Time periodic boundary value Stokes problem in a domain with an outlet to infinity ∗

The Stokes and stationary Navier–Stokes equations with homogeneous boundary condition were intensively studied in domains with outlets to infinity during the last 40 years (see [2, 3, 18, 19, 29, 30] and the literature cited there). In the last 10 years, the special attention was given to problems with nonhomogeneous boundary conditions (see [1, 4–6, 23–28]). Moreover, recently a big progress was obtained in Leray’s problem in bounded and exterior domains [8–14]. On the other hand, the time periodic problem for the Navier–Stokes equations was mainly studied only in the case of homogeneous boundary conditions (see, for example, [15, 20, 21]). The time periodic problems with nonhomogeneous boundary conditions were essentially considered by H. Morimoto [22] and T. Kobayashi [7]. However, they investigated the problem only in domains with compact boundaries. A wide review and study of periodic problems could be found in the habilitation thesis of M. Kyed [16]. In this paper, we consider the time periodic Stokes system with nonhomogeneous boundary condition


Introduction
The Stokes and stationary Navier-Stokes equations with homogeneous boundary condition were intensively studied in domains with outlets to infinity during the last 40 years (see [2,3,18,19,29,30] and the literature cited there).In the last 10 years, the special attention was given to problems with nonhomogeneous boundary conditions (see [1,[4][5][6][23][24][25][26][27][28]).Moreover, recently a big progress was obtained in Leray's problem in bounded and exterior domains [8][9][10][11][12][13][14].On the other hand, the time periodic problem for the Navier-Stokes equations was mainly studied only in the case of homogeneous boundary conditions (see, for example, [15,20,21]).The time periodic problems with nonhomogeneous boundary conditions were essentially considered by H. Morimoto [22] and T. Kobayashi [7].However, they investigated the problem only in domains with compact boundaries.A wide review and study of periodic problems could be found in the habilitation thesis of M. Kyed [16].
In this paper, we consider the time periodic Stokes system with nonhomogeneous boundary condition u(x, t) − ν∆u(x, t) + ∇p(x, t) = f (x, t), (x, t) ∈ Ω × (0, 2π), div u(x, t) = 0, (x, t) ∈ Ω × (0, 2π), u(x, t) = ϕ(x), (x, t) ∈ ∂Ω × (0, 2π), in a two dimensional multiply connected unbounded domain Ω.Here the vector valued function u(x, t) is the unknown velocity field, the scalar function p(x, t) is the pressure of the fluid, while the vector valued functions ϕ(x) and f (x, t) denote the given boundary value and the external force, ν is the viscosity constant of the given fluid.
Let F (inn) = Γ1 ϕ • n dS and F (out) = Λ ϕ • n dS be the fluxes of the boundary value ϕ over the inner and the outer boundary, respectively.Then This condition is natural, because we consider incompressible fluid.
In this paper, we prove the existence and uniqueness of a weak solution to problem (1) in a domain with an outlet to infinity Ω (see Fig. 1).Notice that this result is the first step to study the nonlinear time periodic Navier-Stokes problem in such domains.

Notation and preliminaries
Vector valued functions are denoted by bold letters, while function spaces for scalar and vector valued functions are denoted in the same way.
We use the symbols c, C, c j , C j , j = 1, 2, . . ., to denote constants whose numerical values are unessential to our considerations.In such case, c, C may have different values in single computations.
Let G be a domain in R n .As usual, C ∞ (G) denotes the set of all infinitely differentiable functions defined on Ω, and C ∞ 0 (G) is the subset of all functions from C ∞ (G) having compact supports in Ω.For a given nonnegative integer k and q > 1, L q (Ω) and W k,q (Ω) indicate the usual Lebesgue and Sobolev spaces, while W k−1/q,q (∂Ω) is the trace space on ∂Ω of functions from W k,q (Ω).Denote by J ∞ 0 (Ω) the set of all solenoidal (div u = 0) vector fields u from C ∞ 0 (Ω).By H(Ω) we indicate the space formed as the closure of J ∞ 0 (Ω) in the Dirichlet norm u H(Ω) = ∇u L 2 (Ω) generated by the scalar product Definition 1.By a weak solution of problem (1) we understand a solenoidal vector field u with ∇u, u t ∈ L 2 (0, 2π; L 2 (Ω)) satisfying the boundary condition u| ∂Ω = ϕ, the time periodicity condition u(x, 0) = u(x, 2π) and the integral identity Later, we will use the notion of the regularized distance.
Lemma 1. (See [31].)Let M be a closed set in R 2 .Denote by ∆ M (x) the regularized distance from the point x to the set M. Function ∆ M (x) is infinitely differentiable in R 2 \ M, and the following estimates hold, where d G (x) = dist(x, G) is the distance from x to M, positive constants a 1 , a 2 and a 3 are independent of M.

Construction of the extension of the boundary value
We start with the construction of a suitable extension A of the boundary value ϕ.Then we can reduce a nonhomogeneous condition to the homogeneous one.Since the boundary https://www.mii.vu.lt/NA value ϕ is independent of time, the extension of the boundary value could be constructed using the similar ideas as in [5].Additionally, we need to estimate the term ∆A .We construct the extension A in the following form: where B (inn) extends the boundary value ϕ from the inner boundary Γ 1 , and B (out) extends ϕ from the outer boundary Γ 0 .

Construction of the extension B (inn)
First, we construct a vector field b (inn) such that Let ∆ γ+ and ∆ ∂D∩∂Ω be the regularized distances from a point x ∈ D to the line γ + = {x: x 1 = 0, x 2 > R 0 } and the boundary ∂D ∩ ∂Ω, respectively.Define in D a Hopf's-type cut-off function where Ψ is a smooth monotone function, 0 Ψ 1, where d 0 is a positive number such that dist(γ + , ∂D ∩ ∂Ω) d 0 , and a 1 , a 2 are positive constants from the estimates of the regularized distance (see Lemma 1).
Proof.The proof of the lemma follows directly from the definition of the function ξ(x), properties of the regularized distance and the fact that supp ∇ξ(x) is contained in the set where ∆ ∂D∩∂Ω (x) (∆ γ+ (x)).
Let us define on ∂Ω another vector field given by ( 5) and b (inn) # defined as following: Nonlinear Anal.Model.Control, 23(6):866-888 Then according to the properties of the vector fields b and b (inn) # , we get In order to extend h 1 into Ω 0 , first, we define the solenoidal vector field b(inn) where H ∈ W 2,3 (Ω 0 ) satisfies the following boundary conditions: Then we extend where the support of Hopf's-type smooth cut-off function κ is contained in the neighborhood of ∈ W 2,2 (Ω 0 ) and satisfies the following estimate: where the constant c depends only on the domain Ω 0 (see [17]).
Then we define a vector field which satisfies the following condition: Therefore, the function h 0 can be extended inside Ω in the form b (inn) 0 where E(x) ∈ W 2,2 (Ω 0 ), (∂E(x)/∂x 2 , −∂E(x)/∂x 1 ) = h 0 , the support of Hopf's-type smooth cut-off function χ is contained in the neighborhood of Γ 1 (see [17]).Finally, we put The properties of the extension B (inn) we formulate in the following lemma.
Lemma 4. The vector field B (inn) is solenoidal, (Ω) and satisfies the following estimates:

Construction of the extension B (out)
Take any point x (1) ∈ Λ ⊂ Γ 0 .Let γ be a smooth simple curve, which intersects ∂Ω at the point x (1) , and where γ is a semi-infinite line lying in D, γ 0 is a finite simple curve connecting γ and the point x (1) .Assume that inf x∈γ, y∈∂Ω\Λ |x − y| d 0 .
Define a Hopf's-type cut-off function where functions Ψ and are defined by ( 3) and ( 4), respectively.
Proof.The proof follows directly from the definition of ζ(x), properties of the regularized distance and the fact that supp ∇ζ(x) is contained in the set where Let us introduce the vector field where ζ(x) = ζ(x) above the curve γ, and ζ(x) = 0 under the curve γ.Lemma 6.The vector field b (out) is infinitely differentiable and solenoidal, vanishes near the set ∂Ω \ Λ and in a small neighborhood of the curve γ.The following estimates hold: Proof.Estimates ( 14)-( 17) could be proved in the same way as in Lemma 3. Due to the construction of b (out) , we get (18): and h can be extended (see [17]) inside Ω in the form b (out) 0 where E(x) ∈ W 2,2 (Ω 0 ), (∂E(x)/∂x 2 ; −∂E(x)/∂x 1 )| Λ = h and χ is a Hopf's cut-off function such that χ = 1 on Λ, supp χ is contained in a small neighborhood of Λ.Finally, we put The properties of the extension B (out) are formulated in the following lemma.
Lemma 7. The vector field (Ω) and satisfies the following estimates: Therefore, we have constructed the extension A = B (inn) + B (out) of the boundary value ϕ.The properties of A are given in the following theorem.
4 Solvability of problem (1) ( We look for the solution of (1) in the form where A is the suitable extension of the boundary value ϕ constructed in the previous section.Then problem (1) is reduced to the problem with homogeneous boundary condition and now we look for the new unknown velocity field v.
Let us denote the following space: , where L 2 1 (Ω) is weighted space with the norm https://www.mii.vu.lt/NADefinition 2. Let f ∈ L 2 per (0, 2π; L 2 1 (Ω)).By a weak solution of problem ( 23) we understand a solenoidal vector field v with ∇v, v t ∈ L 2 (0, 2π; L 2 (Ω)) satisfying the homogeneous boundary condition v| ∂Ω = 0, the time periodicity condition v(x, 0) = v(x, 2π) and the integral identity: for all η ∈ L 2 (0, 2π; J ∞ 0 (Ω)).Theorem 2. Assume that the domain Ω ⊂ R 2 has one outlet to infinity, boundary value dx 2 /g 3 (x 2 ) < +∞, then problem (1) has a unique weak solution u = A + v satisfying the following estimate: Proof.We start with the choosing a family of bounded domains Ω k , i.e., where The existence of a unique solution v satisfying the integral identity ( 24) could be proved by three following steps.Firstly, we prove the existence of the approximate solution v (k,N ) to the problem Secondly, we show the convergence of the approximate weak solution v (k,N ) to the weak solution v (k) , which satisfies Nonlinear Anal.Model.Control, 23 (6):866-888 Finally, passing to a limit as k → +∞, we get the existence of a weak solution v to problem (23).
Consider problem (26).It is well known that every 2π-periodic function in L 2 (0, 2π) could be written as Fourier sieries: Let f (N ) be a partial sum of (28).
We look for the approximate solution (v (k,N ) , p (k,N ) ) in the form In order to prove the existence of the approximate solution, we need to prove the existence of Fourier coefficients a n , n = 0, 1, . . ., N .To do this, we substitute ( 28)-( 30) into problem (26), and by collecting the coefficients of sin and cos functions we obtain the following stationary problems: Notice that (31) is the Stokes system with homogeneous boundary condition and the existence of a weak solution of ( 31) is well known (see [17]).
In order to prove the existence of a unique solution to problem (32), we multiply (32) 1 by η ∈ H(Ω k ) and (32) 2 by ξ ∈ H(Ω k ).Then by integrating by parts over Ω k we obtain the following system: https://www.mii.vu.lt/NA To prove the existence of the unique solution of (33), we use Fredholm alternative by reducing (33) to the system of operator equations where B is linear completely continuous operator.
Then we consider homogeneous operator equations n (x) and summing up the equations, we obtain Then it follows that According to Fredholm alternative, we obtained that (32) has a unique solution.Therefore, the existence and uniqueness of the approximate solution v (k,N ) to problem (26) is proved.
In order to prove the convergence of an approximate solution v (k,N ) (x, t) to v (k) (x, t) in bounded domains Ω k , we need to obtain the estimates for the norms of v (k,N ) (x, t).
To do this, we multiply equation (26) 1 by v (k,N ) (x, t), and after integrating by parts over Ω k , we get from (34) it follows that Integration with respect to time variable t from 0 till 2π yields Using the periodicity condition v (k,N ) (x, 0) = v (k,N ) (x, 2π), we derive Notice that we need to get estimates with the constant independent of the domain Ω k .To do this, we rewrite equation (35) as follows: https://www.mii.vu.lt/NABy Cauchy-Schwarz inequality, Since, due to Poincaré-Friedrichs inequality, we have that from (36) we obtain Dividing both sides by ν( 2π 0 , we rewrite the last estimate as follows: where the constant C is independent of the domain Ω k .Due to Theorem 1, we estimate the norm ∇A 2 L 2 (Ω k ) : According to the fact that ∂Ω) , from (37), using (38), we get where C is independent of Ω k .Let us get the estimate for the norm of the term v (k,N ) t . Multiplying equation (26) 1 by v (k,N ) t (x, t) and after integrating by parts over Ω k , we arrive at Since According to the fact that it follows from (41) using (42) the following estimate: where C 1 is independent of Ω k .For the fixed k, from estimates (39), (43) we conclude that {∇v (k,N ) } and {v (k,N ) t } are bounded sequences in the space L 2 (0, 2π; L 2 (Ω k )).Hence there exists a subsequence {v (k,Nm) } such that {∇v (k,Nm) } and {v (k,Nm) t } are converging weakly to {∇v (k) } and {v (k) t } in the space L 2 (0, 2π; L 2 (Ω k )).Moreover, {f (N ) } converges to {f } in the space L 2 (0, 2π, L 2 (Ω k )).For the approximate solution, the following integral identity holds: (44) Thus, v (k) are weak solutions of problem (27) in bounded domains Ω k .Finally, we will get the solution in whole domain Ω.Since the estimates we got for the approximate solution v (k,N ) remain valid for the limit solution v (k) , using estimates (39) and (43), we have: where constant c is independent of domain Ω k .Since +∞ 1 1/g 3 (x 2 ) dx 2 < +∞, the right-hand side of estimate (45) is bounded by a constant independent of k.So {∇v (k) } and {v (k) t } are bounded sequences in the space L 2 (0, 2π; L 2 (Ω k )).Therefore, there exists a subsequence {v (km) } such that {∇v (km) } and {v (km) t } converge weakly to {∇v} and {v t } as k m → +∞ in the space L 2 (0, 2π; L 2 (Ω)).Taking in integral identity (44) an arbitrary test function η with a compact support, we can pass to a limit as k → +∞.As a result, we get for the limit function v integral identity (24).
The uniqueness is obtained by standard way assuming that (23) has two weak solutions w 1 and w 2 , which satisfy the integral identities Making a difference of the last two integral identities, we get Notice that both terms are positive.Therefore, we have 2 dx dt = 0.