cdot \nabla_{x} Ψ dx dt - \int_{0}^{T} \int_{γ} c_{in} θ_{*} \partial_{t} Ψ d x_{1} dt \\ - \int_{0}^{T} \int_{γ} α_{in} \partial_{x_{1}} u_{* 1} \partial_{t} Ψ d x_{1} dt + \int_{0}^{T} \int_{γ} κ_{in} \partial_{x_{1}} θ_{*} \partial_{x_{1}} Ψ d x_{1} dt \\ - \int_{Ω} c θ_{0} Ψ (0) dx - \int_{Ω} α ({div}_{x} u_{*} (0)) Ψ (0) dx - \int_{γ} c_{in} θ_{0} Ψ (0) d x_{1} \\ - \int_{γ} α_{in} \partial_{x_{1}} (u_{* 1} (0)) Ψ (0) d x_{1} = \int_{0}^{T} \int_{Ω} g Ψ dxdt . \end{matrix}

(52)

Now, we study the questions of higher regularity of $u_{*}$ and $θ_{*}$ in $t$ and of attainment of the initial data by $θ_{*}$ . We start with establishing the necessary estimates for the Galerkin approximations $\partial_{t} u_{m}$ , $\partial_{t} u_{m 1}$ , and $\partial_{t} θ_{m}$ .

Step 4. The uniform estimates in $m$ for $\partial_{t} u_{m}$ , $\partial_{t} u_{m 1}$ , and $\partial_{t} θ_{m}$ . For each $k = 1, \dots, m$ , we differentiate equation (38a) in $t$ , which is a legal procedure since $a_{mk}, b_{mk} \in C^{1} [0, T]$ , multiply the both sides by $\partial_{t} a_{mk}$ , and then sum up over $k$ to get:

\begin{matrix} \int_{Ω} (2 μ | ε_{x} (\partial_{t} u_{m} (t)) |^{2} + λ | {div}_{x} \partial_{t} u_{m} (t {) |}^{2}) dx - \int_{Ω} α \partial_{t} θ_{m} (t) {div}_{x} \partial_{t} u_{m} (t) dx \\ + \int_{γ} E_{in} | \partial_{x_{1}} \partial_{t} u_{m 1} (t) |^{2} dx \\ - \int_{γ} α_{in} \partial_{t} θ_{m} (t) \partial_{x_{1}} \partial_{t} u_{m 1} (t) d x_{1} = \int_{Ω} \partial_{t} f (t) \cdot \partial_{t} u_{m 1} (t) dx . \end{matrix}

(53)

For each $k = 1, \dots, m$ , we multiply the both sides of equation (38b) by $\partial_{t} b_{mk}$ and then sum up over $k$ to get:

\begin{matrix} \int_{Ω} (c | \partial_{t} θ_{m} (t {) |}^{2} + κ \frac{1}{2} \partial_{t} | \nabla_{x} θ_{m} (t {) |}^{2}) dx + \int_{Ω} α ({div}_{x} \partial_{t} u_{m} (t)) \partial_{t} θ_{m} (t) dx \\ + \int_{γ} (c_{in} | \partial_{t} θ_{m} (t {) |}^{2} + κ_{in} \frac{1}{2} \partial_{t} | \partial_{x_{1}} θ_{m} (t {) |}^{2}) d x_{1} \\ + \int_{γ} α_{in} (\partial_{x_{1}} \partial_{t} u_{m 1} (t)) \partial_{t} θ_{m} (t) d x_{1} = \int_{Ω} g (t) \partial_{t} θ_{m} (t) dx . \end{matrix}

(54)

We join equations (53) and (54) and then integrate the result in $t$ on $(0, τ) (\forall τ \in (0, T])$ . After this, with the help of the Korn inequality, the $ε - version$ of Young’s inequality, and the Cauchy–Bunyakovsky and Poincaré–Friedrichs inequalities, by the standard techniques we derive the uniform (in $m$ ) bounds:

\begin{matrix} ∥ \partial_{t} u_{m} ∥_{L^{2} (0, T; H_{0}^{1} {(Ω)}^{2})}^{2} \leq c, ∥ \partial_{t} θ_{m} ∥_{L^{2} (0, T; L^{2} (Ω))}^{2} \leq c, ∥ θ_{m} ∥_{L^{\infty} (0, T; H_{0}^{1} (Ω))}^{2} \leq c, \\ ∥ \partial_{t} u_{m 1} ∥_{L^{2} (0, T; H_{0}^{1} (γ))}^{2} \leq c, ∥ \partial_{t} θ_{m} ∥_{L^{2} (0, T; L^{2} (γ))}^{2} \leq c, ∥ θ_{m} ∥_{L^{\infty} (0, T; H_{0}^{1} (γ))}^{2} \leq c, \end{matrix}

(55)

where the constant $c$ is independent of $m$ and depends on $μ_{*}$ , $λ_{*}$ , $α_{*}$ , $c_{*}$ , $κ_{*}$ , $∥ \partial_{t} f ∥_{L^{2} (0, T; L^{2} {(Ω)}^{2})}$ , $∥ g ∥_{L^{2} (0, T; L^{2} (Ω))}$ and on the constants in the Korn and the Poincaré–Friedrichs inequalities.

Step 5. The higher regularity of $\partial_{t} u_{*}$ and $\partial_{t} θ_{*}$ . On the strength of the weak lower semi-continuity property of the norms, as $m \to \infty$ , from equation (55) we derive exactly the same bounds for $\partial_{t} u_{*}$ and $\partial_{t} θ_{*}$ . Thus:

\begin{matrix} \partial_{t} u_{*} \in L^{2} (0, T; H_{0}^{1} {(Ω)}^{2}), \partial_{t} u_{* 1} \in L^{2} (0, T; H_{0}^{1} (γ)), \\ \partial_{t} θ_{*} \in L^{2} (0, T; L^{2} (Ω) \cap L^{2} (γ)) . \end{matrix}

(56)

Step 6. Attainment of the initial data. Let us discuss the question about whether the limit function $θ_{*}$ attain the initial data $θ_{0}$ and how.

Since $\partial_{t} θ_{*} \in L^{2} (0, T; L^{2} (Ω))$ , one has $θ_{*} \in C ([0, T]; L^{2} (Ω))$ . This means that the trace $θ_{*} (\cdot, 0)$ on the section ${t = 0}$ is well-defined. More certainly:

∥ θ_{*} (\cdot, t) - θ_{*} (\cdot, 0) ∥_{L^{2} (Ω)} \underset{t \to 0 +}{\to} 0 .

(57)

It remains to show that $θ_{*} (\cdot, 0) = θ_{0} (\cdot)$ on $Ω$ . In order to do this, we revisit (50b). In equation (50b), we integrate by parts in $t$ and pass to the limit, first, as $m \to \infty$ , and then, as $l \to \infty$ . Thus, we arrive at the integral equality:

\begin{matrix} - \int_{0}^{τ} \int_{Ω} c θ_{*} \partial_{t} Ψ dxdt - \int_{0}^{τ} \int_{Ω} α ({div}_{x} u_{*}) \partial_{t} Ψ dxdt + \int_{0}^{τ} \int_{Ω} κ \nabla_{x} θ_{*} \cdot \nabla_{x} Ψ dxdt \\ - \int_{0}^{τ} \int_{γ} c_{in} θ_{*} \partial_{t} Ψ d x_{1} dt - \int_{0}^{τ} \int_{γ} α_{in} \partial_{x_{1}} u_{* 1} \partial_{t} Ψ d x_{1} dt \\ + \int_{0}^{τ} \int_{γ} κ_{in} \partial_{x_{1}} θ_{*} \partial_{x_{1}} Ψ d x_{1} dt + \int_{Ω} c θ_{*} (τ) Ψ (τ) dx - \int_{Ω} c θ_{0} Ψ (0) dx \\ + \int_{Ω} α ({div}_{x} u_{*} (τ)) Ψ (τ) dx - \int_{Ω} α ({div}_{x} u_{*} (0)) Ψ (0) dx \\ + \int_{γ} c_{in} θ_{*} (τ) Ψ (τ) d x_{1} - \int_{γ} c_{in} θ_{0} Ψ (0) d x_{1} + \int_{γ} α_{in} \partial_{x_{1}} (u_{* 1} (τ)) Ψ (τ) d x_{1} \\ - \int_{γ} α_{in} \partial_{x_{1}} (u_{* 1} (0)) Ψ (0) d x_{1} = \int_{0}^{τ} \int_{Ω} g Ψ dxdt . \end{matrix}

(58)

On the strength of equation (56), we note that ${div}_{x} u_{*} \in C ([0, T]; L^{2} (Ω))$ and $\partial_{x} u_{* 1} \in C ([0, T]; L^{2} (γ))$ . Due to this and the other already established regularity properties of $u_{*}$ and $θ_{*}$ , passing to the limit in equation (58) as $τ \to 0 +$ , we get:

lim_{τ \to 0 +} [\int_{Ω} c (θ_{*} (τ) Ψ (τ) - θ_{0} Ψ (0)) dx + \int_{γ} c_{in} (θ_{*} (τ) Ψ (τ) - θ_{0} Ψ (0)) d x_{1}] = 0 .

Since $Ψ$ is sufficiently arbitrary, this yields that:

θ_{*} (τ) \underset{τ \to 0 +}{\to} θ_{0} weakly in L^{2} (Ω) \cap L^{2} (γ) .

(59)

Thus, $θ_{*} (\cdot, 0) = θ_{0} (\cdot)$ in $Ω$ , which along with equation (57) yields equation (29c).

Step 7. Completion of the proof of Theorem 1. In Remark 3, we have already noticed that equation (29a) holds true for $(u_{*}, θ_{*})$ . Furthermore, since $\partial_{t} θ_{*}$ and $\partial_{t} u_{*}$ satisfy equation (56), we can integrate by parts in $t$ in equation (52). After this, we take $Ψ : = ξ (x) ϕ (t)$ and derive equation (29b) due to the arbitrariness of $ϕ$ .

By this, we have shown that the pair $(u_{*}, θ_{*})$ is a solution of Problem B_var, which completes the proof of Theorem 1.

4.2. Stability and uniqueness of the solution to Problem B_var

Assume $(u_{*}^{'}, θ_{*}^{'})$ and $(u_{*}^{''}, θ_{*}^{''})$ are the two solutions of Problem B_var corresponding to respective given data $(θ_{0}^{'}, f^{'}, g^{'})$ and $(θ_{0}^{''}, f^{''}, g^{''})$ . Subtract equation (29a) with $(u_{*}^{'}, θ_{*}^{'}, f^{'})$ from equation (29a) with $(u_{*}^{''}, θ_{*}^{''}, f^{''})$ . Analogously, subtract equation (29b) with $(u_{*}^{'}, θ_{*}^{'}, g^{'})$ from equation (29b) with $(u_{*}^{''}, θ_{*}^{''}, g^{''})$ . In the resulting equalities, we take $v : = \partial_{t} (u_{*}^{''} - u_{*}^{'})$ and $ξ = θ_{*}^{''} - θ_{*}^{'}$ , denote:

U_{*} : = u_{*}^{''} - u_{*}^{'}, Θ_{*} = θ_{*}^{''} - θ_{*}^{'}, F = f^{''} - f^{'}, G = g^{''} - g^{'},

for briefness, and thus get:

\begin{matrix} \frac{d}{dt} \int_{Ω} (μ | ε_{x} (U_{*} (t {)) |}^{2} + \frac{λ}{2} | {div}_{x} U_{*} (t {) |}^{2}) dx - \int_{Ω} α Θ_{*} (t) {div}_{x} \partial_{t} U_{*} (t) dx \\ + \frac{d}{dt} \int_{γ} \frac{E_{in}}{2} | \partial_{x_{1}} U_{* 1} (t) |^{2} d x_{1} - \int_{γ} α_{in} Θ_{*} (t) \partial_{x_{1}} \partial_{t} U_{* 1} (t) d x_{1} = \\ = \int_{Ω} F (t) \cdot \partial_{t} U_{*} (t) dx \end{matrix}

(60)

and

\begin{matrix} \frac{d}{dt} \int_{Ω} \frac{c}{2} | Θ_{*} (t) |^{2} dx + \int_{Ω} κ | \nabla_{x} Θ_{*} (t) |^{2} dx + \int_{Ω} α Θ_{*} (t) {div}_{x} \partial_{t} U_{*} (t) dx \\ + \frac{d}{dt} \int_{γ} \frac{c_{in}}{2} | Θ_{*} (t) |^{2} d x_{1} + \int_{γ} κ_{in} | \partial_{x_{1}} Θ_{*} (t) |^{2} d x_{1} \\ + \int_{γ} α_{in} Θ_{*} (t) \partial_{x_{1}} \partial_{t} U_{* 1} (t) d x_{1} = \int_{Ω} G (t) Θ_{*} (t) dx . \end{matrix}

(61)

Now, we join equations (60) and (61), cancel the like terms, and integrate in $t$ on $(0, τ)$ to establish the integral equality:

\begin{matrix} \int_{Ω} (μ | ε_{x} (U_{*} (τ)) |^{2} + \frac{λ}{2} | {div}_{x} U_{*} (τ {) |}^{2}) dx \\ + \int_{γ} \frac{E_{in}}{2} | \partial_{x_{1}} U_{* 1} (τ) |^{2} d x_{1} + \int_{Ω} \frac{c}{2} | Θ_{*} (τ) |^{2} dx \\ + \int_{0}^{τ} \int_{Ω} κ | \nabla_{x} Θ_{*} (t) |^{2} dxdt + \int_{γ} \frac{c_{in}}{2} | Θ_{*} (τ) |^{2} d x_{1} \\ + \int_{0}^{τ} \int_{γ} κ_{in} | \partial_{x_{1}} Θ_{*} (t) |^{2} d x_{1} dt = \\ = \int_{Ω} (μ | ε_{x} (U_{*} (0)) |^{2} + \frac{λ}{2} | {div}_{x} U_{*} (0) |^{2}) dx + \int_{γ} \frac{E_{in}}{2} | \partial_{x_{1}} U_{* 1} (0) |^{2} d x_{1} \\ + \int_{Ω} \frac{1}{2} c | {θ^{''}}_{0} - {θ^{'}}_{0} |^{2} dx + \int_{γ} \frac{1}{2} c_{in} | {θ^{''}}_{0} - {θ^{'}}_{0} |^{2} d x_{1} + \int_{0}^{τ} \int_{Ω} G Θ_{*} dxdt \\ + \int_{Ω} F (τ) \cdot U_{*} (τ) dx - \int_{Ω} F (0) \cdot U_{*} (0) dx - \int_{0}^{τ} \int_{Ω} \partial_{t} F \cdot U_{*} dxdt . \end{matrix}

(62)

Here, we apply the Korn and the Poincaré–Friedrichs inequalities in the left-hand side and the $ε - version$ of Young’s inequality in the right-hand side to get the inequality:

\begin{matrix} ∥ U_{*} (τ) ∥_{H_{0}^{1} {(Ω)}^{2}}^{2} + ∥ U_{* 1} (τ) ∥_{H_{0}^{1} (γ)}^{2} \\ + ∥ Θ_{*} (τ) ∥_{L^{2} (Ω)}^{2} + ∥ Θ_{*} (τ) ∥_{L^{2} (γ)}^{2} + ∥ \nabla_{x} Θ_{*} ∥_{L^{2} (0, τ; L^{2} {(Ω)}^{2})}^{2} + ∥ \partial_{x_{1}} Θ_{*} ∥_{L^{2} (0, τ; L^{2} (γ))}^{2} \\ \leq C_{*} (∥ F ∥_{L^{\infty} (0, τ; L^{2} {(Ω)}^{2})}^{2} + ∥ G ∥_{L^{2} (0, τ; L^{2} (Ω))}^{2} + ∥ \partial_{t} F ∥_{L^{2} (0, τ; L^{2} {(Ω)}^{2})}^{2} \\ + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (Ω)}^{2} + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (γ)}^{2} + ∥ U_{*} (0) ∥_{H_{0}^{1} {(Ω)}^{2}}^{2} + ∥ U_{* 1} (0) ∥_{H_{0}^{1} (γ)}^{2}), \\ \forall τ \in (0, T], \end{matrix}

(63)

where the positive constant $C_{*}$ does not depend on $τ$ , $U_{*}$ , $Θ_{*}$ , $F$ , and $G$ .

Now, we estimate $U_{*} (0)$ by means of equation (29a). More precisely, we insert $t = 0$ , $Θ_{*} (0) : = θ_{0}^{''} - θ_{0}^{'}$ , and $v : = U_{*} (0)$ into equation (29a), and using the same techniques, as in the procedure of derivation of equation (63) from equation (62), we establish the inequality:

\begin{matrix} ∥ U_{*} (0) ∥_{H_{0}^{1} {(Ω)}^{2}}^{2} + ∥ U_{* 1} (0) ∥_{H_{0}^{1} (γ)}^{2} \\ \leq C_{* *} (∥ F (0) ∥_{L^{2} {(Ω)}^{2}}^{2} + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (Ω)}^{2} + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (γ)}^{2}), \end{matrix}

(64)

where the constant $C_{* *}$ does not depend on $U_{*}$ , $Θ_{*}$ , $F$ , and $G$ . Combining equations (63) and (64), we arrive at the following result.

Theorem 2. Assume $(u_{*}^{'}, θ_{*}^{'})$ is a solution of Problem B_var corresponding to an initial function $θ_{0}^{'}$ and the right-hand sides $f^{'}$ and $g^{'}$ and $(u_{*}^{''}, θ_{*}^{''})$ is a solution of Problem B_var corresponding to an initial function $θ_{0}^{''}$ and the right-hand sides $f^{''}$ and $g^{''}$ .

There exists a constant $C_{★} > 0$ , which depends on $α^{*}$ , $μ_{*}$ , $μ^{*}$ , $λ_{*}$ , $λ^{*}$ , $c_{*}$ , $c^{*}$ , and on the constants from the Korn and the Poincaré–Friedrichs inequalities and is independent of $τ \in (0, T]$ , $u_{*}^{'}$ , $θ_{*}^{'}$ , $θ_{0}^{'}$ , $f^{'}$ , $g^{'}$ , $u_{*}^{''}$ , $θ_{*}^{''}$ , $θ_{0}^{''}$ , $f^{''}$ , and $g^{''}$ , such that the estimate:

\begin{matrix} ∥ {u^{''}}_{*} (τ) - {u^{'}}_{*} (τ) ∥_{H_{0}^{1} {(Ω)}^{2}}^{2} + ∥ {u^{''}}_{* 1} (τ) - {u^{'}}_{* 1} (τ) ∥_{H_{0}^{1} (γ)}^{2} \\ + ∥ {θ^{''}}_{*} (τ) - {θ^{'}}_{*} (τ) ∥_{L^{2} (Ω)}^{2} + ∥ {θ^{''}}_{*} (τ) - {θ^{'}}_{*} (τ) ∥_{L^{2} (γ)}^{2} \\ + ∥ \nabla_{x} ({θ^{''}}_{*} (τ) - {θ^{'}}_{*} (τ)) ∥_{L^{2} (0, τ; L^{2} {(Ω)}^{2})}^{2} + ∥ \partial_{x_{1}} ({θ^{''}}_{*} (τ) - {θ^{'}}_{*} (τ)) ∥_{L^{2} (0, τ; L^{2} (γ))}^{2} \\ \leq C_{★} (∥ f^{''} - f^{'} ∥_{L^{\infty} (0, τ; L^{2} {(Ω)}^{2})}^{2} + ∥ g^{''} - g^{'} ∥_{L^{2} (0, τ; L^{2} (Ω))}^{2} \\ + ∥ \partial_{t} (f^{''} - f^{'}) ∥_{L^{2} (0, τ; L^{2} {(Ω)}^{2})}^{2} + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (Ω)}^{2} + ∥ {θ^{''}}_{0} - {θ^{'}}_{0} ∥_{L^{2} (γ)}^{2}), \\ \forall τ \in (0, T], \end{matrix}

holds true.

In particular, the solution of Problem B_var corresponding to the given triple $(θ_{0}, f, g)$ is unique.

Footnotes

Acknowledgements

The authors thank the anonymous reviewers for their valuable remarks and recommendations that helped improve the earlier version of this paper.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Russian Science Foundation (grant no. 22-21-00627).

ORCID iD

Evgeny M Rudoy

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A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

Abstract

Keywords

1. Introduction

2. Preliminaries: the quasi-static model in thermoelasticity

3. Derivation of the model with a thin inclusion

3.1. Refinement of the geometric structure and appropriate adaptation of the formulation of Problem Avar

3.2. Formal asymptotic analysis

3.3. Setting the limit problem

4. Well-posedness of the model with a thin inclusion

4.1. Existence of a solution to Problem Bvar

4.2. Stability and uniqueness of the solution to Problem Bvar

Footnotes

Acknowledgements

Funding

ORCID iD

References

3.1. Refinement of the geometric structure and appropriate adaptation of the formulation of Problem A_var

4.1. Existence of a solution to Problem B_var

4.2. Stability and uniqueness of the solution to Problem B_var