On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains
aa r X i v : . [ m a t h . A P ] A ug ON THE MIXED PROBLEM FOR THE SEMILINEAR DARCY-FORCHHEIMER-BRINKMANPDE SYSTEM IN BESOV SPACES ON CREASED LIPSCHITZ DOMAINS
ROBERT GUTT, MIRELA KOHR, SERGEY E. MIKHAILOV, AND WOLFGANG L. WENDLAND
Abstract.
The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semi-linear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces on a bounded Lipschitz domain in R , with p ina neighborhood of 2. This system is obtained by adding the semilinear term | u | u to the linear Brinkman equation.First, we provide some results about equivalence between the Gagliardo and non-tangential traces, as well as betweenthe weak canonical conormal derivatives and the non-tangential conormal derivatives. Various mapping and invertibilityproperties of some integral operators of potential theory for the linear Brinkman system, and well posedness results forthe Dirichlet and Neumann problems in L p -based Besov spaces on bounded Lipschitz domains in R n ( n ≥
3) are alsopresented. Then, employing integral potential operators, we show the well-posedness in L -based Sobolev spaces for themixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in R n ( n ≥ L p -based Sobolev spaces. Next we use thewell-posedness result in the linear case combined with a fixed point theorem in order to show the existence and uniquenessfor a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkmansystem in L p -based Besov spaces, with p ∈ (2 − ε, ε ) and some parameter ε > Introduction
Boundary integral methods are a powerful tool to investigate linear elliptic boundary value problems that appearin various areas of science and engineering (see, e.g., [4, 18, 22, 45, 62]). Among many valuable contributions in thefield we mention the well-posedness result of the Dirichlet problem for the Stokes system in Lipschitz domains in R n ( n ≥
3) with boundary data in L -based Sobolev spaces, which have been obtained by Fabes, Kenig and Verchota in[23] by using a layer potential analysis. Also, Mitrea and Wright [61] obtained the well-posedness results for Dirichlet,Neumann and transmission problems for the Stokes system on arbitrary Lipschitz domains in R n ( n ≥ L . Their results extended the results of [23] from the Euclidean settingto the case of compact Riemannian manifolds. Continuing the study of [62], Dindo˘s and Mitrea [22] developed a layerpotential analysis to obtain existence and uniqueness results for the Poisson problem for the Stokes and Navier-Stokessystems on C domains, but also on Lipschitz domains in compact Riemannian manifolds. Medkov´a in [45] studiedvarious transmission problems for the Brinkman system.Due to many practical applications, the mixed problems for elliptic boundary value problems on smooth and Lipschitzdomains have been also intensively investigated. Let us mention that Mitrea and Mitrea in [57] have proved sharp well-posedness results for the Poisson problem for the Laplace operator with mixed boundary conditions of Dirichlet andNeumann type on bounded Lipschitz domains in R whose boundaries satisfy a suitable geometric condition introducedby Brown [7], and with data in Sobolev and Besov spaces. Brown et al. [9] have obtained the well-posedness result of the Mathematics Subject Classification.
Primary 35J25, 42B20, 46E35; Secondary 76D, 76M.
Key words and phrases.
Semilinear Darcy-Forchheimer-Brinkman system; mixed Dirichlet-Neumann problem; L p -based Besov spaces;layer potential operators; Neumann-to-Dirichlet operator; existence and uniqueness.The work of M. Kohr was supported by the ”Scientific Grant for Excellence in Research”, GSCE–30259/2015, of the Babe¸s-BolyaiUniversity. The work of S.E. Mikhailov and W.L. Wendland was supported by the grant EP/M013545/1DSM: ”Mathematical Analysis ofBoundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK.. mixed Dirichlet-Neumann problem for the Stokes system on creased Lipschitz domains in R n ( n ≥ R (see also [14] for the mixed problems with cracks and [48] for united boundary-domain integral equations). Aninteresting boundary integral equation method for a mixed boundary value problem of the biharmonic equation hasbeen developed in [11].Boundary integral methods combined with fixed point theorems have been focused on the analysis of boundary valueproblems for linear elliptic systems with nonlinear boundary conditions and for nonlinear elliptic systems with various(linear or nonlinear) boundary conditions. Recently, the authors in [33] have used a boundary integral method toobtain existence results for a nonlinear problem of Neumann-transmission type for the Stokes and Brinkman systems onLipschitz domains in Euclidean setting and with boundary data in various L p , Sobolev, or Besov spaces. The techniquesof layer potential theory for the Stokes and Brinkman systems was used in [36] to analyze Poisson problems for semilineargeneralized Brinkman systems on Lipschitz domains in R n with Dirichlet or Robin boundary conditions and given data inSobolev and Besov spaces. Boundary value problems of Robin type for the Brinkman and Darcy-Forchheimer-Brinkmansystems in Lipschitz domains in Euclidean setting have been investigated in [35] (see also [34, 37]). An integral potentialmethod for transmission problems with Lipschitz interface in R for the Stokes and Darcy-Forchheimer-Brinkman systemsand data in weighted Sobolev spaces has been recently obtained in [32]. Transmission problems for the Navier-Stokesand Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds have been recentlyanalyzed in [39]. Well-posedness results for semilinear elliptic problems on Lipschitz domains in compact Riemannianmanifolds have been obtained by Dindo˘s and Mitrea in [21]. Let us also mention that Russo and Tartaglione in [67, 68]used a double-layer integral method in order to obtain existence results for boundary problems of Robin type for theStokes and Navier-Stokes systems in Lipschitz domains in Euclidean setting with data in Sobolev spaces. Maz’ya andRossmann [42] obtained Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedraldomains. Taylor, Ott and Brown in [70] studied Lp -mixed Dirichlet-Neumann problem for the Laplace equation in a abounded Lipschitz domain in R n with general decomposition of the boundary.In this paper we analyze the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces on a bounded Lipschitz domain in R , when the given boundary data belongto L p spaces, with p in a neighborhood of 2. This system is obtained by adding the semilinear term | u | u to thelinear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and non-tangentialtraces, as well as between the weak canonical conormal derivatives and the non-tangential conormal derivatives. Variousmapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, andwell posedness results for the Dirichlet and Neumann problems in L p -based Besov spaces on bounded Lipschitz domainsin R n ( n ≥
3) are also presented. Based on these results we show the well-posedness result for the mixed problem ofDirichlet-Neumann type for the Brinkman system in a bounded domain in R n ( n ≥
3) with given data in L -basedSobolev spaces. Further, by using some stability results of Fredholm and invertibility properties, we extend the well-posedness property to the case of boundary data in L p -based Sobolev spaces, with p ∈ (cid:16) n − n +1 − ε, ε (cid:17) ∩ (1 , + ∞ ),for some ε >
0. The main idea for showing this property is the invertibility of an associated Neumann-to-Dirichletoperator, inspired by the approach developed by Mitrea and Mitrea in [57]. Next we use the well-posedness result in thelinear case combined with a fixed point theorem in order to show the existence and uniqueness in L p -based Besov spacesfor a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkmansystem in a Lipschitz domain in R , when the boundary data belong to some L p spaces, with p ∈ (2 − ε, ε ) and someparameter ε >
0. The motivation of this work is based on some practical applications, where the semilinear Darcy-Forchheimer-Brinkman system describes the motion of viscous incompressible fluids in porous media. A suggestiveexample is given by a sandstone reservoir filled with oil, or the convection of a viscous fluid in a porous medium located
IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 3 in a bounded domain, where a part of the boundary is in contact with air and the remaining part is a solid surface orthe interface with another immiscible material or fluid. All these problems are well described by the Brinkman system,the semilinear Darcy-Forchheimer-Brinkman system, or by the Darcy-Forchheimer-Brinkman system, the latter of thesesystems containing both the nonlinear convective term ( u · ∇ ) u and the semilinear term | u | u . For further details werefer the reader to the book by Nield and Bejan [65] (see also the theoretical and numerical approach in [25, 26]).It is supposed that the methods presented in this paper can be developed further, to analyze also the nonlinearboundary-domain integro-differential equations, e.g., the ones formulated in [49, 50] for some quasi-linear boundaryvalue problems. 2. Functional setting and useful results
The purpose of this section is to provide main notions and results used in this paper. We recall the definition of abounded Lipschitz domain and give a short review of the involved Sobolev, Bessel potential and Besov spaces. Also wepresent the main properties of the layer potential operators for the Stokes and Brinkman systems in Lipschitz domainsin R n .For any point x = ( x , x , . . . , x n ) ∈ R n , we use the representation x = ( x ′ , x n ), where x ′ ∈ R n − and x n ∈ R . First,we recall the definition of Lipschitz domain (cf., e.g., [58, Definition 2.1]). Definition 2.1.
A nonempty, open, bounded subset Ω of ⊂ R n ( n ≥ is called a bounded Lipschitz domain if for any x ∈ ∂ Ω there exist some constants r, h > and a coordinate system in R n , ( y , . . . , y n ) = ( y ′ , y n ) ∈ R n − × R , whichis isometric to the canonical one and has origin at x , along with a Lipschitz function ϕ : R n − → R , such that thefollowing property holds. If C ( r, h ) denotes the open cylinder (cid:8) y = ( y ′ , y n ) ∈ R n − × R : | y ′ | < r, | y n | < h (cid:9) ⊆ R n , then Ω ∩ C ( r, h ) = { y = ( y ′ , y n ) ∈ R n − × R : | y ′ | < r and ϕ ( y ′ ) < y n < h } . (2.1)In view of the Definition 2.1, condition (2.1) implies that ∂ Ω = ∂ Ω and the characterization (cf. [58, (2.4)-(2.6)]) ∂ Ω ∩ C ( r, h ) = { y = ( y ′ , y n ) ∈ R n − × R : | y ′ | < r and y n = ϕ ( y ′ ) } , ( R n \ Ω) ∩ C ( r, h ) = { y = ( y ′ , y n ) ∈ R n − × R : | y ′ | < r and − h < y n < ϕ ( y ′ ) } . (2.2) Let all along the paper, Ω + denote a bounded Lipschitz domain with a connected boundary ∂ Ω , and Ω − := R n \ Ω + denote the corresponding exterior domain. Unless stated otherwise, it will be also assumed that n ≥ . Let κ = κ ( ∂ Ω) > the non-tangential maximal operator of an arbitraryfunction u : Ω ± → R is defined by M ( u )( x ) := { sup | u ( y ) | : y ∈ D ± ( x ) , x ∈ ∂ Ω } , (2.3)where D ± ( x ) ≡ D κ ; ± ( x ) := { y ∈ Ω ± : dist( x, y ) < κ dist(y , ∂ Ω) , x ∈ ∂ Ω } , (2.4)are non-tangential approach cones located in Ω + and Ω − , respectively (see, e.g., [61]). Moreover, u ± nt ( x ) := lim D ± ∋ y → x u ( y ) (2.5)are the non-tangential limits of u with respect to Ω ± at x ∈ ∂ Ω. Note that if M ( u ) ∈ L p ( ∂ Ω) for one choice of κ , where p ∈ (1 , ∞ ), then this property holds for arbitrary choice of κ (see, e.g., [47, p. 63]). For the sake of brevity, we use thenotation D ± ( x ) instead of D κ ; ± ( x ). We often need the property below (cf. [64, page 80], [75, Theorem 1.12]; see also[55, Lemma 2.2]). Lemma 2.2. If Ω ⊂ R n is a Lipschitz domain, then there exists a sequence of C ∞ domains Ω j approximating Ω(Ω j → Ω as j → ∞ ) in the following sense: ( i ) Ω j ⊂ Ω , and there exists a covering of ∂ Ω with finitely many coordinate cylinders (atlas) that also form afamily of coordinate cylinders for ∂ Ω j , for each j . Moreover, for each such cylinder C ( r, h ) , if ϕ and ϕ j are thecorresponding Lipschitz functions whose graphs describe the boundaries of ∂ Ω and ∂ Ω j , respectively, in C ( r, h ) ,then k∇ ϕ j k L ∞ ( R n − ) ≤ k∇ ϕ k L ∞ ( R n − ) and ∇ ϕ j → ∇ ϕ pointwise a.e. R. GUTT, M. KOHR, S.E. MIKHAILOV, AND W.L. WENDLAND ( ii ) There exist a sequence of Lipschitz diffeomorphisms Φ j : ∂ Ω → ∂ Ω j such that the Lipschitz constants of Φ j , Φ − j are uniformly bounded in j . ( iii ) There is a constant κ > such that for all j ≥ and all x ∈ ∂ Ω , we have Φ j ( x ) ∈ D + ( x ) ≡ D κ ; ± ( x ) , where D + ( x ) ≡ D κ ; ± ( x ) is the non-tangential approach cone with vertex at x . Moreover, lim j →∞ | Φ j ( x ) − x | = 0 uniformly in x ∈ ∂ Ω , (2.6)lim j →∞ ν ( j ) (Φ j ( x )) = ν ( x ) for a.e. x ∈ ∂ Ω , and in every space L p ( ∂ Ω) , p ∈ (1 , ∞ ) , (2.7) where ν ( j ) is the outward unit normal to ∂ Ω j , and ν is the outward unit normal to ∂ Ω . ( iv ) There exist some positive functions ω j : ∂ Ω → R ( the Jacobian related to Φ j , j ∈ N ) bounded away from zeroand infinity uniformly in j , such that, for any measurable set A ⊂ ∂ Ω , ´ A ω j dσ = ´ Φ j ( A ) dσ j . In addition, lim j →∞ ω j = 1 a.e. on ∂ Ω and in every space L p ( ∂ Ω) , p ∈ (1 , ∞ ) . Lemma 2.2 implies that the Lipschitz characters of the domains Ω j are uniformly controlled by the Lipschitz characterof Ω. The meaning of Lipschitz character of a Lipschitz domain is given below (cf., e.g., [58, p. 22]). Definition 2.3.
Let Ω ⊂ R n be a Lipschitz domain. Let {C k ( r k , h k ) : 1 ≤ k ≤ N } (with associated Lipschitzfunctions { ϕ k : 1 ≤ k ≤ N } ) be an atlas for ∂ Ω, i.e., a finite collection of cylinders covering the boundary ∂ Ω.Having fixed such an atlas of ∂ Ω, the
Lipschitz character of Ω is defined as the set consisting of the numbers N ,max {k∇ ϕ k k L ∞ ( R n − ) : 1 ≤ k ≤ N } , min { r k : 1 ≤ k ≤ N } , and min { h k : 1 ≤ k ≤ N } .2.1. Sobolev and Besov spaces and related results.
In this subsection we assume n ≥
2. We denote by D ( R n ) := C ∞ comp ( R n ) the space of infinitely differentiable functions with compact support in R n and by D ( R n , R n ) := C ∞ comp ( R n , R n )the space of infinitely differentiable vector-valued functions with compact support in R n . Also, let E (Ω ± ) := C ∞ (Ω ± )denote the space of infinitely differentiable functions and let D (Ω ± ) := C ∞ comp (Ω ± ) be the space of infinitely differentiablefunctions with compact support in Ω ± , equipped with the inductive limit topology. Let E ′ ( R n ) and D ′ ( R n ) be theduals of E ( R n ) and D ( R n ), respectively, i.e., the spaces of distributions on R n . The spaces E ′ (Ω ± ) and D ′ (Ω ± ) can besimilarly defined.Let F denote the Fourier transform defined on the space of tempered distributions to itself, and F − be its inverse.For p ∈ (1 , ∞ ), L p ( R n ) is the Lebesgue space of (equivalence classes of) measurable, p th integrable functions on R n ,and L ∞ ( R n ) is the space of (equivalence classes of) essentially bounded measurable functions on R n . For s ∈ R , the L p -based Bessel potential spaces H sp ( R n ) and H sp ( R n , R n ) are defined by H sp ( R n ) := { f : ( I − △ ) s f ∈ L p ( R n ) } = { f : J s f ∈ L p ( R n ) } , (2.8) H sp ( R n , R n ) := n ˜f = ( f , f , . . . , f n ) : f i ∈ H sp ( R n ) , j = 1 , . . . , n o , (2.9)where J s : S ′ ( R n ) → S ′ ( R n ) is the Bessel potential operator of order s defined by J s f = F − ( ρ s F f ) with ρ ( ξ ) = (1 + | ξ | ) (2.10)(see, e.g., [44, Chapter 3]). Note that H sp ( R n ) is a Banach space with respect to the norm k f k H sp ( R n ) = k J s f k L p ( R n ) = kF − ( ρ s F f ) k L p ( R n ) . (2.11)For integer s ≥
0, the spaces H sp ( R n ) coincide with the Sobolev spaces W sp ( R n ).The Bessel potential spaces H sp (Ω) and e H sp (Ω) are defined by H sp (Ω) := { f ∈ D ′ (Ω) : ∃ F ∈ H sp ( R n ) such that F | Ω = f } , (2.12) e H sp (Ω) := (cid:8) f ∈ H sp ( R n ) : supp f ⊆ Ω (cid:9) , (2.13) If X is a topological space, then X ′ denotes its dual. IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 5 and the Bessel potential spaces H sp (Ω , R n ) and e H sp (Ω , R n ) are defined as the spaces of vector-valued functions (distribu-tions) whose components belong to the spaces H sp (Ω) and e H sp (Ω), respectively (see, e.g., [44]). For any s ∈ R , C ∞ (Ω)is dense in H sp (Ω) and the following duality relations hold (see [29, Proposition 2.9], [24, (1.9)], [63, (4.14)]) (cid:0) H sp (Ω) (cid:1) ′ = e H − sp ′ (Ω) , H − sp ′ (Ω) = (cid:16) e H sp (Ω) (cid:17) ′ . (2.14)Here and further on p, p ′ ∈ (1 , ∞ ) are related as 1 p + 1 p ′ = 1 . Replacing Ω by Ω − in (2.12) and (2.13), one obtains the Bessel potential spaces H sp (Ω − ), e H sp (Ω − ).For p ∈ (1 , ∞ ) and s ∈ ( − , H sp ( ∂ Ω) can be defined by using the space H sp ( R n − ), a partition of unity and pull-pack. In addition, H − sp ′ ( ∂ Ω) = (cid:0) H sp ( ∂ Ω) (cid:1) ′ . We can also equivalently define H p ( ∂ Ω) = L p ( ∂ Ω) as the Lebesgue space of measurable, p th power integrable functions on ∂ Ω. In addition, H p ( ∂ Ω)coincides, with equivalent norm, with the Sobolev space W p ( ∂ Ω) := n f ∈ L p ( ∂ Ω) : k f k W p ( ∂ Ω) < ∞ o , k f k W p ( ∂ Ω) := k f k L p ( ∂ Ω) + k∇ tan f k L p ( ∂ Ω) . (2.15)Here the weak tangential gradient of a function f locally integrable on ∂ Ω is ∇ tan f := (cid:0) ν k ∂ τ kj f (cid:1) ≤ j ≤ n , where ∂ τ kj f is defined in the weak form as (cf. e.g., [61, (2.9)]) h ∂ τ kj f, φ i ∂ Ω := −h f, ∂ τ kj φ i ∂ Ω for any φ ∈ D ( R n ) with ∂ τ kj φ := ν k ( ∂ j φ ) | ∂ Ω − ν j ( ∂ k φ ) | ∂ Ω , j, k = 1 , . . . , n, and ν = ( ν , . . . , ν n ) is the outward unit normal to Ω, which exists at almostevery point on ∂ Ω. If f is defined and smooth enough in the vicinity of ∂ Ω, then by integrating by parts it is possibleto show that the weak definition coincides with the strong one, given by ∂ τ kj f := ν k ( ∂ j f ) | ∂ Ω − ν j ( ∂ k f ) | ∂ Ω .Now, for s ∈ R and p, q ∈ (1 , ∞ ), denote by B sp,q ( R n ) the scale of Besov spaces in R n , see Appendix A. Similar to(2.12) and (2.13), the Besov spaces B sp,q (Ω) and B sp,q (Ω , R n ) are defined by B sp,q (Ω) := { f ∈ D ′ (Ω) : ∃ F ∈ B sp,q ( R n ) such that F | Ω = f } , (2.16) B sp,q (Ω , R n ) := (cid:8) f = ( f , f , . . . , f n ) : f i ∈ B sp,q (Ω) , j = 1 , . . . , n (cid:9) , (2.17) e B sp,q (Ω) := (cid:8) f ∈ B sp,q ( R n ) : supp f ⊆ Ω (cid:9) . (2.18)For s ∈ [0 ,
1] and p, q ∈ (1 , ∞ ), the Sobolev and Besov spaces H sp ( ∂ Ω) and B sp,q ( ∂ Ω) on the boundary ∂ Ω can be definedby using the spaces H sp ( R n − ) and B sp,q ( R n − ), a partition of unity and the pull-backs of the local parametrization of ∂ Ω.In addition, we note that H − sp ( ∂ Ω) = (cid:0) H sp ′ ( ∂ Ω) (cid:1) ′ and B − sp,q = (cid:0) B sp ′ ,q ′ ( ∂ Ω) (cid:1) ′ , where p ′ , q ′ ∈ (1 , ∞ ) such that p + p ′ = 1and q + q ′ = 1 (for further details about boundary Sobolev and Besov spaces see, e.g., [61, p. 35]).A useful result for the problems we are going to investigate in this paper is the following trace lemma (see [30, ChapterVIII, Theorems 1,2], [29, Theorem 3.1] and also [18, Lemma 3.6] for the case p = 2 and a discussion on the criticalsmoothness index s = 1). Lemma 2.4.
Assume that Ω ⊂ R n is a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω bethe corresponding exterior domain. Let p, q ∈ (1 , ∞ ) and s ∈ (0 , . Then there exist linear and continuous Gagliardotrace operators γ ± : H s + p p (Ω ± ) → B sp,p ( ∂ Ω) and γ ± : B s + p p,q (Ω ± ) → B sp,q ( ∂ Ω) , respectively, such that γ ± f = f | ∂ Ω forany f ∈ C ∞ (Ω ± ) . These operators are surjective and have ( non-unique ) linear and continuous right inverse operators γ − ± : B sp,p ( ∂ Ω) → H s + p p (Ω ± ) and γ − ± : B sp,q ( ∂ Ω) → B s + p p,q (Ω ± ) , respectively. Lemma 2.4 holds also for vector-valued and matrix-valued functions f . If f is such that γ + f = γ − f , we will oftenwrite γf .We have the following trace equivalence assertion. R. GUTT, M. KOHR, S.E. MIKHAILOV, AND W.L. WENDLAND
Theorem 2.5.
Assume that Ω ⊂ R n is a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω be the corresponding exterior domain. Let p, q ∈ (1 , ∞ ) , and let u ∈ B s + p p,q (Ω ± ) or u ∈ H s + p p (Ω ± ) for some s > . Thenthe Gagliardo trace γ + u is well defined on ∂ Ω and, moreover, ( i ) if the pointwise non-tangential trace u ± nt exists a.e. on ∂ Ω , then u ± nt = γ ± u ; ( ii ) if the pointwise non-tangential trace u ± nt exists a.e. on ∂ Ω and s ∈ (0 , then u ± nt = γ ± u ∈ B sp,q ( ∂ Ω) ; ( iii ) if u ± nt ∈ H sp ( ∂ Ω) for some s ∈ (0 , , then γ ± u ∈ H sp ( ∂ Ω) as well.Proof. Item (i) for 0 < s < s ≥ γ ± u = u ± nt still appliesby an imbedding argument. Item (ii) and (iii) follow from item (i) and the well known imbedding γ ± u ∈ B sp,q ( ∂ Ω) for s ∈ (0 , (cid:3) Further on, h· , ·i Ω ′ will denote the dual form between corresponding dual spaces defined on a set Ω ′ . For furtherdetails about Sobolev, Bessel potential and Besov spaces, we refer the reader to, e.g., [1, 27, 44, 72, 73].2.2. The Brinkman system and conormal derivatives in Bessel-potential and Besov spaces.
In this subsectionwe also assume n ≥
2. For a couple ( u , π ), and a real number α ≥
0, let us consider the linear Brinkman system (in theincompressible case) L α ( u , π ) = f , div u = 0 , (2.19)where the Brinkman operator is defined as L α ( u , π ) := △ u − α u − ∇ π. (2.20)When α = 0, the Brinkman operator becomes the Stokes operator.Now, for ( u , π ) ∈ C (Ω ± , R n ) × C (Ω ± ), such that div u = 0 in Ω ± , we define the classical conormal derivatives(tractions) for the Brinkman (or the Stokes) system, t c ± α ( u , π ), by using the well-known formula t c ± ( u , π ) := ( γ ± σ ( u , π )) ν , (2.21)where σ ( u , π ) := − π I + 2 E ( u ) (2.22)is the stress tensor, E ( u ) is the strain rate tensor (symmetric part of ∇ u ), and ν = ν + is the outward unit normal to Ω + ,defined a.e. on ∂ Ω. Then for any function ϕ ∈ D ( R n , R n ) we obtain by integrating by parts the first Green identity, ± (cid:10) t c ± ( u , π ) , ϕ (cid:11) ∂ Ω =2 h E ( u ) , E ( ϕ ) i Ω ± + α h u , ϕ i Ω ± − h π, div ϕ i Ω ± + hL α ( u , π ) , ϕ i Ω ± . (2.23)If the non-tangential traces of the stress tensor, σ ± nt ( u , π ) and the normal vector ν exist at a boundary point, thenthe non-tangential conormal derivatives are defined at this point as t ± nt ( u , π ) := σ ± nt ν . (2.24)For s ∈ R and p, q ∈ (1 , ∞ ), we consider the spaces H sp ;div (Ω ± , R n ) = (cid:8) u ± ∈ H sp (Ω ± , R n ) : div u = 0 in Ω ± (cid:9) , (2.25) B sp,q, div (Ω ± , R n ) := (cid:8) u ± ∈ B sp,q (Ω ± , R n ) : div u = 0 in Ω ± (cid:9) . (2.26)We need also the following spaces (cf. [51, Definition 3.3]). Definition 2.6.
Let Ω be a Lipschitz domain ( bounded or unbounded ) . For s ∈ R , p, q ∈ (1 , ∞ ) and t ≥ − /p ′ , let usconsider the following spaces equipped with the corresponding graphic norms: H s + p ,tp, div (Ω , L α ) := n ( u , π ) ∈ H s + p p (Ω , R n ) × H s + p − p (Ω) : L α ( u , π ) = ˜ f | Ω , ˜ f ∈ e H tp (Ω , R n ) and div u = 0 in Ω o , IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 7 k ( u , π ) k H s + 1 p ,tp, div (Ω , L α ) := k u k H s + 1 pp (Ω , R n ) + k π k H s + 1 p − p (Ω) + k ˜ f k e H tp (Ω , R n ) , B s + p ,tp,q, div (Ω , L α ) := n ( u , π ) ∈ B s + p p,q (Ω , R n ) × B s + p − p,q (Ω) : L α ( u , π ) = ˜ f | Ω , ˜ f ∈ e B tp,q (Ω , R n ) and div u = 0 in Ω o , k ( u , π ) k B s + 1 p ,tp,q, div (Ω , L α ) := k u k B s + 1 pp,q (Ω , R n ) + k π k B s + 1 p − p,q (Ω) + k ˜ f k e B tp,q (Ω , R n ) , where L α ( u , π ) is defined in (2.20) . If t > t , the following continuous embeddings hold, H s + p ,t p, div (Ω , L α ) ֒ → H s + p ,t p, div (Ω , L α ), B s + p ,t p,q, div (Ω , L α ) ֒ → B s + p ,t p,q, div (Ω , L α ).Let D div (Ω , R n ) := (cid:8) v ∈ D (Ω , R n ) : div v = 0 in Ω (cid:9) . Similar to [52, Theorem 6.9], one can prove the followingassertion. Theorem 2.7. If Ω is a Lipschitz domain ( bounded or unbounded ) or Ω = R n , α ≥ , p, q ∈ (1 , ∞ ) , s ∈ R and t > − p ′ ,then D div (Ω , R n ) × D (Ω) is dense in H s + p ,tp (Ω , L α ) and in B s + p ,tp,q (Ω , L α ) . Let p, q ∈ (1 , ∞ ). Let ˚ E ± be the operator of extension of functions defined on Ω ± by zero on R n \ Ω ± . Followingthe proof of Theorem 2.16 in [51], let us define the operator e E ± on H tp (Ω ± ) and B tp,q (Ω ± ) as e E ± := ˚ E ± for 0 ≤ t < p ,and as h e E ± h, v i Ω ± := h h, e E ± v i Ω ± = h h, ˚ E ± v i Ω ± , when − p ′ < t < , for all h ∈ H tp (Ω ± ) , v ∈ H − tp ′ (Ω ± ), or for all h ∈ B tp,q (Ω ± ) , v ∈ B − tp ′ ,q ′ (Ω ± ), respectively. Then, for − /p ′ < t < /p ,evidently e E ± : H tp (Ω ± ) → e H tp (Ω ± ) , e E ± : B tp,q (Ω ± ) → e B tp,q (Ω ± )are bounded linear extension operators. Similar definition and properties hold also for vector fields.Analogously to the corresponding definition for Petrovskii-elliptic systems in [51, Definition 3.6], we can introduce anoperator ˜ L α as follows. Definition 2.8.
Let Ω be a Lipschitz domain ( bounded or unbounded ) , p, q ∈ (1 , ∞ ) , s ∈ R , t ≥ − /p ′ . The operator ˜ L α mapping ( i ) functions ( u , π ) ∈ H s + p ,tp, div (Ω , L α ) to the extension of the distribution L α ( u , π ) ∈ H tp (Ω , R n ) to e H tp (Ω , R n ) or ( ii ) functions ( u , π ) ∈ B s + p ,tp,q, div (Ω; L α ) to the extension of the distribution L α ( u , π ) ∈ B tp,q (Ω , R n ) to e B tp,q (Ω , R n ) ,will be called the canonical extension of the operator L α . Remark 2.9.
Similar to the paragraph following Definition 3.3 in [51], one can prove that the canonical extensionsmentioned in Definition 2.8 exist and are unique. If p, q ∈ (1 , ∞ ), s ∈ R , t ≥ − /p ′ , then k ˜ L α ( u , π ) k e H tp (Ω , R n ) ≤ k ( u , π ) k H s + 1 p ,tp, div (Ω , L α ) , k ˜ L α ( u , π ) k e B tp,q (Ω , R n ) ≤ k ( u , π ) k B s + 1 p ,tp,q, div (Ω , L α ) by definition of the spaces H s + p ,tp, div (Ω , L α ) and B s + p ,tp,q, div (Ω , L α ). Hence the linear operators ˜ L α : H s + p ,tp, div (Ω , L α ) → e H tp (Ω , R n ) and ˜ L α : B s + p ,tp,q, div (Ω , L α ) → e B tp,q (Ω , R n ) are continuous. Moreover, if − /p ′ < t < /p , and Ω is a Lips-chitz domain (bounded or unbounded), then we have the representation ˜ L α := e E + L α , or ˜ L α := e E − L α , respectively, cf.[51, Remark 3.7]. R. GUTT, M. KOHR, S.E. MIKHAILOV, AND W.L. WENDLAND
Formula (2.23) suggests the following definition of the canonical conormal derivative in the setting of Besov spaces, cf.,[18, Lemma 3.2], [36, Lemma 2.2], [51, Definition 3.8, Theorem 3.9], [52, Definition 6.5, Theorem 6.6], [61, Proposition10.2.1]).
Definition 2.10.
Let α ≥ , s ∈ (0 , , p, q ∈ (1 , ∞ ) . Then the canonical conormal derivative operators t ± α are definedon any ( u , π ) ∈ H s + p , − p ′ p, div (Ω ± , L α ) , or ( u , π ) ∈ B s + p , − p ′ p,q, div (Ω ± , L α ) , in the weak sense, by the formula ±h t ± α ( u , π ) , ϕ i ∂ Ω ± := 2 D e E ± E ( u ) , E ( γ − ± ϕ ) E Ω ± + α h e E ± u , γ − ± ϕ i Ω ± − D e E ± π, div( γ − ± ϕ ) E Ω ± + h ˜ L α ( u , π ) , γ − ± ϕ i Ω ± , (2.27) ∀ ϕ ∈ B − sp ′ ,p ′ ( ∂ Ω , R n ) , or ∀ ϕ ∈ B − sp ′ ,q ′ ( ∂ Ω , R n ) , respectively . Note that the canonical conormal derivative operators introduced in Definition 2.10 are different from the generalized conormal derivative operator, cf. [37, Lemma 2.2], [51, Definition 3.1, Theorem 3.2], [52, Definition 5.2, Theorem 5.3].Similar to [51, Theorem 3.9], one can prove the following assertion.
Lemma 2.11.
Under the hypothesis of Definition 2.10, the canonical conormal derivative operators t ± α : H s + p , − p ′ p, div (Ω ± , L α ) → B s − p,p ( ∂ Ω , R n ) , t ± α : B s + p , − p ′ p,q, div (Ω ± , L α ) → B s − p,q ( ∂ Ω , R n ) , are linear, bounded and independent of the choice of the operators γ − ± . In addition, the following first Green identityholds ±h t ± α ( u , π ) , γ + w i ∂ Ω = 2 D e E ± E ( u ) , E ( w ) E Ω ± + α D e E ± u , w E Ω ± − D e E ± π, div w E Ω ± + D ˜ L α ( u , π ) , w E Ω ± (2.28) for all ( u , π ) ∈ H s + p , − p ′ p, div (Ω ± , L α ) , w ∈ H p ′ − sp ′ (Ω ± , R n ) and all ( u , π ) ∈ B s + p , − p ′ p,q, div (Ω ± , L α ) , w ∈ B p ′ − sp ′ ,q ′ (Ω ± , R n ) , and the following second Green identity holds ± (cid:0) h t ± α ( u , π ) , γ + v i ∂ Ω − h t ± α ( v , q ) , γ + u i ∂ Ω (cid:1) = D ˜ L α ( u , π ) , v E Ω ± − D ˜ L α ( v , q ) , u E Ω ± (2.29) for all ( u , π ) ∈ H s + p , − p ′ p, div (Ω ± , L α ) , ( v , q ) ∈ H p ′ − s, − p p ′ , div (Ω ± , R n ) and all ( u , π ) ∈ B s + p , − p ′ p,q, div (Ω ± , L α ) , ( v , q ) ∈ B p ′ − s, − p p ′ ,q ′ (Ω ± , R n ) . Remark 2.12.
Similar to [32, Remark 2.6], we note that by exploiting arguments analogous to those of the proof ofTheorem 3.10 and the paragraph following it in [51], one can see that the canonical conormal derivatives on ∂ Ω canbe equivalently defined as t ± α ( u , π ) = r ∂ Ω t ′± α ( u , π ) . Here t ′± α ( u , π ) is defined by the dual form like (2.27) but only onLipschitz subsets Ω ′± ⊂ Ω ± such that ∂ Ω ⊂ ∂ Ω ′± and closure of Ω ± \ Ω ′± coincides with Ω ± \ Ω ′± (i.e., Ω ′± are some layersnear the boundary ∂ Ω). Moreover, such a definition is well applicable to the functions ( u , π ) from H s + p , − p ′ p, div (Ω ′± , L α )or B s + p , − p ′ p,q, div (Ω ′± , L α ) that are not obliged to belong to H s + p , − p ′ p, div (Ω ± , L α ) or B s + p , − p ′ p,q, div (Ω ± , L α ), respectively. It isparticularly useful for the functions ( u , π ) that belong to H s + p , − p ′ p, div (Ω − , L α ) or B s + p , − p ′ p,q, div (Ω − , L α ) only locally.Now we prove the equivalence between canonical and non-tangential conormal derivatives (as well as classical conormalderivative, when appropriate). Theorem 2.13.
Let n ≥ , α ≥ , and p, q ∈ (1 , ∞ ) . ( i ) Let s > and ( u , π ) ∈ B s + p p,q, div (Ω ± , R n ) × B s − p p,q (Ω ± ) or ( u , π ) ∈ H s + p p, div (Ω ± , R n ) × H s − p q (Ω ± ) . Then the classical conormal derivative t c ± ( u , π ) and the canonicalconormal derivative t ± α ( u , π ) are well defined and t ± α ( u , π ) = t c ± ( u , π ) ∈ L p ( ∂ Ω , R n ) .If, moreover, the non-tangential trace of the stress tensor, σ ± nt ( u , π ) , exists a.e. on ∂ Ω , then the non-tangentialconormal derivative, defined by (2.24) , also exists a.e. on ∂ Ω and t ± nt ( u , π ) = t ± α ( u , π ) = t c ± ( u , π ) ∈ L p ( ∂ Ω , R n ) . IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 9 ( ii ) Let < s ≤ , ( u , π ) ∈ B s + p ,tp,q, div (Ω ± , L α ) or ( u , π ) ∈ H s + p ,tp, div (Ω ± , L α ) , for some t > − p ′ . Let also assume that thenon-tangential maximal function M ( σ ( u , π )) and the non-tangential trace of the stress tensor, σ ± nt ( u , π ) , existand are finite a.e. on ∂ Ω and belong to the space L p ( ∂ Ω , R n × n ) . Then t ± α ( u , π ) = t ± nt ( u , π ) ∈ L p ( ∂ Ω , R n ) .Proof. We will give a proof in the case of a bounded domain Ω + and the Besov spaces. For an unbounded domain Ω − and the Bessel potential spaces the arguments are the same.(i) Let ( u , π ) ∈ B s + p p,q, div (Ω + , R n ) × B s − p p,q (Ω + ) for some p, q ∈ (1 , ∞ ) and s >
1. Evidently, the stress tensor σ ( u , π ) belongs to B s − p p,q (Ω , R n × n ), which for 1 < s < γ − σ ( u , π ) ∈ B s − p,q ( ∂ Ω , R n × n ) ⊂ L p ( ∂ Ω , R n × n ).Taking into account that the unit normal vector to the boundary, ν , belongs to L ∞ ( ∂ Ω , R n ), we obtain by (2.21) that t c+ ( u , π ) ∈ L p ( ∂ Ω , R n ).On the other hand, the inclusion ( u , π ) ∈ B s + p p,q, div (Ω + ) × B s − p p,q (Ω + ) for p, q ∈ (1 , ∞ ) and s > u , π ) ∈ B s + p ,tp,q, div (Ω + , L α ) for t ∈ ( − /p ′ , s − − /p ′ ) and thus the canonical conormal derivative t + α ( u , π ) is well definedand belongs to B s ′ − p,q ( ∂ Ω , R n ) for any s ′ ∈ (0 , < s <
2, the proof that t + α ( u , π ) = t c+ α ( u , π ) ∈ L p ( ∂ Ω , R n ) issimilar to [51, Corollary 3.14] (with evident modification to L p -based spaces), while for s ≥ t + α ( u , π ) = t c+ ( u , π ) ∈ L p ( ∂ Ω , R n ) still stays by imbedding.If, in addition, the non-tangential trace of the stress, σ +nt ( u , π ), exists a.e. on ∂ Ω, then σ +nt ( u , π ) = γ + σ ( u , π ) byTheorem 2.5(i) implying that t +nt ( u , π ) = t + α ( u , π ) = t c+ ( u , π ) ∈ L p ( ∂ Ω , R n ).(ii) Let 0 < s < s = 1 will follow by inclusion. Under the other hypotheses of item (ii), thecanonical conormal derivative, t + α ( u , π ), is well defined on the boundary ∂ Ω and belongs to B s − p,q ( ∂ Ω , R n ). Let { Ω j } j ≥ be a sequence of sub-domains in Ω + that converge to Ω + in the sense of Lemma 2.2, with the corresponding notationsΦ j , ν ( j ) and ω j also introduced there.Similar to the proof of Lemma 3.15 in [51], one can now prove that the canonical conormal derivative on ∂ Ω is alimit of the canonical conormal derivatives on ∂ Ω j , i.e., h t + α,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω = lim j →∞ h t + α,∂ Ω j ( u , π ) , γ ∂ Ω j w i ∂ Ω j forany w ∈ B p ′ − sp ′ ,q ′ (Ω + , R n ).The inclusion ( u , π ) ∈ B s + p ,tp,q, div (Ω + , L α ) means that the couple ( u , π ) satisfies the elliptic Brinkman PDE system(2.19) with a right hand side f ∈ B tp,q (Ω + , R n ), which implies that ( u , π ) ∈ B t +2 p,q, div (Ω j ) × B t +1 p,q (Ω j ).Then γ ∂ Ω j σ ( u , π ) ∈ B t +1 − p p,q ( ∂ Ω j , R n × n ) ⊂ L p ( ∂ Ω j , R n × n ) and t + α,∂ Ω j ( u , π ) = t c+ ∂ Ω j ( u , π ) = γ + ∂ Ω j σ ( u , π ) ν ∈ L p ( ∂ Ω j , R n )by item (i).On the other hand, for a.e. point x ∈ ∂ Ω the non-tangential function M ( σ ( u , π ))( x ) exists and is finite, whichparticularly implies that σ ( u , π ) is well defined and bounded in the approach cones D + ( x ). We can consider σ ( u , π )( x )as strictly defined (by its limit mean values lim r → ffl B ( x,r ) σ ( u , π )( ξ ) dξ in the sense of Jonnson & Wallin [30, p.15],see also [6, Theorem 8.7]); then γ ∂ Ω j σ ( u , π )( y ) = σ ( u , π )( y ) and hence t + α,∂ Ω j ( u , π )( y ) = t c+ ∂ Ω j ( u , π )( y ) = σ ( u , π )( y ) · ν j ( y ) for y ∈ D + ( x ) ∩ ∂ Ω j . In addition t + α,∂ Ω j ( u , π )(Φ j ( x )) = t c+ ∂ Ω j ( u , π )(Φ j ( x )) = σ ( u , π )(Φ j ( x )) · ν (Φ j ( x )) tends to σ +nt ( u , π )( x ) · ν ( x ) = t +nt ,∂ Ω ( u , π )( x ) as j → ∞ for a.e. x ∈ ∂ Ω, for which σ +nt ( u , π )( x ) does exist.Let us now prove that t c+ ∂ Ω j ( u , π )(Φ j ( x )) converges to t +nt ,∂ Ω ( u , π )( x ) not only point-wise for a.e. x ∈ ∂ Ω but also inthe weak sense, i.e., lim j →∞ h t c+ ∂ Ω j ( u , π ) , γ ∂ Ω j w i ∂ Ω j = h t +nt ,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω for any w ∈ B p ′ − sp ′ ,q ′ (Ω + , R n ). We have |h t c+ ∂ Ω j ( u , π ) , γ ∂ Ω j w i ∂ Ω j − h t +nt ,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω | = |h t c+ ∂ Ω j ( u , π ) ◦ Φ j , ω j γ ∂ Ω j w ◦ Φ j i ∂ Ω − h t +nt ,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω |≤ |h t c+ ∂ Ω j ( u , π ) ◦ Φ j − t +nt ,∂ Ω ( u , π ) , ω j γ ∂ Ω j w ◦ Φ j i ∂ Ω | + |h t +nt ,∂ Ω ( u , π ) , ( ω j − γ ∂ Ω j w ◦ Φ j i ∂ Ω | + |h t +nt ,∂ Ω ( u , π ) , γ ∂ Ω j w ◦ Φ j − γ ∂ Ω+ w i ∂ Ω | . (2.30) Let us prove that the summands in the right hand side of (2.30) tend to zero as j → ∞ . To this end, we use theinequality |h t c+ ∂ Ω j ( u , π ) ◦ Φ j − t +nt ,∂ Ω ( u , π ) , ω j γ ∂ Ω j w ◦ Φ j i ∂ Ω |≤ k t c+ ∂ Ω j ( u , π ) ◦ Φ j − t +nt ,∂ Ω ( u , π ) k L p ( ∂ Ω) k ω j γ ∂ Ω j w ◦ Φ j k L p ′ ( ∂ Ω) . (2.31)We have, | t c+ ∂ Ω j ( u , π )(Φ j ( x )) − t +nt ,∂ Ω ( u , π )( x ) | ≤ M ( σ ( u , π ))( x ) + | t +nt ,∂ Ω ( u , π )( x ) | , (2.32)the both terms in the right hand side of (2.32) belong to L p ( ∂ Ω) and t c+ ∂ Ω j ( u , π ) ◦ Φ j − t +nt ,∂ Ω ( u , π ) → ∂ Ω. Then the Lebesgue dominated convergence theorem implies that the first multiplier in the right hand side of(2.31) tends to zero. Since γ ∂ Ω j w ∈ B − sp ′ ,q ′ ( ∂ Ω j , R n ) ⊂ L − sp ′ ( ∂ Ω j , R n ) and γ ∂ Ω j w ◦ Φ j → γ ∂ Ω+ w (cf. [64, Chapter 2,Theorem 4.5]), the second multiplier in the right hand side of (2.31) is bounded and hence the whole right hand side of(2.31) tends to zero. The second summand in the right hand side of (2.30) tends to zero since ω j →
1, and the third,again, because γ ∂ Ω j w ◦ Φ j → γ ∂ Ω+ w .Combining this with the previous argument, we obtain, h t + α,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω = lim j →∞ h t c+ ∂ Ω j ( u , π ) , γ ∂ Ω j w i ∂ Ω j = h t +nt ,∂ Ω ( u , π ) , γ ∂ Ω+ w i ∂ Ω ∀ w ∈ B p ′ − sp ′ ,q ′ (Ω + , R n )Taking w = γ − ϕ , this gives h t + α,∂ Ω ( u , π ) , ϕ i ∂ Ω = h t +nt ,∂ Ω ( u , π ) , ϕ i ∂ Ω for any ϕ ∈ B − sp ′ ,q ′ ( ∂ Ω , R n ), i.e., t + α ( u , π ) = t +nt ( u , π ), and since t +nt ( u , π ) = σ +nt ( u , π ) ν ∈ L p ( ∂ Ω , R n ), this completes the proof of item (ii) for 0 < s <
1, while for s = 1 the statement follows by inclusion. (cid:3) Remark 2.14.
Due to Remark 2.12, Theorem 2.13 will still valid for Ω − if the functions belong to the correspondingspaces only locally, i.e., if ( u , π ) ∈ B s + p p,q, div , loc (Ω − , R n ) × B s − p p,q, loc (Ω − ) in item (i) and ( u , π ) ∈ B s + p ,tp,q, div , loc (Ω − , L α ) initem (ii). 3. Integral potentials for the Brinkman system
This section is devoted to the main properties of Newtonian and layer potentials for the Brinkman system.3.1.
Newtonian potential for the Brinkman system.
Let α > G α and Π thefundamental velocity tensor and the fundamental pressure vector for the Brinkman system in R n ( n ≥ G αjk ( x ) = 1˜ ω n (cid:26) δ jk | x | n − A ( α | x | ) + x j x k | x | n A ( α | x | ) (cid:27) , Π k ( x ) = 1˜ ω n x k | x | n (3.1)where A ( z ) and A ( z ) are defined by A ( z ) := (cid:0) z (cid:1) n − K n − ( z )Γ (cid:0) n (cid:1) + 2 (cid:0) z (cid:1) n K n ( z )Γ (cid:0) n (cid:1) z − z , A ( z ) := nz − (cid:0) z (cid:1) n +1 K n +1 ( z )Γ (cid:0) n (cid:1) z , (3.2) K κ is the Bessel function of the second kind and order κ ≥
0, Γ is the Gamma function, and ˜ ω n is the area of the unitsphere in R n . The fundamental solution of the Stokes system, ( G , Π), which corresponds to α = 0, is given by (see, e.g.,[74, (1.12)]) G jk ( x ) = 12˜ ω n (cid:26) n − δ jk | x | n − + x j x k | x | n (cid:27) , Π k ( x ) = 1˜ ω n x k | x | n . (3.3)Next we use the notations G α ( x , y ) = G α ( x − y ) and Π( x , y ) = Π( x − y ). Then( △ x − α I ) G α ( x , y ) − ∇ x Π( x , y ) = − δ y ( x ) I , div x G α ( x , y ) = 0 , ∀ y ∈ R n , (3.4)where δ x is the Dirac distribution with mass in y , and the subscript x added to a differential operator refers to theaction of that operator with respect to the variable x . IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 11
The fundamental stress tensor S α has the components S αijℓ ( x , y ) = − Π j ( x , y ) δ iℓ + ∂ G αij ( x , y ) ∂x ℓ + ∂ G αℓj ( x , y ) ∂x i , (3.5)where δ jk is the Kronecker symbol. Let Λ α be the fundamental pressure tensor with components Λ αjk . Then for fixed i and k , the pair ( S αijk , Λ αik ) satisfies the Brinkman system in R n if x = y , i.e., △ x S αijk ( x , y ) − αS αijk ( x , y ) − ∂ Λ αik ( y , x ) ∂x j = 0 ,∂S αijk ( x , y ) ∂x j = 0 (3.6)The components Λ αjk ( x , y ) are given by (see, e.g., [74, (2.18)])Λ αik ( x , y ) = 1 ω n (cid:26) − ( y i − x i ) 2 n ( y k − x k ) | y − x | n +2 + 2 δ ik | y − x | n − α n −
2) 1 | y − x | n − δ ik (cid:27) . (3.7)For α = 0, we use the notations S ijk := S ijk and Λ ik := Λ ik .Let ∗ denote the convolution product. Let us consider the velocity and pressure Newtonian potential operators forthe Brinkman system,( N α ; R n ϕ ) ( x ) := − ( G α ∗ ϕ ) ( x ) = − (cid:10) G α ( x , · ) , ϕ (cid:11) R n , ( Q α ; R n ϕ ) ( x ) = ( Q R n ϕ ) ( x ) := − ( Π ∗ ϕ ) ( x ) = − (cid:10) Π ( x , · ) , ϕ (cid:11) R n , (3.8)where the fundamental tensor G α is presented through its components in (3.1). Note that the Fourier transform of G α -components is given by b G αkj ( ξ ) = (2 π ) − n | ξ | + α (cid:18) δ kj − ξ k ξ j | ξ | (cid:19) . (3.9)Then we have the following property (cf. [43, Theorem 3.10] in the case n = 3, s = 0). Lemma 3.1.
Let α > . Then for all p, q ∈ (1 , ∞ ) and s ∈ R the following linear operators are continuous N α ; R n : H sp ( R n , R n ) → H s +2 p ( R n , R n ) , (3.10) N α ; R n : B sp,q ( R n , R n ) → B s +2 p,q ( R n , R n ) , (3.11) Q R n : H sp ( R n , R n ) → H s +1 p, loc ( R n ) , (3.12) Q R n : B sp,q ( R n , R n ) → B s +1 p,q, loc ( R n ) . (3.13) Proof.
Let ϕ ∈ H sp ( R n , R n ). By (2.11), k N α ; R n ϕ k H s +2 p ( R n , R n ) = (cid:13)(cid:13) F − (cid:0) ρ s +2 F ( N α ; R n ϕ ) (cid:1)(cid:13)(cid:13) L p ( R n , R n ) , (3.14)where ρ is the weight function given by (2.10). In addition, we note that F ( N α ; R n ϕ ) = F ( G α ∗ ϕ ) = b G α b ϕ (3.15)and hence by (3.14), k N α ; R n ϕ k H s +2 p ( R n , R n ) = (cid:13)(cid:13)(cid:13) F − (cid:16) ρ s +2 b G α b ϕ (cid:17)(cid:13)(cid:13)(cid:13) L p ( R n , R n ) = (cid:13)(cid:13) F − ( b m F ( J s ϕ )) (cid:13)(cid:13) L p ( R n , R n ) . (3.16)In view of (3.9), the matrix-function b m := ρ b G α has the components b m kj ( ξ ) = (2 π ) − n | ξ | | ξ | + α (cid:18) δ kj − ξ k ξ j | ξ | (cid:19) , k, j = 1 , . . . , n, and is smooth everywhere except the origin and uniformly bounded in R n × R n . Hence it is a Fourier multiplier in L p ( R n ) (cf. Theorem 2 in Appendix of [54]), i.e., there exists a constant M >
0, (which depends on p but is independentof ϕ ) such that k N α ; R n ϕ k H s +2 p ( R n , R n ) ≤ M k J s ϕ k L p ( R n , R n ) = M k ϕ k H sp ( R n , R n ) . and thus k N α ; R n k H sp ( R n , R n ) → H s +2 p ( R n , R n ) ≤ M, while operator (3.10) is continuous.Moreover, by formula (A.12) we have the interpolation property (cid:0) H s p ( R n , R n ) , H s p ( R n , R n ) (cid:1) θ,q = B sp,q ( R n , R n ) , (cid:0) H s +2 p ( R n , R n ) , H s +2 p ( R n , R n ) (cid:1) θ,q = B s +2 p,q ( R n , R n ) , (3.17)where s = (1 − θ ) s + θs . Then by continuity of operator (3.10), we obtain that operator (3.11) is also continuous for p, q ∈ (1 , ∞ ) and any s ∈ R .Let us now show the continuity of operators (3.12) and (3.13). To this end, we note that the pressure Newtonianpotential operator for the Brinkman system coincides with the one for the Stokes system and for any ϕ ∈ D ( R n , R n )can be written as Q R n ϕ = div N △ ; R n ϕ , (3.18)where ( N △ ; R n ϕ ) ( x ) := − ( G △ ∗ ϕ ) ( x ) , (3.19)and G △ ( x , y ) := − n − ω n | x − y | n − is the fundamental solution of the Laplace equation in R n . Therefore, themapping properties of the pressure Newtonian potential are provided by those of the harmonic Newtonian potential N △ ; R n . Since N △ ; R n is a pseudodifferential operator of order − R n , the following operator is continuous, N △ ; R n : H sp ( R n ) → H s +2 p, loc ( R n ) , ∀ s ∈ R , p ∈ (1 , ∞ ) . (3.20)Then by (3.18) and (3.20) we deduce the continuity property of the pressure Newtonian potential operator in (3.12).By using an interpolation argument as for (3.11), we also obtain continuity of operator (3.13). (cid:3) Let α ≥ p ∈ (1 , ∞ ) be given. The Newtonian velocity and pressure potential operators of the Brinkman systemin Lipschitz domains Ω ± are defined as N α ;Ω = r Ω N α ; R n ˚ E ± and Q Ω ± = r Ω ± Q R n ˚ E ± . (3.21)Recall that ˚ E ± is the operator of extension of vector fields defined in Ω ± by zero on R n \ Ω ± , and r Ω ± is the restrictionoperator from R n to Ω ± . The operators ˚ E ± : L p (Ω ± , R n ) → L p ( R n , R n ) and r Ω ± : H p ( R n , R n ) → H p (Ω ± , R n ) are linearand continuous. In addition, the volume potential operator N α ; R n : L p ( R n , R n ) → H p ( R n , R n ) is linear and continuousas well, for any p ∈ (1 , ∞ ) (cf., e.g., [43, Theorem 3.10], [20, Lemma 1.3] and Lemma 3.1). Therefore, the velocityNewtonian potential operators N α ;Ω ± : L p (Ω ± , R n ) → H p (Ω ± , R n ) , p ∈ (1 , ∞ ) , (3.22)are continuous operators. A similar argument yields the continuity of the Newtonian pressure potential operators Q Ω + : L p (Ω + , R n ) → H p (Ω + ) , Q Ω − : L p (Ω − , R n ) → H p, loc (Ω − ) , p ∈ (1 , ∞ ) . (3.23)Next, in view of (A.5), (A.6) and the first inclusion in (A.8) we obtain the inclusions H p ( R n , R n ) = W p ( R n , R n ) ֒ → W p p ( R n , R n ) = B p p,p ( R n , R n ) ֒ → B p p,p ∗ ( R n , R n ) , ∀ p ≥ , p ∗ = max { p, } , (3.24)which are continuous. Then relations (3.22) and (3.24) imply also the continuity of the velocity Newtonian potentialoperator N α ;Ω ± : L p (Ω ± , R n ) → B p p,p ∗ (Ω ± , R n ) , p ∈ (1 , ∞ ) . (3.25) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 13
A similar argument yields the continuity property of the pressure Newtonian potential operator Q α ;Ω + : L p (Ω + , R n ) → B p p,p ∗ (Ω + ) , Q α ;Ω − : L p (Ω − , R n ) → B p p,p ∗ , loc (Ω − ) , p ∈ (1 , ∞ ) . (3.26)In addition, due to (3.21), we have the relations △ N α ;Ω ± f − α N α ;Ω ± f − ∇Q Ω ± f = f , div N α ;Ω ± f = 0 in Ω ± . (3.27)This leads us to the following assertion. Corollary 3.2.
Let α > , p ∈ (1 , ∞ ) , and p ∗ = max { p, } . Then the Brinkman Newtonian potentials satisfy equations (6.46) and the following operators are continuous ( N α ;Ω + , Q Ω + ) : L p (Ω + , R n ) → H , p, div (Ω + , L α ) , ( N α ;Ω − , Q Ω − ) : L p (Ω − , R n ) → H , p, div , loc (Ω − , L α ) , (3.28)( N α ;Ω + , Q Ω + ) : L p (Ω + , R n ) → B , p,p ∗ , div (Ω + , L α ) , ( N α ;Ω − , Q Ω − ) : L p (Ω − , R n ) → B , p,p ∗ , div , loc (Ω − , L α ) . (3.29) Remark 3.3.
Let f ± ∈ L p (Ω ± , R n ) for some p ∈ (1 , ∞ ) , and p ∗ = max { p, } . Then Corollary . , Lemmas . , . and Remark . imply that γ ± (cid:0) N α ;Ω ± f ± (cid:1) ∈ B sp,p ∗ ; ν ( ∂ Ω , R n ) , t ± α (cid:0) N α ;Ω ± f ± , Q Ω ± f ± (cid:1) ∈ B s − p,p ∗ ( ∂ Ω , R n ) , ∀ s ∈ (0 , . (3.30) Moreover, due to (3.22) , the first equality in (3.24) , Theorem . , and [10, Theorem 5] , these inclusions can beimproved to the following ones γ ± (cid:0) N α ;Ω ± f ± (cid:1) ∈ H p ; ν ( ∂ Ω , R n ) , t ± α (cid:0) N α ;Ω ± f ± , Q Ω ± f ± (cid:1) = t c ± (cid:0) N α ;Ω ± f ± , Q Ω ± f ± (cid:1) ∈ L p ( ∂ Ω , R n ) . (3.31)In (3.30), (3.31) and further on, the following space notations are used for p ∈ (1 , ∞ ), q ∈ (1 , ∞ ], s ∈ (0 , ν to the Lipschitz domain Ω + ⊂ R n , L p ; ν ( ∂ Ω , R n ) := (cid:26) v ∈ L p ( ∂ Ω , R n ) : ˆ ∂ Ω v · ν dσ = 0 (cid:27) , H sp ; ν ( ∂ Ω , R n ) := (cid:26) v ∈ H sp ( ∂ Ω , R n ) : ˆ ∂ Ω v · ν dσ = 0 (cid:27) ,B sp,q ; ν ( ∂ Ω , R n ) := (cid:26) v ∈ B sp,q ( ∂ Ω , R n ) : ˆ ∂ Ω v · ν dσ = 0 (cid:27) . (3.32)3.2. Layer potentials for the Brinkman system.
For a given density g ∈ L p ( ∂ Ω , R n ), the velocity single-layerpotential for the Brinkman system, V α g , and the corresponding pressure single-layer potential , Q s g , are given by( V α g )( x ) := hG α ( x , · ) , g i ∂ Ω , ( Q s g )( x ) := h Π( x , · ) , g i ∂ Ω , x ∈ R n \ ∂ Ω . (3.33)Let h ∈ H p ( ∂ Ω , R n ) be a given density. Then the velocity double-layer potential , W α ; ∂ Ω h , and the corresponding pressure double-layer potential , Q dα ; ∂ Ω h , are defined by( W α h ) j ( x ) := ˆ ∂ Ω S αijℓ ( x , y ) ν ℓ ( y ) h i ( y ) dσ y , ( Q dα h )( x ) := ˆ ∂ Ω Λ αjℓ ( x , y ) ν ℓ ( y ) h j ( y ) dσ y , ∀ x ∈ R n \ ∂ Ω , (3.34)where ν ℓ , ℓ = 1 , . . . , n , are the components of the outward unit normal ν to Ω + , which is defined a.e. (with respect tothe surface measure σ ) on ∂ Ω. Note that the definition of the double layer potential in [69, (3.9)] differs from definition(3.34) due to different conormal derivatives used in [69, (1.14)] and in formula (2.22) of our paper.The single- and double-layer potentials can be also defined for any g ∈ B s − p,q ( ∂ Ω , R n ) and h ∈ B sp,q ( ∂ Ω , R n ), respec-tively, where s ∈ (0 ,
1) and p, q ∈ (1 , ∞ ). For α = 0 (i.e., for the Stokes system) we use the notations Vg , Q s g , Wh and Q d h for the corresponding single- and double-layer potentials.In view of equations (3.4) and (3.6), the pairs ( V α g , Q s g ) and ( W sα h , Q dα h ) satisfy the homogeneous Brinkmansystem in Ω ± , ( △ − α I ) V α g − ∇Q s g = , div V α g = 0 in R n \ ∂ Ω , (3.35)( △ − α I ) W α h − ∇Q s h = , div W α h = 0 in R n \ ∂ Ω . (3.36) The direct value of the double layer potential W α ; ∂ Ω h on the boundary is defined in terms of Cauchy principal valueby ( K α h ) k ( x ) := p . v . ˆ ∂ Ω S αjkℓ ( y , x ) ν ℓ ( y ) h j ( y ) dσ y a.e. x ∈ ∂ Ω . (3.37) Lemma 3.4.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ≥ and p ∈ (1 , ∞ ) . There exist some constants C i > , i = 1 , . . . , , depending only on p , α and the Lipschitzcharacter of Ω + , such that the following properties hold: k M ( ∇ V α g ) k L p ( ∂ Ω) + k M ( V α g ) k L p ( ∂ Ω) + k M ( Q s g ) k L p ( ∂ Ω) ≤ C k g k L p ( ∂ Ω , R n ) , ∀ g ∈ L p ( ∂ Ω , R n ) , (3.38) k M ( V α g ) k L p ( ∂ Ω) ≤ C k g k H − p ( ∂ Ω , R n ) , ∀ g ∈ H − p ( ∂ Ω , R n ) , (3.39) k M ( W α h ) k L p ( ∂ Ω) ≤ C k h k L p ( ∂ Ω , R n ) , ∀ h ∈ L p ( ∂ Ω , R n ) , (3.40) k M ( ∇ W α h ) k L p ( ∂ Ω) + k M ( W α h ) k L p ( ∂ Ω) + k M (cid:0) Q dα h (cid:1) k L p ( ∂ Ω) ≤ C k h k H p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.41) Moreover, the following estimates hold for the non-tangential traces that exist at almost all points of ∂ Ω : k ( V α g ) ± nt k L p ( ∂ Ω , R n ) , k ( ∇ V α g ) ± nt k L p ( ∂ Ω , R n ) , k ( Q s g ) ± nt k L p ( ∂ Ω , R n ) ≤ C k g k L p ( ∂ Ω , R n ) , ∀ g ∈ L p ( ∂ Ω , R n ) , (3.42) k ( V α g ) ± nt k L p ( ∂ Ω) ≤ C k g k H − p ( ∂ Ω , R n ) , ∀ g ∈ H − p ( ∂ Ω , R n ) , (3.43) k ( W α g ) ± nt k L p ( ∂ Ω , R n ) ≤ C k h k L p ( ∂ Ω , R n ) , ∀ h ∈ L p ( ∂ Ω , R n ) , (3.44) k ( W α h ) ± nt k L p ( ∂ Ω , R n ) , k ( ∇ W α h ) ± nt k L p ( ∂ Ω , R n ) , k ( Q dα h ) ± nt k L p ( ∂ Ω , R n ) ≤ C k h k H p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.45) Proof.
In the case α = 0, inequalities (3.38)-(3.41) follow from [61, Propositions 4.2.3 and 4.2.8].In the case α >
0, Inequality (3.38) has been obtained in [69, Lemma 3.2]. In addition, inequality (3.39) follows bythe same arguments as in the proof of its counterpart in the case α = 0 (cf. [61, (4.61)]). Indeed, if g ∈ H − p ( ∂ Ω , R n ),then there exist g = ( g , . . . , g n ) , g rℓ = ( g rℓ ;1 , . . . , g rℓ ; n ) ∈ L p ( ∂ Ω , R n ), r, ℓ = 1 , . . . , n , such that g k = g k + n X r,ℓ =1 ∂ τ rℓ g rℓ ; k , k g k k L p ( ∂ Ω) + n X r,ℓ =1 k g rℓ ; k k L p ( ∂ Ω) ≤ k g k k H − p ( ∂ Ω) , k = 1 , . . . , n, (3.46)(cf. [61, Corollary 2.1.2 and relation (4.65)]), where ∂ τ rℓ = ν r ∂ ℓ − ν ℓ ∂ r are the tangential derivative operators. Hence,integrating by parts,( V α g ) j ( x ) = ˆ ∂ Ω G αjk ( x − y ) g k ( y ) dσ y − n X k =1 n X r,ℓ =1 ˆ ∂ Ω (cid:0) ∂ τ rℓ (cid:0) G αjk ( x − y ) (cid:1)(cid:1) g rℓ ; k ( y ) dσ y , ∀ x ∈ R n \ ∂ Ω (3.47)(cf. [61, (4.66)] for α = 0). Inequality (3.39) immediately follows from equality (3.47) and the estimates in (3.38) and(3.46).Let us now show inequality (3.40) for α > W α was given in [69, Theorem 3.5]). First, we note that Lemma 4.1 in [46] (see also [69, Theorem 2.5]) impliesthat there exists a constant c α = c α (Ω + , α ) > |∇G α ( x , y ) − ∇G ( x , y ) | ≤ c α | x − y | − n , ∀ x , y ∈ Ω + , x = y . (3.48)Then, in view of formula (3.5) and equality Π α = Π, there exists a constant C = C (Ω + , α ) > | S αijk ( y , x ) − S ijk ( y , x ) | ≤ (cid:12)(cid:12)(cid:12) ∂ G αij ( y , x ) ∂y k − ∂ G ij ( y , x ) ∂y k (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ G αkj ( y , x ) ∂y i − ∂ G ( y , x ) ∂y i (cid:12)(cid:12)(cid:12) ≤ C | x − y | − n , ∀ x , y ∈ Ω + , x = y . (3.49)Inequality (3.49) and [47, Proposition 1] (applied to the integral operator W α − W whose kernel is ( S α ( y , x ) − S ( y , x )) ν ( y ))show that there exists a constant C = C ( ∂ Ω , p, α ) > k M (( W α − W ) h ) k L p ( ∂ Ω) ≤ C k h k L p ( ∂ Ω , R n ) , ∀ h ∈ L p ( ∂ Ω , R n ) . (3.50) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 15
Moreover, by [61, (4.56)], there exists a constant C = C ( ∂ Ω , p ) > k M ( Wh ) k L p ( ∂ Ω) ≤ C k h k L p ( ∂ Ω , R n ) , ∀ h ∈ L p ( ∂ Ω , R n ) , (3.51)and then, by (3.50) and (3.51), we obtain inequality (3.40).Let us now show inequality (3.41) for α >
0. According to the second formula in (3.34) and formula (3.7) the kernelof the Brinkman double-layer pressure potential operator Q dα is given byΛ αjk ( x , y ) ν k ( y ) = 1˜ ω n (cid:26) − n ( y j − x j )( y k − x k ) ν k ( y ) | y − x | n +2 + 2 ν j ( y ) | y − x | n − α n −
2) 1 | y − x | n − ν j ( y ) (cid:27) . (3.52)For α = 0, (3.52) reduces to the kernel of the Stokes double-layer pressure potential operator Q d . Therefore, | Λ αjk ( x , y ) ν k ( y ) − Λ jk ( x , y ) ν k ( y ) | ≤ α ˜ ω n ( n −
2) 1 | y − x | n − , ∀ x ∈ Ω + , y ∈ ∂ Ω , x = y . (3.53)Then according to [47, Proposition 1] applied to the operator Q dα − Q d , there exists a constant C = C ( ∂ Ω , p, α ) suchthat (cid:13)(cid:13) M (cid:0)(cid:0) Q dα − Q d (cid:1) h (cid:1)(cid:13)(cid:13) L p ( ∂ Ω) ≤ C k h k L p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.54)In view of [61, Proposition 4.2.8], the Stokes double-layer pressure potential operator Q d satisfies the inequality (cid:13)(cid:13) M (cid:0) Q d h (cid:1)(cid:13)(cid:13) L p ( ∂ Ω) ≤ C k h k H p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) , (3.55)with a constant C ≡ C ( ∂ Ω , p ) >
0. Then by (3.54) and (3.55) there exists a constant C ≡ C ( ∂ Ω , p, α ) > (cid:13)(cid:13) M (cid:0) Q dα h (cid:1)(cid:13)(cid:13) L p ( ∂ Ω) ≤ C k h k H p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.56)Next, we show that there exists a constant c = c (Ω + , p, α ) > k M ( ∇ W α h ) k L p ( ∂ Ω) ≤ c k h k H p ( ∂ Ω , R n ) , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.57)To this end, we use expressions (3.34) and (3.5) for the Brinkman double layer potential W α h to obtain for any h ∈ H p ( ∂ Ω , R n ), ∂ r ( W α h ) j ( x ) = − ˆ ∂ Ω (cid:8) ν ℓ ( y ) (cid:0) ∂ r ∂ ℓ G αjk (cid:1) ( y − x ) + ν ℓ ( y ) ( ∂ r ∂ j G αℓk ) ( y − x ) − ν j ( y ) ( ∂ r Π k ) ( y − x ) (cid:9) h k ( y ) dσ y = − ˆ ∂ Ω (cid:8) ∂ τ ℓr ( y ) (cid:0) ∂ ℓ G αjk (cid:1) ( y − x ) + ∂ τ ℓr ( y ) ( ∂ j G αℓk ) ( y − x ) − ∂ τ jr ( y ) Π k ( y − x ) (cid:9) h k ( y ) dσ y − ˆ ∂ Ω (cid:8) ν r ( y ) △G αjk ( y − x ) + ν r ( y ) ( ∂ ℓ ∂ j G αℓk ) ( y − x ) − ν r ( y ) ( ∂ j Π k ) ( y − x ) (cid:9) h k ( y ) dσ y = ˆ ∂ Ω (cid:8)(cid:0) ∂ ℓ G αjk (cid:1) ( y − x ) ( ∂ τ ℓr h k ) ( y ) + ( ∂ j G αℓk ) ( y − x ) ( ∂ τ ℓr h k ) ( y ) − Π k ( y − x ) (cid:0) ∂ τ jr h k (cid:1) ( y ) (cid:9) dσ y − α ˆ ∂ Ω ν r ( y ) G αjk ( y − x ) h k ( y ) dσ y , j, r = 1 , . . . , n, (3.58)where ∂ j := ∂∂x j . We also employed the following integration by parts formula, which holds for any p ∈ (1 , ∞ ) (cf. [61,(2.16)]), ˆ ∂ Ω f (cid:0) ∂ τ jk g (cid:1) dσ = ˆ ∂ Ω (cid:0) ∂ τ kj f (cid:1) gdσ, ∀ f ∈ H p ( ∂ Ω) , ∀ g ∈ H p ′ ( ∂ Ω) , (3.59)where p + p ′ = 1. The last integral in (3.58) follows from equations (3.4), which, in particular, yield that( △ y − α I ) G α ( y − x ) − ∇ y Π( y − x ) = 0 , div y G α ( y − x ) = 0 , ∀ x ∈ R n \ ∂ Ω , y ∈ ∂ Ω . (3.60) In the case α = 0, formula (3.58) has been obtained in [61, (4.84)].Now, from formula (3.58) and its counterpart corresponding to α = 0, we obtain for all j, r = 1 , . . . , n , ∂ r ( W α h ) j = ∂ r ( Wh ) j + ∂ ℓ (( V α − V ) ( ∂ τ ℓr h )) j + ∂ j (( V α − V ) ( ∂ τ ℓr h )) ℓ − α ( V α ( ν r h )) j , ∀ h ∈ H p ( ∂ Ω , R n ) . (3.61)Further, by using estimate (4.86) in [61, Proposition 4.2.8] for the Stokes double layer potential, Wh , property (3.38) forthe Brinkman and Stokes single layer potentials involved in formula (3.61), and continuity of the tangential derivativeoperators ∂ τ jk : H p ( ∂ Ω) → L p ( ∂ Ω), we obtain inequality (3.57), as asserted (see also [38, (3.35)]).Finally, inequalities (3.40), (3.56) and (3.57) imply inequality (3.41).For any n ≥ ℓ ≥
0, there exists a constant C = C ( n, ℓ, α ) > (cid:12)(cid:12) ∇ ℓ x G α ( x ) (cid:12)(cid:12) ≤ C (1 + α | x | ) | x | n − ℓ , (3.62)holds and implies that |G α ( x − y ) | ≤ C | x − y | − n , with some constant C = C ( n, α ) >
0. Then in view of [47,Proposition 1], for any g ∈ L p ( ∂ Ω , R n ) there exist the non-tangential limits of the Brinkman single layer potential V α g at almost all points of ∂ Ω. Moreover, the existence of the non-tangential limits of ∇ V α g at almost all pointsof ∂ Ω follows immediately from [69, Lemma 3.3]. For Q s g such a result is valid since the Brinkman pressure singlelayer potential coincides with the Stokes pressure single layer potential, for which the result is well known, cf., e.g., [61,Proposition 4.2.2] and [69, Lemma 3.3].If g ∈ H − p ( ∂ Ω , R n ) then the existence of the non-tangential limits of V α g a.e. on ∂ Ω follows from formula (3.47)and the corresponding statement for the existence of non-tangential limits for a single layer potential and the gradienta single layer potential with a density in L p ( ∂ Ω , R n ).Now let h ∈ L p ( ∂ Ω , R n ). Then the existence of the non-tangential limits of the Brinkman double layer potential W α h at almost all points of ∂ Ω follows easily from the case α = 0. Indeed, estimate (3.49) and [47, Proposition 1] implythat the difference( W α h ) j ( x ) − ( Wh ) j ( x ) = ˆ ∂ Ω (cid:0) S αijk ( y − x ) − S ijk ( y − x ) (cid:1) ν k ( y ) h i ( y ) dσ y = ˆ ∂ Ω (cid:26)(cid:18) ∂ G αij ( y − x ) ∂y k − ∂ G ij ( y − x ) ∂y k (cid:19) + (cid:18) ∂ G αkj ( y − x ) ∂y i − ∂ G kj ( y − x ) ∂y i (cid:19)(cid:27) ν k ( y ) h i ( y ) dσ y , x ∈ Ω ± (3.63)has non-tangential limits (cid:16) ( W α h ) j − ( Wh ) j (cid:17) ± nt ( x ) at almost all points x ∈ ∂ Ω. On the other hand, according to[61, Proposition 4.2.2] there exist the non-tangential limits of the Stokes double layer potential Wh at almost all points x of ∂ Ω. Therefore, the non-tangential limits of the Brinkman double layer potential W α h exist as well at almost allpoints x of ∂ Ω.Now let h ∈ H p ( ∂ Ω , R n ). Then the existence of the non-tangential limits of ∇ W α h at almost all points of ∂ Ωfollows from their existence in the case α = 0 (cf. [61, (4.91)]), formula (3.61), and the statement for the existence ofnon-tangential limits for a single layer potential and the gradient a single layer potential with a density in L p ( ∂ Ω , R n ),while the existence of non-tangential limits of Q α h a.e. on ∂ Ω is provided by the corresponding result in the case α = 0(cf. [61, (4.85)]) and [47, Proposition 1] applied to the complementary term (cid:0) Q dα − Q d (cid:1) h = αV △ ( h · ν ), which by (3.52)is the Laplace single layer potential with density α h · ν ∈ L p ( ∂ Ω).Finally, note that inequalities (3.42)-(3.45) follow from inequalities (3.38)-(3.41) and the estimate k f ± nt k L p ( ∂ Ω) ≤k M ( f ) k L p ( ∂ Ω) , whenever f has the property that both f ± nt and M ( f ) exist a.e on ∂ Ω (see [16, Remark 9]). (cid:3)
The mapping properties of layer potential operators for the
Stokes system (i.e., for α = 0) in Bessel-potential andBesov spaces on bounded Lipschitz domains, as well as their jump relations across a Lipschitz boundary, are well known,cf., e.g., [23], [27], [61, Theorem 10.5.3], [62, Theorem 3.1, Proposition 3.3]. The main properties of layer potentialoperators for the Brinkman system are collected below (some of them are also available in [22, Proposition 3.4], [32,Lemma 3.4], [33, Lemma 3.1], [62, Theorem 3.1], [69, Theorems 3.4 and 3.5]). IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 17
Theorem 3.5.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let p, q ∈ (1 , ∞ ) , α > , and p ∗ := max { p, } . Let t ≥ − p ′ be arbitrary, where p + p ′ = 1 . ( i ) Then the following operators are linear and continuous, V α | Ω + : L p ( ∂ Ω , R n ) → B p p,p ∗ ;div (Ω + , R n ) , Q s | Ω + : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ) , (3.64) (cid:0) V α | Ω + , Q s | Ω + (cid:1) : L p ( ∂ Ω , R n ) → B p ,tp,p ∗ ;div (Ω + , L α ) , (3.65) V α | Ω + : H − p ( ∂ Ω , R n ) → B p p,p ∗ ;div (Ω + , R n ) , Q s | Ω + : H − p ( ∂ Ω , R n ) → B − p p,p ∗ (Ω + ) , (3.66) (cid:0) V α | Ω + , Q s | Ω + (cid:1) : H − p ( ∂ Ω , R n ) → B p ,tp,p ∗ ;div (Ω + , L α ) , (3.67) W α | Ω + : H p ( ∂ Ω , R n ) → B p p,p ∗ ;div (Ω + , R n ) , Q dα (cid:12)(cid:12) Ω + : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ) , (3.68) (cid:0) W α | Ω + , Q dα | Ω + (cid:1) : H p ( ∂ Ω , R n ) → B p ,tp,p ∗ ;div (Ω + , L α ) . (3.69) W α | Ω + : L p ( ∂ Ω , R n ) → B p p,p ∗ ;div (Ω + , R n ) , Q dα (cid:12)(cid:12) Ω + : L p ( ∂ Ω , R n ) → B p − p,p ∗ (Ω + ) , (3.70) (cid:0) W α | Ω + , Q dα | Ω + (cid:1) : L p ( ∂ Ω , R n ) → B p ,tp,p ∗ ;div (Ω + , L α ) . (3.71)( ii ) Moreover, the following operators are also linear and continuous for s ∈ (0 , , V α : B s − p,q ( ∂ Ω , R n ) → B s + p p,q ;div ( R n , R n ) , Q s : B s − p,q ( ∂ Ω , R n ) → B s + p − p,q ;loc ( R n ) , (3.72) V α | Ω + : B s − p,q ( ∂ Ω , R n ) → B s + p p,q ;div (Ω + , R n ) , ( Q s ) | Ω + : B s − p,q ( ∂ Ω , R n ) → B s + p − p,q (Ω + ) , (3.73) (cid:0) V α | Ω + , Q s | Ω + (cid:1) : B s − p,q ( ∂ Ω , R n ) → B s + p ,tp,q, div (Ω + , L α ) , (3.74) W α | Ω + : B sp,q ( ∂ Ω , R n ) → B s + p p,q ;div (Ω + , R n ) , Q dα | Ω + : B sp,q ( ∂ Ω , R n ) → B s + p − p,q (Ω + ) , (3.75) (cid:0) W α | Ω + , Q dα | Ω + (cid:1) : B sp,q ( ∂ Ω , R n ) → B s + p ,tp,q ;div (Ω + , L α ) , (3.76) V α | Ω − : B s − p,q ( ∂ Ω , R n ) → B s + p p,q ;div (Ω − , R n ) , Q s | Ω − : B s − p,q ( ∂ Ω , R n ) → B s + p − p,q ;loc (Ω − ) , (3.77) (cid:0) V α | Ω − , Q s | Ω − (cid:1) : B s − p,q ( ∂ Ω , R n ) → B s + p ,tp,q ;div;loc (Ω − , L α ) , (3.78) W α | Ω − : B sp,q ( ∂ Ω , R n ) → B s + p p,q ;div;loc (Ω − , R n ) , Q d | Ω − : B sp,q ( ∂ Ω , R n ) → B s + p − p,q ;loc (Ω − ) , (3.79) (cid:0) W α | Ω − , Q dα | Ω − (cid:1) : B sp,q ( ∂ Ω , R n ) → B s + p ,tp,q, div;loc (Ω − , L α ) . (3.80)( iii ) The following relations hold a.e. on ∂ Ω , (cid:0) V α g (cid:1) +nt = (cid:0) V α g (cid:1) − nt =: V α g , ∀ g ∈ H − p ( ∂ Ω , R n ); (3.81)12 h + ( W α h ) +nt = − h + ( W α h ) − nt =: K α h , ∀ h ∈ L p ( ∂ Ω , R n ); (3.82) − g + t +nt ( V α g , Q s g ) = 12 g + t − nt ( V α g , Q s g ) =: K ∗ α g , ∀ g ∈ L p ( ∂ Ω , R n ); (3.83) t +nt (cid:0) W α h , Q dα h (cid:1) = t − nt (cid:0) W α h , Q dα h (cid:1) =: D α h , ∀ h ∈ H p ( ∂ Ω , R n ); (3.84) where K ∗ α is the transpose of K α ; ∂ Ω , and the following boundary integral operators are linear and bounded, V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) , K α : H p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) , (3.85) V α : H − p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) , K α : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) , (3.86) K ∗ α : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) , D α : H p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) . (3.87) For h ∈ B sp,q ( ∂ Ω , R n ) and g ∈ B s − p,q ( ∂ Ω , R n ) , s ∈ (0 , , the following relations hold a.e. on ∂ Ω , γ + (cid:0) V α g (cid:1) = γ − (cid:0) V α g (cid:1) =: V α g , (3.88)12 h + γ + ( W α h ) = − h + γ − ( W α h ) =: K α h , (3.89) − g + t + α ( V α g , Q s g ) = 12 g + t − α ( V α g , Q s∂ Ω g ) =: K ∗ α g , (3.90) t + α (cid:0) W α h , Q dα h (cid:1) = t − α (cid:0) W ) α h , Q dα ; ∂ Ω h (cid:1) =: D α h , (3.91) and the following operators are linear and continuous, V α : B s − p,q ( ∂ Ω , R n ) → B sp,q ( ∂ Ω , R n ) , K α : B sp,q ( ∂ Ω , R n ) → B sp,q ( ∂ Ω , R n ) , (3.92) K ∗ α : B s − p,q ( ∂ Ω , R n ) → B s − p,q ( ∂ Ω , R n ) , D α : B sp,q ( ∂ Ω , R n ) → B s − p,q ( ∂ Ω , R n ) . (3.93) Proof. (i) First of all, we remark that all range spaces of the velocity vector-valued layer potential operators in (3.64)-(3.80) are divergence-free due to the second relations in (3.35)-(3.36). Further, let us note that by (3.33) and (3.8) thesingle layer potential can be presented as (cf. [18, (4.1)]), V α g = h γ G α ( x , · ) , g i ∂ Ω = hG α ( x , · ) , γ ′ g i R n = N α ; R n ◦ γ ′ g (3.94)for any g ∈ B s − p,q ( ∂ Ω , R n ), p, q ∈ (1 , ∞ ) and s ∈ (0 , γ ′ : B s − p,q ( ∂ Ω , R n ) → B s − − p ′ p,q ;comp ( R n , R n ) isadjoint to the trace operator γ : B − s + p ′ p ′ ,q ′ ;loc ( R n , R n ) → B − sp ′ ,q ′ ( ∂ Ω , R n ) and they both are continues due to Lemma 2.4.Next, we show the continuity of the first operator in (3.64) in the case α > Brinkman system). To thisend, we split the Brinkman single-layer potential operator into two operators, as V α = V + V α ;0 , where V α ;0 is thecomplementary single-layer potential operator, i.e., V α ;0 := V α − V = N α ;0; R n ◦ γ ′ ◦ ι, (3.95)where the imbedding operator ι : L p ( ∂ Ω , R n ) ֒ → B s − p,p ∗ ( ∂ Ω , R n ) is continuous for any s ∈ (0 ,
1) and p ∈ (1 , ∞ ). Inaddition, N α ;0; R n := N α ; R n − N R n is a pseudodifferential operator of order − G α ;0 := G α − G (seeformula (2.27) in [33]), and hence the linear operator N α ;0; R n : B s − − p ′ p,p ∗ ;comp ( R n , R n ) → B s +3 − p ′ p,p ∗ ;loc ( R n , R n ) (3.96)is continuous for any s ∈ (0 ,
1) and p ∈ (1 , ∞ ), where p ′ = 1 − p , and B s − − p ′ p,p ∗ ;comp ( R n , R n ) is the space of distributions in B s − − p ′ p,p ∗ ( R n , R n ) with compact supports. Then formula (3.95) and the continuity of the involved operators imply thatthe operators V α ;0 : L p ( ∂ Ω , R n ) → B s +2+ p p,p ∗ ;loc ( R n , R n ) , ( V α ;0 ) | Ω + : L p ( ∂ Ω , R n ) → B s +2+ p p,p ∗ (Ω + , R n )are continuous as well. Now, the continuity of the embedding B s +2+ p p,p ∗ (Ω + , R n ) ֒ → B p p,p ∗ (Ω + , R n ) for any s ∈ (0 , V α ;0 : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) (3.97)is a continuous operator, even compact.Moreover, the Stokes single layer potential operator V : L p ( ∂ Ω , R n ) → L p (Ω + , R n ) is continuous (cf., e.g., the mappingproperty (10.73) in [61] and the continuity of the embeddings L p ( ∂ Ω , R n ) ֒ → B s − p,p ∗ ( ∂ Ω , R n ) and B s + p p,p ∗ (Ω + , R n ) ֒ → L p (Ω + , R n ) for any s ∈ (0 , ∇G of the integral operator ∇ V satisfies the relations ∇G ∈ C ∞ ( R n \ { } ) , ( ∇G )( − x ) = − ( ∇G )( x ) , ( ∇G )( λ x ) = λ − ( n − ( ∇G )( x ) , ∀ λ > . (3.98) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 19
Then, in view of [58, Proposition 2.68], there exists a constant C ≡ C (Ω + , p ) > k∇ Vg k B pp,p ∗ (Ω + , R n × n ) ≤ C k g k L p ( ∂ Ω , R n ) , ∀ g ∈ L p ( ∂ Ω , R n ) . (3.99)Consequently, there exists a constant C ≡ C (Ω + , p ) > k Vg k B
1+ 1 pp,p ∗ (Ω + , R n ) = k Vg k L p (Ω + , R n ) + k∇ Vg k B pp,p ∗ (Ω + , R n × n ) ≤ C k g k L p (Ω + , R n ) , ∀ g ∈ L p (Ω + , R n ) , (3.100)which shows that the Stokes single layer potential operator V : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) (3.101)is also continuous (cf., e.g., [62, Theorem 7.1, (3.33)], see also [23] for p = 2). This mapping property and the continuityof operator (3.97) show that the Brinkman single layer operator V α : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) is continuous, aswell.Let us show the continuity of the second operator in (3.64). To this end, we note that the Stokes single layer pressurepotential Q s f with a density f = ( f , . . . , f n ) ∈ L p ( ∂ Ω , R n ) can be written as( Q s f ) ( x ) = (div V △ f ) ( x ) , ∀ x ∈ R n \ ∂ Ω , (3.102)where V △ g is the harmonic single layer potential with density g ∈ L p ( ∂ Ω), given by( V △ g )( x ) := − n − ω n ˆ ∂ Ω | x − y | n − g ( y ) dσ y , x ∈ R n \ ∂ Ω . (3.103)Then the continuity of the single layer pressure potential potential operator Q s : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ) for any p ∈ (1 , ∞ ) is a direct consequence of Proposition 4.23 in [59]. Note that Proposition 2.68 in [58] applies as well, andshows the desired continuity of the single layer pressure potential operator in (3.64) (see also [62, Theorem 3.1, (3.30)]).Thus, we have proved the continuity of the operators in (3.64).Continuity of the first operator in (3.66) follows from the continuity of operators involved in the right hand side ofequality (3.47). Continuity of the second operator in (3.66) follows from equality (3.102), which is valid also for any f ∈ H − p ( ∂ Ω , R n ), and by the continuity of the harmonic single layer potential operator V △ from H − p ( ∂ Ω) to B p p,p ∗ (Ω + ).Indeed, for any f ∈ H − p ( ∂ Ω) there exist f , f rℓ ∈ L p ( ∂ Ω), r, ℓ = 1 , . . . n , such that f = f + P nr,ℓ =1 ∂ τ rℓ f rℓ (see (3.46)).Then by using the integration by parts formula (3.59), we obtain that( V △ f )( x ) = ˆ ∂ Ω G △ ( x − y ) f ( y ) dσ y − n X r,ℓ =1 ˆ ∂ Ω (cid:0) ∂ τ rs, y G △ ( x , y ) (cid:1) f rℓ ( y ) dσ y , ∀ x ∈ R n \ ∂ Ω , (3.104)where G △ ( x , y ) is the fundamental solution of the Laplace equation in R n ( n ≥ V △ : L p ( ∂ Ω) → B p p,p ∗ (Ω + ) (see, e.g.,[59, Proposition 4.23] and property (3.49) in [62, Proposition 3.3]), there exists a constant C such that k V △ f k B
1+ 1 pp,p ∗ (Ω + ) = k V △ f k L p (Ω + ) + k∇ V △ f k B pp,p ∗ (Ω + , R n ) ≤ C k f k L p (Ω + ) , ∀ f ∈ L p (Ω + ) . (3.105)Thus, the operator ∇ V △ : L p ( ∂ Ω) → B p p,p ∗ (Ω + , R n ) is also continuous. Finally, by continuity of this operator and ofthe operator V △ : L p ( ∂ Ω) → B p p,p ∗ (Ω + ) and also by the second relation in (3.46), we obtain from (3.104) continuity ofthe operator V △ : H − p ( ∂ Ω) → B p p,p ∗ (Ω + ) and, accordingly, continuity of the second operator in (3.66).Let us now show the continuity of the first operator in (3.68). To this end, we notice that the Brinkman double-layer potential operator can be written as W α = W + W α ;0 , where W α ;0 is the complementary double layer potentialoperator, i.e., W α ;0 := W α − W = K α ;0 ◦ γ ′ ◦ N (3.106) (see [33, Eq. (3.31)]), where the operator N : H p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ⊗ R n ) ֒ → B − sp,p ∗ ( ∂ Ω , R n ⊗ R n ) , N h ( x ) := ν ( x ) ⊗ h ( x ), is continuous for any s ∈ (0 , K α ;0 is a pseudodifferential operator of order − S α ;0 := S α − S (cf., e.g., [33, (2.27)]), and hence the operator K α ;0 : B − − s + p p,p ∗ ;comp ( R n , R n ⊗ R n ) → B − s + p p,p ∗ ;loc ( R n , R n ) , ( K α ;0 T ) j ( x ) := (cid:10)(cid:0) S αjiℓ − S jiℓ (cid:1) ( · , x ) , T iℓ (cid:11) R n , ∀ T ∈ B − − s + p p,p ∗ ;comp ( R n , R n ⊗ R n ) , (3.107)is also linear and continuous for any s ∈ (0 , B − − s + p p,p ∗ ;comp ( R n , R n ⊗ R n ) is the space of all distributions in B − − s + p p,p ∗ ( R n , R n ⊗ R n ) having compact support in R n . In addition, the trace operator γ : B s + p ′ p ′ ,p ∗′ ;loc ( R n ⊗ R n ) → B sp ′ ,p ∗′ ( ∂ Ω , R n ⊗ R n ) (acting on matrix valued functions) and its adjoint γ ′ : B − sp,p ∗ ( ∂ Ω , R n ⊗ R n ) → B − s − p p,p ∗ ;comp ( R n , R n ⊗ R n )are continuous (see the proof of [18, Theorem 1]). Then formula (3.106) and the continuity of the involved operatorsimply that the operators W α ;0 : H p ( ∂ Ω , R n ) → B − s + p p,p ∗ ;loc ( R n , R n ) , ( W α ;0 ) | Ω + : H p ( ∂ Ω , R n ) → B − s + p p,p ∗ (Ω + , R n )are continuous as well. Now, the continuity of the embedding B p − sp,p ∗ (Ω + , R n ) ֒ → B p p,p ∗ (Ω + , R n ) for any s ∈ (0 , W α ;0 : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) (3.108)is a continuous operator, even compact. Let us now show that the Stokes double-layer potential operator W : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) (3.109)is continuous as well. In the setting of Riemannian manifolds and for double layer potentials for second order ellipticequations, this continuity property follows from [63, Theorem 8.5], but we will provide a direct proof here in the contextof Euclidean setting. To this end, we use the following characterization of the space H p ( ∂ Ω) h ∈ H p ( ∂ Ω) ⇐⇒ h ∈ L p ( ∂ Ω) , ∂ τ jk h ∈ L p ( ∂ Ω) , j, k = 1 , . . . , n (3.110)(cf., e.g., [61, (2.11)]), and recall that the tangential derivative operators ∂ τ jk : H p ( ∂ Ω) → L p ( ∂ Ω) are continuous. Inaddition, consider the operator V jk defined as( V jk g ) ( x ) := ˆ ∂ Ω G jk ( x − y ) g ( y ) dσ y , x ∈ R n \ ∂ Ω . (3.111)We have proved that the Stokes single layer potential operator (3.101) is continuous for any p ∈ (1 , ∞ ) (see also [62,Theorem 3.1, (3.33)]). Consequently, the operators V jk : L p ( ∂ Ω) → B p p,p ∗ (Ω + ) (3.112)are continuous as well, for all j, k = 1 , . . . , n . Recall that the operator V △ : L p ( ∂ Ω) → B p p,p ∗ (Ω + ) is also linear andcontinuous. Finally, we mention the following formula (cf. [61, (4.84)]) ∂ r ( Wh ) j = − ∂ ℓ V jk ( ∂ τ ℓr h k ) − ∂ j V ℓk ( ∂ τ ℓr h k ) − ∂ k V △ (cid:0) ∂ τ jr h k (cid:1) in R n \ ∂ Ω , (3.113)which holds for every h ∈ H p ( ∂ Ω , R n ) and j, r = 1 , . . . , n , where h j is the j -th component of h . Then by using thecontinuity of operator (3.112) and properties (3.110) and (3.113), we deduce that the operators ∂ r ( W ) j : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ) , r, j = 1 , . . . , n (3.114)are continuous. By [61, Proposition 10.5.1, (10.68)], the operator W : H p ( ∂ Ω , R n ) → L p (Ω + , R n ) is also contin-uous (as its range is a subspace of the space H s + p p (Ω + , R n ) for any s ∈ (0 , H p ( ∂ Ω , R n ) ֒ → B sp,p ( ∂ Ω , R n ) (due IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 21 to formula (A.12)), and B s + p p,p (Ω + , R n ) ֒ → L p (Ω + , R n )). Consequently, the Stokes double layer potential operator W : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) is continuous, as asserted. This mapping property combined with the continuity ofoperator (3.108) implies the continuity of the first operator in (3.68).Continuity of the second operator in (3.68) follows from similar arguments. To this end, let us mention the usefulformula Q d g = div( W △ g ), where the harmonic double layer potential operator W △ : H p ( ∂ Ω) → B p p,p ∗ (Ω + ) is continuous(cf., e.g., [59, Proposition 4.23, (2.120), (4.96)]). Thus, the continuity of the Stokes double layer pressure potentialoperator Q d : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ) immediately follows. This property and continuity of the complementarydouble layer potential operator Q dα ;0 := Q dα − Q d : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ), where (cf. [69, (3.10)]) Q dα ;0 h = αV △ ( h · ν ) , (3.115)yield the continuity of the Brinkman double layer pressure potential operator Q dα = Q d + Q dα ;0 : H p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + ).Continuity of the first operator in (3.70) for the case α = 0 is an immediate consequence of [58, Proposition 2.68]applied to the integral operator whose kernel is given by the fundamental stress tensor S . Moreover, by using againformulas (3.106) and (3.107) we can see that the operator W α ;0 : L p ( ∂ Ω , R n ) → B p p,p ∗ (Ω + , R n ) is continuous. Therefore,for α > Q d g = div( W △ g ), and continuity of the harmonic double layer potential operator W △ : L p ( ∂ Ω) → B p p,p ∗ (Ω + ), e.g., again by [58, Proposition 2.68], along with continuity of the complementary doublelayer potential operator Q dα ;0 : L p ( ∂ Ω , R n ) → B p − p,p ∗ (Ω + ).Mapping properties (3.65), (3.67) and (3.69) are implied by the ones just above them and by the first relations in(3.35)-(3.36).( ii ) Now, relation (3.94), continuity of the operator γ ′ : B s − p,q ( ∂ Ω , R n ) → B s − p p,q ( R n , R n ) (cf. Lemma 2.4), and con-tinuity of the Newtonian potential operator N α ; R n : B s − p p,q ( R n , R n ) → B s + p p,q ( R n , R n ) (see (3.11)) imply the continuityof the first operator in (3.72) and thus of the first operators in (3.73) and (3.77). Continuity of the second operator in(3.72) follows by similar arguments based on the equalities Q s = Q R n ◦ γ ′ , and implies also continuity of the secondoperators in (3.73) and (3.77) (cf. [61, Proposition 10.5.1]).Further, let us mention that relations (3.106) and (3.107) imply that the operator W α ;0 : B sp,q ( ∂ Ω , R n ) → B s + p p,q (Ω + , R n )is continuous for all p ∈ (1 , + ∞ ) and s ∈ (0 , W | Ω + : B sp,q ( ∂ Ω + , R n ) → B s + p p,q (Ω + , R n ) (see [61, Proposition 10.5.1]) implies the con-tinuity of the first operator in (3.75). The continuity of the second operator in (3.75) can be similarly obtained. Othermapping properties of layer potentials mentioned in (3.72) and (3.79), follow with similar arguments to those for (3.64)and (3.68). We omit the details for the sake of brevity (see also the proof of [32, Lemma 3.4]).( iii ) Equality (3.81) for g ∈ L p ( ∂ Ω , R n ) can be obtained by using inequality (3.62) and [47, Proposition 1] (see also[69, Theorems 3.4]). Since (cid:0) V α g (cid:1) +nt and (cid:0) V α g (cid:1) − nt are well defined for g ∈ H − p ( ∂ Ω , R n ) due to Lemma 3.4(iii), inequality(3.43) and the density argument then imply equality (3.81) also for g ∈ H − p ( ∂ Ω , R n ). Formulas (3.82) and (3.83) followby using arguments similar to those for the trace formulas (3.11) and (3.18) in [69]. To this end, we first prove theformulas( ∂ j ( V αik g )) (cid:12)(cid:12) ± nt ( x ) = ± ν j ( x ) ( δ ik − ν i ( x ) ν k ( x )) g ( x ) + p . v . ˆ ∂ Ω ∂ j G αik ( x − y ) g ( y ) dσ y a.a. x ∈ ∂ Ω (3.116)for any g ∈ L p ( ∂ Ω) and all i, k = 1 , . . . , n , where the function V αik g is defined as in (3.111) with G αjk instead of G jk .Indeed, formula (3.116) has been proved in [61, (4.50)] in the case α = 0. Moreover, the estimate [69, (2.27)] of the kernel ∇ x G αjk ( x ) − ∇ x G jk ( x ) and [47, Proposition 1] imply that there exist the non-tangential limits of the complementary potential ∂ j V αik g − ∂ j V ik g at almost all points of ∂ Ω, and( ∂ j ( V αik g ) − ∂ j ( V ik g )) | ± nt ( x ) = p . v . ˆ ∂ Ω ( ∂ j G αik − ∂ j G ik ) ( x − y ) g ( y ) dσ y a.a. x ∈ ∂ Ω , (3.117)which implies (3.116) also for α = 0. Moreover, formula (3.116) yields for any f ∈ L p ( ∂ Ω , R n ) that( ∂ j ( V α f )) (cid:12)(cid:12) ± nt ( x ) = ± ν j ( x ) { f ( x ) − f k ( x ) ν k ( x ) ν ( x ) } + p . v . ˆ ∂ Ω ∂ j G α ( x − y ) f ( y ) dσ y a.a. x ∈ ∂ Ω (3.118)(cf. [61, (4.54)] for α = 0 and [69, Lemma 3.3] for α > Q s f ) (cid:12)(cid:12) ± nt ( x ) = ∓ ν k ( x ) f k ( x ) + p . v . ˆ ∂ Ω Π k ( x − y ) f k ( y ) dσ y a.a. x ∈ ∂ Ω (3.119)(cf. [61, (4.42)], [69, Lemma 3.3]). Then formulas (3.82) and (3.83) follow from formulas (2.22), (2.24), (3.5), (3.34),(3.118) and (3.119).Formula (3.84) follows from formula (3.61) and (3.115) together with [61, Proposition 4.2.9] (i.e., the counterpart ofthe trace formula (3.84) corresponding to the case α = 0).Continuity of operators (3.74), (3.76), (3.78), (3.80) is implied by the continuity of the operators just above them andby the first relations in (3.35) and (3.36).Now, we note that formula V α = V + V α ;0 , continuity of the Stokes single layer operator V : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n )(cf. [61, Proposition 4.2.5]), and continuity of the complementary operator V α ;0 : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) (cf. [33,Theorem 3.4(b)]) imply continuity of the first operator in (3.85). Continuity of the second operator in (3.85) and of theoperators in (3.87) similarly follows from [61, Propositions 4.2.7 - 4.2.10] and [33, Theorem 3.4(b)]. In addition, formula(3.47) and the first relation in (3.46) yield the following equality( V α g ) j ( x ) = ˆ ∂ Ω G αjk ( x − y ) g k ( y ) dσ y − n X k =1 n X r,ℓ =1 p . v . ˆ ∂ Ω (cid:0) ∂ τ rℓ (cid:0) G αjk ( x − y ) (cid:1)(cid:1) g rℓ ; k ( y ) dσ y a.a x ∈ ∂ Ω , (3.120)for any g ∈ H − p ( ∂ Ω , R n ) (cf., e.g., [61, (4.69)] for α = 0). Then the continuity of the first operator in (3.86) immediatelyfollows (see also [61, Proposition 4.2.5 (iii)] for α = 0). Continuity of the Stokes double layer operator K : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) (cf., e.g., [61, Corllary 4.2.4]) and the continuity of the reminder operator K α − K : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n )(see [33, Theorem 3.4 (b)]) show the continuity of the second operator in (3.86). Continuity of the traces and conormalderivatives of the layer potentials involved in (3.88)-(3.91) and hence continuity of the boundary operators (3.92), (3.93)immediately follow from the mapping properties of the layer potentials in item (ii) and Lemmas 2.4, 2.11.Finally, the jump relations given by the first equalities in (3.88)-(3.91) follow from formulas (3.81)-(3.84), togetherwith the density of the embeddings H p ( ∂ Ω , R n ) ֒ → B sp,q ( ∂ Ω , R n ) and L p ( ∂ Ω , R n ) ֒ → B s − p,q ( ∂ Ω , R n ), and equivalenceresults in Theorems 2.5(i) and 2.13(i) for traces and conormal derivatives. (cid:3) Let us mention the following useful result.
Lemma 3.6.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + . ( i ) If p ∈ (1 , ∞ ) , α ∈ (0 , ∞ ) , g ∈ L p ( ∂ Ω , R n ) and h ∈ H p ( ∂ Ω , R n ) , then γ ± ( V α g ) = ( V α g ) ± nt ∈ H p ; ν ( ∂ Ω , R n ) , (3.121) γ ± ( W α h ) = ( W α h ) ± nt ∈ H p ; ν ( ∂ Ω , R n ) , (3.122) t ± α ( V α g , Q s g ) = t ± nt ( V α g , Q s g ) ∈ L p ( ∂ Ω , R n ) , (3.123) t ± α (cid:0) W α h , Q dα h (cid:1) = t ± nt (cid:0) W α h , Q dα h (cid:1) ∈ L p ( ∂ Ω , R n ) (3.124) with the corresponding norm estimates. IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 23 ( ii ) If p, q ∈ (1 , ∞ ) , s ∈ (0 , , α ∈ (0 , ∞ ) , g ∈ B s − p,q ( ∂ Ω , R n ) and h ∈ B sp,q ( ∂ Ω , R n ) , then γ ± ( V α g ) = ( V α g ) ± nt ∈ B sp,q ; ν ( ∂ Ω , R n ) , (3.125) γ ± ( W α h ) = ( W α h ) ± nt ∈ B sp,q ; ν ( ∂ Ω , R n ) (3.126) with the corresponding norm estimates.Proof. Let first g ∈ L p ( ∂ Ω , R n ) and h ∈ H p ( ∂ Ω , R n ), p ∈ (1 , ∞ ). Then, according to Lemma 3.4(ii,v), the right handsides of the equalities in (3.121)-(3.124) exist almost everywhere on ∂ Ω in the sense of non-tangential limit, while Theorem3.5(i) yields that ( V α g , Q s g ) , (cid:0) W α h , Q dα h (cid:1) ∈ B p ,tp,p ∗ ;div (Ω + , L α ) and ( V α g , Q s g ) , (cid:0) W α h , Q dα h (cid:1) ∈ B p ,tp,p ∗ ;div , loc (Ω − , L α )for any t ≥ − p ′ . Moreover, Theorem 3.5 (iii) and the divergence theorem applied to the single layer potentials V α g and W α h in the domain Ω + yield that ( V α g ) ± nt ∈ H p ; ν ( ∂ Ω , R n ) , t ± nt ( V α g , Q s g ) ∈ L p ( ∂ Ω , R n ), for any g ∈ L p ( ∂ Ω , R n ),while ( W α h ) ± nt ∈ H p ; ν ( ∂ Ω , R n ) , t ± nt (cid:0) W α g , Q d g (cid:1) ∈ L p ( ∂ Ω , R n ), for any h ∈ H p ( ∂ Ω , R n ), with the corresponding normestimates. Hence Theorems 2.5(i) and 2.13(ii) along with Remark 2.14 imply relations (3.121)-(3.124).For p, q ∈ (1 , ∞ ) and s ∈ (0 , g ∈ B s − p,q ( ∂ Ω , R n ) ⊂ H − p ( ∂ Ω , R n ), h ∈ B sp,q ( ∂ Ω , R n ) ⊂ L p ( ∂ Ω , R n ) and,according to Lemma 3.4(iii,iv), the right hand sides of the equalities in (3.125) and (3.126) exist almost everywhere on ∂ Ω, while Theorem 3.5(ii) yields that V α g , W α h ∈ B s + p p,q ;div (Ω + ). Hence Theorem 2.5(i) implies relations (3.125) and(3.126). (cid:3) We will further need the following integral representation (the third Green identity) for the homogeneous Brinkmansystem solution.
Lemma 3.7.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ∈ (0 , ∞ ) , p, q ∈ (1 , ∞ ) and s ∈ (0 , . If the the pair ( u , π ) satisfies the system △ u − α u − ∇ π = , div u = 0 in Ω + (3.127) and ( u , π ) ∈ H s + p p (Ω + , R n ) × H s − − p p (Ω + ) , or ( u , π ) ∈ B s + p p,q (Ω + , R n ) × B s − − p p,q (Ω + ) , then u ( x ) = V α (cid:0) t + α ( u , π ) (cid:1) ( x ) − W α ( γ + u ) ( x ) , ∀ x ∈ Ω + . (3.128) Proof.
Let B ( y , ǫ ) ⊂ Ω be a ball of a radius ǫ around a point y ∈ Ω + and let G αk ( x ) = ( G αk ( x ) , . . . , G αkn ( x )), k = 1 , . . . , n ,where ( G α , Π) is the fundamental solution of the Brinkman system in R n (see (3.1) and (3.2)). Applying the secondGreen identity (2.29) in the domain Ω + \ B ( y , ǫ ) to ( u , π ) and to the fundamental solution ( G αk ( · − y ) , Π k )( · − y ) andtaking the limit as ǫ →
0, we obtain (3.128). (cid:3)
Next, we show the counterpart of the integral representation formula (3.128) written in terms of the non-tangentialtrace and conormal derivative.
Lemma 3.8.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α > and p ∈ (1 , ∞ ) be given constants. Assume that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) , there exist the non-tangential limits of u , ∇ u and π at almost all points of the boundary ∂ Ω , and that the pair ( u , π ) satisfies the homogeneous Brinkman system △ u − α u − ∇ π = , div u = 0 in Ω + . (3.129) Then u satisfies also the following integral representation formula u ( x ) = V α (cid:0) t +nt ( u , π ) (cid:1) ( x ) − W α (cid:0) u +nt (cid:1) ( x ) , ∀ x ∈ Ω + . (3.130) Proof.
We use arguments similar to the ones in [61, Proposition 4.4.1] for the Stokes system. In the case of a smoothbounded domain Ω ⊂ R n and for u ∈ C (Ω + , R n ), π ∈ C (Ω + ), formula (3.130) follows easily from the integration byparts, cf. e.g. (3.128). Now consider a sequence of sub-domains { Ω j } j ≥ in Ω + that contain the point x ∈ Ω + andconverges to Ω + in the sense of Lemma 2.2. Then formula (3.130) holds for each of the domains Ω j and by the LebesgueDominated Convergence Theorem (applied again after the change of variable as in Lemma 2.2 that reduces the integralover ∂ Ω j to an integral over ∂ Ω) letting j → ∞ , we obtain (3.130) for the Lipschitz domain Ω + as well. (cid:3) Invertibility of related integral operators
Lemma 4.1.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ∈ (0 , ∞ ) and ≤ s ≤ . Then the following operators are isomorphisms, I + K ∗ α : H − s ( ∂ Ω , R n ) → H − s ( ∂ Ω , R n ) , (4.1)12 I + K α : H s ( ∂ Ω , R n ) → H s ( ∂ Ω , R n ) . (4.2) Proof.
Isomorphism property of operator (4.1) for s = 0 follows from [46, Proposition 7.1] (see also [69, Lemma 5.1]).By duality this also implies the isomorphism property of operator (4.2) for s = 0.Let us now remark that for α = 0 and 0 < s ≤
1, operator (4.2) is a Fredholm operator with index zero (cf., e.g., [61,Proposition 10.5.3 and Theorem 5.3.6]), while the operator K α ;0 := K α − K : H s ( ∂ Ω , R n ) → H s ( ∂ Ω , R n ) is compact(cf., e.g., [33, Theorem 3.4]), implying that for α > < s ≤
1, (4.2) is a Fredholm operator with index zero aswell. Then by Lemma 2.4 and the invertibility property of operator (4.2) for s = 0 we obtain the equalitiesKer (cid:26) I + K α : H s ( ∂ Ω , R n ) → H s ( ∂ Ω , R n ) (cid:27) = Ker (cid:26) I + K α : H ( ∂ Ω , R n ) → H ( ∂ Ω , R n ) (cid:27) = { } , < s ≤ , (4.3)which show invertibility and hence isomorphism property of operator (4.2) for α > < s ≤ α > < s ≤ (cid:3) We will often need the following two intervals, R ( n, ε ) = (cid:18) n − n + 1 − ε, ε (cid:19) ∩ (1 , + ∞ ) , R ( n, ε ) = ( (2 − ε, + ∞ ) if n = 3 , (cid:16) − ε, n − n − + ε (cid:17) if n > , (4.4)which are particular cases of a more general interval R θ ( n, ε ) = ( (2 − ε, + ∞ ) if n = 3 and θ = 1 , (cid:16) n − n +1 − θ − ε, n − n − − θ + ε (cid:17) ∩ (1 , + ∞ ) if n > ≤ θ < . (4.5) Lemma 4.2.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ∈ (0 , ∞ ) . Thenthere exists ε = ε ( ∂ Ω) > such that for any p ∈ R ( n, ε ) and p ′ ∈ R ( n, ε ) , see (4.4) , the following operators areisomorphisms, I + K ∗ α : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) , (4.6)12 I + K ∗ α : H − p ′ ( ∂ Ω , R n ) → H − p ′ ( ∂ Ω , R n ) , (4.7)12 I + K α : L p ′ ( ∂ Ω , R n ) → L p ′ ( ∂ Ω , R n ) , (4.8)12 I + K α : H p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) . (4.9) If Ω + is of class C , then the above invertibility properties hold for all p, p ′ ∈ (1 , ∞ ) .Proof. By [61, Theorem 9.1.11] there exists a parameter ε > p ∈ R ( n, ε ),12 I + K ∗ : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) (4.10)is a Fredholm operator with index zero. Then compactness of the operator K ∗ α ;0 := K ∗ α − K ∗ : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n )for any p ∈ (1 , ∞ ) (see [33, Theorem 3.4(b)]), imply that operator (4.6) is Fredholm with index zero as well, for any IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 25 p ∈ R ( n, ε ). In addition, a density argument based on Lemma 2.4 and the invertibility property of operator (4.1) inthe case s = 0, show that operator (4.6) is an isomorphism for p = 2 and hence for any p ∈ R ( n, ε ).Similarly, by [61, Theorem 9.1.3] there exists a parameter (for the sake of brevity, we use the same notation as above) ε > p ∈ R ( n, ε ) the operator12 I + K : H p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) (4.11)is Fredholm with index zero. Then compactness of the complementary operator K α ;0 := K α − K : H p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) for any p ∈ (1 , ∞ ) (see [33, Theorem 3.4(b)]), implies that operator (4.9) is Fredholm with index zero aswell, for any p ∈ R ( n, ε ). In addition, a density argument based on Lemma 2.4 and the invertibility property foroperator (4.2) in the case s = 1, show that operator (4.9) is an isomorphism for p = 2 and hence for any p ∈ R ( n, ε ).Isomorphism property of operators (4.7) and (4.8) then follow by duality and isomorphism property of operators (4.9)and (4.6), respectively, for p ′ = pp − .If Ω + is of class C , then operator (4.11) is Fredholm with index zero for any p ∈ (1 , ∞ ), cf., e.g., [67, Remark 3.1],and the the rest of the proof holds true for any p, q ∈ (1 , ∞ ). (cid:3) Lemmas 4.2, A.1 and B.1 (ii) and an interpolation argument (provided by the complex and real interpolation theory)imply the following assertion.
Corollary 4.3.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω , and α ∈ (0 , ∞ ) .Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R s ( n, ε ) and p ′ ∈ R − s ( n, ε ) , cf. (4.5) , the following operatorsare isomorphisms I + K α : H sp ′ ( ∂ Ω , R n ) → H sp ′ ( ∂ Ω , R n ) , s ∈ [0 , , (4.12)12 I + K ∗ α : H − sp ( ∂ Ω , R n ) → H − sp ( ∂ Ω , R n ) , s ∈ [0 , , (4.13)12 I + K α : B sp ′ ,q ( ∂ Ω , R n ) → B sp ′ ,q ( ∂ Ω , R n ) , s ∈ (0 , , q ∈ (1 , ∞ ) , (4.14)12 I + K ∗ α : B − sp,q ( ∂ Ω , R n ) → B − sp,q ( ∂ Ω , R n ) , s ∈ (0 , , q ∈ (1 , ∞ ) . (4.15) If Ω + is of class C , then the properties hold for all p, p ′ ∈ (1 , ∞ ) . Next we show the following invertibility result (see also [46, Proposition 7.2] in the case p = 2 and s = 0). Lemma 4.4.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ∈ (0 , ∞ ) . Then there exists a number ε = ε ( ∂ Ω) > such that the operators − I + K α : L p ′ ; ν ( ∂ Ω , R n ) → L p ′ ; ν ( ∂ Ω , R n ) , (4.16) − I + K ∗ α : L p ( ∂ Ω , R n ) / R ν → L p ( ∂ Ω , R n ) / R ν , (4.17) − I + K α : H p ; ν ( ∂ Ω , R n ) → H p ; ν ( ∂ Ω , R n ) , (4.18) − I + K ∗ α : H − p ′ ( ∂ Ω , R n ) / R ν → H − p ′ ( ∂ Ω , R n ) / R ν (4.19) are isomorphisms for all p ∈ R ( n, ε ) and p ′ ∈ R ( n, ε ) ( cf. (4.4)) .If the domain Ω is of class C , the above properties hold for all p, p ′ ∈ (1 , ∞ ) .Proof. In the case α = 0, operator (4.16) is an isomorphism (cf. [61, Corollary 9.1.12]), and hence a Fredholm operatorwith index zero for any p ′ ∈ R ( n, ε ). Moreover, the operator K α − K is compact on the space L p ′ ( ∂ Ω , R n ) (see [33, Theorem 3.4(b)]), and its range is a subset of L p ′ ; ν ( ∂ Ω , R n ). Indeed, by using the formula( K α − K ) h = (cid:18) − I + K α (cid:19) h − (cid:18) − I + K (cid:19) h = γ + W α h − γ + Wh , the equations div W α h = 0 and div Wh = 0 in Ω + , and then, the divergence theorem and the trace formulas (3.82), wededuce that ( K α − K ) h ∈ L p ′ ; ν ( ∂ Ω , R n ) for any h ∈ L p ′ ; ν ( ∂ Ω , R n ). Therefore, the operator K α − K : L p ′ ; ν ( ∂ Ω , R n ) → L p ′ ; ν ( ∂ Ω , R n ) is compact, and then operator (4.16) is Fredholm with index zero for any p ′ ∈ R ( n, ε ). On the otherhand, by a similar reasoning (cf., e.g., [61, Theorem 9.1.3] and [33, Theorem 3.4 (b)]), operator (4.18) is Fredholm withindex zero as well, for any p ∈ R ( n, ε ).We show now that operators (4.16) and (4.18) are also injective. Let us start from operator (4.18) with p = 2.Let h ∈ H ν ( ∂ Ω , R n ) be such that (cid:0) − I + K α (cid:1) h = . Thus, γ + W α h = , and by applying the Green formula(2.28) to the double layer velocity and pressure potentials W α h and Q dα h in Ω + , we deduce that W α h = and Q dα h = c ∈ R in Ω + . According to formula (3.84), we obtain that t − nt (cid:0) W α h , Q dα h (cid:1) = t +nt (cid:0) W α h , Q dα h (cid:1) = − c ν ,and then the relation γ − W α h = h ∈ H ν ( ∂ Ω , R n ) shows that h t − nt (cid:0) W α h , Q dα h (cid:1) , γ − W α h i ∂ Ω = 0. Finally, therelations W α h ( x ) = O ( | x | − n ) and Q d h = O ( | x | − n ) as | x | → ∞ (see, e.g., [74, Lemma 2.12, (2.76)]), and the Greenformula (2.28) applied to W α h and Q dα h in Ω − imply that W α h = and Q dα h = in Ω − . Then the trace formula(3.82) yields that h = . Consequently, operator (4.18) with p = 2 is injective. Then Lemma 2.4 implies that operator(4.16) with p ′ = 2 is injective as well. Applying Lemma 2.4 again, we now obtain that operator (4.18) with p ∈ R ( n, ε )and operator (4.16) with p ′ ∈ R ( n, ε ) are injective, and according to their Fredholm property, these operators are alsoisomorphisms. Operators (4.17) and (4.19) are then isomorphisms by duality.If Ω is of C class, then for all p, p ′ ∈ (1 , ∞ ) operators (4.16) and (4.17) are Fredholm with index zero due tocompactness of the operators K and K ∗ on the corresponding spaces (cf., e.g., [22, Eq. (3.51) in the proof of Proposition3.5]), and [33, Theorem 3.4 (b)]. Then the previous paragraph implies that operators (4.16)-(4.19) are isomorphisms for p, p ′ ∈ (1 , ∞ ). (cid:3) Lemmas 4.4, A.1 and B.1(ii) by interpolation imply the following result (see also [46, Proposition 7.2] for p = 2 and s = 0). Corollary 4.5.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ∈ (0 , ∞ ) . Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R s ( n, ε ) and p ′ ∈ R − s ( n, ε ) ( cf. (4.5)) , thefollowing operators are isomorphisms, − I + K α : H sp ′ ; ν ( ∂ Ω , R n ) → H sp ′ ; ν ( ∂ Ω , R n ) , s ∈ [0 , , (4.20) − I + K ∗ α : H − sp ( ∂ Ω , R n ) / R ν → H − sp ( ∂ Ω , R n ) / R ν , s ∈ [0 , , (4.21) − I + K α : B sp ′ ,q ; ν ( ∂ Ω , R n ) → B sp ′ ,q ; ν ( ∂ Ω , R n ) , s ∈ (0 , , q ∈ (1 , ∞ ) , (4.22) − I + K ∗ α : B − sp,q ( ∂ Ω , R n ) / R ν → B − sp,q ( ∂ Ω , R n ) / R ν , s ∈ (0 , , q ∈ (1 , ∞ ) . (4.23) If Ω + is of class C , then the properties hold for all p, p ′ ∈ (1 , ∞ ) . In the case α = 0, the result, corresponding to the next one, has been obtained in [61, Theorem 9.1.4, Corollary 9.1.5](see also [62, Theorem 6.1]). Lemma 4.6.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ∈ (0 , ∞ ) . Then there exists a number ε > such that for any p ∈ R ( n, ε ) and p ′ ∈ R ( n, ε ) , see (4.4) , thefollowing Brinkman single layer potential operators are isomorphisms V α : L p ( ∂ Ω , R n ) / R ν → H p ; ν ( ∂ Ω , R n ) , (4.24) V α : H − p ′ ( ∂ Ω , R n ) / R ν → L p ′ ; ν ( ∂ Ω , R n ) . (4.25) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 27 If Ω + is of class C , then the above invertibility properties hold for all p, p ′ ∈ (1 , ∞ ) .Proof. First, we note that for any f ∈ L p ( ∂ Ω , R n ) the inclusion V α f ∈ H p ( ∂ Ω , R n ) follows by Theorem 3.5(iii). Moreover,the inclusion V α f ∈ H p ; ν ( ∂ Ω , R n ) follows from the equation div V α f = 0 in Ω + , the divergence theorem and relation(3.88). On the other hand, there exists a number ε > V : L p ( ∂ Ω , R n ) / R ν → H p ; ν ( ∂ Ω , R n )is an isomorphism for any p ∈ R ( n, ε ) (cf. [61, Theorem 9.1.4]), which implies that V : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is aFredholm operator with index zero for the same range of p . Thus, the Brinkman single layer potential operator V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) (4.26)is a Fredholm operator of index zero for any p ∈ R ( n, ε ), as follows from the equality V α = V + V α ;0 , where V α ;0 := V α − V : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is a compact operator (cf. [33, Lemma 3.1]). Then by Lemma 2.4, we obtain theequality Ker (cid:8) V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) (cid:9) = Ker (cid:8) V α : L ( ∂ Ω , R n ) → H ( ∂ Ω , R n ) (cid:9) , (4.27)for each p ∈ R ( n, ε ).Moreover, by considering a density ϕ ∈ L ( ∂ Ω , R n ) such that V α ϕ = on ∂ Ω, by applying the Green identity (2.28)to the single layer velocity and pressure potentials u = V α ϕ and π = Q s ϕ , and by using Theorem 3.5, we deducethat u = and π = c ∈ R in Ω + . In addition, the behavior at infinity of the single layer potentials, u ( x ) = O ( | x | − n ), σ ( u , π )( x ) = O ( | x | − n ) as | x | → ∞ (see, e.g., [46, Section 4]), yields that the Green identity (2.28) applies also to thefields u and π in the exterior domain Ω − and yields u = , π = 0 in Ω − . Then by formulas (3.83) ϕ = c ν . On theother hand, the divergence theorem and the second equation in (3.4) imply that ( V α ν ) j ( x ) = ˆ Ω + ∂ G αjk ( x − y ) ∂y k dy = 0 , and accordingly that V α ν = . Thus, we obtain the equalityKer (cid:8) V α : L ( ∂ Ω , R n ) → H ( ∂ Ω , R n ) (cid:9) = R ν . Therefore, by (4.27) the codimension of the range of the operator V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is equal to one.Moreover, Range ( V α ; ∂ Ω ) ⊆ H p ; ν ( ∂ Ω , R n ), as follows from the divergence theorem and the second equation in (3.4).Since H p ; ν ( ∂ Ω , R n ) is a subspace of codimension one in H p ( ∂ Ω , R n ), we conclude that the range of the operator V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is just H p ; ν ( ∂ Ω j , R n ). Then the Fundamental quotient theorem for linear continuous mapsimplies V α : L p ( ∂ Ω , R n ) / R ν → H p ; ν ( ∂ Ω , R n ) is an isomorphism for any p ∈ R ( n, ε ), as asserted.Since the operator V α is self-adjoint, duality shows that operator (4.25) is also an isomorphism for any q ∈ (1 , ∞ )such that q = pp − . Note that for the same range of q , the Stokes single layer potential operator V : H − q ( ∂ Ω , R n ) / R ν → L q ; ν ( ∂ Ω , R n ) is an isomorphism as well (see [61, Corollary 9.1.5] for α = 0).If Ω + is of class C , then the operator V : H − q ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) is Fredholm with index zero for any q ∈ (1 , ∞ )(cf., e.g., [67, Remark 3.1]; see also [28, Proposition 4.1]). By duality, we deduce that operator (4.26) is Fredholmwith index zero as well for any p ∈ (1 , ∞ ) whenever α = 0. In view of [33, Theorem 3.4], the complementary operator V α − V : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is compact (even in the case of a Lipschitz domain). Therefore, the operator V α : L p ( ∂ Ω , R n ) → H p ( ∂ Ω , R n ) is Fredholm with index zero for any p ∈ (1 , ∞ ). Then the rest of the proof holds truefor any p, q ∈ (1 , ∞ ). (cid:3) Lemmas 4.6, A.1 and B.1(ii) and an interpolation argument imply the following assertion (see also [67, Remark 3.1]in the case of a C domain). Corollary 4.7.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω and let Ω − := R n \ Ω + .Let α ∈ (0 , ∞ ) and p ∈ R s ( n, ǫ ) , see (4.5) . Then there exists ε = ε ( ∂ Ω) > such that the following operators areisomorphisms, V α : H − sp ( ∂ Ω , R n ) / R ν → H − sp ; ν ( ∂ Ω , R n ) , s ∈ [0 , , (4.28) V α : B − sp,q ( ∂ Ω , R n ) / R ν → B − sp,q ; ν ( ∂ Ω , R n ) , s ∈ (0 , , q ∈ (1 , ∞ ) . (4.29) If Ω + is of class C , then the property holds for any p ∈ (1 , ∞ ) . The Dirichlet and Neumann problems for the Brinkman system
The Dirichlet problem for the Brinkman system.
Let us consider the Dirichlet problem for the homogeneousBrinkman system, △ u − α u − ∇ π = , div u = 0 in Ω + , (5.1) u +nt = h on ∂ Ω , (5.2)and show the following assertion (cf. [69, Theorem 5.5] for p = 2 and the boundary data in the space L ν ( ∂ Ω , R n ); for α = 0 see also [61, Corollary 9.1.5, Theorems 9.1.4, 9.2.2 and 9.2.5] and [62, Theorem 7.1]). The Dirichlet boundarycondition (5.2) is understood in the sense of non-tangential limit at almost all points of ∂ Ω. Theorem 5.1.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω + . Let α ∈ (0 , ∞ ) , p ∈ (1 , ∞ ) , and p ∗ := max { p, } . ( i ) Let h ∈ H p ; ν ( ∂ Ω , R n ) . Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R ( n, ε ) , the Dirichlet problem (5.1) - (5.2) has a solution ( u , π ) such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) and there exist the non-tangential limitsof u , ∇ u and π at almost all points of the boundary ∂ Ω . Moreover, there exists a constant C = C ( ∂ Ω , p, α ) > such that k M ( u ) k L p ( ∂ Ω) + k M ( ∇ u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) ≤ C k h k H p ( ∂ Ω , R n ) , (5.3) k u +nt k L p ( ∂ Ω) + k∇ u +nt k L p ( ∂ Ω) + k π +nt k L p ( ∂ Ω) ≤ C k h k H p ( ∂ Ω , R n ) . (5.4) In addition, u ∈ B p p,p ∗ (Ω + , R n ) , π ∈ B p p,p ∗ (Ω + ) and k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C k h k H p ( ∂ Ω , R n ) . ( ii ) Let h ∈ L p ; ν ( ∂ Ω , R n ) . Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R ( n, ε ) the Dirichlet problem (5.1) - (5.2) has a solution ( u , π ) such that M ( u ) ∈ L p ( ∂ Ω) . Moreover, there exists a constant C > such that k M ( u ) k L p ( ∂ Ω) ≤ C k h k L p ( ∂ Ω , R n ) . (5.5) In addition, u ∈ B p p,p ∗ (Ω + , R n ) and k u k B pp,p ∗ (Ω + , R n ) ≤ C k h k L p ( ∂ Ω , R n ) . ( iii ) Let < s < and h ∈ H sp ; ν ( ∂ Ω , R n ) . Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R − s ( n, ǫ ) ( cf. (4.5)) , the Dirichlet problem (5.1) - (5.2) ( where the Dirichlet condition (5.2) is considered in the Gagliardo tracesense ) has a solution u ∈ B s + p p,p ∗ (Ω + , R n ) , π ∈ B s + p − p,p ∗ (Ω + ) , and there exists a constant C > such that k u k B s + 1 pp,p ∗ (Ω + , R n ) + k π k B s + 1 p − p,p ∗ (Ω + ) ≤ C k h k H sp ( ∂ Ω , R n ) . In each of the cases ( i ) , ( ii ) and ( iii ) , the solution is unique up to an arbitrary additive constant for the pressure π , andcan be expressed in terms of the following double layer velocity and pressure potentials u = W α (cid:18) − I + K α (cid:19) − h ! , π = Q dα (cid:18) − I + K α (cid:19) − h ! in Ω + . (5.6) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 29
Proof.
According to Lemmas 3.4, 4.4 and Theorem 3.5(iii), the functions given by (5.6) provide a solution of the Dirichletproblem (5.1)-(5.2), which satisfies the corresponding norm estimates mentioned in items ( i ) − ( ii ). For 0 < s < (cid:0) − I + K α (cid:1) − h ∈ H sp ( ∂ Ω , R n ) ֒ → B sp,p ∗ ( ∂ Ω , R n ) with corresponding normestimates, which by (3.40), (3.75) and (3.82) proves the desired solution properties.We will now prove uniqueness of the solution of the Dirichlet problem (5.1)-(5.2) satisfying the conditions in item ( ii ),by modifying arguments in the proofs of [61, Theorem 5.5.4] and [62, Theorem 7.1]. Let ( u , π ) be a solution of thehomogeneous version of the Dirichlet problem (5.1)-(5.2) such that M ( u ) ∈ L p ( ∂ Ω) and u satisfies the homogeneousboundary condition in the sense of non-tangential limit at almost all points of the boundary ∂ Ω. Let x ∈ Ω + and let { Ω j } j ≥ be a sequence of C ∞ sub-domains in Ω + that contain x and converge to Ω + in the sense described in Lemma2.2. Let G αk ( x ) = ( G αk ( x ) , . . . , G αkn ( x )), k = 1 , . . . , n , where ( G α , Π) is the fundamental solution of the Brinkman systemin R n (see (3.1) and (3.2)). Then for each Ω j and any k = 1 , . . . , n , the functions v j and q j given by v j x = W jα (cid:16) h ′ ( j ) (cid:17) , q j x = Q j ; dα (cid:16) h ′ ( j ) (cid:17) in R n \ ∂ Ω j , h ′ ( j ) = (cid:18) − I + K jα (cid:19) − ( G αk ( x − · ) | ∂ Ω j ) , (5.7)satisfy the system (cid:26) △ v j x − α v j x − ∇ q j x = , div v j x = 0 in Ω j , ( v j x ) +nt = G αk ( x , · ) | ∂ Ω j . (5.8)Here W jα := W α ; ∂ Ω j and Q j ; dα := Q dα ; ∂ Ω j are the double layer velocity and pressure potential operators correspondingto ∂ Ω j , while K jα : H p ′ ( ∂ Ω j , R n ) → H p ′ ( ∂ Ω j , R n ) is the corresponding double layer integral operator. Indeed, G αk ( x −· ) | ∂ Ω j ∈ H p ; ν ( j ) ( ∂ Ω j , R n ) and, in view of Lemma 4.4, the operator − I + K jα : H p ′ ; ν ( j ) ( ∂ Ω j , R n ) → H p ′ ; ν ( j ) ( ∂ Ω j , R n ) isan isomorphism for any p ′ ∈ (1 , ∞ ) since Ω j is a smooth domain.Note that the operator − I + K α : H p ′ ; ν ( ∂ Ω , R n ) → H p ′ ; ν ( ∂ Ω , R n ) is an isomorphism for any p ′ ∈ R ( n, ε ) (seeLemma 4.4), i.e., for any p ′ such that p ′ = 1 − p , where p ∈ R ( n, ε ). After performing a change of variable as in Lemma2.2, the operator − I + K jα defined on ∂ Ω j can be identified with an operator T jα acting on functions defined on ∂ Ω.Then, employing the arguments, e.g., similar to those in the last paragraph in p.116 in [61], which are based on [61,Lemmas 11.9.13 and 11.12.2], and taking into account [47, Proposition 1] (see also [23, Theorems 3.8 (iv) and 4.15]),one can show that the sequence of operators T jα converges to the operator T α := − I + K α in the operator norm andthe sequence of the inverses of the operators T jα converges to the inverse of the operator T α in the operator norm. Hencethe operator norms k (cid:0) − I + K jα (cid:1) − k H p ′ ( ∂ Ω j , R n ) are bounded uniformly in j , implying that there exist some constants C , C ′ depending only on p , n , α and the Lipschitz character of Ω + (thus, C does not depend on j ) such that k h ′ ( j ) k H p ′ ( ∂ Ω j , R n ) ≤ C k G αk ( x , · ) k H p ′ ( ∂ Ω j , R n ) ≤ C ′ ( k M ( G αk ( x , · )) k L p ′ ( ∂ Ω) + k M ( ∇ G αk ( x , · )) k L p ′ ( ∂ Ω) ) , (5.9)where the non-tangential maximal operator M is considered with respect to a regular family of cones truncated at aheight smaller than the distance from x to ∂ Ω (cf. [75, Theorem 1.12], see also Lemma 2.2). Further, by consideringthe change of variable y j := Φ j ( y ) as in Lemma 2.2, the double-layer potential representations (5.7) become v j x ; ℓ ( x ) = ˆ ∂ Ω j S αiℓs ( y j , x ) ν s ( y j ) h ′ ( j ) i ( y j ) dσ y j = ˆ ∂ Ω S αiℓs (Φ j ( y ) , x ) ν s (Φ j ( y )) H ′ ( j ) i ( y ) dσ y , (5.10) q j x ( x ) = ˆ ∂ Ω j Λ αis ( y j , x ) ν s ( y j ) h ′ ( j ) i ( y j ) dσ y j = ˆ ∂ Ω Λ αis (Φ j ( y ) , x ) ν s (Φ j ( y )) H ′ ( j ) i ( y ) dσ y , ∀ x ∈ Ω j , (5.11)where H ′ ( j ) ( y ) := h ′ ( j ) (Φ j ( y )) ω j ( y ) , y ∈ ∂ Ω , y ( j ) = ( y ( j )1 , . . . , y ( j ) n ), h ′ ( j ) = ( h ′ ( j )1 , . . . , h ′ ( j ) n ), H ′ ( j ) = ( H ′ ( j )1 , . . . , H ′ ( j ) n ),and ω j is the Jacobian of Φ j : ∂ Ω → ∂ Ω j .In view of (5.9) and of the uniform boundedness of { ω j } j ≥ , there exists a constant C > p , n and the Lipschitz character of Ω + ) such that k H ′ ( j ) k H p ′ ( ∂ Ω , R n ) ≤ C k h ′ ( j ) k H p ′ ( ∂ Ω j , R n ) ≤ C ′ C ( k M ( G αk ( x , · )) k L p ′ ( ∂ Ω) + k M ( ∇ G αk ( x , · )) k L p ′ ( ∂ Ω) ) , ∀ j ≥ . (5.12) Hence { H ′ ( j ) } j ≥ is a bounded sequence in H p ′ ( ∂ Ω , R n ), and, thus, there exists a subsequence, still denoted as thesequence, and a function H ′ ∈ H p ′ ( ∂ Ω , R n ), such that H ′ ( j ) → H ′ weakly in H p ′ ( ∂ Ω , R n ). By this property and letting j → ∞ in (5.10)-(5.11), we obtain v j x ( x ) → v x ( x ) = W α H ′ ( x ) , q j x ( x ) → q x ( x ) = Q dα H ′ ( x ) pointwise for any x ∈ Ω + . Moreover, in view of Lemma 3.4 (where the constants depend only on the Lipschitz character of Ω + ), appliedto ∂ Ω j , and (5.9), we obtain the inequality k M ( ∇ v j x ) k L p ′ ( ∂ Ω j ) + k M ( q j x ) k L p ′ ( ∂ Ω j ) ≤ C k h ′ ( j ) k ≤ C ′ C (cid:16) k M ( G αk ( x , · )) k L p ′ ( ∂ Ω) + k M ( ∇ G αk ( x , · )) k L p ′ ( ∂ Ω) (cid:17) , (5.13)with a constant C depending only on p , n and the Lipschitz character of Ω + .In addition, the pair (cid:16) G α ; jk ( x , · ) , π jk ( x , · ) (cid:17) given by G α ; jk ( x , · ) := G αk ( x − · ) − v j x , π jk ( x , · ) := Π k ( x − · ) − q j x (5.14)defines the Green function of the Brinkman system in Ω j and its corresponding pressure vector, i.e., it satisfies for each x ∈ Ω j the following relations −∇ π jk ( x , y ) + △ G α ; jk ( x , y ) − α G α ; jk ( x , y ) = − δ y ( x ) I , div y G α ; jk ( x , y ) = 0 in Ω j , G α ; jk ( x , y ) = , y ∈ ∂ Ω j . (5.15)Hence, for each Ω j and any k = 1 , . . . , n , we obtain the relations D △ G α ; jk ( x , · ) − α G α ; jk ( x , · ) − ∇ π jk ( x , · ) , u E Ω j = u k ( x ) . (5.16)Then by (5.15) and (5.16) we obtain that u k ( x ) = ˆ ∂ Ω j t c+ ( G α ; jk ( x , · ) , π jk ( x , · )) · u dσ j . (5.17)By (5.14) and (5.13), there exists a constant C depending only on α , p , n and the Lipschitz character of Ω + such that k M ( ∇ G α ; jk ( x , · )) k L p ′ ( ∂ Ω j ) + k M ( π jk ( x , · )) k L p ′ ( ∂ Ω j ) ≤ C ( k M ( G αk ( x , · )) k L p ′ ( ∂ Ω) + k M ( ∇ G αk ( x , · )) k L p ′ ( ∂ Ω) ) , Since also M ( u ) ∈ L p ( ∂ Ω) and ( u ) +nt = on ∂ Ω, then the Lebesgue Dominated Convergence Theorem (applied againafter the change of variable as in Lemma 2.2 that reduces the integral over ∂ Ω j to an integral over ∂ Ω) implies thatthe right hand side in (5.17) tends to zero as ∂ Ω j tends to ∂ Ω and hence u k ( x ) = 0. Because x is an arbitrary pointin Ω + , we conclude that u = in Ω + , and by the first equation in (5.1), π is a constant pressure, as asserted. Thiscompletes the proof of the uniqueness in item (ii).Let us show also the uniqueness result for item ( i ). To do so, assume that ( u , π ) is a solution of the homogeneousversion of the Dirichlet problem (5.1) such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω), there exist the non-tangential limitsof u , ∇ u and π at almost all points of the boundary ∂ Ω, and u satisfies the homogeneous Dirichlet boundarycondition in the sense of non-tangential limit at almost all points of ∂ Ω. Then the Green representation formula u = V α (cid:0) t +nt ( u , π ) (cid:1) − W α (cid:0) u (cid:1) in Ω + (cf. Lemma 3.8) reduces to u = V α (cid:0) t +nt ( u , π ) (cid:1) in Ω + , and, by consideringthe non-tangential trace, we obtain that V α (cid:0) t +nt ( u , π ) (cid:1) = on ∂ Ω. Thus, t +nt ( u , π ) ∈ R ν (see Lemma 4.6), andhence u = in Ω + , while the Brinkman equation (5.1) shows that π = 0 in Ω + (up to an additive constant pressure).This completes the proof of the statement in item ( i ).Next we show for s ∈ (0 ,
1) the uniqueness of a solution to the Dirichlet problem (5.1)-(5.2), in the hypothesisof item ( iii ). To this end, let ( u , π ) ∈ B s + p p,p ∗ (Ω + , R n ) × B s + p p,p ∗ (Ω + ) denote a solution of the homogeneous versionof the Dirichlet problem (5.1)-(5.2). By Lemmas 2.4, 2.11 and Theorem 2.5 we obtain that γ + u = u = 0 and t + α ( u , π ) ∈ B s − p,p ∗ ( ∂ Ω , R n ). Then for s ∈ (0 , u , π )implies that γ + V α ( t + α ( u , π )) = on ∂ Ω. Hence by (3.88) and (4.29) we obtain that t + α ( u , π ) ∈ R ν . Since V α ν = inΩ + , we deduce that u = in Ω + , and by the Brinkman equation (5.1) π = 0 (up to an additive constant). (cid:3) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 31
Note that for p = 2, Theorem 5.1 (ii) has been obtained by Z. Shen in [69, Theorem 5.5] by using another doublelayer potential approach.The following regularity result has been obtained in [61, Theorem 4.3.1] and [62, Theorem 7.1] in the case of theStokes system (i.e., for α = 0). We prove a similar result in the case of the Brinkman system (i.e., for α >
0) by usingthe main ideas of the proof of [62, Theorem 7.1] (see also [56, (2.95), Remark V p. 37], [16, Theorem 2], [35, Lemma3.3], [45]).
Theorem 5.2.
Let Ω + ⊂ R n be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ≥ , p ∈ (1 , ∞ ) and p ∗ := max { p, } . Assume that a pair ( u , π ) satisfies the homogeneous Brinkman system (5.1) . Then the followingproperties hold. ( i ) There exists ε = ε ( ∂ Ω) > such that for any p ∈ (2 − ε, ∞ ) , the condition M ( u ) ∈ L p ( ∂ Ω) implies that thereexists the non-tangential limit of u almost everywhere on ∂ Ω and u +nt ∈ L p ; ν ( ∂ Ω , R n ) . Moreover, k u +nt k L p ( ∂ Ω , R n ) ≤ C k M ( u ) k L p ( ∂ Ω) , k u k B pp,p ∗ (Ω + , R n ) ≤ C ′ k M ( u ) k L p ( ∂ Ω) , (5.18) with some constants C ≡ C ( ∂ Ω , p, α ) > , C ′ ≡ C ′ ( ∂ Ω , p, α ) > . ( ii ) There exists ε = ε ( ∂ Ω) > such that for any p ∈ R ( n, ε ) ∪ (2 , ∞ ) , the assumption M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) implies that there exist the non-tangential limits of u , ∇ u , π almost everywhere on ∂ Ω , and that u +nt ∈ H p ; ν ( ∂ Ω , R n ) and t +nt ( u , π ) ∈ L p ( ∂ Ω , R n ) . In addition, there exist some constants C ≡ C ( ∂ Ω , p, α ) > , C ′ ≡ C ′ ( ∂ Ω , p, α ) > such that k u +nt k H p ( ∂ Ω , R n ) + k t +nt ( u , π ) k L p ( ∂ Ω , R n ) ≤ C (cid:0) k M ( u ) k L p ( ∂ Ω) + k M ( ∇ u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) (cid:1) , (5.19) k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C ′ (cid:0) k M ( u ) k L p ( ∂ Ω) + k M ( ∇ u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) (cid:1) . (5.20) Proof. (i) We will use arguments similar to the ones in the proof of [16, Lemma 8]. First, let { Ω j } j ≥ be a sequenceof sub-domains in Ω + that converge to Ω + in the sense described in Lemma 2.2, with the corresponding notations Φ j , ν ( j ) and ω j also introduced there. Due to ellipticity of the homogeneous Brinkman system in Ω + , we have ( u , π ) ∈ C ∞ (Ω + , R n ) × C ∞ (Ω + ). Now, let h ( j ) := u | ∂ Ω j . Then ( u j , π j ) := ( u | Ω j , π | Ω j ) satisfies the homogeneous Brinkmansystem in Ω j and the Dirichlet boundary condition u j | ∂ Ω j = h ( j ) on ∂ Ω j , where h ( j ) ∈ L p ; ν ( j ) ( ∂ Ω j , R n ). The solutionof such a problem is unique, up to an additive constant for the pressure (see, e.g., Theorem 5.1).According to Lemma 4.4 applied to the smooth domain Ω j , such a solution can be expressed in terms of the doublelayer potential u j = W α ; ∂ Ω j h ′ ( j ) , π j = Q dα ; ∂ Ω j h ′ ( j ) , with a density h ′ ( j ) ∈ L p ; ν ( j ) ( ∂ Ω j , R n ) satisfying the equation (cid:0) − I + K jα (cid:1) h ′ ( j ) = h ( j ) , where K jα := K α ; ∂ Ω j is associated (as in (3.89)) with the double layer potential W α ; ∂ Ω j defined on L p ; ν ( j ) ( ∂ Ω j , R n ), and, in view of Lemma 4.4, the operator − I + K jα : L p ; ν ( j ) ( ∂ Ω j , R n ) → L p ; ν ( j ) ( ∂ Ω j , R n ) isan isomorphism for any p ∈ (1 , ∞ ).Note that the operator − I + K α : L p ; ν ( ∂ Ω , R n ) → L p ; ν ( ∂ Ω , R n ) is an isomorphism for any p ∈ R ( n, ε ) (see Lemma4.4). After performing a change of variable as in Lemma 2.2, the operator − I + K jα defined on ∂ Ω j can be identifiedwith an operator T jα acting on functions defined on ∂ Ω. Then, employing the arguments, e.g., similar to those in thelast paragraph in p.116 in [61], which are based on [61, Lemmas 11.9.13 and 11.12.2], and taking into account [47,Proposition 1] (see also [23, Theorems 3.8 (iv) and 4.15]), one can show that the sequence of operators T jα convergesto the operator T α := − I + K α in the operator norm and the sequence of the inverses of the operators T jα convergesto the inverse of the operator T α in the operator norm for p ∈ R ( n, ε ). Hence, if p ∈ R ( n, ε ), the operator norms k (cid:0) − I + K jα (cid:1) − k L p ( ∂ Ω j , R n ) are bounded uniformly in j , implying that there exists a constant c depending only on p , n , α , and the Lipschitz character of Ω + (thus, not depending on j ) such that k h ′ ( j ) k pL p ( ∂ Ω j , R n ) ≤ c k h ( j ) k pL p ( ∂ Ω j , R n ) = c k u k pL p ( ∂ Ω j , R n ) = c ˆ ∂ Ω j | u ( y j ) | p dσ y j = c ˆ ∂ Ω | u (Φ j ( y )) | p ω j ( y ) dσ y ≤ c ˆ ∂ Ω | M ( u ( y )) | p dσ y = c k M ( u ) k pL p ( ∂ Ω , R n ) , (5.21) Recall that we have approximated the domain Ω + with a sequence of smooth domains Ω j with uniform Lipschitzcharacters from inside, and we have employed here the change of variable y j := Φ j ( y ), y ∈ ∂ Ω , y j ∈ ∂ Ω j , and ω j is theJacobian of Φ j : ∂ Ω → ∂ Ω j (cf. Lemma 2.2). Hence the constants c and c depend only on p , n , α , and the Lipschitzcharacter of Ω + .Further, the double-layer potential W α ; ∂ Ω j h ′ ( j ) becomes u ℓ ( x ) = ˆ ∂ Ω j S αiℓs ( y j , x ) ν s ( y j ) h ′ ( j ) i ( y j ) dσ y j = ˆ ∂ Ω S αiℓs (Φ j ( y ) , x ) ν s (Φ j ( y )) H ′ ( j ) i ( y ) dσ y , ∀ x ∈ Ω j , (5.22)where H ′ ( j ) ( y ) := h ′ ( j ) (Φ j ( y )) ω j ( y ) , h ′ ( j ) = ( h ′ ( j )1 , . . . , h ′ ( j ) n ), H ′ ( j ) = ( H ′ ( j )1 , . . . , H ′ ( j ) n ).In view of (5.21) and of the uniform boundedness of { ω j } j ≥ , there exist some constants c , c > + and p ) such that ˆ ∂ Ω | H ′ ( j ) ( y ) | p dσ y ≤ c ˆ ∂ Ω j | u ( y j ) | p dσ y j ≤ c ′ ˆ ∂ Ω | M ( u ( y )) | p dσ y , ∀ j ≥ . (5.23)Hence { H ′ ( j ) } j ≥ is a bounded sequence in L p ( ∂ Ω , R n ), and, thus, there exists a subsequence, still denoted as thesequence, and a function H ′ ∈ L p ( ∂ Ω , R n ), such that H ′ ( j ) → H ′ weakly in L p ( ∂ Ω , R n ). By this property and letting j → ∞ in (5.22), we obtain u = W α H ′ in Ω + . According to Lemma 3.4(i,iv), there exists the non-tangential limit u +nt = ( W α H ′ ) +nt of u at almost all points of ∂ Ω, and by estimates (3.40) and (5.23), we obtain that k u +nt k L p ( ∂ Ω , R n ) = k ( W α H ′ ) +nt k L p ( ∂ Ω , R n ) ≤ c k H ′ k L p ( ∂ Ω , R n ) ≤ c lim inf j →∞ k H ′ ( j ) k L p ( ∂ Ω , R n ) ≤ c k M ( u ) k L p ( ∂ Ω) , (5.24)where the constants c , c > j . Moreover, the divergence theorem shows that u +nt = ( W α H ′ ) +nt ∈ L p ; ν ( ∂ Ω , R n ). Estimate (5.18) is provided by the representation u = W α H ′ , by continuity of operator (3.70), and byestimates (5.24). This completes the proof of item (i) for any p ∈ R ( n, ε ).Let us now consider item (i) for any p > n > ∈ R ( n, ε ) and L p ( ∂ Ω) ⊂ L ( ∂ Ω) particularly imply that for such p there exist non-tangential limits of u almost everywhere on ∂ Ω.Implementing now, e.g., [58, Proposition 3.29] completes the proof for any p > u and π satisfy the Brinkman system and that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω). As in the proofof item (i), we consider again a sequence of smooth domains { Ω j } j ∈ N , such that Ω j ⊆ Ω + and Ω j → Ω + as j → ∞ .As we already mentioned, ( u j , π j ) := ( u | Ω j , π | Ω j ) ∈ C ∞ (Ω j , R n ) × C ∞ (Ω j ). Thus, h ( j ) := u | ∂ Ω j ∈ C ∞ ( ∂ Ω j , R n ) ⊂ H p ( ∂ Ω j , R n ) and h ( j ) ∈ L p ; ν ( j ) ( ∂ Ω j , R n ), for any j ∈ N . Then the pair ( u j , π j ) ∈ C ∞ (Ω j , R n ) × C ∞ (Ω j ) satisfies theBrinkman system in Ω j with the Dirichlet boundary condition u j | ∂ Ω j = h ( j ) ∈ H p ; ν ( j ) ( ∂ Ω j , R n ). The solution of such aproblem is unique up to an additive constant pressure (see Theorem 5.1(i)) and can be expressed in terms of a doublelayer potential as in item (i), but now with a density in H p ; ν ( j ) ( ∂ Ω j , R n ). Proceeding similar to the proof of item (i),we prove item (ii). (cid:3) Remark 5.3.
The condition requiring the existence of the non-tangential limits of u , ∇ u and π at almost all pointsof the boundary ∂ Ω in Lemma 3.8 is particularly satisfied if p ∈ R ( n, ε ) ∪ (2 , ∞ ) with ε > p , the condition is implied by the inclusions M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) and by the Brinkmansystem (3.129).Having in view Theorem 5.1(iii), we are now able to consider the Poisson-Dirichlet problem for the Brinkman system, (cid:26) △ u − α u − ∇ π = f , div u = 0 in Ω + γ + u = h on ∂ Ω (5.25)with the Dirichlet datum for the Gagliardo trace γ + u (see also [61, Theorem 10.6.2] for α = 0). Theorem 5.4.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ∈ (0 , ∞ ) and < s ≤ . Then there exists ε = ε ( ∂ Ω) > such that for any p ∈ R − s ( n, ǫ ) ( cf. (4.5)) , the Dirichlet problem (5.25) with f ∈ L p (Ω + , R ) and h ∈ H sp ; ν ( ∂ Ω , R n ) has a solution ( u , π ) ∈ B s + p p,p ∗ (Ω + , R n ) × B s + p − p,p ∗ (Ω + ) , which is unique IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 33 up to an arbitrary additive constant for the pressure π , where p ∗ = max { , p } . In addition, there exists a constant C = C ( s, p, Ω + ) > such that k u k B s + 1 pp,p ∗ (Ω + , R n ) + k π k B s + 1 p − p,p ∗ (Ω + ) / R ≤ C ( k f k L p (Ω + , R n ) + k h k H sp ( ∂ Ω , R n ) ) . Proof. If f = , the existence of a solution of the problem (5.25) for 0 < s < s = 1 it follows from Theorems 5.1 (i) and 2.5 (iii).If f = , we will look for a solution of problem (5.25) in the form u = N α ;Ω + f + v , π = Q Ω + f + q, (5.26)where the Newtonian velocity and pressure potentials N α ;Ω + f and Q Ω + f are defined by (3.21). By Remark 3.3, △ N α ;Ω + f − α N α ;Ω + f − ∇Q Ω+ f = f , div N α ;Ω ± f = 0 in Ω + , ( N α ;Ω + f , Q Ω + f ) ∈ B p,p ∗ (Ω + , R n ) × B p,p ∗ (Ω + ) , γ + ( N α ;Ω + f ) ∈ H p ; ν ( ∂ Ω , R n ) , t + α (cid:0) N α ;Ω + f , Q Ω ± f (cid:1) ∈ L p ( ∂ Ω , R n ) . Then problem (5.25) reduces to the one for the corresponding homogeneous Brinkman system, (cid:26) △ v − α v − ∇ q = , div v = 0 in Ω + ,γ + v = h , (5.27)where h := h − γ + (cid:0) N α ;Ω + f (cid:1) ∈ H sp ; ν ( ∂ Ω , R n ) , already discussed in the first paragraph of the proof. Therefore,there exists a solution ( u , π ) ∈ B s + p p,p ∗ (Ω + , R n ) × B s + p − p,p ∗ (Ω + ) of the Poisson problem (5.25), which satisfies the assertedestimate.Let us prove the uniqueness of the solution to the Poisson problem (5.25) for 0 < s <
1. To do so, we consider asolution ( u , π ) ∈ B s + p p,p ∗ (Ω , R ) × B s + p − p,p ∗ (Ω) of the homogeneous version of the problem (5.25). Let us take the traceof the Green representation formula (3.128) for ( u , π ). Since γ + u = , we obtain the equation V α (cid:0) t + α ( u , π ) (cid:1) = on ∂ Ω , for t + α ( u , π ) ∈ B s − p,p ∗ ( ∂ Ω), which by Corollary 4.7 has a one-dimensional set of solutions, t + α ( u , π ) = c ν , where c ∈ R . Substituting this back into the Green representation formula (3.128) we obtain u = c V α ν = in Ω + (cf.the arguments in the proof of Lemma 4.6), and by the homogeneous Brinkman equation, π is an arbitrary constant.Finally, uniqueness for 0 < s < s = 1. (cid:3) The Neumann problem for the Brinkman system.
Using an argument similar to the one for the Robinboundary value problem for the Brinkman system in [35], we obtain in this section the well-posedness of the Neumannproblem for the linear Brinkman system, △ u − α u − ∇ π = 0 , in Ω + , div u = 0 in Ω + , t +nt ( u , π ) = g on ∂ Ω (5.28)in L p − based Bessel potential and Besov spaces for some ε >
0, and extend the results obtained in the case p = 2 andfor a conormal derivative given by ∂ u ∂n := − π ν + ∂ u ∂ ν , in [69, Theorem 5.3] (see also [61, Theorem 5.5.2] in the case α = 0). Note that the Neumann boundary condition in (5.28) is understood in the sense of non-tangential limit almosteverywhere on ∂ Ω. Theorem 5.5.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ∈ (0 , ∞ ) . Thenthere exists ǫ > , such that for any p ∈ R ( n, ǫ ) ( see (4.4)) , and for any given datum g ∈ L p ( ∂ Ω , R n ) , the Neumannproblem (5.28) has a unique solution ( u , π ) such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) . The solution can be representedby the single layer velocity and pressure potentials u = V α (cid:18) I + K ∗ α (cid:19) − g ! , π = Q s (cid:18) I + K ∗ α (cid:19) − g ! . (5.29) Moreover, ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , and there exist some constants C M , C and C ′ depending only on Ω + , α , and p such that k M ( ∇ u ) k L p ( ∂ Ω) + k M ( u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) ≤ C M k g k L p ( ∂ Ω , R n ) , (5.30) k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C k g k L p ( ∂ Ω , R n ) , (5.31) k γ + u k H p ( ∂ Ω , R n ) + k t + α ( u , π ) k L p ( ∂ Ω , R n ) ≤ C ′ k g k L p ( ∂ Ω , R n ) . (5.32) Proof.
We use an argument similar to that for [23, Theorem 4.15] (see also [62, Theorem 3.1, Proposition 3.3]). ByLemma 4.2 there exists ǫ > I + K ∗ α : L p ( ∂ Ω , R n ) → L p ( ∂ Ω , R n ) is an isomorphism for p ∈ R ( n, ǫ ).Along with Lemma 3.4, Theorem 3.5 and Lemma 3.6 this implies that representation (5.29) gives a solution of problem(5.28) that belongs to the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) and satisfies estimates (5.30)-(5.32).In order to show the uniqueness assertion, we assume that ( u , π ) is a solution of the homogeneous version of (5.28)such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) and satisfies the Neumann condition almost everywhere on ∂ Ω in the senseof non-tangential limit. Then the Green representation formula (3.130) gives, u = V α (cid:0) t +nt ( u , π ) (cid:1) − W α (cid:0) u (cid:1) = − W α (cid:0) u (cid:1) in Ω + , (5.33)which, combined with formulas (3.82), leads to the boundary integral equation (cid:18) I + K α (cid:19) u = on ∂ Ω . (5.34)Here u ∈ H p ( ∂ Ω , R n ) due to Lemma 3.4(i). Then invertibility of operator (4.9) in Lemma 4.2 implies that u = on ∂ Ω and thus, by (5.33), u = in Ω + . Moreover, by the homogeneous Neumann condition satisfied by ( u , π ), weobtain that π = 0 in Ω + . This concludes the proof of uniqueness of the solution of the Neumann problem (5.28), andhence the proof of the theorem. (cid:3) Having in view Theorem 5.5, we are now able to consider the Poisson-Neumann problem for the Brinkman system, (cid:26) △ u − α u − ∇ π = f , div u = 0 in Ω + t + α ( u , π ) = g on ∂ Ω (5.35)with the Neumann datum for the canonical conormal derivative t + α ( u , π ) (see also [62, Theorem 10.6.4] for α = 0). Theorem 5.6.
Let Ω + ⊂ R n ( n ≥ be a bounded Lipschitz domain with connected boundary ∂ Ω . Let α ∈ (0 , ∞ ) . Thenthere exists ε = ε ( ∂ Ω) > such that for any p ∈ R ( n, ǫ ) ( cf. (4.4)) , the Neumann problem (5.35) with f ∈ L p (Ω + , R ) and g ∈ L p ( ∂ Ω , R n ) has a unique solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , where p ∗ = max { , p } . In addition,there exists a constant C = C ( p, Ω + ) > such that k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C ( k f k L p (Ω + , R n ) + k g k L p ( ∂ Ω , R n ) ) , k γ + u k H p ( ∂ Ω , R n ) ≤ C ( k f k L p (Ω + , R n ) + g k L p ( ∂ Ω , R n ) ) . Moreover, if f = , then M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) and there exists a constant C M > such that k M ( u ) k L p ( ∂ Ω) + k M ( ∇ u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) ≤ C M k g k L p ( ∂ Ω , R n ) . Proof. If f = , there exists a solution of problem (5.35) given by the solution of the corresponding problem (5.28) withthe non-tangential conormal derivative in the Neumann condition, whose existence is provided by Theorem 5.5 togetherwith the asserted estimate. Here we rely also on the equivalence of the conormal derivatives, t + α ( u , π ) = t +nt ( u , π ), dueto Theorem 2.13.If f = , we will look for a solution of problem (5.35) in the form u = N α ;Ω + f + v , π = Q Ω + f + q, (5.36) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 35 where the Newtonian velocity and pressure potentials N α ;Ω + f and Q Ω + f are defined by (3.21). According to Remark 3.3,we obtain the relations △ N α ;Ω + f − α N α ;Ω + f − ∇Q Ω+ f = f , div N α ;Ω ± f = 0 in Ω + , ( N α ;Ω + f , Q Ω + f ) ∈ B p,p ∗ (Ω + , R n ) × B p,p ∗ (Ω + ) , γ + ( N α ;Ω + f ) ∈ H p ; ν ( ∂ Ω , R n ) , t + (cid:0) N α ;Ω + f , Q Ω ± f (cid:1) ∈ L p ( ∂ Ω , R n ) . Then problem (5.35) reduces to the problem for the corresponding homogeneous Brinkman system, (cid:26) △ v − α v − ∇ q = , div v = 0 in Ω + , t + α ( u , π ) = g on ∂ Ω , (5.37)where g := g − t + α (cid:0) N α ;Ω ± f ± , Q Ω ± f ± (cid:1) ∈ L p ( ∂ Ω , R n ) , already discussed in the first paragraph of the proof. Therefore,there exists a solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) of the Poisson problem (5.35), which satisfies all the assertedestimates.Let us prove uniqueness of the solution to the Poisson problem (5.35). Indeed, let us consider a solution ( u , π ) ∈ B p p,p ∗ (Ω , R ) × B p p,p ∗ (Ω) of the homogeneous version of problem (5.35). Let us take the trace of the Green representationformula (3.128) for ( u , π ), considered for any s ∈ (0 , t + α ( u , π ) = , we obtain the equation γ + u = 12 γ + u − K α γ + u on ∂ Ω , with the unknown γ + u ∈ B sp,p ∗ ( ∂ Ω , R n ), which, by Corollary 4.3, has only the trivial solution. Substituting this backto the Green representation formula (3.128) we obtain u = in Ω + . Then the Brinkman system implies π = c ∈ R ,and taking again into account that t + α ( u , π ) = , we obtain π = 0 in Ω + , as asserted. (cid:3) The mixed Dirichlet-Neumann problem for the Brinkman system
In this section we show the well-posedness of the mixed
Dirichlet-Neumann boundary value problem for the Brinkmansystem △ u − α u − ∇ π = , div u = 0 in Ω + , u +nt | S D = h , t +nt ( u , π ) | S N = g , (6.1)on a bounded, creased Lipschitz domain Ω + ⊂ R n ( n ≥
3) with connected boundary ∂ Ω, which is decomposed intotwo disjoint admissible patches S D and S N (see Definition 6.3), ·| S D is the operator of restriction from H sp ( ∂ Ω , R n ) to H sp ( S D , R n ), and ·| S N is defined similarly. We show that for h ∈ H p ( S D , R n ) and g ∈ L p ( S N , R n ) given and for somerange of p , there exists a unique solution ( u , π ) of the mixed problem (6.1), such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω),and the Dirichlet and Neumann boundary conditions in (6.1) are satisfied in the sense of non-tangential limits at almostall points of S D and S N , respectively. Moreover, we will show that ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ).We consider also a counterpart mixed problem △ u − α u − ∇ π = , div u = 0 in Ω + γ + u | S D = h , t + α ( u , π ) | S N = g , (6.2)where, unlike the mixed problem setting (6.1), the trace is considered in the Gagliardo sense and the conormal derivativein the canonical sense. We will show that for h ∈ H p ( S D , R n ) and g ∈ L p ( S N , R n ) given and for some range of p ,there exists a unique solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) of problem (6.2). Moreover, we will obtain that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω).The corresponding mixed problems for the Poisson-Brinkman system, i.e., with non-zero right hand side of theBrinkman system, will be also considered.
Creased Lipschitz domains.
Next, we recall the definition of admissible patch (cf., e.g., [57, Definition 2.1], [9]).
Definition 6.1.
Let Ω ⊂ R n ( n ≥ be a Lipschitz domain. Let S be an open set of ∂ Ω , such that for any x ∈ ∂S there exists a new orthogonal system obtained from the original one by a rigid motion with x as the origin and with theproperty that one can find a cube Q = Q × Q × · · · × Q n ⊂ R n centered at and two Lipschitz functions (cid:26) Φ : Q ′ := Q × . . . × Q n − → Q n , Φ(0) = 0 , Ψ : Q ′′ := Q × . . . × Q n − → Q , Ψ(0) = 0 , (6.3) such that S ∩ Q = { ( x ′ , Φ( x ′ )) : x ′ ∈ Q ′ , Ψ( x ′′ ) ≤ x } , (cid:0) ∂ Ω \ S (cid:1) ∩ Q = { ( x ′ , Φ( x ′ )) : x ′ ∈ Q ′ , Ψ( x ′′ ) ≥ x } , (6.4) ∂S ∩ Q = { (Ψ( x ′′ ) , x ′′ , Φ(Ψ( x ′′ ) , x ′′ )) : x ′′ ∈ Q ′′ } . Such a set S is called an admissible patch of ∂ Ω . Definition 6.1 shows that if S ⊂ ∂ Ω is an admissible patch then ∂ Ω \ S is also an admissible patch (cf., e.g., [57]).Next, we recall the definition of a creased Lipschitz graph domain (cf. [57, Definition 2.2]). Definition 6.2.
Let Ω ⊂ R n ( n ≥ be an open, connected set. Suppose that S D , S N ⊂ ∂ Ω are two non-empty, disjointadmissible patches such that S D ∩ S N = ∂S D = ∂S N and S D ∪ S N = ∂ Ω . The set Ω is a creased Lipschitz graph domainif the following conditions are satisfied: ( a ) There exists a Lipschitz function φ : R n − → R such that Ω = { ( x ′ , x n ) ∈ R n : x n > φ ( x ′ ) } . ( b ) There exists a Lipschitz function
Ψ : R n − → R such that S N = { ( x , x ” , x n ) ∈ R n : x > Ψ( x ”) } ∩ ∂ Ω , (6.5) S D = { ( x , x ” , x n ) ∈ R n : x < Ψ( x ”) } ∩ ∂ Ω . (6.6)( c ) There exist some constants δ D , δ N ≥ , δ D + δ N > with the property that ∂φ∂x ≥ δ N a.e. on S N , ∂φ∂x ≤ − δ D a.e. on S D . (6.7)Let us now refer to a creased bounded Lipschitz domain (cf. [57, Definition 2.3]). Definition 6.3.
Assume that Ω ⊂ R n is a bounded Lipschitz domain with connected boundary ∂ Ω , and that S D , S N ⊂ ∂ Ω are two non-empty, disjoint admissible patches such that S D ∩ S N = ∂S D = ∂S N and S D ∪ S N = ∂ Ω . Then Ω is creasedif ( a ) There exist m ∈ N , a > and z i ∈ ∂ Ω , i = 1 , . . . , m , such that ∂ Ω ⊂ ∪ mi =1 B a ( z i ) , where B a ( z i ) is the ball ofradius a and center at z i . ( b ) For any point z i , i = 1 , . . . , m , there exist a coordinate system { x , . . . , x n } with origin at z i and a Lipschitzfunction φ i from R n − to R such that the set Ω i := { ( x ′ , x n ) ∈ R n : x n > φ i ( x ′ ) } , whose boundary ∂ Ω i admitsthe decomposition ∂ Ω i = S D i ∪ S N i , is a creased Lipschitz graph domain in the sense of Definition . , and Ω ∩ B a ( z i ) = Ω i ∩ B a ( z i ) , S D ∩ B a ( z i ) = S D i ∩ B a ( z i ) , S N ∩ B a ( z i ) = S N i ∩ B a ( z i ) . (6.8)The geometric meaning of Definitions 6.2 and 6.3 is that S D and S N are separated by a Lipschitz interface ( S D ∩ S N is a creased collision manifold for D ) and that S D and S N meet at an angle which is strictly less than π (cf., e.g., [7, 57]).A main property of a (bounded or graph) creased Lipschitz domain is the existence of a function ϕ ∈ C ∞ (Ω , R n ) andof a constant δ > ϕ · ν > δ a.e. on S N , ϕ · ν < − δ a.e. on S D , (6.9) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 37 i.e., the scalar product ϕ · ν , between ϕ and the unit normal ν , changes the sign when moving from S D to S N (cf.,e.g., [8, (1.122)], [9, (2.2)]). For such a domain, Brown [7] showed that the mixed problem for the Laplace equation hasa unique solution whose gradient belongs to L ( ∂ D ) when the Dirichlet datum belongs to H ( S D ) and the Neumanndatum to L ( S N ). For the same class of domains, well-posedness of the mixed problem for the Laplace equation in arange of L p − based spaces has been obtained in [57].6.2. Mixed Dirichlet-Neumann problem for the Brinkman system with boundary data in L -based spaces. Mitrea and Mitrea in [57] have proved sharp well-posedness results for the Poisson problem for the Laplace operator withmixed boundary conditions of Dirichlet and Neumann type on bounded creased Lipschitz domains in R n ( n ≥ L -based spaces on creased Lipschitz domains in R n ( n ≥ Dirichlet-Robin problem for the Brinkman system in a creased Lipschitz domain with boundary data in L -based spaces has beenrecently proved in [35, Theorem 6.1]. Using the main ideas of that proof, we show in this section well-posedness of themixed Dirichlet-Neumann boundary value problem for the Brinkman system in L -based Bessel potential spaces definedon a bounded, creased Lipschitz domain Ω + . Theorem 6.4.
Assume that Ω + ⊂ R n ( n ≥ is a bounded, creased Lipschitz domain with connected boundary ∂ Ω ,which is decomposed into two disjoint admissible patches S D and S N . Then the mixed problem (6.1) with given data ( h , g ) ∈ H ( S D , R n ) × L ( S N , R n ) has a unique solution ( u , π ) such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L ( ∂ Ω) . Moreover, ( u , π ) ∈ H (Ω + , R n ) × H (Ω + ) , and there exist some constants C M and C depending only on S D , S N and α such that k M ( ∇ u ) k L ( ∂ Ω) + k M ( u ) k L ( ∂ Ω) + k M ( π ) k L ( ∂ Ω) ≤ C M (cid:16) k h k H ( S D , R n ) + k g k L ( S N , R n ) (cid:17) , (6.10) k u k H (Ω + , R n ) + k π k H (Ω + ) ≤ C (cid:16) k h k H ( S D , R n ) + k g k L ( S N , R n ) (cid:17) . (6.11) Proof.
First, we note that if a couple ( u , π ) satisfies the Brinkman system (6.1) and the conditions M ( u ) , M ( ∇ u ) , M ( π ) ∈ L ( ∂ Ω), then, taking into account that B , (Ω + , R n ) = H (Ω + , R n ), B , (Ω + ) = H (Ω + ), Theorem 5.2(ii) implies that( u , π ) ∈ H ,t , div (Ω , L α ) for any t ≥ − , while γ + u = u +nt and t + α ( u , π ) = t +nt ( u , π ) by Theorems 2.5 and 2.13.Let us show that the mixed boundary value problem (6.1) has at most one L -solution. Indeed, if a couple ( u (0) , π (0) )satisfies the homogeneous problem associated to (6.1), and moreover ( u (0) , π (0) ) ∈ H , , div (Ω , L α ), then by the first Greenidentity (2.28), we obtain the equality D t + α ( u (0) , π (0) ) , γ + u (0) E ∂ Ω = 2 D E ( u (0) ) , E ( u (0) ) E Ω + + α D u (0) , u (0) E Ω + , (6.12)where the left-hand side vanishes, due to the homogeneous boundary conditions satisfied by γ + u (0) = u (0)+nt and t + α ( u (0) , π (0) ) = t +nt ( u (0) , π (0) ) on S D and S N , respectively. Then by (6.12) we immediately obtain that u (0) = and π (0) = 0 in Ω + .Next, we consider the operator S α : L ( ∂ Ω , R n ) → H ( S D , R n ) × L ( S N , R n ) , S α Ψ := (cid:18) ( V α Ψ ) (cid:12)(cid:12) S D , (cid:18)(cid:18) I + K ∗ α (cid:19) Ψ (cid:19) (cid:12)(cid:12)(cid:12) S N (cid:19) (6.13)(cf. [35, (6.6)-(6.8)]), and show that this is an isomorphism, which will yield the well-posedness of the mixed problem(6.1). To this end, we note that S α can be written as S α = S + S α ;0 , where S : L ( ∂ Ω , R n ) → H ( S D , R n ) × L ( S N , R n ) , S Ψ := (cid:18) ( V Ψ ) (cid:12)(cid:12) S D , (cid:18)(cid:18) I + K ∗ (cid:19) Ψ (cid:19) (cid:12)(cid:12)(cid:12) S N (cid:19) , (6.14) S α ;0 : L ( ∂ Ω , R n ) → H ( S D , R n ) × L ( S N , R n ) , S α ;0 Ψ := (cid:16) ( V α ;0 Ψ) (cid:12)(cid:12) S D , (cid:0) K ∗ α ;0 Ψ (cid:1) (cid:12)(cid:12) S N (cid:17) . (6.15) Here V α ;0 : L ( ∂ Ω , R n ) → H ( ∂ Ω , R n ) and K ∗ α ;0 : L ( ∂ Ω , R n ) → L ( ∂ Ω , R n ) are the complementary layer potentialoperators defined as V α ;0 Ψ := V α Ψ − V Ψ and K ∗ α ;0 Ψ := K ∗ α Ψ − K ∗ Ψ . (6.16)The operator S defined in (6.14) is an isomorphism and this property is equivalent with the well-posedness result of themixed Dirichlet-Neumann problem for the Stokes system on a creased Lipschitz domain with Dirichlet and Neumannboundary data in L -based spaces (cf. the proof of [9, Theorem 6.3]), when the BVP solution is looked for in the form ofthe Stokes single layer potential. In addition, the continuity of the restriction operators from H ( ∂ Ω , R n ) to H ( S D , R n )and from L ( ∂ Ω , R n ) to L ( S N , R n ), respectively, as well as the compactness of the complementary operators in (6.16)(cf. [33, Theorem 3.4]) imply that the operator S α ;0 in (6.15) is compact as well. Therefore, the operator S α in(6.13) is Fredholm with index zero. This operator is also injective. Indeed, if Ψ (0) ∈ L ( ∂ Ω , R n ) satisfies the equation S α Ψ (0) = 0 then the single layer velocity and pressure potentials u (0) := V α Ψ (0) and π (0) := Q s Ψ (0) will determinea solution of the homogeneous mixed problem associated to (6.1), such that ( u (0) , π (0) ) ∈ H (Ω + , R n ) × H (Ω + )and M ( u (0) ) , M ( ∇ u (0) ) , M ( π (0) ) ∈ L ( ∂ Ω). Then u (0) = and π (0) = 0 in Ω + , as shown above. Consequently, t +nt ( u (0) , π (0) ) = a.e. on ∂ Ω, which, in view of (3.83), can be written as (cid:18) I + K ∗ α (cid:19) Ψ (0) = . Moreover, the invertibility of the operator I + K ∗ α : L ( ∂ Ω , R n ) → L ( ∂ Ω , R n ) (see Lemma 4.2) shows that Ψ (0) = .Consequently, operator (6.13) is an isomorphism, as asserted. Then the fields u = V α (cid:0) S − α ( h , g ) (cid:1) , π = Q s (cid:0) S − α ( h , g ) (cid:1) (6.17)determine the unique solution of the mixed Dirichlet-Neumann problem (6.1). According to Lemma 3.4, Theorem 3.5and (6.17), the solution belongs to the space H (Ω + , R n ) × H (Ω + ) and satisfies the estimate (6.10) with some constant C M > S D , S N and α , as well as estimate (6.11) with the constant C = ( k V α k + k Q s k ) kS − α k . (cid:3) Mixed Dirichlet-Neumann problem for the Brinkman system with data in L p -spaces. Next, we extendthe results established in Theorem 6.4, to L p -based spaces with p in some neighborhood of 2, for the mixed Dirichlet-Neumann problem for the Brinkman system (6.1), with the boundary data ( h , g ) ∈ H p ( S D , R n ) × L p ( S N , R n ). Wewill obtain the well-posedness result in the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ), where p ∗ = max { , p } .We further need the space e H p ( S , R n ) := (cid:8) Φ ∈ L p ( ∂ Ω , R n ) : supp Φ ⊆ S (cid:9) , S ⊂ ∂ Ω . (6.18) • The Neumann-to-Dirichlet operator for the Brinkman system.
As in the work [57], devoted to the mixedDirichlet-Neumann problem for the Laplace equation in a creased Lipschitz domain, we consider the Neumann-to-Dirichlet operator Υ nt; α , which associates to g ∈ L p ( ∂ Ω , R n ), the restriction of the non-tangential trace u +nt to thepatch S D , where ( u , π ) is the unique L p -solution of the Neumann problem (5.28) for the Brinkman system with thenon-tangential conormal derivative g . Thus, ( u , π ) satisfies the Neumann condition almost everywhere on ∂ Ω in thesense of non-tangential limit, as well as the conditions M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω), andΥ nt; α g = u +nt | S D . (6.19)Similarly, we consider the Neumann-to-Dirichlet operator Υ α , which associates to g ∈ L p ( ∂ Ω , R n ), the restriction of thetrace γ + u to the patch S D , where ( u , π ) is the unique solution of the Neumann problem (5.35) for the Brinkman systemwith f = and the canonical conormal derivative g , i.e.,Υ α g = γ + u | S D . (6.20)A way to extend the well-posedness result in Theorem 6.4 to L p -based spaces is to show the invertibility of the Neumann-to-Dirichlet operator Υ nt; α on such spaces. An intermediary step to obtain this property is given by the following result. IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 39
Lemma 6.5.
Let Ω + ⊂ R n ( n ≥ be a bounded, creased Lipschitz domain with connected boundary ∂ Ω which isdecomposed into two disjoint admissible patches S D and S N . Let α ∈ (0 , ∞ ) . Then there exists ε = ε ( ∂ Ω) > such thatfor any p ∈ R ( n, ε ) the following properties hold. ( i ) The operators Υ nt; α and Υ α coincide and are given by Υ nt; α =Υ α = V α ◦ (cid:18) I + K ∗ α (cid:19) − ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S D . (6.21)( ii ) The mixed Dirichlet-Neumann problem (6.1) with given data ( h , g ) ∈ H p ( S D , R n ) × L p ( S N , R n ) has a uniquesolution ( u , π ) , such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) , if and only if the operator Υ nt; α : e H p ( S D , R n ) → H p ( S D , R n ) (6.22) is an isomorphism. ( iii ) The mixed Dirichlet-Neumann problem (6.2) with given data ( h , g ) ∈ H p ( S D , R n ) × L p ( S N , R n ) has a uniquesolution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) if and only if the operator Υ α : e H p ( S D , R n ) → H p ( S D , R n ) (6.23) is an isomorphism.Moreover, when the solution ( u , π ) in item ( ii ) or ( iii ) exists, then it belongs to the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) and there exist some constants C M ≡ C M ( α, p, S D , S N ) , C ≡ C ( α, p, S D , S N ) and C ′ ≡ C ′ ( α, p, S D , S N ) such that k M ( ∇ u ) k L p ( ∂ Ω) + k M ( u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) ≤ C M (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.24) k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , p ∗ = max { , p } , (6.25) k γ + u k H p ( ∂ Ω , R n ) + k t + α ( u , π ) k L p ( ∂ Ω , R n )) ≤ C ′ (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) . (6.26) Proof. (i) By Theorem 5.5, there exists ε = ε ( ∂ Ω) > p ∈ R ( n, ε ) the Neumann problem (5.28) hasa unique solution, and it can be expressed in form (5.29). Then due to Theorem 3.5 and Lemma 3.6 we deduce that theoperator (6.19) has the expression (6.21) and is continuous, due to the continuity of both operators in the right-handside of (6.21).(ii) First, we assume that problem (6.1) is well-posed and show the invertibility of operator (6.22).In order to prove the injectivity property of this operator, we consider a function g ∈ e H p ( S D , R n ), such thatΥ nt; α g = . Denoting by ( u , π ) the unique L p -solution of the Neumann problem (5.28) for the homogeneous Brinkmansystem with boundary datum g ∈ e H p ( S D , R n ) on ∂ Ω, in view of (6.19), we have u +nt | S D = Υ nt; α g = , (6.27)and (cid:26) △ u − α u − ∇ π = , div u = 0 in Ω + , t +nt ( u , π ) = g on ∂ Ω . (6.28)In addition, ( u , π ) satisfies the conditions M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω), and the Neumann condition holds almosteverywhere on ∂ Ω in the sense of non-tangential limit.According to relation (6.27) and the inclusion g ∈ e H p ( S D , R n ), we have u | S D = on S D , t +nt ( u , π ) | S N = on S N , (6.29)and hence by the assumed well-posedness of the mixed Dirichlet-Neumann problem (6.1), we deduce that u = and π = 0 in Ω + . Thus, g = t +nt ( u , π ) = on ∂ Ω, which implies that the operator Υ α is injective. We show that the operator Υ nt; α is also surjective. Due to the assumed well posedness of the mixed Dirichlet-Neumann problem (6.1), for any Dirichlet datum h ∈ H p ( S D , R n ) on S D and the Neumann datum g ≡ on S N , thereexists a unique L p -solution, ( u , π ), of this problem. In particular, we deduce that the vector field g := t +nt ( u , π ) ∈ L p ( ∂ Ω , R n ) belongs to e H p ( S D , R n ), due to definition (6.18). In addition, the uniqueness result in Theorem 5.5 showsthat ( u , π ) is the unique solution of the Neumann problem for the Brinkman system in Ω + with the Neumann datum g ∈ e H p ( S D , R n ) ⊂ L p ( ∂ Ω , R n ). Then by definition (6.19) of the operator Υ nt; α , we obtain that Υ nt; α g = u +0 , nt | S D = h . Consequently, for a given h ∈ H p ( S D , R n ) there exists g ∈ e H p ( S D , R n ) such that Υ nt; α g = h . This shows that theoperator Υ nt; α is surjective, and thus, it is an isomorphism, as asserted.Next, we show the converse result, i.e., that the invertibility of the operator Υ nt; α implies the well-posedness of themixed Dirichlet-Neumann problem (6.1). Let us first show uniqueness of the solution to problem (6.1). To this end, weassume that ( u (0) , π (0) ) is an L p -solution of the homogeneous version of (6.1). Hence, g (0) := t +nt ( u (0) , π (0) ) ∈ e H p ( S D , R n )since t +nt ( u (0) , π (0) ) | S N = , implying that ( u (0) , π (0) ) is (by Theorem 5.5) the unique solution of the Neumann problemfor the Brinkman system with Neumann datum g (0) on ∂ Ω. Then by (6.19), Υ nt; α g (0) = u (0)+nt | S D = , and injectivityof Υ nt; α implies that g (0) = . Hence t +nt ( u (0) , π (0) ) = on ∂ Ω and Theorem 5.5 implies that u = , π = 0 in Ω + .This concludes the proof of uniqueness of the solution to the mixed problem (6.1).To show existence of an L p -solution to the mixed problem (6.1), let us consider such a problem with arbitraryboundary data ( h , g ) ∈ H p ( S D , R n ) × L p ( S N , R n ). Also let G ∈ e H p ( S N , R n ) be such that G | S N = g . (6.30)Then by Theorem 5.5 there exists a unique L p -solution ( v , q ) of the Neumann problem (5.28) with the Neumann datum G , such that there exist the non-tangential limits of u , ∇ u , π at almost all points of ∂ Ω, M ( v ) , M ( ∇ v ) , M ( q ) ∈ L ( ∂ Ω),and satisfies the Neumann boundary condition in the sense of non-tangential limit at almost all points of ∂ Ω. Note that v can be expressed in terms of a single-layer potential with a density in the space L p ( ∂ Ω , R n ), and hence v +nt ∈ H p ( ∂ Ω , R n )(see Lemma 3.6).On the other hand, the invertibility of the operator Υ nt; α : e H p ( S D , R n ) → H p ( S D , R n ) assures that the equationΥ nt; α g = (cid:0) h − v +nt | S D (cid:1) ∈ H p ( S D , R n ) (6.31)has a unique solution g ∈ e H p ( S D , R n ) ⊂ L p ( ∂ Ω , R n ). Next, let ( u , π ) be the unique L p -solution of the Neumannproblem (5.28) with the Neumann datum g . Also let( u , π ) := ( v + u , q + π ) . (6.32)Then we obtain the relations u +nt | S D = v +nt | S D + u | S D = (cid:0) h − Υ nt; α g (cid:1) + Υ nt; α g = h , (6.33) t +nt ( u , π ) | S N = t +nt ( v , q ) | S N + t +nt ( u , π ) | S N = G | S N + g | S N = g , (6.34)where the last equality follows from (6.30) and the inclusion g ∈ e H p ( S D , R n ). Moreover, the estimates (6.24) and(6.25) corresponding to item (ii) are due to (6.32) and the mapping properties of the pairs ( v , q ) and ( u , π ) given byTheorem 5.5. Consequently, the mixed Dirichlet-Neumann problem (6.1) is well-posed and estimates (6.24)-(6.26) holdtrue.The proof for item (iii) of the lemma and estimates (6.24)-(6.26) follow from similar arguments as those for item (ii),by refering to Theorems 5.4 and 5.6 instead of Theorems 5.1 and 5.5. (cid:3) By combining Theorem 6.4 and Lemma 6.5, we are now able to obtain the well-posedness results for the mixedDirichlet-Neumann problem (6.1) with boundary data in L p -based Bessel potential spaces and with p in a neighborhoodof 2, which is the main result of this section. Recall that p ∗ = max { , p } . IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 41
Theorem 6.6.
Assume that Ω + ⊂ R n ( n ≥ is a bounded, creased Lipschitz domain with connected boundary ∂ Ω which is decomposed into two disjoint admissible patches S D and S N . Then there exists a number ε > such that forany p ∈ (2 − ε, ε ) and for all given data ( h , g ) ∈ H p ( S D , R n ) × L p ( S N , R n ) the following properties hold. ( i ) The mixed Dirichlet-Neumann problem for the Brinkman system (6.1) has a unique solution ( u , π ) , such that M ( u ) , M ( ∇ u ) , M ( π ) ∈ L p ( ∂ Ω) . Moreover, ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , and there exist some constants C M ≡ C M ( α, p, S D , S N ) > , C ≡ C ( α, p, S D , S N ) > and C ′ ≡ C ′ ( α, p, S D , S N ) > such that k M ( ∇ u ) k L p ( ∂ Ω) + k M ( u ) k L p ( ∂ Ω) + k M ( π ) k L p ( ∂ Ω) ≤ C M (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.35) k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.36) k γ + u k H p ( ∂ Ω , R n ) + k t + α ( u , π ) k L p ( ∂ Ω , R n )) ≤ C ′ (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) . (6.37)( ii ) The mixed Dirichlet-Neumann problem for the Brinkman system (6.2) has a unique solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) . Moreover, the solution satisfies estimates (6.35) - (6.37) .Proof. (i) By Theorem 6.4 the mixed Dirichlet-Neumann problem (6.1) is well-posed for p = 2. Then by Lemma 6.5 (ii)and Theorem 5.5 for p = 2, the operator Υ nt; α : e H ( S D , R n ) → H ( S D , R n ) is an isomorphism. Moreover, by LemmaA.1, the sets { e H p ( S D , R n ) } p ≥ and { H p ( S D , R n ) } p ≥ are complex interpolation scales. Then by the stability of theinvertibility property given in Lemma 2.2, there exists a number ε >
0, such that the operator Υ nt; α : e H p ( S D , R n ) → H p ( S D , R n ) is an isomorphism as well, for any p ∈ (2 − ε , ε ). Finally, by choosing the parameter ε := min { ǫ, ε } > ǫ is the parameter in Theorem 5.5, and by using again Lemma 6.5 (ii), we deduce the well-posedness result of themixed Dirichlet-Neumann problem (6.1) and estimates (6.35)-(6.37), whenever p ∈ (2 − ε, ε ).(ii) Let ε be as in the proof of item (i). Let p ∈ (2 − ε, ε ). Then Lemma 6.5 (i) implies that Υ α = Υ nt; α , and henceΥ α : e H p ( S D , R n ) → H p ( S D , R n ) is an isomorphism, and by Lemma 6.5 (ii) the mixed Dirichlet-Neumann problem (6.2)is well posed and estimates (6.35)-(6.37) hold. (cid:3) Remark 6.7.
Under the conditions of Theorem . , the solution ( u , π ) of the mixed Dirichlet-Neumann problem (6.1) can be expressed by the single layer velocity and pressure potentials u = V α (cid:0) S − α ( h , g ) (cid:1) , π = Q s∂ Ω (cid:0) S − α ( h , g ) (cid:1) , (6.38) where the operator S α : L p ( ∂ Ω , R n ) → H p ( S D , R n ) × L p ( S N , R n ) , S α Ψ := (cid:18) ( V α Ψ ) (cid:12)(cid:12) S D , (cid:18)(cid:18) I + K ∗ α (cid:19) Ψ (cid:19) (cid:12)(cid:12)(cid:12) S N (cid:19) (6.39) is an isomorphism. Indeed, as shown in the proof of Theorem . , the operator S α : L ( ∂ Ω , R n ) → H ( S D , R n ) × L ( S N , R n ) is an isomorphism, and then, by using again Lemma A.1 and Lemma . , we can extend the isomorphismproperty of the operator (6.39) to L p -spaces, with p in a neighborhood of , which can be chosen to coincide with that inTheorem . . Poisson problem of mixed Dirichlet-Neumann type for the Brinkman system with data in L p -basedspaces. Having in view Theorem 6.6, we are now able to consider the well-posedness of the following Poisson problem ofmixed Dirichlet-Neumann type for the Brinkman system in a creased Lipschitz domain Ω + , with data in some L p -basedspaces, △ u − α u − ∇ π = f ∈ L p (Ω + , R ) , div u = 0 in Ω + γ + u | S D = h ∈ H p ( S D , R n ) t + α ( u , π ) | S N = g ∈ L p ( S N , R n ) . (6.40) Remark 6.8. (i) By a solution of the Poisson problem of mixed Dirichlet-Neumann type (6.40) we mean a pair( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ), where p ∗ = max { , p } , which satisfies the non-homogeneous Brinkman system inΩ + , the Dirichlet boundary condition on S D in the Gagliardo trace sense, and the Neumann boundary condition on S N in the canonical sense described in Definition 2 . u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ), p ∈ (1 , ∞ ), solves the non-homogeneous Brinkman system in thefirst line of (6.40) with f ∈ L p ( ∂ Ω , R n ), then ( u , π ) ∈ B p , p,p ∗ , div (Ω + ; L α ) by Definition 2.6. Hence, by Lemma 2.4, Defini-tion 2.10, Lemma 2.11 and the embeddings B p p,p ∗ (Ω + , R n ) ֒ → B s + p p,p ∗ (Ω + , R n ), B p , p,p ∗ , div (Ω + ; L α ) ֒ → B s + p , − p ′ p,p ∗ , div (Ω + ; L α ),for any 0 < s <
1, the trace γ + u and canonical conormal derivative t + α ( u , π ) are well defined and belong to B sp,p ∗ ( ∂ Ω , R n )and B s − p,p ∗ ( ∂ Ω , R n ), respectively. Thus, the boundary conditions in (6.40) are well defined as well. In what follows, weshow that the sharper inclusions, γ + u ∈ H p ( ∂ Ω , R n ) and t + α ( u , π ) ∈ L p ( ∂ Ω , R n ), hold if the spaces of the given boundarydata in the boundary conditions are those mentioned in (6.40). Theorem 6.9.
Assume that Ω + ⊂ R n ( n ≥ is a bounded, creased Lipschitz domain with connected boundary ∂ Ω , andthat ∂ Ω is decomposed into two disjoint admissible patches S D and S N . Then there exists a number ε > such that forany p ∈ (2 − ε, ε ) and for all given data ( f , h , g ) ∈ L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ) the Poisson problemof mixed Dirichlet-Neumann type (6.40) has a solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) that can be represented inthe form u = N α ;Ω + f + V α (cid:0) S − α ( h , g ) (cid:1) , π = Q Ω + f + Q s∂ Ω (cid:0) S − α ( h , g ) (cid:1) , (6.41) where S α : L p ( ∂ Ω , R n ) → H p ( S D , R n ) × L p ( S N , R n ) is the isomorphism defined in (6.39) , and h := h − γ + (cid:0) N α ;Ω + f (cid:1) | S D ∈ H p ( S D , R n ) , g := g − t + α (cid:0) N α ;Ω + f , Q α ;Ω + f (cid:1) | S N ∈ L p ( S N , R n ) . (6.42) Moreover, the solution ( u , π ) is unique in the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , and there exist some constants C ≡ C ( α, p, S D , S N ) > and C ′ ≡ C ′ ( α, p, S D , S N ) > such that the following inequalities hold k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C (cid:16) f k L p (Ω + , R n ) + k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.43) k γ + u k H p ( ∂ Ω , R n ) + k t + α ( u , π ) k L p ( ∂ Ω , R n ) ≤ C ′ (cid:16) f k L p (Ω + , R n ) + k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) . (6.44) In addition, there exists a linear continuous operator A p : L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ) → B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) delivering this solution, i.e., A p ( f , h , g ) = ( u , π ) .Proof. Let ε > p ∈ (2 − ε, ε ). We will look for a solution of problem (6.40) in the form u = N α ;Ω + f + v , π = Q Ω + f + q, (6.45)where the Newtonian velocity and pressure potentials N α ;Ω + f and Q Ω + f are defined by (3.21). By properties (3.23)-(3.26), Corollary 3.2 and Remark 3.3, we obtain that △ N α ;Ω + f − α N α ;Ω + f − ∇Q Ω + f = f , div N α ;Ω ± f = 0 in Ω + , (6.46)( N α ;Ω + f , Q Ω + f ) ∈ H p (Ω + , R n ) × H p (Ω + ) ֒ → B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , (6.47) γ + N α ;Ω + f ∈ H p ( ∂ Ω , R n ) , t + α ( N α ;Ω + f , Q Ω + f ) ∈ L p ( ∂ Ω , R n ) , (6.48) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 43 where γ + is the Gagliardo trace operator from H p (Ω + , R n ) to H p ( ∂ Ω , R n ). Then the mixed Poisson problem (6.40)reduces to the mixed problem for the corresponding homogeneous system, △ v − α v − ∇ q = , div v = 0 in Ω + ,γ + v | S D = h ∈ H p ( S D , R n ) , t + α ( v , q ) | S N = g ∈ L p ( S N , R n ) , (6.49)where h ∈ H p ( S D , R n ) and g ∈ L p ( S N , R n ) are given by (6.42), and these inclusions follow from (6.47).By Theorem 6.6(ii), there exists a unique solution ( v , q ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) of problem (6.49), and itsatisfies the following estimates k v k B
1+ 1 pp,p ∗ (Ω + , R n ) + k q k B pp,p ∗ (Ω + ) ≤ c (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.50) k γ + v k H p ( ∂ Ω , R n ) + k t + α ( v , q ) k L p ( ∂ Ω , R n )) ≤ c ′ (cid:16) k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (6.51)with some constants c ≡ c ( α, p, S D , S N ) > c ′ ≡ c ′ ( α, p, S D , S N ) > v = V α (cid:0) S − α ( h , g ) (cid:1) , q = Q s∂ Ω (cid:0) S − α ( h , g ) (cid:1) , (6.52)where S α : L p ( ∂ Ω , R n ) → H p ( S D , R n ) × L p ( S N , R n ) is the isomorphism defined by (6.39), determine the unique solutionof problem (6.49). Moreover, in view of Theorem 3.5 (i) and Lemma 3.6, the pair ( v , q ) given by (6.52) belongs indeedto the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ),Therefore, there exists a solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) of the mixed Poisson problem (6.40), whichis given by representation (6.41) and satisfies estimates (6.43) and (6.44). The uniquness result of such a solutionfollows from Theorem 6.6 (ii). Moreover, linearity and continuity of the Newtonian potential operators (3.25), (3.26)and estimate (6.50) imply the continuity of the operator A p delivering such a solution. (cid:3) Mixed Dirichlet-Neumann problem for the semilinear Darcy-Forchheimer-Brinkman system inBesov spaces
Next we consider the mixed Dirichlet-Neumann problem for the semilinear Darcy-Forchheimer-Brinkman system △ u − α u − β | u | u − ∇ π = f , div u = 0 in Ω + . (7.1)Such a nonlinear system describes flows in porous media saturated with viscous incompressible fluids (see, e.g., [65,p.17]), and the constants α, β > n = 3.A numerical study of a mixed Dirichlet-Neumann problem for system (7.1) in the particular case of a two-dimensionalsquare cavity driven by a moving wall has been obtained in [26]. Well-posedness and numerical results for an extendednonlinear system, called the Darcy-Forchheimer-Brinkman system, where both semilinear and nonlinear terms | u | u and( u ·∇ ) u are involved, have been obtained in [25], and boundary value problems of Robin type for the Darcy-Forchheimer-Brinkman system with data in L -based Bessel potential (Sobolev) spaces have been studied in [34, 35].In what follows, we extend an existence and uniqueness result obtained in [35, Theorem 7.1] for the mixed problem(7.3) with the given data in L -based Sobolev spaces, to the case of L p -based Bessel potential spaces, i.e., when thegiven boundary data ( h , g ) belong to the space H p ( S D , R n ) × L p ( S N , R n ), with p ∈ (2 − ε, ε ), and the parameter ε > Theorem 7.1.
Assume that Ω + ⊂ R is a bounded creased Lipschitz domain with connected boundary ∂ Ω , and that ∂ Ω is decomposed into two disjoint admissible patches S D and S N . Let α, β > be given constants. Then there exists anumber ε > such that for any p ∈ (2 − ε, ε ) and p ∗ = max { , p } , there exist two constants ζ p ≡ ζ p (Ω + , α, β, p ) > and η p ≡ η p (Ω + , α, β, p ) > with the property that for all given data ( f , h , g ) ∈ L p (Ω + , R ) × H p ( S D , R ) × L p ( S N , R ) satisfying the condition k h k H p ( S D , R ) + k g k L p ( S N , R ) + k f k L p (Ω + , R ) ≤ ζ p , (7.2) the mixed Dirichlet-Neumann problem for the semilinear Darcy-Forchheimer-Brinkman system △ u − α u − β | u | u − ∇ π = f , div u = 0 in Ω + ,γ + u | S D = h on S D t + α ( u , π ) | S N = g on S N (7.3) has a unique solution ( u , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , which satisfies the inequality k u k B
1+ 1 pp,p ∗ (Ω + , R n ) ≤ η p . (7.4) Moreover, γ + u ∈ H p ( ∂ Ω , R n ) , t + α ( u , π ) ∈ L p ( ∂ Ω , R n ) and the solution depends continuously on the given data, whichmeans that there exists some constants C ∗ ≡ C ∗ (Ω + , α, β, p ) > and C ′∗ ≡ C ∗ (Ω + , α, β, p ) > such that k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C ∗ (cid:16) k f k L p (Ω + , R n ) + k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) , (7.5) k γ + u k H p ( ∂ Ω , R n ) + k t + α ( u , π ) k L p ( ∂ Ω , R n )) ≤ C ′∗ (cid:16) k f k L p (Ω + , R n ) + k h k H p ( S D , R n ) + k g k L p ( S N , R n ) (cid:17) . (7.6) Proof.
We use the arguments similar to those in the proof of [32, Theorem 5.2] devoted to transmission problems withLipschitz interface in R n for the Stokes and Darcy-Forchheimer-Brinkman systems in L − based Sobolev spaces.According to (A.7) and the second formula in (A.8), for n ≤ p > /
2, we obtain the following continuousembeddings, B p p,p ∗ (Ω + , R n ) ֒ → B p, min { p, (2 p ) ′ } (Ω + , R n ) ֒ → H p (Ω + , R n ) = L p (Ω + , R n ) . (7.7)Now, by (7.7) and the H¨older inequality we obtain the estimates k | v | w k L p (Ω + , R n ) ≤ k v k L p (Ω + , R n ) k w k L p (Ω + , R n ) ≤ c ′ k v k B
1+ 1 pp,p ∗ (Ω + , R n ) k w k B
1+ 1 pp,p ∗ (Ω + , R n ) , ∀ v , w ∈ B p p,p ∗ (Ω + , R n ) , (7.8)with some constants c ′ k ≡ c ′ k (Ω + , p ) > k = 0 ,
1, implying that | v | w ∈ L p (Ω + , R n ) , ∀ v , w ∈ B p p,p ∗ (Ω + , R n ) . Next, for a given fixed v ∈ B p p,p (Ω + , R n ), we consider the linear Poisson problem of mixed type for the Brinkmansystem △ v − α v − ∇ π = f + β | v | v in Ω + ,γ + v | S D = h ∈ H p ( S D , R n ) , t + α (cid:0) v , π (cid:1) | S N = g ∈ L p ( S N , R n ) , (7.9)with the unknown fields ( v , π ) ∈ B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ).Let 2 − ε < p < ε , where ε > − ε > . Then by Theorem 6.9, problem(7.9) with given data ( f + β | v | v , h , g ) ∈ L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ) has a unique solution (cid:0) v , π (cid:1) := ( U ( v ) , P ( v )) = A p ( f + β | v | v , h , g ) ∈ X p , (7.10)where the linear and continuous operator A p : Y p → X p has been defined in Theorem 6.9, and X p := B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ) , Y p := L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ) . (7.11)Hence, for fixed data ( f , h , g ) ∈ L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ), the nonlinear operators( U , P ) : B p p,p ∗ (Ω + , R n ) → X p (7.12) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 45 defined in (7.10), are continuous and bounded, we obtain, (cid:13)(cid:13)(cid:0) U ( w ) , P ( w ) (cid:1)(cid:13)(cid:13) X p ≤ C k ( f + β | w | w , h , g ) k Y p ≤ C (cid:16) k ( f , h , g ) k L p (Ω + , R n ) × H p ( S D , R n ) × L p ( S N , R n ) + β k | w | w k L p (Ω + , R n ) (cid:17) ≤ C k ( f , h , g ) k Y p + CC k w k B
1+ 1 pp,p ∗ (Ω + , R n ) , ∀ w ∈ B p p,p ∗ (Ω + , R n ) , (7.13) k γ + U ( w ) k H p ( ∂ Ω , R n ) + k t + α (cid:0) U ( w ) , P ( w ) (cid:1) k L p ( ∂ Ω , R n )) ≤ C ′ k ( f , h , g ) k Y p + C ′ C k w k B
1+ 1 pp,p ∗ (Ω + , R n ) . (7.14)where C := c ′ β >
0, and c ′ ≡ c ′ (Ω + , p ) > C can be taken as C = kA p k L ( Y p , X p ) . In addition, in view of (7.10) and due to the definition of A p , we obtain that (cid:0) v , π (cid:1) = ( U ( v ) , P ( v ))and satisfy (7.9). Therefore, if we show that the nonlinear operator U has a fixed point u ∈ B p p,p ∗ (Ω + , R n ), i.e., such that U ( u ) = u , then u together with the pressure function π = P ( u ) determine a solution of the nonlinear mixed problem(7.3) in the space X p . In order to show the existence of such a fixed point, we introduce the constants ζ p := 316 C C > , η p := 14 C C > B η p := ( w ∈ B p p,p ∗ (Ω + , R n ) : k w k B
1+ 1 pp,p ∗ (Ω + , R n ) ≤ η p ) , (7.16)and assume that the given data satisfy the inequality k ( f , h , g ) k Y p ≤ ζ p . (7.17)Then by (7.13), (7.15)-(7.17) we deduce that k ( U ( w ) , P ( v )) k X p ≤ C C = η p , ∀ w ∈ B η p . (7.18)Consequently, U maps B η p into B η p .Moreover, we now prove that U is a contraction on B η p . Indeed, by using the expression of U given in (7.10), thelinearity and continuity of the operator A p , and inequality (7.8), we obtain that kU ( v ) − U ( w ) k B
1+ 1 pp,p ∗ (Ω + , R n ) ≤ kA p ( β | v | v − β | w | w , , ) k B
1+ 1 pp,p ∗ (Ω + , R n ) ≤ Cβ k | v | v − | w | w k L p (Ω + , R n ) = Cβ k ( | v | − | w | ) v + | w | ( v − w ) k L p (Ω + , R n ) ≤ Cc ′ β (cid:16) k v k B
1+ 1 pp,p ∗ (Ω + , R n ) + k w k B
1+ 1 pp,p ∗ (Ω + , R n ) (cid:17) k v − w k B
1+ 1 pp,p ∗ (Ω + , R n ) ≤ η p CC k v − w k B
1+ 1 pp,p ∗ (Ω + , R n ) = 12 k v − w k B
1+ 1 pp,p ∗ (Ω + , R n ) , ∀ v , w ∈ B η p , (7.19)see also (7.13). Then the Banach-Caccioppoli fixed point theorem implies that there exists a unique fixed point u ∈ B η p of U , i.e., U ( u ) = u . Moreover, u and the pressure function π = P ( u ), given by (7.10), determine a solution of thesemilinear problem (7.3) in the space B p p,p ∗ (Ω + , R n ) × B p p,p ∗ (Ω + ). In addition, since the solution satisfies the condition u ∈ B η , by inequality (7.13) we obtain the estimate k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C k ( f , h , g ) k Y p + 14 k u k B
1+ 1 pp,p ∗ (Ω + , R n ) , (7.20) implying that k u k B
1+ 1 pp,p ∗ (Ω + , R n ) + k π k B pp,p ∗ (Ω + ) ≤ C k ( f , h , g ) k Y p , (7.21)which is just the inequality (7.5) with the constant C ∗ = 43 C = 43 kA − p k L ( Y p , X p ) . Similarly, (7.14) and (7.21) lead to(7.6) with the constant C ′∗ = 43 C ′ .Next, we prove the uniqueness of the semilinear mixed problem (7.3) solution ( u , π ) ∈ X p , that satisfies inequality(7.4), when the given data satisfy conditions (7.2). Assume that ( u ′ , π ′ ) ∈ X p is another solution of problem (7.3), whichsatisfies inequality (7.4), implying u ′ ∈ B η p . Then U ( u ′ ) ∈ B η p , where ( U ( u ′ ) , P ( u ′ )) are given by (7.10) and satisfy(7.9) with v replaced by u ′ . Then by (7.3) and (7.21) (both written in terms of ( u ′ , π ′ )) we obtain the linear mixedDirichlet-Neumann problem △ ( U ( u ′ ) − u ′ ) − α ( U ( u ′ ) − u ′ ) − ∇ ( P ( u ′ ) − π ′ ) = in Ω + , ( γ + ( U ( u ′ ) − u ′ )) | S D = on S D , ( t + α ( U ( u ′ ) − u ′ , P ( u ′ ) − π ′ )) | S N = on S N , (7.22)and γ + ( U ( u ′ ) − u ′ ) ∈ H p ( ∂ Ω + , R n ) , t + α ( U ( u ′ ) − u ′ , P ( u ′ ) − π ′ ) ∈ L p ( ∂ Ω + , R n ). This problem has only the trivial solu-tion in the space X p (see Theorem 6.9), i.e., U ( u ′ ) = u ′ , P ( u ′ ) = π ′ . Thus, u ′ is a fixed point of U . Since U : B η p → B η p is a contraction, it has a unique fixed point in B η p , which has been already denoted by u . Consequently, u ′ = u , and,in addition, π ′ = π . (cid:3) AppendicesAppendix A. Besov spaces in R n Let µ = ( µ , . . . , µ n ) be an arbitrary multi-index in Z n + of length | µ | := µ + · · · + µ n , and let ∂ µ := ∂ | µ | ∂x µ · · · ∂x µ n n . Next we recall the definition of Besov spaces in R n (cf., e.g., [61, Section 11.1]). By Ξ one denotes the collection of allsets { ξ j } ∞ j =0 of Schwartz functions with the following property:(i) There are some constants b, c, d > ξ ) ⊂ { x : | x | ≤ b } , supp( ξ j ) ⊂ { x : 2 j − c ≤ | x | ≤ j +1 d } , j = 1 , , . . . (A.1)(ii) Let µ be an arbitrary multi-index in R n . Then there exists a constant c ∂ Ω > x ∈ R n sup j ∈ N j | µ | | ∂ µ ξ j ( x ) | ≤ c ∂ Ω . (A.2)(iii) The following equality holds ∞ X j =0 ξ j ( x ) = 1 , ∀ x ∈ R n . (A.3)Let s ∈ R , p, q ∈ (0 , ∞ ). Then for a sequence { ξ j } ∞ j =0 ⊂ Ξ, the Besov space B sp,q ( R n ) is defined by B sp,q ( R n ) := f ∈ S ′ ( R n ) : k f k B sp,q ( R n ) := (cid:16) ∞ X j =1 k sj F − ( ξ j F f ) k qL p ( R n ) (cid:17) q < ∞ , (A.4)where f is the Fourier transform and S ′ ( R n ) denotes the space of tempered distributions in R n . Note that the abovedefinition of the Besov space B sp,q ( R n ) is independent of the choice of the set { ξ j } ∞ j =0 ⊂ Ξ, which means that anothersequence in Ξ leads to the same space with an equivalent norm. In particular, for any s ∈ R , the Besov space B s , ( R n ) IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 47 coincides with the Sobolev space H s ( R n ), i.e., B s , ( R n ) = H s ( R n ). Moreover, denoting by W sp ( R n ) the Sobolev-Slobodeckij spaces (defined in the classical way through their norms), we have the relations (see, e.g., [72], [5]) W sp ( R n ) = B sp,p ( R n ) , s ∈ R \ Z , (A.5) W kp ( R n ) = H kp ( R n ) , k ∈ Z . (A.6)Let s , s ∈ R , 1 < p ≤ p < ∞ be such that s − np < s − np , and 0 < q , q ≤ ∞ . Then the embedding B s p ,q ( R n ) ֒ → B s p ,q ( R n ) (A.7)is continuous (cf. [72, Theorem in Section 2.7.1 and Proposition 2(ii) in Section 2.3.2], [66, Remark 2 in Section 2.2.3]).Note that R n in (A.7) can be replaced by a domain Ω ∈ R n .Let us also recall the following useful inclusions between Besov spaces and Bessel potential spaces. Assume that1 ≤ q ≤ q ≤ ∞ , 1 ≤ p, q ≤ ∞ and s < s < s . Let p ′ denote the conjugate exponent of p , i.e., p ′ = 1 − p . Then wehave the following continuous embeddings, B sp,q ( R n ) ֒ → B sp,q ( R n ) , B sp, min { p,p ′ } ( R n ) ֒ → H sp ( R n ) ֒ → B sp, max { p,p ′ } ( R n ) , (A.8) B s , ( R n ) = H s ( R n ) , B s p, ∞ ( R n ) ֒ → H sp ( R n ) ֒ → B s p, ( R n ) , (A.9)(cf., e.g., [3, Chapter 6], [71, (3.2)], [62, (4.19)]), which imply the continuity of the embedding B s p,q ( R n ) ֒ → B s p,q ( R n ) . (A.10)These embeddings hold also when R n is replaced by a bounded Lipschitz domain (see [3, Chapter 6], [73, (8)]).The scales of Bessel potential and Besov spaces can be obtained by the method of complex interpolation. Indeed,if s , s ∈ R , s = s , p , p ∈ (1 , + ∞ ), q , q ∈ (1 , + ∞ ) and θ ∈ (0 , (cid:2) H s p ( R n ) , H s p ( R n ) (cid:3) θ = H sp ( R n ) , (cid:2) B s p ,q ( R n ) , B s p ,q ( R n ) (cid:3) θ = B sp,q ( R n ) , (A.11)where s = (1 − θ ) s + θs , p = − θp + θp and q = − θq + θq .Moreover, the scale of Besov spaces can be also obtained by using the method of real interpolation of Sobolev spaces.Indeed, for p, q ∈ (1 , + ∞ ), s = s , and θ ∈ (0 , (cid:0) H s p ( R n ) , H s p ( R n ) (cid:1) θ,q = B sp,q ( R n , R n ) , (A.12)where s = (1 − θ ) s + θs (cf., e.g., [1, Theorem 14.1.5], [24, p. 329], [29], [57, (5.38)], [72], [5, Theorem 3.1]).Formulas (A.11) and (A.12) remain true if R n is replaced by a Lipschitz domain (cf., e.g., [5, Theorem 3.2, Remark3.3]).For the following property we refer the reader to, e.g., [57, relations (3.11) and Proposition 4.2]. Lemma A.1.
Let Ω ⊂ R n be a bounded Lipschitz domain. Let S ⊂ ∂ Ω be an admissible patch. If p , p ∈ (1 , ∞ ) , s , s ∈ [0 , or s , s ∈ [ − , , and θ ∈ (0 , , then the following complex and real interpolation properties hold [ H s p ( ∂ Ω) , H s p ( ∂ Ω)] θ = H sp ( ∂ Ω) , [ H s p ( S ) , H s p ( S )] θ = H sp ( S ) , [ e H s p ( S ) , e H s p ( S )] θ = e H sp ( S ) , (A.13)( H s p ( ∂ Ω) , H s p ( ∂ Ω)) θ,q = B sp,q ( ∂ Ω) , ( H s p ( S ) , H s p ( S )) θ,q = B sp,q ( S ) , [ e H s p ( S ) , e H s p ( S )] θ,q = e B sp,q ( S ) , (A.14) where p = − θp + θp and s = (1 − θ ) s + θs . In (A.14) also s = s and q ∈ (1 , ∞ ] . Appendix B. Some general assertions on interpolation theory and continuous operators
Let us consider two compatible couples of Banach spaces, X , X and Y , Y . Let X θ and Y θ be interpolation spaceswith respect to X , X and Y , Y , according to [3, Definition 2.4.1]. If A j : X j → Y j , j = 0 , A | X ∩ X = A | X ∩ X ) then they induce the operator A + : X + X → Y + Y , such that A + x := A x + A x , for any x ∈ X + X , where x = x + x , x j ∈ X j , and k A + k ≤ max( k A k , k A k ), cf. [3, Section X θ ⊂ X + X and the operator A θ := A + | X θ is linear and continuous. In the following assertionwe consider some cases when the interpolation preserves isomorphism properties of operators. Lemma B.1.
Let X , X and Y , Y be two compatible couples of Banach spaces. Let X θ and Y θ be interpolation spaceswith respect to X , X and Y , Y . Let A j : X j → Y j , j = 0 , , be linear continuous compatible operators that areisomorphisms. Let A θ : X θ → Y θ be the operator induced by A j . ( i ) If the operators R j : Y j → X j , inverse to the operators A j : X j → Y j , j = 0 , , respectively, are compatible (i.e., R | Y ∩ Y = R | Y ∩ Y ), then A θ : X θ → Y θ is an isomorphism. ( ii ) If X ⊂ X , then A θ : X θ → Y θ is an isomorphism. ( iii ) If there exist linear subspaces X ∗ ⊂ X ∩ X and Y ∗ ⊂ Y ∩ Y such that Y ∗ is dense in Y ∩ Y and the operator A ∗ := A | X ∗ = A | X ∗ : X ∗ → Y ∗ is an isomorphism, then A θ : X θ → Y θ is an isomorphism.Proof. Let us prove item (i). Since the inverse operators R j : Y j → X j are compatible, they induce a continuousoperator R + : Y + Y → X + X , such that R + y := R y + R y , for any y ∈ Y + Y , where y = y + y , y j ∈ Y j , andcontinuous operator R θ = R + | Y θ : Y θ → X θ . Let us show that the operator R θ is inverse to A θ . Indeed, any x ∈ X θ canbe represented as x = x + x , where x j ∈ X j , and then R θ A θ x = R + A + x = R + A + ( x + x ) = R + ( A x + A x ) = R A x + R A x = x + x = x. Similarly, any y ∈ Y θ can be represented as y = y + y , where y j ∈ Y j , and then A θ R θ y = A + R + y = A + R + ( y + y ) = A + ( R y + R y ) = A R y + A R y = y + y = y. This proves that R θ : Y θ → X θ is the operator inverse to A θ : X θ → Y θ and hence the latter one is an isomorphism.To prove item (ii) we remark that the inclusion X ⊂ X , the compatibility of the operators A j : X j → Y j , j = 0 ,
1, andthe invertibility of the operator A : X → Y imply that Y ⊂ Y . Then the invertibility of the operator A : X → Y implies R | Y = R , i.e., the compatibility of the inverse operators to the operators A j : X j → Y j , j = 0 ,
1, whichreduces item (ii) to item (i).Let us prove item (iii). If A j : X j → Y j , j = 0 ,
1, are isomorphisms then there exist continuous inverse operators R j : Y j → X j , j = 0 ,
1. Let us prove that R j are compatible operators. Let R ∗ : Y ∗ → X ∗ be the inverse to the operator A ∗ := A | X ∗ = A | X ∗ : X ∗ → Y ∗ . Then for any ψ ∈ Y ∗ , there exists φ ∈ X ∗ such that ψ = A ∗ φ = A φ = A φ . Hence R ∗ ψ = φ = R ψ = R ψ , i.e., R ∗ = R | Y ∗ = R | Y ∗ .Due to the density of Y ∗ in Y ∩ Y , for any y ∈ Y ∩ Y there exists a sequence { ψ i } ∞ i =1 ⊂ Y ∗ converging to y in Y ∩ Y and hence in Y and in Y . Then R ∗ ψ i ∈ X ∗ ⊂ X ∪ X and due to continuity of the operators R j : Y j → X j , j = 0 , i →∞ R ∗ ψ i = lim i →∞ R j ψ i = R j y in X j for j = 0 ,
1, which implies R | Y ∩ Y = R | Y ∩ Y , i.e., the inverse operators R j : Y j → X j , j = 0 , (cid:3) Note that item (iii) of Lemma B.1 is available in [24, Lemma 8.4] for the cases, when the image and domain spacescoincide, i.e, X j = Y j , under the additional assumptions that X ∗ = Y ∗ is a Banach space.Let us give the following useful result in the complex interpolation theory (cf., e.g., [12, Theorem 2.7, Corollary 2.8]and the references therein, see also [44, Appendix B]). Lemma 2.2.
Let X , X and Y , Y be two compatible couples of Banach spaces and A j : X j → Y j , j = 0 , , be twocontinuous compatible linear operators. Let X θ := [ X , X ] θ and Y θ := [ Y , Y ] θ denote the complex interpolation spacesof X , X and Y , Y , respectively, for each θ ∈ (0 , . If there exists a number θ ∈ (0 , such that A θ : X θ → Y θ is an isomorphism, then there exists ε > such that the operator A θ : X θ → Y θ is an isomorphism as well, for any θ ∈ ( θ − ε, θ + ε ) . Remark 2.3.
The extension of Lemma 2.2 to the case of two compatible couples of quasi-Banach spaces, X , X and Y , Y , such that X + X and Y + Y are analytically convex can be found in [61, Theorem 11.9.24] and the referencestherein. Note that any Banach space is analytically convex ( cf., e.g., [61, p. 223] ) . Finally, let us mention the following useful result (cf, e.g., [61, Lemma 11.9.21]).
IXED PROBLEM FOR DARCY-FORCHHEIMER-BRINKMAN PDE SYSTEM 49
Lemma 2.4.
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