A Time-Dependent Wave-Thermoelastic Solid Interaction

# A Time-Dependent Wave-Thermoelastic Solid Interaction

G. C. Hsiao, T. Sánchez-Vizuet, F.–J.Sayas & R. J. Weinacht
Department of Mathematical Sciences, University of Delaware, USA.
{ghsiao, fjsayas, weinacht}@udel.edu
Courant Institute of Mathematical Sciences, New York University, USA.
tonatiuh@cims.nyu.edu
July 30, 2019
This paper is dedicated to Wolfgang L. Wendland
on the occasion of his 80th birthday.
###### Abstract

This paper presents a combined field and boundary integral equation method for solving the time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid-structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analyzed with Lubich’s approach for time-dependent boundary integral equations. Existence and uniqueness results are established in terms of time-domain data with the aid of Laplace-domain techniques. Galerkin semi-discretization approximations are derived and error estimates are obtained. A full discretization based on the Convolution Quadrature method is also outlined. Some numerical experiments in 2D are also included in order to demonstrate the accuracy and efficiency of the procedure.

Key words: Fluid-structure interaction, Coupling BEM-FEM, Kirchhoff representation formula, Retarded potential, Time-domain boundary integral equation, Variational formulation, Wave scattering, Convolution quadrature.

Mathematics Subject Classifications: 35J20, 35L05, 45P05, 65N30, 65N38. âº

## 1 Introduction

The mathematical study of the thermodynamic response of a linearly elastic solid to mechanical strain dates back at least to Duhamel’s 1837 pioneering work [10] on thermoelastic materials where he proposed the constitutive relation linking the temperature variations and elastic strains with the thermoelastic stress now known as Duhamel-Neumann law [7, 32].

Kupradze’s encyclopedic works [23] can be considered the standard reference for a modern mathematical treatment of the purely thermoelastic problem. The dynamic problem is dealt with in more recent works like [34, 37] where the matrix of fundamental solutions for the dynamic equations is revisited, while [20, 21] provide generalized Kirchhoff-type formulas for thermoelastic solids.

In the case of the scattering of thermoelastic waves, major theoretical contributions have been made by Çakoni and Dassios in [6]. The unique solvability of a boundary integral formulation is established in [5] and the interaction of elastic and thermoelastic waves is explored for homogeneous materials in [8]. The study of time-harmonic interaction between a scalar field and a thermoelastic solid has been the subject of works like [27] where the interface is taken to be a plane, or [26, 22] where time-harmonic scattering by bounded obstacles is considered.

In this paper, we present a combined field and boundary integral method for a time-dependent fluid-thermoelastic solid interaction problem. The approach here is a generalization of the method employed by the authors for treating time-dependent fluid -structure interaction problems in [18, 17]. The present communication is an improvement over those previous efforts in the sense that it considers a more general constitutive law that accounts for the coupling between elastic and thermal effects. To our knowledge no attempt has been made to investigate with rigorous justifications the time-dependent acoustic scattering by a thermoelastic obstacle.

The setting is introduced in Section 2, along with the physical assumptions and the constitutive relation under consideration leading to the time-domain system of governing equations. The problem is then recast in Section 3 where the Laplace domain system is transformed into an equivalent integro-differential non-local problem that will be formulated variationally for discretization later on. The question of existence and uniqueness of the solutions to the non-local problem is dealt with on Section 4. The error analysis of the proposed discretization is addressed in Section 5, where semi-discrete error estimates for spatial discretization are provided.

The final Section 6 discusses the computational considerations related to the numerical solution of the discrete problem. A full discretization using Convolution Quadrature (CQ) in time is outlined and the coupling of boundary and finite elements for the spatial discretization is discussed. Convergence experiments in 2D are performed for test problems in both frequency and time domains as a demonstration of the applicability of the formulation which remains valid also in 3D. For time discretization both second order backward differentiation formula (BDF2) and Trapezoidal Rule CQ are used, providing evidence that the approximation is stable and of second order globally. Time-domain illustrative experiments using the proposed formulation are included.

In closing, we remark that for homogeneous thermoelastic solid medium, a pure boundary integral equation formulation may be adapted as in the fluid-structure interaction problem [17]. We will pursue these investigations in a separate communication.

## 2 Formulation of the problem

Consider a thermoelastic solid with constant density in an undeformed reference configuration and at thermal equilibrium at temperature . Under the action of external forces the body will be subject to internal stresses that will induce local variations of temperature. Reciprocally, if a heat source induces a change in temperature, the body will react by dilating or contracting and this will create internal stresses and deformations. We will denote by the elastic deformation with respect to the reference configuration and by the variation of temperature with respect to the equilibrium temperature. In the classical linear theory [23, 24], the coupling between the mechanic strain and the thermal gradient is modeled by the Duhamel-Neumann law which defines the thermoelastic stress and the thermoelastic heat flux (also known as free energy) as functions of the elastic displacement and the variation in temperature by

 σ:= Cε(U)−ζΘI, F:= −η∂U∂t+κ∇Θ.

In the previous expressions

 ε(U):=12(∇U+(∇U)t)

is the elastic strain tensor, is the identity matrix, is the thermal diffusivity coefficient, which from physical principles [14] is required to be positive, is the product of the volumetric thermal expansion coefficient and the bulk modulus of the material, and is given by the relation

 η=Θ0ζ/cvol.

Here the volumetric heat capacity is the ratio between the thermal diffusivity and the thermal conductivity, and it can also be expressed as the product of the mass density and the specific heat capacity. For the case of homogeneous isotropic material that we are considering, the elastic stiffness tensor is given by

 Cijkl:=λδijδkl+μ(δikδjl+δilδjk),

where the constants and are Lamé’s second parameter and the shear modulus respectively, and is Kronecker’s delta.

We are concerned with a time-dependent direct scattering problem in fluid-thermoelastic solid interaction, which can be simply described as follows: an acoustic wave propagates in a fluid domain of infinite extent in which a bounded thermoelastic body is immersed. Throughout the paper, we let be the bounded domain in occupied by the thermoelastic body with a Lipschitz boundary and we let be its exterior, occupied by a compressible fluid. The problem is then to determine the scattered velocity potential in the fluid domain, the deformation of the solid and the variation of the temperature in the obstacle. It is assumed that .

The governing equations of the displacement field and temperature field are the thermo-elastodynamic equations:

 ρΣ∂2U∂t2−Δ∗U+ζ∇Θ= 0 in Ω−×(0,T), (2.1) 1κ∂Θ∂t−ΔΘ+η∂∂t(∇⋅U) = 0 in Ω−×(0,T), (2.2)

where is a given positive final time, and as usual the symbol is the Lamé operator defined by

 Δ∗U:=μΔU+(λ+μ)∇(∇⋅U).

We remark that if the thermal effect is neglected () Duhamel-Neumann’s law reduces to the usual expression for Hooke’s law of the classical theory for an arbitrary isotropic medium (see, e.g. [7, 23]). In the thermoelastic medium, the given physical constants , are assumed to satisfy the inequalities:

 ρΣ>0,μ>0,3λ+2μ>0,ζη>0,κ>0.

In the fluid domain , we consider a barotropic and irrotational flow of an inviscid and compressible fluid with density as in [18]. The formulation can be presented in terms of a scalar potential such that the scattered velocity field and the pressure are given by

 V=−∇VandP=ρf∂V∂t.

Then we arrive at the wave equation

 1c2∂2V∂t2−ΔV=0inΩ+×(0,T) (2.3)

where is the sound speed.

On the interface between the solid and the fluid we have the transmission conditions

 σ(U,Θ)−n= −ρf(∂V∂t+∂Vinc∂t)+n on Γ×(0,T), (2.4a) ∂U−∂t⋅n= −(∂V∂n+∂Vinc∂n)+ on Γ×(0,T), (2.4b) ∂Θ∂n−= 0 on Γ×(0,T), (2.4c)

where is the exterior unit normal to , and denotes the given incident field, which is assumed to be supported away from at . Here and in the sequel, we adopt the notation that denotes the limit of the function on from respectively. Regarding the transmission conditions we remark that from the physical point of view, equation (2.4a) enforces the equilibrium of pressure at the solid-fluid interface, the condition (2.4b) expresses the continuity of the normal component of the velocity field, and (2.4c) refers to a thermally insulated body. We assume the causal initial conditions

 U(x,t)=∂U(x,t)∂t=0,Θ(x,t)=0 forx∈Ω−,t≤0, (2.5a) V(x,t)=∂V∂t(x,t)=0, forx∈Ω+,t≤0. (2.5b)

We will study the the time-dependent scattering problem consisting of the partial differential equations (2.1)-(2.3) together with the transmission conditions (2.4a)-(2.4c) and the homogeneous initial conditions (2.5a)-(2.5b).

## 3 Reduction to a nonlocal problem

In order to apply Lubich’s approach as in the case of fluid-structure interaction [18, 17], we first need to transform the initial-boundary transmission problem (2.1)-(2.4c) in the Laplace domain. Then we reduce the corresponding problem to a nonlocal boundary value problem. We begin with the Laplace transform for a restricted class of distributions. Let be a Banach space and denote the Schwartz class of functions. We say that is a causal tempered distribution with values in if it is a continuous linear map such that

 F(φ)=0∀φ∈S(R) such that supp\leavevmode\nobreak φ⊂(−∞,0).

For such a distribution and

 s∈C+:={s∈C:Re\leavevmode\nobreak s>0},

the Laplace transform of can be defined in a natural way by

 f(s)=L{F}(s):=∫∞0e−stF(t)dt,

where the integral must be understood in the sense of Bochner [35]. We remark that the Laplace transform can be defined for a much broader class of distributions [4, 9], but this restricted class suffices for the current application.

Let then

 u:=u(x,s)=L{U(x,t)},θ:=θ(x,s)=L{Θ(x,t)},v:=v(x,s)=L{V(x,t)}.

Then the initial-boundary transmission problem consisting of (2.1) - (2.4c) in the Laplace transformed domain becomes the following transmission boundary value problem:

 −Δ∗u+ρΣs2u+ζ∇θ= 0 inΩ−, (3.1a) −Δθ+sκ\leavevmode\nobreak θ+s\leavevmode\nobreak η\leavevmode\nobreak ∇⋅u= 0 inΩ−, (3.1b) −Δv+s2c2\leavevmode\nobreak v= 0 inΩ+, (3.1c) σ(u,θ)−n+ρfsv+n= −ρfsvincn onΓ, (3.1d) s\leavevmode\nobreak u−⋅n+∂v∂n+= −∂vinc∂n onΓ, (3.1e) ∂θ∂n−= 0 onΓ. (3.1f)

We remark that (3.1) is an exterior scattering problem for which normally a radiation condition is needed in order to guarantee the uniqueness of the solution. However, in the present case no additional radiation condition is required and global behavior at infinity suffices.

To derive a proper nonlocal boundary problem, as usual, we begin via Green’s third identity with the representation of the solutions of (3.1c) in the form:

 v=D(s)ϕ−S(s)λinΩ+, (3.2)

where and are the Cauchy data for in (3.1c) and and are the simple-layer and double-layer potentials, respectively defined by

 S(s)λ(x):= ∫ΓEs/c(x,y)λ(y)dΓy, x∈R3∖Γ, (3.3) D(s)ϕ(x):= ∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy, x∈R3∖Γ. (3.4)

Here

 Es/c(x,y)=e−s/c\leavevmode\nobreak |x−y|4π|x−y|

is the fundamental solution of the operator in (3.1c). By standard arguments in potential theory, we have the relations for the the Cauchy data and :

 (ϕλ)=(12I+K(s)−V(s)−W(s)(12I−K(s))′)(ϕλ)onΓ. (3.5)

Here and are the four basic boundary integral operators familiar from potential theory [19] such that

 V(s)λ(x):= ∫ΓEs/c(x,y)λ(y)dΓy x∈Γ, K(s)ϕ(x):= ∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy x∈Γ, K′(s)λ(x):= ∫Γ∂∂nxEs/c(x,y)λ(y)dΓy x∈Γ, W(s)ϕ(x):= −∂∂nx∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy x∈Γ.

By using the transmission condition (3.1e), we obtain from the second boundary integral equation in (3.5),

 −su−⋅n+W(s)ϕ−(12I−K(s))′λ=∂vinc∂nonΓ (3.6)

while the first boundary integral equation in (3.5) is simply

 (12I−K(s))ϕ+V(s)λ=0onΓ. (3.7)

With the Cauchy data and as new unknowns, the partial differential equation (3.1c) in may be eliminated. This leads to a nonlocal boundary value problem in for the unknowns satisfying the partial differential equations (3.1a), (3.1b), and the boundary integral equations (3.6), (3.7) together with the conditions (3.1d) and (3.1f) on .

Here and in the sequel let and denote trace operators of the functions and their normal derivatives from inside and outside , respectively. We will use the symbol interchangeably to denote the scalar, vector, or Frobenius inner products of functions defined on the open set , while the angled brackets will be reserved for pairings between elements of the trace space and its dual. All the forms will be kept linear and conjugation will be done explicitly when needed. Finally, the space should be understood as the Cartesian product of copies of the standard scalar Sobolev space endowed with the natural product norm.

Let us first consider the unknowns . Then multiplying (3.1a) by the testing function and integrating by parts, we obtain the weak formulation of (3.1a):

 a(u,v;s)−⟨σ(u,θ)n,γ−v⟩Γ−ζ(θ,∇⋅v)Ω−=0, (3.8)

where is the bilinear form defined by

 a(u,v;s):=(Cε(u),ε(v))Ω−+s2ρΣ(u,v)Ω−. (3.9)

In terms of the transmission condition (3.1d), we obtain from (3.8)

 a(u,v;s)−ζ(θ,∇⋅v)Ω−+ρfs⟨ϕ\leavevmode\nobreak n,γ−v⟩Γ=−sρf⟨vincn,γ−v⟩Γ. (3.10)

Similarly, multiplying (3.1b) by the test function , integrating by parts and making use of the condition (3.1f), we have

 b(θ,ϑ;s)+s\leavevmode\nobreak η(∇⋅u,ϑ)Ω−=0 (3.11)

with

 b(θ,ϑ;s):=(∇θ,∇ϑ)Ω−+sκ(θ,ϑ)Ω−. (3.12)

Now let

 As: H1(Ω−) ⟶ (H1(Ω−))′ u ⟼ a(u,⋅;s) Bs: H1(Ω−) ⟶ (H1(Ω−))′ θ ⟼ b(θ,⋅;s)

be the operators associated to the bilinear forms (3.9) and (3.12), respectively. Then from (3.10), (3.12), (3.6), and (3.7), the nonlocal problem may be formulated as a system of operator equations: Given data find such that

 A⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠:=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s\leavevmode\nobreak η\leavevmode\nobreak divBs00−s\leavevmode\nobreak n⊤γ−0W(s)−(12I−K(s))′0012I−K(s)V(s)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜⎝d1d2d3d4⎞⎟ ⎟ ⎟⎠. (3.13)

In the above expression denotes the transpose (row) of the outward unit normal (column) to , and data is given by

 d1=−s\leavevmode\nobreak ρfγ−′(γ+vincn),d2=0,d3=∂+nvinc,d4=0. (3.14)

We have made use of the product spaces:

 X:= H1(Ω−)×H1(Ω−)×H1/2(Γ)×H−1/2(Γ), X′:= (H1(Ω−))′×(H1(Ω−))′×H−1/2(Γ)×H1/2(Γ)

(i. e., is the dual of ). Our aim is to show that equation (3.13) has a unique solution in . We will do this in the next section.

## 4 Existence and uniqueness results

Before considering the existence and uniqueness results, we first discuss the invertibility of the operator in (3.13). We begin with the definitions of the following energy norms:

 |||u|||2|s|,Ω−:= (Cε(u),ε(¯u))Ω−+ρΣ∥su∥2Ω−, u∈H1(Ω−), (4.1) |||θ|||2|s|,Ω−:= ∥∇θ∥2Ω−+κ−1∥√|s|θ∥2Ω−, θ∈H1(Ω−), (4.2) |||v|||2|s|,Ω+:= ∥∇v∥2Ω++c−2∥sv∥2Ω+, v∈H1(Ω+). (4.3)

For the complex Laplace parameter we will denote

 σ:=Re\leavevmode\nobreak s,σ––:=min{1,σ},

and will make use of the following equivalence relations for the norms

 σ––|||u|||1,Ω−≤|||u||||s|,Ω−≤|s|σ––|||u|||1,Ω−, (4.4) √σ––|||θ|||1,Ω−≤|||θ||||s|,Ω+≤√|s|σ––|||θ|||1,Ω−, (4.5) σ––|||v|||1,Ω+≤|||v||||s|,Ω+≤|s|σ––|||v|||1,Ω+, (4.6)

which can be obtained from the inequalities:

 min{1,σ}≤min{1,|s|},andmax{1,|s|}min{1,σ}≤|s|,\leavevmode\nobreak ∀s∈C+.

We remark that the norms and are equivalent to and , respectively, and so is the energy norm equivalent to the norm of by the second Korn inequality [11].

For the invertibility of , let us introduce the diagonal matrix :

 Z=⎛⎜ ⎜ ⎜ ⎜⎝10000ζ(η|s|)−10000ρf0000ρf⎞⎟ ⎟ ⎟ ⎟⎠ (4.7)

and consider the modified operator

 B:=ZA=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s|s|−1\leavevmode\nobreak ζ\leavevmode\nobreak divζ\leavevmode\nobreak (η\leavevmode\nobreak |s|)−1Bs00−s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−0ρf\leavevmode\nobreak W(s)−ρf(12I−K(s))′00ρf(12I−K(s))ρfV(s)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (4.8)

It will be clear that in order to show the invertibility of it suffices to prove that of . By the Gaussian elimination procedure (as in [25]), a simple computation shows that can be decomposed in the form:

 B=˜PCP, (4.9)

where

 C=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s|s|−1\leavevmode\nobreak ζ\leavevmode\nobreak divζ\leavevmode\nobreak (η\leavevmode\nobreak |s|)−1Bs00−s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−0ρf\leavevmode\nobreak (W(s)+CΓ(s))0000ρfV(s)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,
 ˜P=⎛⎜ ⎜ ⎜ ⎜⎝I0000I0000I−(12I−K(s))′V−1(s)000I⎞⎟ ⎟ ⎟ ⎟⎠andP=⎛⎜ ⎜ ⎜ ⎜⎝I0000I0000I000V−1(s)(12I−K(s))I⎞⎟ ⎟ ⎟ ⎟⎠

with .

Since both and are invertible, the invertibility of will imply that of , but as in the time-dependent fluid-structure interaction [18], the operator matrix is indeed invertible. In fact, it is not difficult to show that the operator matrix is strongly elliptic [19, 33] in the sense that

 ≥ C(ζ,η,ρf)σσ––3|s|2∥(v,ϑ,ψ,χ)∥2X (4.10)

for all , where is a constant depending only on the physical parameters and on the geometry of , and is the matrix defined by

 Z(s):=⎛⎜ ⎜ ⎜ ⎜⎝¯s/|s|000010000¯s/|s|0000s/|s|⎞⎟ ⎟ ⎟ ⎟⎠.

The action of amounts only for a rotation that reveals the elliptic nature of the original system. As such, the analyticity of is immaterial, since only the inverse Laplace transform of is sought for. As for the proof of (4.10), we will just point out that it is simple to show that

 Re{ −¯s/|s|(ζ\leavevmode\nobreak (div)′ϑ,¯v)Ω−+¯s/|s|(s\leavevmode\nobreak ρfγ−′ψ\leavevmode\nobreak n,¯v)Ω− +s/|s|(ζ\leavevmode\nobreak div\leavevmode\nobreak v,¯ϑ)Ω−−¯s/|s|⟨s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−v,¯ψ⟩Γ}= 0. (4.11)

Further details are omitted, since a similar proof will be repeated when we discuss the existence and uniqueness results for the solution of the nonlocal problem (3.13).

We now return to the solutions of the modified system of equations (4.8) from (3.13):

 B⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠:=ZA⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠=Z⎛⎜ ⎜ ⎜⎝d1d2d3d4⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜⎝d1ζ/η\leavevmode\nobreak |s|−1\leavevmode\nobreak d2ρfd3ρfd4⎞⎟ ⎟ ⎟ ⎟⎠. (4.12)

Suppose that is a solution of (4.12). Let

 v:=D(s)ϕ−S(s)λinR3∖Γ. (4.13)

Then is the solution of the transmission problem:

 −Δv+(s/c)2v=0inR3∖Γ (4.14)

satisfying the following jump relations across ,

 [[γv]]:=γ+v−γ−v=ϕ∈H1/2(Γ),[[∂nv]]:=∂+nv−∂−nv=λ∈H−1/2(Γ).

First, from (4.14) we see that

 Asu−ζ\leavevmode\nobreak (div)′θ+sρfγ−′\leavevmode\nobreak [[γv]]n= d1 inΩ−, (4.15) s|s|ζ\leavevmode\nobreak divu+ζη1|s|Bs\leavevmode\nobreak θ= ζη1|s|d2 inΩ−, (4.16) −s\leavevmode\nobreak ρfn⋅γ−u−ρf∂+nv= ρf\leavevmode\nobreak d3 onΓ, (4.17) −ρfγ−v= ρf\leavevmode\nobreak d4 onΓ. (4.18)

Since , this means that is a solution of the homogeneous Dirichlet problem for the partial differential equation (4.14) in . Hence by the uniqueness of the the solution, we obtain in . Consequently, we have

 [[γv]]=γ+v=ϕand[[∂nv]]=∂+nv=λ. (4.19)

Next, we consider the variational formulation of the problem for equations (4.14) , (4.15) and (4.16) together with the boundary conditions (4.17) and (3.1f). We seek a solution

 (u,θ,v)∈H=H1(Ω−)×H1(Ω−)×H1(Ω+)

with the corresponding test functions in the same function space. To derive the variational equations, we should keep in mind that the variational formulation should be formulated not in terms of the Cauchy data and directly, but only through the jumps and as indicated.

Let , we introduce the bilinear form

 cΩ+(f,g;s):=(∇f,∇g)Ω++(s/c)2(f,g)Ω+,

and its associated operator

 Cs,Ω+: H1(Ω+) ⟶ (H1(Ω+))′ f ⟼ cΩ+(f,⋅;s)

Note that in these definitions the domain of integration is indicated explicitly. Using this notation we can use the first Green formula for equation (4.14) and condition (4.17) to obtain

 ρf(Cs,Ω+v,w)Ω+= −ρf⟨∂+nv,γ+w⟩ = ρf⟨d3,γ+w⟩Γ+ρf⟨s\leavevmode\nobreak γ−u⋅n,γ+w⟩Γ. (4.20)

Together with the weak formulations of (4.15) and (4.16), we arrive at the following variational formulation: Find satisfying

 (Asu,v)Ω−−ζ(θ,divv)Ω−+sρf⟨γ+vn,γ−v⟩Γ= (d1,v)Ω− ∀\leavevmode\nobreak v∈H1(Ω−), ζ|s|η(Bsθ,\leavevmode\nobreak ϑ)Ω−+s|s|ζ\leavevmode\nobreak (div\leavevmode\nobreak u,ϑ)Ω−= ζ|s|η(d2,\leavevmode\nobreak ϑ)Ω− ∀\leavevmode\nobreak ϑ∈H1(Ω−), (4.21) ρf(Cs,Ω+v,w)Ω+−s\leavevmode\nobreak ρf⟨γ−u,\leavevmode\nobreak γ+v\leavevmode\nobreak n⟩Γ= ρf⟨d3,γ+w⟩Γ ∀\leavevmode\nobreak w∈H1(Ω+).

We remark that by construction, it can be shown that as in [35] this variational problem is equivalent to the transmission problem defined by (4.14), (4.15), (4.16), and (4.17). The latter is equivalent to the nonlocal problem defined by (4.12), which is equivalent to (3.13). Consequently, the variational problem (4.21) is equivalent to the nonlocal problem (3.13). Hence for the existence of the solution of (3.13), it is sufficient to show the existence of the solution of (4.21).

We have the following basic results.

###### Theorem 4.1.

The variational problem (4.21) has a unique solution . Moreover, the following estimate holds:

 (|||u|||2|s|,Ω−+|||θ|||2|s|,Ω−+|||v|||2|s|,Ω+)1/2≤c0|s|3/2σσ––3/2\leavevmode\nobreak ∥(d1,0,d3,0)∥X′, (4.22)

where is a constant depending only on the physical parameters .

###### Proof.

Starting with the system (4.21), a simple computation shows that

 Re{¯s|s|(Asu,¯¯¯¯u)Ω−+ζη|s|(Bsθ,¯¯¯θ)Ω− +¯s|s|ρf(Cs,Ω+v,¯¯¯v)Ω+} = Re{¯¯¯s|s|(d1,¯¯¯¯u)Ω−+ζη1|s|(d2,¯¯¯θ)Ω−+¯¯¯s|s|ρf⟨d3,¯¯¯¯¯¯¯¯¯γ+v⟩Γ}. (4.23)

On the other hand, it is not hard to verify that

 Re{¯s|s|\leavevmode\nobreak (Asu,¯¯¯¯u)Ω−}= σ|s|\leavevmode\nobreak |||u|||2|s|,Ω−, (4.24) Re{ζη\leavevmode\nobreak 1|s|(Bsθ,¯¯¯θ)Ω−}≥ ζη\leavevmode\nobreak σ|s|2\leavevmode\nobreak |||θ|||2|s|,Ω−, (4.25) Re{¯s|s|\leavevmode\nobreak ρf\leavevmode\nobreak (