Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor

We obtain well-posedness results in $L_p$-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor, in complementary Lipschitz domains of ${\mathbb R}^n$, $n\ge 3$. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in $L_p$-based weighted Sobolev and Besov spaces. We show that such a mixed variational formulation is well-posed in the space ${\mathcal H}^1_{p}({\mathbb R}^n)^n\times L_p({\mathbb R}^n)$, {$n\geq 3$}, for any $p$ in an open interval containing $2$. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system. Various mapping properties of these operators are also obtained. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of the nonlinear transmission problem for the anisotropic Stokes and Navier-Stokes systems in $L_p$-based weighted Sobolev spaces, whenever the given data are small enough.


Introduction
A powerful tool in the analysis of boundary value problems for partial differential equations is played by the layer potential methods.Mitrea and Wright [50] used them to obtain well-posedness results for the main boundary value problems for the constant-coefficient Stokes system in Lipschitz domains in R n in Sobolev, Bessel potential, and Besov spaces (see also [9, Proposition 4.5]  for an unsteady exterior Stokes problem).The authors in [34] obtained mapping properties of the constant-coefficient Stokes and Brinkman layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators (see also [35,36]).
The methods of layer potential theory play also a significant role in the study of elliptic boundary problems with variable coefficients.Mitrea and Taylor [49,Theorem 7.1] used the technique of layer potentials to prove the well-posedness of the Dirichlet problem for the Stokes system in L p -spaces on arbitrary Lipschitz domains in a compact Riemannian manifold.Dindos and Mitrea [24,Theorems 5.1,5.6,7.1,7.3]used a boundary integral approach to show well-posedness results in Sobolev and Besov spaces for Poisson problems of Dirichlet type for the Stokes and Navier-Stokes systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds.A layer potential analysis of pseudodifferential operators of Agmon-Douglis-Nirenberg type in Lipschitz domains on compact Riemannian manifolds has been developed in [39].The authors in [37] used a layer potential approach and a fixed point theorem to show well-posedness of transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds.Choi and Lee [21] proved the well-posedness in Sobolev spaces for the Dirichlet problem for the Stokes system with non-smooth coefficients in a Lipschitz domain Ω ⊂ R n (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations (VMO) with respect to all variables.Choi and Yang [22] established existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in the space R d , d ≥ 3, when the weak solutions of the system are locally Hölder continuous.
Alliot and Amrouche [3] developed a variational approach to show the existence of weak solutions for the exterior Stokes problem in weighted Sobolev spaces (see also [5,29]).The authors in [40] developed a variational approach in order to analyze Stokes and Navier-Stokes systems with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds (see also [41]).
An alternative integral approach, which reduces various boundary value problems for variablecoefficient elliptic partial differential equations to boundary-domain integral equations (BDIEs), by means of explicit parametrix-based integral potentials, was explored e.g., in [16][17][18]48].Equivalence of BDIEs to the boundary problems and invertibility of BDIE operators in L 2 and L pbased Sobolev spaces have been analyzed in these works.Localized boundary-domain integral equations based on a harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients have been also developed, see [19] and the references therein.
Brewster et al. in [11] used a variational approach to show well-posedness results for Dirichlet, Neumann and mixed problems for higher order divergence-form elliptic equations with L ∞ coefficients in locally (ǫ, δ)-domains and in Besov and Bessel potential spaces.Sayas and Selgas in [54] developed a variational approach for the constant-coefficient Stokes layer potentials, by using the technique of Nédélec [51].Bȃcut ¸ȃ, Hassell and Hsiao [9] developed a variational approach for the constant-coefficient Brinkman single layer potential and analyzed the time-dependent exterior Stokes problem with Dirichlet condition in R n , n = 2, 3. Barton [8] used the Lax-Milgram Lemma to construct layer potentials for strongly elliptic operators in general settings.
Throughout this paper, we use the Einstein convention on summation in repeated indices from 1 to n, and the standard notation ∂ α for the first order partial derivative with respect to the variable x α , α = 1, . . ., n.Let Ľ be a second order differential operator in divergence form, where A = A αβ 1≤α,β≤n is the viscosity coefficient fourth order tensor, and for fixed α and β A αβ = A αβ (x) are n × n matrix-valued functions on R n , such that We will further shorten (1.2) as A ∈ L ∞ (R n ) n 4 .We assume that the boundedness condition a αβ ij (x) ≤ c A and the strong ellipticity condition hold for almost any x ∈ R n , with a constant c A > 0 (cf.[11, (7.23)], [20, (1.1)]).
Let u be an unknown vector field for velocity, π be an unknown scalar field for pressure, and f be a given vector field for distributed forces, defined on an open set D ⊂ R n with the compact boundary ∂D.Then the equations L(u, π) := ∂ α A αβ ∂ β u − ∇π = f , div u = 0 in D (1.4) determine the Stokes system with L ∞ tensor viscosity coefficient.Let λ ∈ L ∞ (R n ).Then the nonlinear system is called the anisotropic Navier-Stokes system with L ∞ viscosity tensor A = A αβ 1≤α,β≤n .The systems (1.4) and (1.5) can describe flows of viscous incompressible fluids with anisotropic viscosity tensor, and the viscosity tensor A is related to the physical properties of such a fluid (see [20,25,52]).Our goal is to treat transmission problems for the Stokes and Navier-Stokes systems (1.4) and (1.5) in R n \ ∂Ω, where ∂Ω is a Lipschitz boundary.Then we have to add adequate conditions at infinity by setting our problems in weighted Sobolev spaces.
Remark 1.1.In the isotropic case (see [25]), with µ ∈ L ∞ (R n ), we assume that there exists a constant c µ > 0, such that c −1 µ ≤ µ ≤ c µ a.e. in R n .In such a case, the operator L given by (1.4) takes the form if div u = 0.The tensor āαβ ij given by (1.6) satisfies the second (ellipticity) condition in (1.3)only for symmetric matrices ξ.On the other hand, for any u and π, L(u, π) given by (1.7) can be also represented as where x ∈ R n and for any ξ = (ξ iα ) 1≤i,α≤n ∈ R n×n .Hence the ellipticity condition (1.3) is satisfied for any matrices, and our analysis is also applicable to the isotropic Stokes system.Note that a αβ ij ∂ β u j = µ∂ α u i can be associated with the viscous part of the pseudostress µ∂ α u i − δ αi π, cf., e.g., [14].The approaches based on the pseudostress formulation have been intensively used in the study of viscous incompressible fluid flows due to their ability to avoid the symmetry condition that appears in the approaches based on the standard stress formulation (see, e.g., [14,15]).

Preliminary results
Let further on in the paper Ω + := Ω be a bounded Lipschitz domain in R n (n ≥ 3) with connected boundary ∂Ω.Let Ω − := R n \ Ω + .Let E± denote the operator of extension by zero outside Ω ± .

Standard L p -based Sobolev spaces and related results
For p ∈ (1, ∞), L p (R n ) denotes the Lebesgue space of (equivalence classes of) measurable, p th integrable functions on R n , and L ∞ (R n ) denotes space of (equivalence classes of) essentially bounded measurable functions on R n .For any p ∈ (1, ∞), the conjugate exponent p ′ is given by Given a Banach space X , its topological dual is denoted by X ′ .The duality pairing of two dual spaces defined on a subset ) denote the space of infinitely differentiable functions with compact support in Ω ′ , equipped with the inductive limit topology.Let D ′ (Ω ′ ) denote the corresponding space of distributions on Ω ′ , i.e., the dual space of ) n are the spaces of vector-valued functions with components in H 1 p (Ω ′ ) and H 1 p (Ω ′ ), respectively, and similar extensions to the vector-valued functions or distributions are assumed to all other spaces introduced further.The Sobolev space H 1 p (Ω ′ ) can be identified with the closure H1 p (Ω ′ ) of D(Ω ′ ) in H 1 p (Ω ′ ) (see, e.g., [33], and [44,Theorem 3.33] for p = 2).For p ∈ (1, ∞) and s ∈ (0, 1), the boundary Besov space B s p,p (∂Ω) can be defined by means of the method of real interpolation, B s p,p (∂Ω) = L p (∂Ω), H ∂Ω).For further properties of standard Sobolev and Besov spaces we refer the reader to [33,44,50,57].
Lemma 2.1.Let Ω + be a bounded Lipschitz domain of R n with connected boundary ∂Ω, and let Ω − := R n \Ω be the corresponding exterior domain.If p ∈ (1, ∞), then there exist a linear bounded trace operator γ ± : The operator γ ± is surjective and has a (non-unique) linear and bounded right inverse γ −1 ± : ∂Ω) is also well defined and bounded.
∂Ω) n is linear and continuous, and definition (2.15) does not depend on the choice of a right inverse In addition, the first Green identity The proof follows with similar arguments as those for [36, Lemma 2.2] (see also [46, Definition 3.1, Theorem 3.2], [47]).We omit the details for the sake of brevity.
For (u and denote the jump of the corresponding conormal derivatives by (2.17) Lemma 2.5 implies the following result.

Conormal derivative for the adjoint system
The formally adjoint operator L * is defined by where Note that our notation A * αβ coincides with the notation (A βα ) ⊤ in [20].Evidently, the coefficients of L * also satisfy conditions (1.3) with the same constant c.
, the classical conormal derivative operator T * c± associated with L * is defined by For more general functions v and q, we can introduce, similar to Definition 2.4, the notion of formal and generalized conormal derivatives associated with ∂Ω) n is linear and continuous, and definition (2.21) does not depend on the choice of a right inverse γ −1 ± : In addition, the following first Green identity holds for any Lemma 2.8 implies the following analogue of Lemma 2.6.
Let v and q be the couples {v + , v − } and {q + , q − }.Then
Theorem 2.10.Let X and M be two real Hilbert spaces.Let a(•, •) : ) with some constant β > 0. Then the mixed variational problem with the unknown (u, p) ∈ X × M, is well-posed, which means that (2.28) has a unique solution (u, p) in X × M and there exists a constant C > 0 depending on β and c a , such that We will also need the following result (see [26, Theorem A.56, Remark 2.7]).
Lemma 2.11.Let X and M be reflexive Banach spaces.Let b(•, •) : X × M → R be a bounded bilinear form.Let B : X → M ′ and B * : M → X ′ be the linear bounded operators given by where •, • := X ′ •, • X denotes the duality pairing of the dual spaces X ′ and X.The duality pairing between M ′ and M is also denoted by •, • .Then the following assertions are equivalent: (i) There exists a constant β > 0 such that b(•, •) satisfies the inf-sup condition (2.27).
(ii) The map B : X/V → M ′ is an isomorphism and Bw M ′ ≥ β w X/V , for any w ∈ X/V.
3. Volume and layer potential operators for the L ∞ coefficient Stokes system in L p -based Sobolev and Besov spaces In the sequel, Ω + ⊂ R n (n ≥ 3) is a bounded Lipschitz domain with connected boundary ∂Ω, and Ω − := R n \ Ω.

Weak solution of the Stokes system with
The main role in our analysis is played by the following result (see also [38,Lemma 4.1] for p = 2).
Lemma 3.1.Let A satisfy conditions (1.2) and (1.3).Let p ∈ (1, ∞), and a R n : Then there exists p * ∈ (2, ∞) such that for any p ∈ R(p * , n), where and for all given data ξ ∈ H Proof.Inequalities (1.3) combined with the Hölder inequality imply that there exists a constant Thus, the bilinear form a R n : Let us first prove the lemma for p = 2.To do so, we intend to use Theorem 2.10, which requires the coercivity of the bilinear form a R n (•, Indeed, the strong ellipticity condition (1.3) and the property that the semi-norm is a norm on 3) and (2.4) with p = 2), imply that there exists a constant Inequalities (3.6) and (3.7) show that the bilinear form a R n : Moreover, the boundedness of the operator div : divergence free vector fields has the following characterization In view of the isomorphism property of the operator , Lemma 2.5]), there exists a constant c 2 > 0 such that for any q ∈ L 2 (R n ) there exists v ∈ H 1 (R n ) n satisfying the equation −div v = q and the inequality Consequently, the bilinear form b R n (•, (see also Lemma 2.11(ii), and [54, Proposition 2.4] for n = 2, 3).Then Theorem 2.10, with and note that ) be the operator defined on any (u, π) ∈ X p (R n ) in the weak form by Hence, establishing the existence of a solution to the variational problem (3.4) is equivalent to showing that the operator T R n : ) is an isomorphism (see also [11,Proposition 7.2], [30,Theorem 5.6], and [53, Theorem 3.1] for the standard Stokes system).
The linear operator T R n : To show that it is also a isomorphism for p in an open interval containing 2, we proceed as follows.
Consequently, whenever condition (3.3) holds and for all given data (ξ, Next we use Lemma 3.1 and show the well-posedness of the L ∞ -coefficient Stokes system in the space [38,Theorem 4.2] for p = 2 with A(x) = µ(x)I, [42, Proposition 2.9] and [2, Theorem 3] for p ∈ (1, n) in the constant-coefficient case).
) 3).For f ∈ H −1 p (R n ) n , we define the Newtonian velocity and pressure potentials for the L ∞ -coefficient Stokes system, by setting Proof.First, we note that the last condition in (3.12) is understood in the sense of distributions, as in Definition 2.4.Next, we show that the transmission problem (3.12) has the following equivalent mixed variational formulation: where a R n and b R n are the bilinear forms given by (3.1) and (3.2).First, assume that the pair (u ψ , π ψ ) ∈ H 1 p (R n ) n × L p (R n ) satisfies the transmission problem (3.12).Then formula (2.18) shows that the same pair satisfies also the first equation in (3.13).The second equation of the mixed variational formulation (3.13) follows from the fact that u ψ ∈ H 1 p (R n ) n satisfies the second equation in (3.12).Conversely, assume that the pair (u ) is a solution of the mixed variational formulation (3.13).In view of the density of the space D(R n ) n in H 1 p ′ (R n ) n , and by choosing in the first equation of the system (3.13)any v ∈ C ∞ (R n ) n with compact support in Ω ± (and, thus, γv = 0), we obtain the variational equation , which yields the first equation in (3.12).The second equation in (3.12) follows immediately from the second equation in (3.13), the property that the operator div : [2, Proposition 2.1], see also [54,Proposition 2.4] for p = 2), and the duality between the spaces L p (R n ) and L p ′ (R n ).The assumption u ψ ∈ H 1 p (R n ) n implies the first transmission condition in (3.12).Using again formula (2.18), the first equation in (3.13), and Lemma 2.1, we obtain the relation [T(u ψ , π ψ )] − ψ, Φ ∂Ω = 0, for any Φ ∈ B 1 p p ′ ,p ′ (∂Ω) n and hence the second transmission condition in (3.12).
In addition, the continuity of the trace operator γ : p ′ ,p ′ (∂Ω) n and of its adjoint According to Lemma 3.1 there exists p * ∈ (2, ∞), such that for any p as in (3.3) and for ), which depends continuously on ψ.Moreover, the equivalence between problems (3.12) and (3.13) shows that (u ψ , π ψ ) ∈ H 1 p (R n ) n × L p (R n ) is the unique solution of the transmission problem (3.12).
The next result can be proved by the arguments similar to those in the proof of Theorem 3.5, mainly based on the Green formula (2.22).Theorem 3.6.Let A satisfy conditions (1.2) and (1.3).Then there exists p * ∈ (2, ∞), such that for any p ′ ∈ R(p * , n), cf.(3.3), and for any and there exists and the boundary operators where (u ψ , π ψ ) is the unique solution of the transmission problem The well-posedness of the transmission problem (3.12) proved in Theorem 3.5, definitions (3.17) and the transmission conditions in (3.12) imply the following assertion (cf.[54, Propositions 5.2 and 5.3], [34, Lemma A.4, (A.10), (A.12)] and [50, Theorem 10.5.3] for A = I).
For any ψ ∈ B − 1 p p,p (∂Ω) n , the following jump relations hold a.e. on ∂Ω By using Theorem 3.6 we can also define the single layer potential operators, V * ∂Ω and Q s * ∂Ω , of the adjoint Stokes system (3.15).Definition 3.9.Let A satisfy conditions (1.2) and (1.3).Let p * ∈ (2, ∞) be as in Theorem 3.6 we define the single layer velocity and pressure potentials with the density ψ * for the adjoint Stokes operator L * defined in (2.20), with coefficients A, by setting and the operators V * ∂Ω : B where (v ψ * , π ψ * ) is the unique solution of the transmission problem Lemma 3.10.Let A satisfy conditions (1.2) and (1.3).Let p * ∈ (2, ∞) be as in Theorem 3.5 Proof.Formulas (3.22) follow with arguments similar to those for (3.20).By definition, the couple Moreover, the second formulas in (3.20) and (3.22) imply that For a given operator T : X → Y , we denote by Ker {T : X → Y } := {x ∈ X : T (x) = 0} the null space of T .Let ν denote the outward unit normal to Ω, which exists a.e. on ∂Ω, and let span{ν} := {cν : c ∈ R}.For p ∈ (1, ∞), consider the space V ∂Ω ν = 0 a.e. on ∂Ω , (3.30) In addition, for any p Proof.First, note that Theorem 3.5 implies that the transmission problem (3.12) with the Similarly, where V * ∂Ω : B n is the single layer operator for the adjoint Stokes system (3.15)(see Definition 3.9).By using formula (3.23) for the densities ψ ∈ B and the second relation in (3.34), we obtain relation (3.31).Next we determine the kernel of the single layer operator in case p = 2.To do so, we assume that ψ 0 ∈ Ker V ∂Ω : the transmission problem (3.12) with given datum ψ 0 .According to formula (2.19) and the assumption that γu ψ 0 = 0 a.e. on ∂Ω, we obtain that In addition, assumption Then by (3.33) we conclude that (3.32) holds also for any p ∈ [2, p * ) ∩ [2, n).

solution of the homogeneous version of problem (3.41). Then the first transmission condition implies that u
) is a solution of the homogeneous version of the transmission problem (3.12), which, in view of Theorem 3.5, has only the trivial solution.
The arguments similar to the ones for Theorem 3.5 imply that problem (3.41) has the following equivalent variational formulation: The existence of the bounded right inverses γ −1 ± : ∂Ω) n given, there is w ϕ ∈ H 1 p (Ω ± ) n , such that [γw ϕ ] = −ϕ on ∂Ω.Thus, v ϕ := u ϕ −w ϕ has no jump across ∂Ω, and hence v ϕ ∈ H 1 p (R n ) n (see also [11,Theorem 5.13]).Moreover, problem (3.42) reduces to the variational problem with the unknown are the bounded bilinear forms given by (3.1) and (3.2), respectively.In addition, conditions (1.2) show the boundedness of the linear forms Therefore, Lemma 3.1 shows that the variational problem (3.43) has a unique solution ), and due to the equivalence between problems (3.41) and (3.42), it is also the unique solution of the problem (3.41) Theorem 3.14 leads to the following definition of the double layer operator for the nonsmoothcoefficient Brinkman system (1.4) (cf.[54, p. 77] for the constant-coefficient Stokes system in R 3 , and [8, formula (4.5) and Lemma 4.6] for general strongly elliptic differential operators).
and the boundary operators where (u ϕ , π ϕ ) is the unique solution of the transmission problem Theorem 3.14 and Definition 3.15 lead to the next result (see also [50, (10.81), (10.82)] and [54,Propositions 6.2,6.3] for the constant coefficient Stokes system in R 3 , and [8, Lemma 5.8]).(i) The following operators are linear and continuous, K ∂Ω : B Then the Green identities (2.22) and equality (3.51) yield that The second formula in (3.22), the first formula in (3.49), and relation (3.52) lead to equality (3.50).
We now show the following invertibility property of the operator D ∂Ω defined in (3.48) (see [54,Propositions 6.4 and 6.5] in the constant-coefficient case).
For s ∈ [−1, 1], let us define the subspaces H s * * (∂Ω) n := Ψ ∈ H s (∂Ω) n : Ψ, 1 ∂Ω = 0 Lemma 3.17.Let A satisfy conditions (1.2) and (1.3).Then and the following operator is an isomorphism, D ∂Ω : H (ii) Next, we show the invertibility of operator (3.55).First, we note that relations (3.53) imply that this operator is injective on the closed subspace H (see also [54,Proposition 6.5] in the constant coefficient case).To this end, ϕ ∈ H 1 2 0 (∂Ω) n and we apply the first Green identity (2.16) to the pair (u ϕ , π ϕ ) := (W ∂Ω ϕ, Q d ∂Ω ϕ) and w = u ϕ = W ∂Ω ϕ, and use the jump relations (3.49) and conditions (1.3) to obtain the inequality On the other hand, the continuity of the trace operators γ ± : ∂Ω) n and the first in jump relations (3.49) imply that there exists a constant (3.58) Note that the formula  The potentials introduced in the previous sections make the analysis of more general transmission problems for Stokes and Navier-Stokes systems rather elementary.Let us consider the spaces

Poisson problem of transmission type for the anisotropic Stokes system
First, for the given data ( f+ , f− , h, g) in F p , we consider the Poisson problem of transmission type for the anisotropic Stokes system The left-hand side in the last transmission condition in (4.3) is to be understood in the sense of formal conormal derivatives, cf.Definition 2.4.

Poisson problem with transmission conditions for the anisotropic Stokes and Navier-Stokes systems in L p -based weighted Sobolev spaces
In this subsection we restrict our analysis to the cases n = 3 and n = 4, for which some necessary embedding results hold.Next, we consider the following Poisson problem of transmission type for the Stokes and Navier-Stokes systems with E+ the extension by zero outside Ω + , λ ∈ L ∞ (Ω + ), and the left-hand side in the last transmission condition in (4.6) is to be understood in the sense of formal conormal derivatives, cf.Definition 2.4.We will show the following result (see [34,Theorem 5.2] for the Stokes and Navier-Stokes systems in the isotropic constant-coefficient case, A = I. , n there exist two constants, ζ p , η p > 0 depending on Ω + , Ω − , λ, c A , n, and p, with the property that for all given data f+ , f− , h, g ∈ F p satisfying the condition f+ , f− , h, g Fp ≤ ζ p , the transmission problem (4.6) has a unique solution ((u The Sobolev embedding Theorem (cf.Theorem 4.12 in [1]) implies that for any p ∈ n 2 , n , the embeddings are continuous, and by duality the last embedding implies that the embedding Thus, the nonlinear operator I λ;Ω+ : H 1 p (Ω + ) n → H −1 p (Ω + ) is continuous and bounded in the sense of (4.11).
We now construct a nonlinear operator U (p);+ that maps a closed ball B ηp of the space H 1 p;div (Ω + ) n (of divergence-free vector fields in H 1 p (Ω + ) n ) to B ηp and is a contraction on B ηp .Then the unique fixed point of U (p):+ will determine a solution of nonlinear problem (4.6).