In-Flow and Out-Flow Problem for the Stokes System

We investigate the existence and regularity of solutions to the stationary Stokes system and non-stationary Navier–Stokes equations in three dimensional bounded domains with in- and out-lets. We assume that on the in- and out-flow parts of the boundary the pressure is prescribed and the tangential component of the velocity field is zero, whereas on the lateral part of the boundary the fluid is at rest.


Introduction
Let us consider the Stokes system −νΔu + ∇Π = f in Ω, div u = g in Ω, (1) where Ω ⊂ R 3 is a bounded solid with piecewise C 2 -boundary Γ := ∂Ω. Functions u (the velocity) and Π (the pressure) are unknown. Given are f (the external force), g and a positive viscosity coefficient ν. We are interested in the existence and regularity of solutions (u, Π) to (1) when Ω has cylindrical in-and outlets. More specifically, we assume that Γ is the union of Γ in,out and Γ lat , where Γ in,out = 1≤k≤N Γ k in,out . We also assume that for all 1 ≤ k ≤ N the surfaces Γ k in,out are flat and Γ k in,out is orthogonal to Γ lat (see Fig. 1). The latter assumption ensures that we do not need to work in the framework of weighted Sobolev spaces and can rely on the reflection principle.
We need to supplement (1) with proper boundary conditions. On Γ lat we assume that the fluid is at rest. The choice of the boundary conditions on Γ k in,out depends on the problem we would like to model. For some examples of such problems and further motivations originating from real-life situations we refer the reader to the Introduction in e.g. [1,2] and [3,Sect. 3]. The most frequently used boundary conditions are: • prescribed pressure drops (see e.g. [3,4]) where Π k (t) are given functions and Γ k in,out denotes the surface area of Γ k in,out , • prescribed net fluxes (see e.g. [3,5,6]) where F k (t) are given functions, • prescribed pressure and tangential components of the velocity (see e.g. [1,2]) where Π k and a k are given functions. Other boundary conditions that appear in the literature include the so-called artificial boundary conditions (ABCs, see e.g. [7,8]), prescribed normal components of the velocity field on Γ k in,out (see e.g. [9,10]), modified do-nothing boundary conditions (see e.g. [11,12]) and other (see e.g. [13][14][15]). This list, however, is far from being complete but forms a good starting point for further research.
In our work we consider (2) with a k = 0. It can be rewritten as u tan = 0 on Γ, Π = Π k in,out on Γ k in,out , u · n| Γ lat = 0 on Γ lat , where u tan denotes the tangential component of u. We will justify our choice after Theorem 2. Our main result reads: .
Problem (1) + (3) with g = 0 was already considered in [1,2,[16][17][18]. In these papers the authors assume that Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 . Comparing with our notation we see that Γ 1 = Γ lat and Γ 2 = k Γ k in,out . On Γ 1 and Γ 2 slightly more general boundary conditions are assumed, namely u| Γ1 = u 0 and u tan | Γ2 = a, where both u 0 and a are given. On Γ 3 in these papers we have where h and b are prescribed. Assuming that u 0 ∈ H Under slightly stronger assumptions on the boundary data the H 2 -regularity of the solutions was obtained in [19]. Theorem 1 is similar to [ To prove Theorem 1 we will first consider the family of equations with homogeneous boundary condi- where τ ∈ [0, 1] and Π k in,out = 0. In Sect. 3 we will prove the following result: The case when Π k in,out = 0 will be discussed in Sect. 4. There we also demonstrate the proof of Theorem 1. Finally, in Sect. 5 we show an application of the obtained results for the Navier-Stokes equations For the above system we have: The definitions of weak, strong and almost everywhere regular solutions to (5) are in Sect. 5 (see Definitions 5.3, 5.2 and Lemma 5.4). As we observed after Theorem 1 one could generalize Theorem 2 to arbitrary Sobolev spaces by showing W 1 p (Ω)-estimates for solutions to (25). A claim very similar to Theorem 2 (global existence of weak solutions and local-in-time existence of regular solutions) was suggested in [3, Sect. 6] with a remark that its proof would not differ substantially from the case of Dirichlet conditions. Indeed, the main source of computational difficulties is the nonlinear term (v · ∇)v. When we test it with e.g. v, it yields a boundary integral which so far can be only eliminated under additional assumptions on the magnitude of the initial data and certain norms of Π k in,out on Γ k in,out (cf. . This shows the main difference from our result: for obtaining global and regular solutions we only require v 0 H 1 (Ω) to be small. The case of (5) with non-vanishing tangential components of v on all parts of the boundary was formulated and also proved in [20] (see Theorem 5.4) and [21,Theorem 5.1]. For k = 2 problem (5) was studied in e.g. [22,23]. Clearly, when we use the non-linear term in its conservative form, then the boundary integral does not appear. While, as pointed out in [3,Sect. 4], conditions (5) 3,4 are not fulfilled by Poiseuille flow with a constant total pressure, our pressure on in-and out-flow boundaries does not need to be constant and can therefore be chosen as those of Poiseuille flow, which means our solutions do indeed include Poiseuille flow. There is in any case much to be said in favor of prescribing the total pressure Π = p + 1 2 |v| 2 , as in case of the flow of an ideal fluid the total pressure is conserved along flow lines in the absence of gravity, otherwise it is Π + x 3 , if −e 3 is the force of gravity. We are, of course, not discussing an ideal fluid here, but if the inflow on the boundary comes from a tank nearby with fairly slow flow, then the viscosity will not have a very large influence on the total pressure, and we may set it equal to the hydrostatic pressure on the inflow boundary as a reasonable approximation. For further references (e.g. non-stationary Stokes system or stationary Navier-Stokes equations) we refer the reader to the Introduction in [24], where also problem (5) in Ω ⊂ R 2 was studied.
In the subsequent sections, to improve readability we will omit the index k in case of Π in,out and Γ in,out .

Estimates
In this section we discuss some estimates for solutions to (4) with Π = 0 on Γ in,out . We start with an observation about the boundary conditions for u. From (4) 3,5 we infer that u = 0 on Γ lat . To find the condition on Γ in,out we recall that (4) 3 implies that (see e.g. p. 136 in [25]) thus (4) 2,3 give For τ / ∈ {0, 1} we solve (4) 2 for Π and use it in (4) 1 , thus Then, any solution to (9) satisfies Proof. To derive the estimates for the solution to (9) we introduce a partition of unity with proper boundary conditions. It is clear that we can choose this partition of unity in such a way that exactly one of the following cases occurs: 1. supp ζ k ∩ Γ = ∅. This is the whole-space problem since u (k) = 0 on ∂(supp ζ k ∩ Ω). Without loss of generality we may further assume that x 3 ≥ 0, thereby reducing this case to half-space problem with zero Dirichlet boundary condition. Then, Theorem 14.1 [26] implies , In this case we transform supp ζ k ∩ Ω into the half space and use Theorem 14.1 from [26] which eventually yields (10). 3. supp ζ k ∩Γ lat = ∅ but supp ζ k ∩Γ in,out = ∅. Through rotations and translation we reduce this case to x3 . In order to make the right-hand side equal to zero, we may assume that e.g.
This observation suggests the following reflection in such a way that the normal n to Γ lat at x 0 is parallel to e 3 (cf. Fig. 2) (0,0) and e 1 · (ψ y1 (0, y 2 ), ψ y2 (0, y 2 ), −1) = 0, which implies that ψ y1 (0, y 2 ) = 0. Therefore we may reflect U k symmetrically across the plane y 1 = 0, i.e.Ū k = (y 1 , y 2 , ψ(y 1 , y 2 )) y 1 ≥ 0 (−y 1 , y 2 , ψ(y 1 , y 2 )) y 1 < 0 Since U k is of class C 2 , so isŪ k and now we proceed as in Case (2). Summing over k gives for any p ∈ (1, ∞) . To eliminate u W 1 p (Ω) from the right-hand side we note that any weak solution u to (9) satisfies where the boundary integrals vanish due to (9) Thus To complete the proof we use the Gagliardo-Nirenberg inequality The latter condition is satisfied for p ≥ 2. Finally, from the Poincaré and the Young inequalities Taking > 0 small enough and using (11) completes the proof.

Remark 2.2.
To derive (9) we assumed that τ = 0. Since τ > 0 is arbitrary, by continuity Lemma 2.1 remains true when τ = 0. In this case we would have

Remark 2.3. From Lemma 2.1 and (8) it follows that
We easily observe that Lemma 2.1 does not work when τ = 1 in (4). Before we discuss this case, we establish the uniqueness of solutions to (4) for any τ ∈ [0, 1].
Remark 2.4. If (u, Π) is a solution to (4) with Π = 0 on Γ in,out , then it is unique.

Lemma 2.5.
Suppose that f ∈ L p (Ω) and g ∈ W 1 p (Ω). Then any solution to (4) with τ = 1 satisfies Proof. As in the proof of Lemma 2.1 we introduce a partition of unity N k=0 ζ k (x) = 1 on Ω and we write When k is fixed, four cases are possible. There are identical to the cases considered in the proof of Lemma 2.1. However, in cases (3) and (4) we need to define the reflection of the pressure through Γ in,out .
Since Π| Γin,out = 0, we setΠ (13) in the whole space we have the following estimate (cf. [ . By [28, Theorem IV.3.2] we obtain the identical estimate for the half-space case. Summing over k yields . To get rid of u and p from the right-hand side we prove that (Ω) such that they solve (4) with τ = 1, Π m = 0 on Γ in,out and

As system (4) is linear, and the above inequality is homogeneous, the function vector
solves the same system and fulfills the same inequality. Clearly therefore After selecting a subsequence, which we denote by the same symbols as the original one, we have that this subsequence converges weakly to a (v, Ψ,

Existence
The existence of solutions to (1) is obtained in two steps. First, we look into (4) and use homotopy argument. Next, we consider the non-homogeneous problem (see Sect. 4). Let us define the family of operators L τ : By · B1 and · B2 we denote the norms on B 1 and B 2 which are induced by the standard scalar products, respectively. It is easy to check that L τ is injective.   1] . (15) In addition assume there is a function γ : Then, there exists a constant (17) and if L 0 is surjective then L 1 is also surjective.
Proof. Let τ 0 ∈ [0, 1] be fixed. By the injectivity of L τ and (16) the expression is finite for all τ ∈ [0, 1] and we could replace the original γ by it without invalidating (16). Let us assume that γ now equals that supremum. Then for τ 1 , As τ 1 and τ 2 are interchangeable, this implies γ is continuous and therefore has a finite maximum on [0, 1]. Let now L τ0 be surjective. Then L τ0 v = g is equivalent with We define F : . Then, L τ is surjective if and only if F has a fixed point. By the properties of L τ If we take τ close enough to τ 0 , then c 1 c 2 |τ − τ 0 | < 1 and by the Banach Fixed Point Theorem we conclude the surjectivity of L τ . To complete the proof we start with τ 0 = 0 and go to 1 in a finite number of steps as c 1 and c 2 do not depend on τ .

Remark 3.2.
In the above proof we start with τ 0 = 0 which corresponds to the following problem The existence and regularity of solutions (cf. Remark 2.2) follow now from reflection and classical theory.

General Case
Suppose now that Π| Γin,out = Π in,out . We introduce an auxiliary function p : Ω → R such that One can easily show the existence of weak solutions to the above problem. These solutions satisfy Then, the pair (u, ψ), where ψ = Π − p, solves From Proposition 1.1 we get the existence of solution (u, ψ) and the estimate From (20), (21) and the existence of u, ψ and p we get Theorem 1.

Navier-Stokes Equations
As an application of the theory we have presented so far, we will consider the Navier-Stokes equations. Let us denote Π = p + 1 2 |v| 2 and ω = rot v. We first consider the case when Π = 0 on Γ in,out . For a similar problem see e.g. [29,30]. Let us define , ψ| ∂Γin,out = 0 .
By · H we denote the norm on H induced by the standard scalar product (·, ·) on H. Let P : L 2 (Ω) → H be the orthogonal projector. Applying it to (5) we get where A stands for the Stokes operator whose domain of the definition in this case is Assume that f : [λ 1 , ∞) → R. Then we set From the above definition it follows that for λ 1 > 0 we can define for any s ≥ 0 and u ∈ H . Below we show the existence of weak and strong solutions to (5) by analyzing (22). These solutions are defined in Lemma 5.4 and Definition 5.2, respectively. We will use an abstract method from [34], which is an enhanced version of the theorem from [35,Sect. 1.1]. This method is based on the direct application of the next Lemma to (22). First, we need to define almost everywhere regular solutions.
is a set of one-dimensional Lebesgue measure zero (see [34,35]) and u is strong on I k for all k ∈ S. Lemma 5.4. (cf. Theorem in [34]) Let T < +∞. Suppose that A : D(A) → H is positive self-adjoint with compact inverse. Assume that there exists a fixed number μ ∈ (0, 1) and a continuous increasing function g : R + → R + such that the non-linearity F : D(A μ ) × (0, T ) → H has the following properties: (C 1 ) There exists a constant c 1 such that We also have: where α = C 1 − λ 1 , with λ 1 being the lowest eigenvalue of A. Therefore: (A 1 ) If C 1 is sufficiently small, then the solution v(t) will become strong after a time determined by C 1 and v 0 . We easily see that assertions (1)+(2)+(3) ensure that a weak solution v to (22) is an almost everywhere weak regular solution. From the first part of assertion (4) it follows that if a weak solution is small enough, than it becomes the strong solution. The smallness of a weak solutions is guaranteed when the constant C 1 is small enough (see assertion (A1): indeed, if C 1 ≤ λ1 2 , then α ≤ − λ1 2 and (24) 1 implies that for sufficiently large t v(t) H ≤ 2 λ −1 1 C 1 . If C 1 is small enough, we have that v(t) H ≤ 0 .
To use the above Lemma, let [cf. (22)] For the condition (C 2 ) we first note

Now we have
.
The estimate for F 2 (t) is even simpler To verify (C 3 ) we check that for u ∈ D(A) .
Let u = A −1 w. Due to the symmetry of A −1 we have and We have completed the verification of all conditions. Using Lemma 5.4 we get the existence of weak solution, which are regular for almost all times and the existence of short-time strong solutions for any data or long-time strong solutions for small data.