A Nonhomogeneous Boundary Value Problem for Steady State Navier-Stokes Equations in a Multiply-Connected Cusp Domain

: The boundary value problem for the steady Navier–Stokes system is considered in a 2 D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O . The case of a boundary value with nonzero ﬂow rates over connected components of the boundary is studied. It is also supposed that there is a source/sink in O . In this case the solution necessarily has an inﬁnite Dirichlet integral. The existence of a solution to this problem is proved assuming that the ﬂow rates are “sufﬁciently small”. This condition does not require the norm of the boundary data to be small. The solution is constructed as the sum of a function with the ﬁnite Dirichlet integral and a singular part coinciding with the asymptotic decomposition near the cusp point.

We assume that the support of a is separated from the cusp point O, i.e., supp a ⊂ Γ ∩ ∂Ω 0 ∪ ∪ N j=1 Γ j .
be the flow rates of the boundary value a over the outer boundary Γ ∩ ∂Ω 0 and the inner boundaries Γ j , j = 1, . . . , N, where n denotes the unit vector of the outward normal to ∂Ω. By the incompressibility of the fluid it follows that where σ(R) = (−ϕ(R), ϕ(R)) is a cross section of G by the straight line x 2 = R parallel to the x 1 -axis. We assume that the total flux may be nonzero, i.e., F 0 + N ∑ j=1 F j = 0. This nonzero condition means that there is a source or sink in the cusp point O. Then, due to the geometry of the domain, the velocity vector field u necessarily has infinite Dirichlet integral Ω |∇u(x)| 2 dx = ∞ (see, e.g., [1]).
The point source/sink approach is widely used in physics, astronomy and in fluid and aerodynamics. The behaviour of solutions to the Stokes and Navier-Stokes equations in singularly perturbed domains became of growing interest during the last fifty years. There is an extensive literature concerning these issues for various elliptic problems, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In particular, the steady Navier-Stokes equations are studied in a punctured domain Ω = Ω 0 \ {O} with O ∈ Ω 0 assuming that the point O is a sink or source of the fluid [19][20][21] (see also [22] for the review of these results). We also mention the papers [23][24][25] where the existence of a solution (with an infinite Dirichlet integral) to the Navier-Stokes problem with a sink or source in the cusp point O was proved for arbitrary data and the papers [26][27][28] where the asymptotics of a solution to the nonstationary Stokes problem is studied in domains with conical points and conical outlets to infinity.
The existence of singular solutions to the time-periodic and initial boundary value problems for the linear Stokes and the nonlinear Navier-Stokes equations in domains with a cusp point on the boundary were studied in recent papers [29][30][31][32], where the case with a sink/source in the cusp point O was considered. In [23], the existence of a generic stationary solution with infinite Dirichlet integral was proved. However, the behaviour of the solution near the cusp point was not found. The asymptotic decomposition near the cusp point of the solution u to problem (1) was constructed and the existence of a unique solution which is represented as a sum of this decomposition and a vector field belonging to a suitable second order weighted Sobolev space is proved in [1]. In [1], it is assumed that a ∈ W 3/2,2 (∂Ω) and the results are obtained under the condition that the norm a W 3/2,2 (∂Ω) is sufficiently small.
In this paper we extend the results of [1] in two directions: first, we study the case of domains with multiply-connected boundaries and, second, we prove the existence of the solution coinciding near the cusp point with the formal asymptotic decomposition assuming only that the flow rates F 0 , F 1 , . . . , F N of the boundary value a are sufficiently small. The proof is based on the construction of an extension of the boundary value which coincides near the cusp point with the asymptotic decomposition and allows to obtain needed a priori estimates assuming only that flow rates are sufficiently small. Note that in this case the norm of a is not obliged to be small. It is worth to mention the papers [33][34][35] where the nonhomogeneous boundary value problem for the stationary Navier-Stokes equations was studied in bounded domains with multiply-connected boundaries having C 2 -regularity.

Function Spaces
We will use the letter "c" for a generic constant which numerical value or dependence on parameters is unessential to our considerations; "c" may have different values in a single computation. Vector valued functions are denoted by bold letters while function spaces for scalar and vector valued functions are denoted in the same way.
Let D be a bounded domain in R n with Lipschitz boundary. C ∞ (D) denotes the set of all infinitely differentiable in D functions and C ∞ 0 (D) is the subset of all functions from C ∞ (D) with compact supports in D. For given non-negative integers k and q > 1, L q (D) and W k,q (D) denote the usual Lebesgue and Sobolev spaces; Lemma 1 ([36,37], Chapter 1, Lemma 1). Let D ⊂ R 2 be a bounded domain. If u ∈ W 1,2 (D), then the following estimate holds. Moreover, if u ∈W 1,2 (D) then Consider the domain Ω with a cusp point. We introduce a family of subdomains Ω k ⊂ Ω with Lipschitz boundaries: and L is the Lipschitz constant for the function ϕ. We write u ∈ W l,q the integral G |ϕ(x 2 )| 2α−2 |u(x)| 2 dx is finite and the following inequality The proof of this lemma can be found in [32] (see Lemma 2.1).

Formal Asymptotic Decomposition
The formal asymptotic decomposition U [J] , P [J] of the solution (u, p) of problem (1) near the cusp point O was constructed in [1]. It has the form where The asymptotic decomposition U [J] with

Flux Carrier from Inner Boundaries
In this subsection we construct a solenoidal vector function having the flow rates F j on inner components of the boundaries Γ j , j = 1, . . . , N. We call such a function the flux carrier. The construction used below is based on ideas proposed by H. Fujita in [38] for the case of symmetric domains. In [38] such functions are called virtual drains.
First we define some auxiliary functions. Let κ ∈ (0, 1/2) be a parameter. We introduce non-negative, even functions β κ (t) ∈ C ∞ 0 (−∞; +∞) such that Choose one of the domains D j , j = 1, . . . , N, and take two points X j ∈ D j and X 0 j ∈ Γ ∩ ∂Ω 0 such that the line X j X 0 j intersects Γ j and Γ only at one point and, if X j X 0 such that the origin of this coordinate system coincides with the point X j and z 0 > 0 and X j = (0, 0). Let us take a small number µ 0 > 0 and define the strip: where we choose a small number δ j so that the segments {z (j) : z intersect Γ j and Γ only at one point and if intersect other boundaries, then -at even number of points.
In Υ (j) ∩ Ω we define a vector field: Notice that b j (z (j) ) defined on Υ (j) ∩ Ω can be extended by zero into the whole domain Ω, because the bottom of Υ (j) is outside the domain Ω. For the sake of simplicity we keep the same notation for this extension, i.e., in the whole domain Ω we have: We shall show that Let us introduce the domain is a simple connected set (see Figure 2). Since, due to the construction, b j (z (j) ) is solenoidal and b j (z (j) ) z (j) where the vector field n denotes the unit outward normal to ∂Ω on Γ j , while the vector e 2 denotes the unit normal to (10), from the last equality we get (11). Notice that for the case i = j, when Υ (j) does not intersect or touch Γ i , the vector field b j vanishes on Γ i (by construction). Otherwise, if Υ (j) intersects Γ i at even number of points, then flow rates of b j across Γ i are equal to zero: the flow rates of b j over not intersecting parts of Υ (j) ∩ Γ i cancel each other. In order to rewrite vector field b j (z (j) ) in global coordinates let us take the orthogonal . Therefore, the flux carrier from the inner boundaries has the form: Lemma 3. The vector field b (inn) is smooth and solenoidal. Moreover, suppb (inn) ⊂ Ω 0 , and the following estimate holds.

Flux Carrier from the Outer Boundary
The boundary condition u = a is prescribed on Γ ∩ ∂Ω 0 ∪ ∪ N j=1 Γ j . After subtracting the constructed flux carrier b (inn) , which "removes" the fluxes F j from the inner boundaries Γ j , j = 1, . . . , N, we get a modified boundary value a 1 = a − b (inn) | ∂Ω such that supp a 1 ⊂ Γ ∩ ∂Ω 0 ∪ ∪ N j=1 Γ j and the flow rates of a 1 over the inner boundaries Γ j , j = 1, . . . , N, are equal to zero: Γ j a 1 · ndS = 0, and the flow rate of a 1 over the outer boundary Γ ∩ ∂Ω 0 is equal to Now we remove the nonzero flux from the outer boundary Γ ∩ ∂Ω 0 . For this we will need the notion of Stein's regularised distance. Let M be a closed set in R 2 . Stein's regularised distance ∆ M (x) from the point x to the set M is an infinitely differentiable function in R 2 \ M and the following inequalities M) is the distance from x to M. The positive constants a 1 , a 2 and a 3 are independent of M (see [39], Chapter VI, Sections 1 and 2, 167-171, Theorem 2).
Let γ be a smooth simple curve, which intersects the outer boundary at some point x (out) ∈ Γ ∩ ∂Ω 0 , does not intersect or touch any inner boundary Γ j , j = 1, . . . , N, and coincides with the straight line x 1 = 0 in G (see Figure 3).
hold with the constant c 1 dependent only on a 1 , a 2 and d 0 .
The proof of this lemma can be found in [40] (see Lemma 2).
Let us define a vector field where ξ(x) coincides with ξ(x) on the right side of the curve γ and ξ(x) = 0 on the left of γ. By construction, the vector field b (out) (x) is smooth, solenoidal and b (out) (x) ∂Ω\(Γ∩∂Ω 0 ) = 0.
Proof. Since divb (out) = 0, we have Estimate (16) follows from Lemma 4 and properties of the regularised distance.

Extension of a 2
The extension of the boundary value function having zero flux over the boundary was constructed by O.A. Ladyzhenskaya (see [37], Chapter V, Section 4, 127-128). To be more precise, in [37] was proved the following result Lemma 6. Let D ⊂ R 2 be a bounded domain with Lipschitz boundary ∂D, L ⊆ ∂D, meas(L) > 0. Assume that the vector field h ∈ W 1/2,2 (∂D) satisfies the conditions L h · n dS = 0, supp h ⊆ L.
Then h can be extended inside D in the form where is Hopf's type cut-off function, i.e., χ is smooth, χ(x, ε) = 1 on L, supp χ is contained in a small neighborhood of L and The constant c in (19) is independent of ε > 0.
The vector field H ∈ W 1,2 (D) is solenoidal, H ∂D = h, supp H is contained in a small neighbourhood of L and there holds the estimate Moreover, for any ε > 0 the vector field H satisfies the Leray-Hopf inequality with the constant c independent of ε.
Because of the condition (17) we can apply Lemma 6 to a 2 and we obtain the following result.

Lemma 7. There exists a vector field
Moreover, The constant c in (23) is independent of ε > 0 and k.

Construction of Extension Coinciding with Asymptotic Decomposition Near Cusp Point
Now we "glue" the above constructed vector field B = b 0 + b (inn) + b (out) with the asymptotic decomposition U [J] .
Let ζ be a smooth cut-off function such that ζ( where V [J] is the solution of the following problem div Notice that where we used the fact that b (inn) = 0 in ω 4 . Therefore, there exists a solution V [J] ∈W 1,2 (ω 4 ) of problem (25) satisfying the estimate see [41]. Since div b 0 + b (inn) = 0 and supp b 0 + b (inn) ⊂ Ω 4 , from the construction we conclude the following result.

Existence and Uniqueness of Weak Solution
In this section we prove the existence of the weak solution of problem (1). First assume that u, p is a classical solution of (1). Multiplying (1) 1 by the test function η ∈ C ∞ 0 (Ω) and integrating by parts, we obtain We look for the solution u, p in the form where A is the extension of the boundary value a constructed in the previous section, P [j] is defined by (6) 3 and v ∈ H(Ω). Substituting (28) into (27) we obtain where The vector field v will be found as a limit of the sequence {v k }, where v k are weak solutions in the domains Ω k , that is, the vector fields v k ∈ H(Ω k ), satisfy the integral identities then problem (31) admits at least one solution v k ∈ H(Ω k ). There holds the estimate with the constant c independent of k.
Proof. It is well known (see [37]) that integral identity (31) is equivalent to the operator equation with a completely continuous operator B : H(Ω k ) → H(Ω k ), defined by the relation So, the solvability of Equation (34) will follow from the Leray-Schauder theorem provided we prove that the norms of all possible solutions of the operator equations v (λ) are bounded by a constant independent of λ. Operator Equation (35) is equivalent to the identity Taking in (36) To estimate the term λ f, v (λ) k Ω k in the right hand side of (37), we use the representation (24) for the vector field A. We denote , so that A = A 1 + A 2 . Since supp A 1 ⊂ Ω 4 , using estimates (13), (16), (22) and (26), the embedding W 1,2 (Ω 4 ) → L 4 (Ω 4 ) and the definition of a 2 = a − b (inn) + b (out) ∂Ω , we obtain the following inequality Since Thus, by (7), (16) and (26), Further, the straightforward calculations give the equality Because the asymptotic decomposition (U [J] , P [J] ) satisfies Equation (8), by (9), we obtain The integrals J k , k = 2, 3, 4, can be estimated using (7), and we get From (39)- (41) it follows that Substituting this estimate into (37) and choosing ε sufficiently small we obtain The constant c in (42) is independent of k.
Consider now the integral λ In virtue of Lemmas 3 and 5, we have By (26), For the integral λ (7) and (5) yield Finally, using Leray-Hopf's inequality (23), we estimate the integral Estimates (43)-(46) yield the inequality where the constants c 0 and c * are independent of k and λ. Thus, estimate (42) takes the form Choosing ε sufficiently small, say ε = ν 8c 0 and assuming that F 0 = ν 8c * from the last inequality we derive Proof. Let us take the sequence of solutions v k constructed in Theorem 1. Extending v k by zero into Ω \ Ω k we get vector fieldsṽ k ∈ H(Ω). Notice thatṽ k satisfy integral identity (31) in which we can integrate over the domain Ω instead of Ω k . Taking an arbitrary function η ∈ J ∞ 0 (Ω) we can find a number k such that supp η ⊂ Ω k . Since the sequence {ṽ k } is bounded in H(Ω), there exists a subsequence {ṽ k m } which converges weakly in the space H(Ω) and converges strongly in L 4 (Ω k ) for any k, as the embedding H(Ω k ) → L 4 (Ω k ) is compact. Such subsequence can be constructed using Cantor's diagonal argument. Then we can pass to the limit as k m → ∞ in integral identity (31) taking any test function η ∈ J ∞ 0 (Ω). For the limit function v ∈ H(Ω) we obtain the integral identity (29). Obviously, the limit function v obeys estimate (48).
Author Contributions: Both authors contributed equally in this article. Both authors have read and agreed to the published version of the manuscript.