Analysis of Boundary-Domain Integral Equations to the Mixed BVP for a Compressible Stokes System with Variable Viscosity

The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.

The reduction of different boundary value problems for the Stokes system to boundary integral equations in the case of constant viscosity was possible since the fundamental solutions for both, velocity and pressure, are readily available in an explicit form. Such reduction was used not only to analyse the properties of the Stokes system and BVP solutions, but also to solve BVPs by solving numerically the corresponding boundary integral equations.
In this paper we consider the stationary Stokes PDE system with variable viscosity and compressibility, in a bounded domain that models the motion of a laminar compressible viscous fluid, e.g., through a variable temperature field that makes both, viscosity and compressibility depending on coordinates. Reduction of the BVPs for the Stokes system with arbitrarily variable viscosity to explicit boundary integral equations is usually not possible, since the fundamental solution needed for such reduction is generally not available in an analytical form (except for some special dependence of the viscosity on coordinates). Using a parametrix (Levi function) as a substitute of a fundamental solution, in the spirit of [11], [7], it is possible however to reduce such a BVPs to some systems of Boundary-Domain Integral Equations, BDIEs, (cf. e.g. [21,Sect. 18], [23,22], where the Dirichlet, Neumann and Robin problems for some PDEs were reduced to indirect BDIEs) We will extend here the approach developed in [1,16] for a scalar variable-coefficient PDE, and will reduce the mixed boundary value problem for a compressible Stokes system of partial differential equations to two different systems of segregated direct BDIEs expressed in terms of surface and volume parametrix-based potential type operators. A parametrix for a given PDE (or PDE system) is not unique and a special care will be taken to chose a parametrix that leads to te BDIE systems simple enough to be analysed. The mapping properties of the parametrixbased hydrodynamic surface and volume potentials will be obtained and the equivalence and invertibility theorems for the operators associated with the BDIE systems will be proved.
Some preliminary results in this direction were obtained in [19], where we derived BDIE systems for the mixed incompressible Stokes problem in a bounded domain and equivalence between the BVP and BDIE systems was shown, however, invertibility results were not given there.
Note that the paper is mainly aimed not at the mixed boundary-value problem for the Stokes system, which properties are well-known nowadays, but rather at analysis of the BDIE systems per se. The analysis is interesting not only in its own rights but is also to pave the way for studying the corresponding localised BDIEs and analysing convergence and stability of BDIE-based numerical methods for PDEs, cf., e.g., [16,2,6,18,17,26,27,29,32,33].

Preliminaries
Let Ω = Ω + ⊂ R 3 be a bounded and simply-connected domain and let Ω − := R 3 Ω + . We will assume that the boundary ∂Ω is simply-connected, closed and infinitely differentiable. Furthermore, ∂Ω := ∂Ω N ∪ ∂Ω D where both ∂Ω N and ∂Ω D are non-empty, connected disjoint submanifolds of ∂Ω, and the interface between these two submanifolds is also infinitely differentiable.
Let v be the velocity vector field; p the pressure scalar field and µ ∈ C ∞ (Ω) be the variable kinematic viscosity of the fluid such that µ(x) > c > 0. For an arbitrary couple (p, v) the stress tensor operator, σ ij , and the Stokes operator, A j , for a compressible fluid are defined as (2.2) where δ j i is the Kronecker symbol. Henceforth we assume the Einstein summation in repeated indices from 1 to 3. We denote the Stokes operator as A = {A j } 3 j=1 andÅ := A| µ=1 . We will also use the following notation for derivative operators: ∂ j = ∂ x j := ∂ ∂x j with j = 1, 2, 3;
Similar to [15,Theorem 3.12] one can prove the following assertion.
Considering now v ≡ 0 and keeping in mind the Neumann-traction condition (2.10d), it is easy to conclude that p 1 = p 2 .

Parametrix and Remainder
When µ(x) = 1, the operator A becomes the constant-coefficient Stokes operatorÅ, for which we know an explicit fundamental solution defined by the pair of functions (q k ,ů k ), where summation in k is not assumed,ů k j represent components of the incompressible velocity fundamental solution andq k represent the components of the pressure fundamental solution (see e.g. [10], [9], [8]).
Therefore, the couple (q k ,ů k ) satisfies Then by (2.1) the stress tensor of the fundamental solution reads as and the classical boundary traction of the fundamental solution becomes Let us define a pair of functions (q k , u k ) 3 k=1 , Then by (2.1),

2) gives
where is a weakly singular remainder. This implies that (q k , u k ) is a parametrix of the operator A.
Note that the parametrix is generally not unique (cf. [20] for BDIEs based on an alternative parametrix for a scalar PDE). The possibility to factor out µ(x) µ(y) in (3.7)-(3.8) and ∇µ(x) µ(y) in (3.10) is due to the careful choice of the parametrix in form (3.5)-(3.6) and this essentially simplifies the analysis of obtained parametrix-based potentials and BDIE systems further on. Parametrix-based volume and surface potentials Let ρ and ρ be sufficiently smooth scalar and vector function on Ω, e.g., ρ ∈ D(Ω), ρ ∈ D(Ω). Let us define the parametrix-based Newton-type and remainder vector potentials for the velocity, and the scalar Newton-type and remainder potentials for the pressure, for y ∈ R 3 . The integral in (4.3) is understood as a 3D strongly singular integral in the Cauchy sense. The bilinear form in (4.4) should be understood in the sense of distribution, and the equality between (4.3) and (4.4) holds since where Ω ǫ = Ω \B ǫ (y) and B ǫ (y) is the ball of radius ǫ centred in y, which implies that Let us now define the parametrix-based velocity single layer potential, double layer potential and their respective direct values on the boundary, as follows: For the pressure we will need the following single-layer and double layer potentials: It is easy to observe that the parametrix-based integral operators, with the variable coefficient µ, can be expressed in terms of the corresponding integral operators for the constantcoefficient case, µ = 1, marked by, Note that although the constant-coefficient velocity potentialsŮ ρ,V ρ andW ρ are divergence-free in Ω ± , the corresponding potentials Uρ, V ρ and W ρ are not divergence-free for the variable coefficient µ(y). Note also that by (3.1) and (4.1), where is the harmonic Newton potential. Hence (4.14) Moreover, for the constant-coefficient potentials potentials we have the following well-known relations,Å
For the remaining part of the proof, we shall first assume that s ∈ (−1/2, 1/2). In this case, H s (Ω) can be identified with H s (Ω). Hence, the continuity of the operator (4.19) immediately follows from the continuity of (4.18).
Theorem 4.2 Let s > 1/2. The following operators are compact, Proof: The proof of the compactness for the operators R, γ + R and R • immediately follows from Theorem 4.1 and the trace theorem along with the Rellich compact embedding theorem.
To prove compactness of the operators T ± (R • , R), let us consider a function g ∈ H 1 (Ω).
, which implies that both canonical and classical conormal derivatives of (R • g, Rg) are well defined and moreover, similar to [15, Corollary 3.14] and [5, Theorem 2.13], one can prove that they coincide, Proof: The continuity of the operators in (4.33), (4.34) follows from relations (4.10), (4.11) and the continuity of the counterpart operators for the constant coefficient case, see e.g. [9,8].
Let us now define direct values on the boundary of the parametrix-based velocity single layer and double layer potentials and introduce the notations for the conormal derivative of the latter, Here T ± are the canonical derivative (traction) operators for the compressible fluid that are well defined due to continuity of the second operator in (4.35). Similar to the potentials in the domain, we can also express the boundary operators in terms of their counterparts with the constant coefficient µ = 1, Theorem 4.4 Let s ∈ R. Let S 1 and S 2 be two non empty manifolds on ∂Ω with smooth boundaries ∂S 1 and ∂S 2 , respectively. Then the following operators are continuous, Moreover, the following operators are compact, Proof: Continuity of operators in (4.39)-(4.41) follows from relations (4.37)-(4.38) and continuity of the counterpart operators for the constant coefficient case, see e.g. [9,8]. Then compactness of operators (4.42)-(4.44) is implied by the Rellich compactness embedding theorem.
Theorem 4.6 Let τ ∈ H 1/2 (∂Ω). Then, the following jump relation holds: and by continuity of the canonical On the other hand, as m → ∞. This implies (4.46).
Corollary 4.7 Let S 1 be a non empty submanifold of ∂Ω with smooth boundary. Then, the operators are continuous and the operators are compact.
Proof: The continuity of operators in (4.48) follows from Theorems 4.6 and 4.3. The compactness of the operators (4.48) follows from the continuity of the second operators in (4.48) ands the compact embedding H 1/2 (S 1 )֒→H −1/2 (S 1 ).
(i) Let us start from the velocity identity (5.2). For the parametrix, evidently, we have the inclusion (q k , u k ) ∈ H 1,0 (Ω ǫ (y); A). Therefore, we can apply the second Green identity (2.8) in the domain Ω ǫ (y) to (p, v) and to (q k , u k ) to obtain ∂Bǫ(y) Since all the functions in (5.3) are smooth, the canonical conormal derivatives coincide with the classical ones, given by (2.6), and it is easy to show that when ǫ → 0, the first integral in (5.3) tends to 0, the second tends to −v k (y), while integrands in the remaining domain integrals are weakly singular and these integrals tend to the corresponding improper integrals, which leads us to (5.2) for (p, v) ∈ D(Ω) × D(Ω) ⊂ H 1,0 (Ω; A).
(ii) Let us now prove the pressure identity (5.1) for (p, v) ∈ D(Ω) × D(Ω). One can do this using the second Green identity similar to (5.3) but we will employ a slightly different approach. Multiplying equation (2.2) by the fundamental pressure vectorq j (x, y), integrating over the domain Ω and writing it as the bilinear form, which will be then treated in the sense of distributions, we obtain Applying the first Green identity to the first term, we have, and also in the second term q j (·, y), ∂ j p Ω = − ∂ jq j (·, y), p Ω + q j (·, y) , p n j ∂Ω = p(y) + q j (·, y) , p n j ∂Ω , where we took into account that by (3.3) we have ∂ jq j (·, y) , p Ω = −p(y). (5.7) Substituting (5.5) and (5.6) into (5.4) and rearranging terms we get (5.9) Let us now simplify the first term in the right hand side of (5.8) using the symmetry ∂ x iq j (x, y) = ∂ x jq i (x, y) and (3.3). Then, Applying again the first Green identity to the first term in the right hand side of (5.10), we obtain Now, plug (5.11) into (5.10), Now, substitute (5.12), (5.7) and (5.9) into (5.8). As a result, we obtain Rearranging the terms, taking into account that T c+ j (p, v) = T j (p, v), and using the potential operator notations, we obtain (5.1) for (p, v) ∈ D(Ω) × D(Ω).
If the couple (p, v) ∈ H 1,0 (Ω; A) is a solution of the Stokes PDEs (2.9a)-(2.9b) with variable viscosity coefficient, then the third Green identities (5.1) and (5.2) reduce to We will also need the following trace and traction of the third Green identities for (p, v) ∈ H 1,0 (Ω; A) on ∂Ω, Let us now prove the following three assertions that are instrumental for proving the equivalence of the BDIE systems to the mixed BVP.
After multiplying (5.24) by µ and applying relations (4.6) and (4.10), we arrive at Applying the divergence operator to both sides of (5.25) and taking into account that the potentialsŮ,V , andW are divergence free, while forQ we have equation (4.14), we obtain Applying the Stokes operator with µ = 1 to these two equations, by  (5.27) then Ψ * = 0, and Φ * = 0, on ∂Ω.
Proof: Multiplying the second equation in (5.27) by µ and applying relations (4.10), (4.11), we obtainΠ Let us take the trace of the second equation in (5.28) restricting it to S 1 and take the traction with the constant coefficient µ = 1 of both equations in (5.28) restricting it to S 2 . Keeping in mind the jump relations given in Theorem 4.5 and notation (4.45), we arrive at the system of equations where Φ := µΦ * . This BIE system has been studied in [9, Theorem 3.10] (see Theorem 7.1 below) which implies that it has only the trivial solution, Ψ * = 0, Φ = 0.

BDIE system (M22 * )
Let us take equations (5.14) and (5.15) in the domain Ω and restrictions of equations (5.16) and (5.17) to the boundary parts ∂Ω N and ∂Ω D respectively. Substituting there representations (6.1) and considering again the unknown boundary functions ϕ and ψ as formally independent of (segregated from) the unknown domain functions p and v, we obtain the following system where the terms F 0 and F in the right hand side are given by (6.3). Let us denote the right hand side of BDIE system (6.6) as Then Theorems 4.1 and 4.3 imply the inclusion F 22 * ∈ H 1,0 (Ω, A)×H −1/2 (∂Ω D )×H 1/2 (∂Ω N ). Note that the BDIE system (6.6a)-(6.6d) can be split into the BDIE system (M22) of 3 vector equations, (6.6b)-(6.6d), for 3 vector unknowns, v, ψ and ϕ, and the separate equation (6.6a) that can be used, after solving the system, to obtain the pressure, p. However, since the couple (p, v) shares the space H 1,0 (Ω, A), equations (6.6b), (6.6c) and (6.6d) are not completely separate from equation (6.6a).
(iii) Finally, the unique solvability of the BDIE systems (6.2) and (6.6) in item (iii) follows from the unique solvability of the BVP, see Theorem 2.3, and items (i) and (ii).

Boundary Integral equations
When µ ≡ 1, the operator A becomesÅ and R = R • ≡ 0. Consequently, the boundarydomain integral equations system (6.2) can be split into a system of two vector boundary integral equations, and two integral representations, for p and v, where F 0 and F are given by (6.3).
is continuous and continuously invertible.
Theorem 7.1 will be instrumental in proving the following result.
Proof: (i) Let us start from operator (7.4). To this end let us define the operator and consider the new system . Consider now, the last two equations of the system (7.8), Multiplying equation (7.9) by µ and applying relations (4.10) and (4.45) we obtain −r ∂Ω DVψ + r ∂Ω DW (µφ) = µ F 11 3 , (7.11) −r ∂Ω NW ′ψ + r ∂Ω NL (µφ) = F 11 4 . (7.12) This system is uniquely solvable for φ and ψ, since the matrix operator of the left hand side is invertible, cf. Theorem 7.1. Hence v is uniquely determined from the second equation of the system (7.8) and thus also is p from the first equation. This proves the invertibility of the operator M 11 , which implies that M 11 is a Fredholm operator with zero index. Furthermore, the operator is compact due to Theorems 4.2, 4.4 and 4.7. Thus operator (7.4) is also a Fredholm operator with zero index. By virtue of the Equivalence Theorem 6.3 and Remark 6.1, the homogeneous system (M11) has only the trivial solution, hence operator (7.4) is invertible.
(ii) Let us now consider operator (7.3). Let X = (M 11 * ) −1 F 11 * be the solution of system (7.1) with an arbitrary right hand side F 11 is the inverse of operator (7.4). If, moreover, F 11 * ∈ H 1,0 (Ω, A) × H 1/2 (∂Ω D ) × H −1/2 (∂Ω N ), then the first two equations of system (7.1) and the mapping properties of the operators in these equations imply that is also continuous and is an inverse of operator (7.3). Note that the BDIE system (M11 * ) given by (6.2) can be split into the BDIE system (M11), of 3 vector equations (6.2b), (6.2c), (6.2d) for 3 vector unknowns, v, ψ and ϕ, and the scalar equation (6.2a) that can be used, after solving the system, to obtain the pressure, p. Using matrix notation, where (7.14) Theorem 7.3 The operator is continuous and continuously invertible.
Proof: By Theorem 7.2 for operator (7.3), BDIE system (6.2) is uniquely solvable and by Theorem 6.3 it is equivalent to the BVP (2.9), which implies unique solvability of the latter.
In addition, the inverse to operator (7.16) is defined as and is continuous since operator (7.3) is continuously invertible and F 11 * is a continuous function of (f , g, Ψ 0 , Φ 0 ) due to the mapping properties of the operators involved in (6.3) and (6.4).

Operators M 22
* and M 22 BDIE system (6.6) can be written in the matrix form as and X = (p, v, ψ, ϕ). By Theorems 4.1-4.4, the mapping properties of the operators involved in (7.18) imply continuity of the operator Lemma 7.5 Let ∂Ω =S 1 ∪S 2 , where S 1 and S 2 are two non-intersecting simply connected nonempty submanifolds of ∂Ω with infinitely smooth boundaries. For any vector there exists a unique four-tuple Furthermore, the operator is continuous.
Proof: Let us consider system (7.17) with an arbitrary right hand side Continuity of the operators in (7.37)-7.39 implies continuity of operator (7.36).
Proof: Let us consider the operator is also invertible. Thus, operator (7.48) is a zero-index Fredholm operator. Invertibility of this operator then follows from its injectivity implied by Theorem 6.3(iii). The last three vector equations of the system (M22) are segregated from p. Therefore, we can define the new system given by equations (6.6b)-(6.6d) which can be written in the matrix form as M 22 Y = F 22 , (7.57) where Y represents the vector containing the unknowns of the system Y = (v, ψ, φ) ∈ H 1 (Ω) × H −1/2 (∂Ω D ) × H 1/2 (∂Ω N ), and the matrix operator M 22 is given by Following the reasoning similar to the proof of Theorem 7.3, we obtain the following assertion.
Theorem 7.10 The operator is continuous and continuously invertible.