STABILIZATION FOR THE 3D NAVIER-STOKES SYSTEM BY FEEDBACK BOUNDARY CONTROL

We study the problem of stabilization a solution to 3D NavierStokes system given in a bounded domain Ω. This stabilization is carried out with help of feedback control defined on a part Γ of boundary ∂Ω. We assume that Γ is closed 2D manifold without boundary. Here we continuer investigation begun in [6],[7] where stabilization problem for parabolic equation and for 2D Navier-Stokes system was studied.


1.
Introduction. This paper is devoted to study the problem of stabilization a solution v(t, x) to boundary value problem for three-dimensional (3D) Navier-Stokes equations given in a bounded domain Ω ∈ R 3 . This solution is stabilized near a steady-state solution to 3D Navier-Stokes system. We carry out this stabilization with help of control u defined on a part Γ of boundary ∂Ω. Our assumption imposed on Γ is that Γ is a closed two-dimensional surface and therefore Γ is a separate component of ∂Ω.
We require that control u = u(t, x ), t > 0, x ∈ Γ has to possess the following important property: u is a feedback control. This means that for each instant t u(t, ·) is defined by fluid flow velocity vector field v(t, ·) taken at the same instant t and therefore control u can react on unpredictable fluctuations of v suppressing their negative influence to fluid flow. There is a mathematical formalization for this physical notion of feedback, which was proposed long time ago. With help of this formalization number results of stabilization for equations described incompressible fluid flow were obtained: stabilization of 2D Navier-Stokes equation by distributed control supported on the whole Ω and written in an abstract form (Barbu, Sritharan [1]), and stabilization by boundary control of 2D Euler equations for incompressible fluid flow (Coron [3]).
In this paper we use a certain new formalization of feedback notion that we proposed in [6], [7], and that is more adequate (from our point of view) to study stabilization problem by boundary feedback control in the case of parabolic equations and Navier-Stokes system. Moreover, here we develop approach to stabilization problem from [6], [7] in order to get stabilization result for 3D Navier-Stokes equations. The main point of this approach is to construct a special operator that extends solenoidal vector fields from Ω on a certain domain G containing Ω. Extension operator connected with linearized Navier-Stokes equations (i.e. with Oseen system) is constructed below in Section 5 and extension operator corresponding to nonlinear case is built in Section 6. Construction of these extension operators is based on a linear independence property of finite systems of eigen and associated functions for adjoint steady-state Oseen system when these functions are regarded over arbitrary subdomain ω ⊂ G. We prove this property in Section 4 with help of Carleman estimates using some abstract result from [6]. In Section 2 we formulate stabilization problem for Navier-Stokes equation and, besides, we compare two mathematical formalizations of feedback notion: classical one and the formalization that was proposed in [6], [7]. In Section 3 we recall certain well known results, used in the paper.
2. Setting of the problem and the main idea of the method.

Setting of the stabilization problem.
Let Ω ⊂ R 3 be a bounded connected domain with C ∞ -boundary ∂Ω which consists of two nonintersecting parts Γ 0 and Γ: where Γ 0 , Γ are closed subsets of ∂Ω, i.e. Γ 0 , Γ are finite sets of connected C ∞manifolds of dimension 2. We assume that Γ = ∅ but we admit that the set Γ 0 can be empty. We set In space-time cylinder Q we consider the Navier-Stokes equations and boundary conditions v| Σ0 = 0, v| Σ = u, (2.6) where u = (u 1 , u 2 , u 3 ) is a control defined on Σ. We suppose also that a steady-state solution (v(x), ∇p(x)) of Navier-Stokes system with the same right-hand side f (x) as in (2.3) is given: v| Γ0 = 0. (2.8) Let σ > 0 be given. The problem of stabilization with the rate σ is to look for a control u(t, x ), x ∈ Γ such that the solution (v, p) of problem (2.3)-(2.6) with the boundary value u satisfies the inequality v(t, ·) −v (H 1 (Ω)) 2 ce −σt as t → ∞. (2.9) 3D NAVIER-STOKES SYSTEM BY FEEDBACK BOUNDARY CONTROL 291 2.2. Feedback control. Classical definition. Our important additional requirement is that a stabilization problem should be solved with help of feedback control. From physical point of view feedback notion means that control function u(t, ·) at instant t should be defined with help of state function v(t, ·) taken just at the same time moment t. This has to give possibility for control to react on unpredictable fluctuations of the state function in order to suppress all undesirable effects of these fluctuations.
There is well known mathematical formalization of this physical feedback notion which was proposed in control theory for ordinary differential equations. To recall it we rewrite problem (2.3)-(2.6) in a form of abstract ordinary differential vectorvalued equation Suppose that (v,û) ∈ V × U is a given solution of (2.11). Then the problem of stabilization for solution to (2.10) near steady-state solution (v,û) with a prescribed rate σ > 0 is to find u(t, ·) such that the solution v(t, ·) of problem (2.10) with this u(t, ·) substituted into, satisfies the inequality Recall that classical formulation of a stabilization problem by feedback control is as follows: Given steady-state solution (v,û) of (2.10), find a map R : satisfies (2.12). (R does not depend on v 0 ) Remark 2.1. Usual assumption imposed on the exponent σ in this formulation is that σ is a certain positive but it is not an arbitrary positive as in formulation (2.3)-(2.9). But usually a map R is looked for by such a way that boundary value problem (2.13) is well posed, i.e. inequality (2.12) remaines true after small fluctuations of data for this problem .
If stabilization problem by feedback control should be solved only for initial conditions v 0 belonging to a certain neighborhood ofv, then it is called a local stabilization problem. This classical setting of stabilization problem by feedback control was successfully applied not only for controlled ordinary differential equations but also for certain controlled PDE including 2D Navier-Stokes equations with distributed control supported on the whole domain containing liquid (see [1]). Nevertheless this approach did not give yet possibility to solve stabilization problem for general quasi linear parabolic equation or for Navier-Stokes system with feedback control supported on the boundary of the domain as our new setting proposed in [6], [7]. Properties of Euler equations regarded in [3] differ essentially from properties of parabolic equation or Navier-Stokes system.

The main idea of construction.
Construction of feedback control which we proposed in [6], [7] is not included in the framework of Definition 2.1. Let recall this construction.
Let ω ⊂ R 3 be a bounded domain such that We set G = Int(Ω ∪ ω) (2.15) (the notation Int A means, as always, the interior of the set A). We suppose that ∂G is a two-dimensional surface belonging to the smoothness class C ∞ . We extend problem (2.3)-(2.6) from Q = R + × Ω to Θ = R + × G For this end we forget for a while about the second boundary condition in (2.6) and write this extended problem as follows: where right side g(x) is the same as in (2.16). (We show below how to construct such extension.) Note that, actually, w 0 from (2.17) will be a special extension of v 0 in (2.5) from Ω to G : w 0 = Ext σ v 0 . More precisely, w 0 should belong to the stable manifold M σ which is invariant with respect to the semigroup generated by the Navier-Stokes problem (2.16)-(2.18) and which contains solutions w(t, ·) tending to a with the rate σ (as in (2.9)). More detailed definition of M σ will be given in Section 6. We introduce the following space of solenoidal vector fields: where H k (G) is Sobolev space of functions f (x), x ∈ G belonging to L 2 (Ω) together with their derivatives of order not more than k. Definition of H k (G) with fractional or negative k see in [15].
. For vector fields defined on G we introduce the operator γ Ω of restriction on Ω and the operator γ Γ of restriction on Γ: As well known (see, for instance, [15]), operators (2.21) are bounded.
x)) of (2.3)-(2.6) is defined by the equality: where w(t, x) satisfies to (2.16)-(2.18), and γ Ω , γ Γ are operators of restriction of a function defined on G to Ω and to Γ respectively. This definition of feedback control is basic for us: we will use only it below. Now we show connection between Definitions 2.1 and 2.2.
Let S(t, w 0 ) be the semigroup generated by boundary value problem (2.16)-(2.18), i.e. if w(t, ·) is a solution to (2.16)-(2.18) with initial condition w 0 , then S(t, w 0 ) = w(t, ·). Then the map which acts initial condition v 0 from (2.5) to the solution v(t, ·) of (2.3)-(2.5) is defined as follows: Using (2.23) we define the following extension operator from Ω to G: where v 0 is the initial condition of v(t, ·). Note that operator E(t, ·) depends on t because it is defined on the set of functions which belong to the image of the operator S(t, Ext σ ·) and this set depends on t. Now we can define the operator which acts solution v(t, ·) to the correspondent control function u(t, ·): In contrast to operator R from Definition 2.1 operator (2.25) depends on t.
3. Ozeen equations. We begin investigation of stabilization problem from the case of linearized Navier-Stokes equations, i.e. from the Ozeen equations. Note that the results of this section connected with 3D Oseen equations as well as their proofs are absolutely identic to analogous results and proofs for 2D Oseen equations. That is why we give here only their short formulation. Detailed exposition of these results can be found in [7].

Preliminaries.
Let G be domain (2.15). We consider in R + × G the Oseen equation which is written as follows: 3) Moreover, we impose on w the zero Dirichlet boundary condition w| S = 0, (3.4) where S = R + × ∂G. We assume that where, recall, Sobolev spaces V k (G), H k (G) were defined above, and where ν(x) is the vector-field of outer normals to ∂G. In [16] it is established that the operator of orthogonal projection. We consider the Ozeen steady state operator where a(x) is vector-field (3.5). This operator is closed and it has the domain: which is dense in V 0 0 (G). Assuming that spaces in (3.7), (3.9) are complex we denote by ρ(A) the resolvent set of operator A, i.e. the set of λ ∈ C such that the resolvent operator is defined and continuous. Here I is identity operator. Denote by Σ(A) ≡ C 1 \ ρ(A) the spectrum of operator A.
be an eigenvalue of −A, and e = 0, e ∈ ker(λ 0 I + A) be an eigenvector. Vector e k is called associated vector of order k to e if e k satisfies: We say that e, e 1 , e 2 , . . . form a chain of associated vectors. The maximal order m of vectors, associated to e is finite and the number m + 1 is called multiplicity of the eigenvector e.

Definition 3.1. The set of eigenvectors and associated vectors
corresponding to an eigenvalue −λ j is called canonical system if: i) Vectors e (k) (−λ j ), k = 1, 2, . . . , N(−λ j ) form a basis in the space of eigenvectors corresponding to the eigenvalue −λ j . ii) e (1) (−λ j ) is an eigenvector with maximal possible multiplicity.
Besides canonical system (3.16) which corresponds to an eigenvalue −λ j of operator −A we consider a canonical system that corresponds to the eigenvalue −λ j of the adjoint operator −A * . Definition of canonical system (3.17) is absolutely analogous to Definition 3.1 of canonical system (3.16). We define canonical system (3.17) by E * (−λ j ).
Theorem 3.1. Let R k are operators defined in (3.12). Then This assertion follows immediately from one result of Keldysh [12] on structure of the main part of Laurent series for R(λ, −A). The proof of Theorem 3.1 see in [7].
In virtue of Cauchy Theorem we reduce integration over γ in (3.19) to integration over γ σ and integration around poles −λ j from (3.12) for λ j satisfying Reλ j < σ After calculation corresponding residues we transform (3.19) to the equality: Proof see in [6], [7].

unique continuation property.
To solve stabilization problem we will use unique continuation property for solution of adjoint Oseen equation, i.e. for solution of equation . Unique continuation property for the Stokes equations has been established in [9] with help of Carleman estimates derived in [11]. That Stokes equation differs from adjoint Oseen equation indicated above. That is why we give here complete proof of the unique continuation property for a solution of (µ 0 − A) * w = 0. As in [11], [9], to do it we use Carleman estimate, but our technology differ from techniques of [11] and it is close to methods where µ 0 is an eigenvalue of the operator A * and w is a corresponding eigenvector. Note that generally speaking µ 0 is a complex number and w is a complex-valued vector field. As usual, the bar over notation of a complex number means the operation of complex conjugation. By (3.13) and by definition of operator π this equality can be rewritten as follows: Applying to the first equality in (4.1) operator div and taking into account the second equality div v = 0 we get First of all we prove some Carleman estimate for solution of (4.1), (4.2).

Carleman estimate.
We consider the following analog of (4.1),(4.2): (4.4) We suppose also that z(x) and p(x) satisfy on ∂G the following equalities: Let ε > 0 be sufficiently small. Denote (the sign "minus" in lower index in G −ε means that G −ε is a subset of G in contrast of notation Ω ε used below in Theorem 5.1 when Ω ε contains Ω) Let ω be a subdomain compactly embedded to G −ε : ω ⊂⊂ G −ε . We consider a function β(x) ∈ C 2 (G) which has no critical points outside ω , i.e.
We also introduce the function: Then there exists a magnitude λ 0 > 0 such that for each λ > λ 0 the following Carleman estimate holds: where the constant c > 0 does not depend on z, p, f , and λ > λ 0 .
Proof. We reduce our problem to such one that all conditions of Theorem 4.1 are fulfilled. Recall that G −δ = {x ∈ G : dist(x, ∂G) > δ}. We can suppose that for a sufficiently small δ > 0 the set ω in (4.34) satisfies condition ω ⊂⊂ G −δ , i.e. dist(ω, ∂G) > δ. Otherwise we change ω on its a certain open subset.
We reduce now problem (4.1), (4.33) to problem (4.3), (4.5). To do this we consider a function ψ(x) ∈ C ∞ (G) satisfying Let us consider the boundary value problem is the vector field of outer normals to ∂G. As well-known (see, for instance [16]

there exists a solution w(x) of this problem and w(x) ∈ H 4 (G). We introduce the vector field
Note that in virtue of (4.34) the componentp(x) of solution (v,p) to (4.1), (4.33) satisfies ∇p(x) ≡ 0 for x ∈ ω. Sincep(x) is defined from (4.1) to within arbitrary constant we can choose this constant such that , divz(x) = 0, for x ∈ G and z, p satisfy (4.5). Besides, for x ∈ G −δ (z(x), p(x)) = (v(x),p(x)). Therefore if we substitute (z(x), p(x)) in the left side of (4.1), we obtain equations (4.3) with a right side f (x) which satisfies f (x) = 0 for x ∈ G −δ/2 (4.38) As a result we see that the triplet (z, p, f ) satisfies all condition of Theorem 4.1 and therefore estimate (4.10) is true. In virtue of (4.9), (4.34)-(4.37) this estimate implies the following upper bound: which is true for each λ λ 0 and c in (4.39) does not depend on λ λ 0 . Assume that there exists a set Λ ⊂ G −δ of positive Lebesgue measure such that |z(x)| 2 + |p(x)| 2 > 0 for x ∈ Λ. Then (4.39) is not true for sufficiently large λ because f satisfies (4.38), and for β(x) the second equality in (4.7) is true. Hence |z(x)| 2 + |p(x)| 2 = 0 for x ∈ G −δ . In virtue of (4.35),(4.37) the solution (v(x),p(x)) of (4.1) also satisfies the equality Since δ > 0 can be chosen arbitrary small, desired assertion of the Theorem 4.2 has been proved.

On linear independence of ε (k)
l (x, −λ j ). We set some strengthening of well-known result on linear independence of eigenvectors and associated vectors for operator A * which is defined in (3.13). To prove this result we use Theorem 4.2

Theorem 4.3. Consider the set
of canonical systems (3.17) for operator −A * with σ satisfying (3.20).Then for an arbitrary subdomain ω ⊂ G vector fields ε Proof.The main part of Theorem 4.3 is to prove that eigenvectors Since this v(x) with a certain p(x) satisfies (4.1),(4.33), in virtue of Theorem 4.2 equality v(x) = 0, x ∈ ω,imply that v(x) = 0 for x ∈ G. Since by Definition 3.1 eigenvectors ε (k) (−λ j , x) are linear independent on G, the last equality implies that c k = 0, k = 1, ..., N . Note that only in this part of proof we use specific of equation (4.1). The general assertion of Theorem 4.3 is derived from the property proved above with help of some general arguments written in [6].

Stabilization of Oseen equations.
3D NAVIER-STOKES SYSTEM BY FEEDBACK BOUNDARY CONTROL 303 5.1. Setting of the problem. As in section 2 we suppose that Ω ⊂ R 3 is a bounded connected domain with C ∞ -boundary ∂Ω, which is decomposed on two parts: where Γ, Γ 0 are closed subsets of ∂Ω and Γ ∩ Γ 0 = ∅. The case Γ 0 = ∅ is possible. In other words if where ∂Ω j are closed connected components of ∂Ω then (possibly after renumeration of ∂Ω j ) with initial and boundary conditions where a(x) = (a 1 (x), a 2 (x), a 3 (x)) is a solenoidal vector field (div a = 0) and u = (u 1 , u 2 , u 3 ) is a control. Besides solenoidalness of initial condition v 0 (x) we suppose that ∂Ωj (v 0 (x ), n(x ))dx = 0, j = 1, . . . , l. (5.8) Analogously to section 2 stabilization problem for Oseen equations is formulated as follows: Given σ > 0 find a control u on Σ such that the solution v(t, x) of problem (4.2)-(4.5) satisfies the inequality v(t, x) L2(Ω) ce −σt (5.9) where c > 0 depends on v 0 , σ and Γ. Moreover, we require that this control u satisfies the feedback property in the meaning analogous to Definition 2.2: firstly we extend by a special way problem (5.4)-(5.7) (without second equality from (5.7)) to the problem (3.1)-(3.4) defined on a domain G ⊃ Ω, solve the last problem, and after that we define the solution (v, u) of stabilization problem (5.4)-(5.9) by the formula (2.22) Details of this definition will be given simultaneously with the construction of feedback control.

Theorem on extension.
First of all we define the set ω from (2.14), (2.15) which is used to extend the domain Ω to the set G. Note that being closed each component ∂Ω j of ∂Ω separates R 3 on two parts: Ω j− and Ω j+ . By definition points of Ω j− which are close enough to ∂Ω j belongs to Ω. Taking a sufficiently small magnitude κ > 0 we define the set ω as follows: where l is defined in (5.3). Now we define the domain G by the formula: We introduce the following spaces where k is nonnegative integer, and ∂G j , j = 1, ..., J) are all connected components of the boundary ∂G, and n is outer normal to ∂G. Recall that operation of restriction onto the boundary for (u(x), n(x)) is well defined for u ∈ V k (G) with each k 0 (see [16]). Below we use well-known operator rot which is defined by the formula: It is clear that for each gradient vector field ∇p(x), p ∈ H 1 (G) the inclusion ∇p(x) ∈ ker rot is true. Note that, generally speaking, for the functions space V 0 0 (G) defined in (3.6) the inequality V 0 0 (G) ∩ ker rot = {0} holds. Indeed, the following orthogonal decomposition with respect to the scalar product in L 2 (G) is true (see, e.g. [10] and [16, Appendix 1,pp.458-471]): where H c = V 0 0 (G)∩ker rot is a finite-dimensional space of C ∞ -vector fields isomorphic to the space of the first cohomologies of G. H c consists of vector fields ∇p(x), where p(x) are multi-valued functions satisfying ∆p = 0 and (∂p/∂n)| ∂G = 0; for details see [16,Appendix 1]. Now the functions space W 0 (G) is well defined by equality (5.13). For each integer k 0 we define where (H k (G)) 3 is usual Sobolev space of vector valued functions of smoothness k. is an isomorphism.
Introduce now an open subsetΩ = G \ Ω ε/2 . We look for extension 22) where L is operator (5.20) andŵ(x) is a vector field which satisfies: By virtu of (5.17) to establish inclusion E 1 where k = 1, . . . , K. At last, to determineŵ uniquely we suppose that where A = {w : w satisfies (5.23), (5.24)}. (5.25) Step 3.We have to show that there exists unique vector fieldŵ that satisfies (5.25). To do this we define the operator R by the formula: We claim that Im R = R K . Indeed, if this is not true, there exists a vector p = (p 1 , . . . , p K ) = 0 such that This equality implies that Since in virtue of (3.5) ε j (x) ∈ V 3 (G), we get that q(x) ∈ H 4 (Ω). Therefore, equality ε j | ∂G = 0 implies that ∇q| ∂G\Γ0 = 0. As a result we obtain: where c j are constants and ∂ n is the derivative with respect to the vector field n of outer normals to ∂G. Applying to both parts of (5.27) operator div we get that ∆q(x) = 0, x∈Ω (5.29) In virtue of uniqueness of solution for Cauchy problem (5.29), (5.28), q| ωj = c j and therefore (5.27) implies K j=1 p j ε j (x) = 0 x ∈Ω This equality and Lemma 4.2 imply p j = 0, j = 1, . . . , K that contradicts to the assumption (p 1 , . . . , p K ) = 0. Since Im R = R K , the set A of admissible elements for problem (5.25) is not empty. In virtue of definition (5.25) A is a closed convex subset of V 1 0 (G), and (5.25),actually, is the problem to determine the distance from origin to the set A in the Hilbert space V 1 0 (G). As well known, this problem has unique solutionŵ(x).
Step 4. Existence and uniqueness of solution for problem (5.25) implies that the operator E 1 that transforms a vector field v ∈ V 1 (Ω, Γ 0 ) to the solutionŵ of problem (5.25): E 1 v =ŵ, is well defined. To finish the proof of the Theorem we have to show that the operator is linear and bounded. We derive optimality system for minimization problem (5.25) with help of Lagrange principle. As one can see, for instance, in [5], the relation Im R = R K which was proved for the operator R defined in (5.26) guarantees, that we can apply to (5.25) Lagrange principle when Lagrange multiplier before minimized functional equals one. This Lagrange function has a form: By Lagrange principle for solutionŵ of (5.25) there exists a vector p = (p 1 , . . . , p K ) such that for each h(x) ∈ V 1 0 (Ω) In the setΩ we consider the Stokes problem: As well known, for each v ∈ V 0 (Ω) there exists unique solution w ∈ V 1 0 (Ω) ∩ V 2 (Ω) of this problem. The resolving operator of this problem we denote as follows:

Since (5.31) means thatŵ is the solution of the Stokes problem with right side
Substitution of (5.32) into (5.24) yields the linear system of equations: Relations (5.33), (5.32) implies that operator E 1 from(5.30) is linear.
To prove the boundedness of operator (5.30) we show that the matrix A = a kj is positively defined. Note that where w j (x) = (−π∆) −1 Ω1 ε j (x) and : is the sign of scalar product between two tensors. Let α = (α 1 , · · · , α K ), where q j (x) is a harmonic function inΩ (to see this one can apply the operator div to both parts of previous equality).
So, det A = 0. In virtue of (5.31), (5.32) (5.32)), ε = (ε 1 , . . . , ε K ), and therefore operator (5.30) is bounded. Proof. We can assume that σ satisfies to condition (3.20), otherwise we make it a little bit more. We act to initial condition v 0 ∈ V 1 (Ω, Γ 0 ) by the operator E 1 σ from (5.19), (5.22) and by Theorem 5.1 we obtain that 6. Stabilization of 3D Navier-Stokes equations. In this section we study the problem of stabilization a solution to the Navier-Stokes equations which is formulated in subsection 2.1. In particular, the boundary ∂Ω of the space component Ω to space-time cylinder Q = R + × Ω where the Navier-Stokes system is determined, satisfies condition (2.1). We do this stabilization with help of control determined on the part Σ = R + × Γ of the lateral surface to cylinder Q, and we consider only feedback control in the meaning of Definition 2.2.
Since a(x) is steady-state solution of (6.2), S(t, a) = a for each t 0. We can decompose semigroup S(t, w 0 ) in a neighborhood of a in the form where L t w 0 = S w (t, a)w 0 is derivative of S(t, w 0 ) with respect to w 0 at point a, and B(t, w 0 ) is nonlinear operator with respect to w 0 . Differentiability of S(t, w 0 ) is proved, for instance in [2, Ch. 7. Sect. 5]. Therefore Moreover in [2, Ch. 7. Sect. 5] is proved that B (t, w) belongs to class C α with α = 1/2 with respect to w. This means that for each and left side is a continuous function with respect to w 0 .
We study now semigroup L t w 0 = S w (t, a)w 0 of linear operators. First of all note that w(t, x) = L t w 0 is the solution of problem (3.1)- (3.4) in which the coefficient a is the solution of (6.5). Therefore where A is Ozeen operator (3.8).
Generally speaking eigenvalues of operators A and e −At are complex-valued. That is why all spaces in Theorem 6.1 are complex. But to apply obtained results to (nonlinear) Navier-Stokes equation we need to have analogous results for the real spaces of the same type. Actually, for this we have to define the projector of Π + in real spaces. Lemma 6.1. Restriction of operator Π + on the real space V 1 0 (G) can be written in the form where {ε j } is the set of functions (4.42) which are suitably renumbered and renormalized functions (4.41) and {e j } is set of Real and Imaginary parts of functions (3.16) analogously renumbered and renormalized.
The proof of this simple lemma one can find in [6].
To prove this Lemma we first establish analog of Theorem 4.1 for functions (3.16). After that we derive Lemma 6.2 from this Theorem by the same way as Lemma 4.2 was derived from Theorem 4.1.
Using (6.13) we can easily restrict spaces X + and X − as well as operators L + t0 , L − t0 defined in formulation of Theorem 6.1 on the real subspaces of V 1 0 (G). We denote this new real spaces and operators also by X + , X − , L + t0 , L − t0 . This will not lead to misunderstanding because below we do not use their complex analogs.
In a neighborhood of steady-state solution a of (6.5) we establish existence of a manifold M − which is invariant with respect to semigroup S(t, w) (i.e., ∀w ∈ M ∀t > 0, S(t, w) is well defined and for each t > 0, S(t, w) ∈ M − ). This manifold can be represented as the graph: where O is a neighborhood of origin in V 1 0 (G), g : X − ∩ O → X + is an operatorfunction of class C 3/2 and g(0) = 0, g (0) = 0. (6.15) Note that condition (6.15) means that manifold (6.14) is tangent to X − at point a.

Extension operator.
Here we construct extension operator for Navier-Stokes equations. This operator is nonlinear analog of extension operator (5.19) constructed for Ozeen equations.
Recall that the domain Ω and its extension G satisfy (5.10), (5.11). Besides, the space V 1 (Ω, Γ 0 ) is defined in (5.18). where w(x) is a vector field concentrated inΩ = G \ Ω ε/2 which is constructed by v(x). We describe its construction below. At last we define the desired operator Ext σ by the formula Ext σ v = Π − Qz + g(Π − Qz) + a, with z = v − a, (6.20) where Π − = I − Π + , Π + is operator (6.13) of projection on X + = Π + V 1 0 (G), X − = Π − V 1 0 (G), and g : X − → X + is the operator constructed in Theorem 6.2. By definition (6.14) of M − we have Ext σ v ∈ M − . Hence we have to ensure that the equality (Ext σ v)(x) ≡ v(x), x ∈ Ω (6.21) is true, that shows that Ext σ is an extension operator. By (6.13) {e j (x)} generates X + and therefore the map g(u) can be written in the form That is why taking into account (6.13) we can rewrite (6.20) in the form