Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php SIAM J CONTROL OPTIM Vol 53, No 5, pp 3006–3039 c 2015 Society for Industrial and Applied Mathematics BOUNDARY STABILIZATION OF THE NAVIER–STOKES EQUATIONS IN THE CASE OF MIXED BOUNDARY CONDITIONS∗ PHUONG ANH NGUYEN† AND JEAN-PIERRE RAYMOND‡ Abstract We study the boundary feedback stabilization, around an unstable stationary solution, of a two dimensional fluid flow described by the Navier–Stokes equations with mixed boundary conditions The control is a localized Dirichlet boundary control A feedback control law is determined by stabilizing the linearized Navier–Stokes equations around the unstable stationary solution We prove that the linear feedback law locally stabilizes the Navier–Stokes system Thus, we extend results previously known in the case when the boundary of the geometrical domain is regular and when only Dirichlet boundary conditions are present to the case with mixed boundary conditions Key words incompressible Navier–Stokes equations, feedback control, mixed boundary conditions AMS subject classifications 93B52, 93C20, 93D15, 35Q30, 76D55, 76D05, 74D07 DOI 10.1137/13091364X Introduction We consider the Navier–Stokes equations with mixed boundary conditions, of Dirichlet and Neumann types, in two dimensional domains We are interested in stabilizing such fluid flows, in a neighborhood of an unstable stationary solution, by a Dirichlet boundary control This type of problem has already been considered in [3, 4, 41, 43, 45] in the case when only Dirichlet boundary conditions are involved in the model The presence of mixed boundary conditions is the source of several difficulties Indeed, in the case of mixed boundary conditions the solutions to the stationary Stokes equations [36], or the solutions to the stationary Navier–Stokes equations are less regular than in the case when only Dirichlet conditions are present; see, e.g., [37] The existence of solutions to the stationary Navier–Stokes equations is guaranteed only for small data [39, 21] As far as we know, the existence of solutions to the instationary Navier–Stokes equations with mixed boundary conditions, and nonhomogeneous Dirichlet conditions whose normal component is not zero, is not yet studied in the literature We refer to [26] for the case when the normal component of the Dirichlet condition is zero and where there is no junction between Neumann and Dirichlet boundary conditions From the point of view of control theory, the null controllability of the linearized Navier–Stokes equations is nowadays well understood when only Dirichlet conditions are present and when the boundary of the domain occupied by the fluid is regular; see, e.g., [17] As far as we know, there is no similar result in the case of mixed boundary conditions Therefore, the extension of results obtained in [41] to the case of mixed boundary conditions is not trivial ∗ Received by the editors March 20, 2013; accepted for publication (in revised form) July 23, 2015; published electronically September 15, 2015 http://www.siam.org/journals/sicon/53-5/91364.html † International University, Vietnam National University - Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam (npanh@hcmiu.edu.vn) This author was partially supported by VIASM ‡ Universit´ e de Toulouse, UPS, Institut de Math´ ematiques, 31062 Toulouse Cedex, France, and CNRS, Institut de Math´ ematiques, UMR 5219, 31062 Toulouse Cedex, France (jean-pierre.raymond@ math.univ-toulouse.fr) This author was partially supported by the ANR-project CISIFS 09-BLAN0213-03 and by the project ECOSEA from FNRAE 3006 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3007 We are interested in fluid flows modeled by the Navier–Stokes equations in a two dimensional bounded domain Ω, with a boundary Γ = Γd ∪ Γn , when Dirichlet boundary conditions are applied on Γd = Γo ∪ Γi ∪ Γc , while a homogeneous Neumann boundary condition is prescribed on Γn The subset Γc , relatively open in Γd , corresponds to the control zone An inflow boundary condition is prescribed on the subset Γi , while homogeneous boundary conditions are imposed on Γo Precise assumptions on Ω and Γ are stated in section 2.1 We assume that (ws , qs ) ∈ H (Ω; R2 ) × L2 (Ω) is a stationary solution to the equation (ws · ∇)ws − div σ(ws , qs ) = fs , (1.1) on Γc ∪ Γo , ws = σ(ws , qs )n = gs ws = us div ws = in Ω, on Γi , on Γn , where σ(ws , qs ) = ν(∇ws + (∇ws )T ) − qs I is the usual Cauchy stress tensor, ν > is the viscosity of the fluid, the inflow velocity us is applied on Γi , fs and gs are stationary data A particular configuration which satisfies the assumptions stated in section 2.1 corresponds to a fluid flow around a circular cylinder in a rectangular channel, with a Neumann boundary condition at the end of the channel, and a parabolic profile us in the inflow boundary condition; see [1, 2] and Figure Associated with equation (1.1), we consider the controlled Navier–Stokes system ∂w + (w · ∇)w − div σ(w, q) = fs , ∂t w = M u on Σ∞ c = Γc × (0, ∞), (1.2) w = us on Σ∞ i = Γi × (0, ∞), div w = in Q∞ = Ω × (0, ∞), w = on Σ∞ o = Γo × (0, ∞), on Σ∞ n = Γn × (0, ∞), σ(w, q)n = gs w(0) = ws + z0 on Ω Here, M is a truncation function, precisely described later on (see assumption (H4 ) in section 2.1), used to localize the action of the control u on a part of Γc , and z0 is a perturbation whose presence will destabilize the stationary solution ws As noticed in [42] in the case of Dirichlet boundary condition, the initial condition w(0) = ws + z0 has to be understood as w(0) φ dx = Ω Ω (ws + z0 ) φ dx for all φ ∈ {z ∈ L2 (Ω; R2 ) | div z = 0, z · n = on Γd } This is equivalent to Πw(0) = Π(ws + z0 ), where Π is the so-called Leray projector introduced in section 2.2 Since z0 is chosen such that Πz0 = z0 , we shall write Πw(0) = Πws + z0 Our goal is to find a feedback control law K ∈ L(L2 (Ω; R2 ), L2 (Γc ; R2 )) such that the closed loop system ∂w + (w · ∇)w − div σ(w, q) = fs , ∂t w = M K(w − ws ) on Σ∞ c , (1.3) w = us on Σ∞ i , σ(w, q)n = gs div w = in Q∞ , w = on Σ∞ o , on Σ∞ n , Πw(0) = Πws + z0 on Ω, Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3008 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php obeys eωt (w(t) − ws ) H ε (Ω;R2 ) −→ as t tends to infinity for a given decay rate −ω < 0, and for z0 small enough in H ε (Ω; R2 ), and such that div z0 = in Ω and z0 · n = on Γd Similar results have been obtained in the case when only Dirichlet boundary conditions are prescribed on Γ; see [41, 43] The case of Neumann boundary conditions is also studied in [5] The results in [5] cannot be adapted to deal with the case of mixed boundary conditions with junction points between the Dirichlet and Neumann boundary conditions In all these previous results, the feedback control law is determined by looking for a feedback stabilizing the linearized Navier–Stokes equations around the stationary solution (ws , qs ), which is next applied to the nonlinear system Here, we follow the same approach Setting z = w − ws and p = q − qs , the nonlinear system satisfied by (z, p) is ∂z + (ws · ∇)z + (z · ∇)ws + (z · ∇)z − div σ(z, p) = 0, ∂t div z = in Q∞ , (1.4) z = M u on Σ∞ d = Γd × (0, ∞), σ(z, p)n = on Σ∞ n , Πz(0) = z0 on Ω, the Dirichlet boundary condition on Γd × (0, ∞) takes into account the homogeneous ∞ boundary conditions on Σ∞ i ∪Σo because M is supported in Γc ; see assumption (H4 ) The linearized system is (1.5) ∂z + (ws · ∇)z + (z · ∇)ws − div σ(z, p) = 0, ∂t z = M u on Σ∞ d , div z = in Q∞ , σ(z, p)n = on Σ∞ n , Πz(0) = z0 on Ω The plan of the paper is as follows The precise assumptions on the geometrical domain and the boundary conditions are stated at the beginning of section We next study the linearized Navier–Stokes equations with nonhomogeneous boundary conditions in the same section The stabilizability and the feedback stabilization of the linearized Navier–Stokes equations by finite dimensional controls is established in section The local feedback stabilization of the Navier–Stokes equations is proved in section We shall see that the local stabilization result is stated for initial data in 0 (Ω) for any < ε < 1/2 (Vn,Γ (Ω) is the subspace of divergence free H ε (Ω; R2 ) ∩ Vn,Γ d d 2 functions in L (Ω; R ) whose normal trace is zero on Γd ; see section 2.2) Contrary to the case of the two dimensional Navier–Stokes equations with Dirichlet boundary conditions, we cannot hope to have such a result with ε = 0, that is, with initial data (Ω); see Remark 4.3 Thus the fixed point method used in [3, 4, 41, 43, 45] in in Vn,Γ d the space L2 (0, ∞; H (Ω; R2 )) ∩ H 1/2 (0, ∞; L2 (Ω; R2 )), when only Dirichlet boundary conditions are involved, is not appropriate here (Ω) with < ε < 1/2, we are able to By choosing initial data in H ε (Ω; R2 ) ∩ Vn,Γ d prove that the solutions to the closed loop linearized Navier–Stokes equations belong Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3009 to L2 (0, ∞; D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0, ∞; Vn,Γ (Ω))+H (0, ∞; H 3/2+ε0 (Ω; R2 )) d for some ε0 > Roughly speaking, the part of the solution corresponding to the ini0 (Ω) belongs to L2 (0, ∞; D((λ0 I − A)1/2+ε/2 )) ∩ tial condition in H ε (Ω; R2 ) ∩ Vn,Γ d H 1/2+ε/2 (0, ∞; Vn,Γ (Ω)), while the contribution of the feedback control of finite did mension provides the part which belongs to H (0, ∞; H 3/2+ε0 (Ω; R2 )) This result is established in Theorem 3.5 for the nonhomogeneous closed loop linearized Navier– Stokes system, and it is an essential step in the proof of our local feedback stabilization result Let us finally stress that, in the proof of Theorem 3.5, we strongly use the fact that the feedback is of finite dimension, and it depends only on the unstable component of the state variable, and that the unstable component of the state variable belongs to a finite dimensional space of regular functions (functions belonging to H 3/2+ε0 (Ω; R2 )) Let us also mention that additional references on the stabilization of the Navier– Stokes equations may be found in [1, 2, 7, 8, 9, 11, 18, 19, 27] The linearized Navier–Stokes equations 2.1 Assumptions We denote by J = {J1 , · · · , JNJ } the set of corner vertices corresponding to a junction between either two Dirichlet boundary conditions or a Dirichlet and a Neumann boundary condition We have to make some assumptions on the junctions between Γn and Γd (H1 ) Ω is a bounded domain in R2 (in particular Ω is connected), and Γ \ J is a submanifold of class C For any Jk ∈ J , there exists rk > such that {x ∈ R2 | dist (x, Jk ) ≤ rk } ∩ Γ is the union of two segments (H2 ) Γn is either a segment, or a union of a finite number of disjoint segments, or, more generally, a union of a finite number of regular connected components of Γ There is no junction between two segments with Neumann boundary conditions, and the junction between a segment with a Neumann boundary condition and a segment of Γd is a right angle (H3 ) Γd is a union of a finite number of regular connected components of Γ The angles of junctions between two segments with Dirichlet boundary conditions are strictly less than π (H4 ) Γc is a connected component of Γ of class C We also assume that dist(Γc , J ) > We assume that the function M is a nonnegative function, defined on Γd , with support in Γc , taking values in [0, 1], of class C , and positive in a nonempty relatively open subset Γ+ c in Γc (H5 ) We assume that the solution (ws , qs ) to (1.1) belongs to H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω), where < εo < 1/2 is the exponent appearing in Theorem 2.5 We also assume that (ws , qs )|Ωδ,J , with Ωδ,J = {x ∈ Ω | dist(x, J ) > δ}, belongs to H (Ωδ,J ; R2 ) × H (Ωδ,J ) for all δ > We make no explicit assumption on Γi and Γo except that Γi ∪Γo is relatively open in Γd The only assumption we need is actually the regularity condition (ws , qs ) ∈ H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω) stated in (H5 ) We not claim that, for any Ω satisfying (H1 )–(H4 ), any fs ∈ L2 (Ω; R2 ), us ∈ H 3/2 (Γi ; R2 ) ∩ H01 (Γi ; R2 ), gs ∈ H 1/2 (Γn ; R2 ), satisfying some compatibility conditions, (1.1) admits a solution (ws , qs ) in H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω) We claim that there exist configurations for which such solutions exist For that, we refer to the appendix The condition ‘Γc is of class C ’ is used to prove that the solution (φ, ψ) to (3.3) is of class H for φ and H for ψ in a neighborhood of Γ+ c (see (H4 ) for the assumptions on Γ+ ) c Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3010 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND In order to study the linearized Navier–Stokes equations with nonhomogeneous boundary conditions w = M u not satisfying the condition Γc M u · n dx = 0, we have to construct particular solutions, and for that we have to define tubular domains in a very precise way Definition 2.1 An infinite tube of width δ > is a closed connected subset in R2 limited by two simple curves C1∞ and C2∞ of class C , distant from δ, that is such that the distance from any point of C1∞ to C2∞ is δ Such a tube will be denoted by T (δ; C1∞ , C2∞ ) A finite tube of width δ > of extremities S1 and S2 , with S1 ∩ S2 = ∅, is a closed connected subset of an infinite tube T (δ; C1∞ , C2∞ ) limited by two segments S1 and S2 of length δ, orthogonal to C1∞ and C2∞ , that is such that S1 ∩C1∞ and S1 ∩C2∞ contain exactly one point, and S2 ∩ C1∞ and S2 ∩ C2∞ contain exactly one point Such a tube will be denoted by T (δ; C1 , C2 ; S1 , S2 ), where C1 ⊂ C1∞ and C2 ⊂ C2∞ are such that ∂T (δ; C1 , C2 ; S1 , S2 ), the boundary of the finite tube, is equal to C1 ∪ C2 ∪ S1 ∪ S2 Each point x ∈ T (δ; C1 , C2 ; S1 , S2 ) belongs to a unique segment Sx ⊂ T (δ; C1 , C2 ; S1 , S2 ) orthogonal to C1 and to C2 , and a unique curve Cx ⊂ T (δ; C1 , C2 ; S1 , S2 ) parallel to C1 and C2 Thus any point can be localized by its arc length coordinate on Cx , denoted by (x), and its transverse coordinate ρ(x) ∈ [−δ/2, δ/2] in Sx , with the convention that the transverse coordinate of the points belonging to C1 is −δ/2, while the transverse coordinate of points belonging to C2 is δ/2 We also make the convention that the arc length coordinate of points in S1 is zero and that the arc length coordinate of a point x ∈ S2 is the length of the curve Cx To each x ∈ T (δ; C1 , C2 ; S1 , S2 ), we associate τ (x), the unitary tangent vector to Cx at point x oriented in the sense of increasing arc length coordinates We can now state an assumption needed to construct solutions to the linearized Navier–Stokes equations with nonhomogeneous boundary conditions (see Theorem 2.16) (H6 ) There exist a finite tube T (δ; C1 , C2 ; S1 , S2 ) of width δ > 0, and a nonnegative function η belonging to Cc∞ (R), with values in [0, 1], with compact support in (−δ/2, δ/2), and obeying η(0) = 1, such that (i) S2 ⊂ Γn , S1 ⊂ R2 \ Ω; (ii) (T (δ; C1 , C2 ; S1 , S2 ) \ S2 ) ∩ Γ = T (δ; C1 , C2 ; S1 , S2 ) ∩ Γc := Γc,T ; (2.1) (iii) For all x ∈ T (δ; C1 , C2 ; S1 , S2 ), we assume that Cx ∩ Γc,T is reduced to a point, denoted by γc (x), such that Cx ∩ Ω = {x ∈ Cx | (x) ≥ (γc (x))} and (2.2) Γc,T η(ρ(x)) τ (x) · n(x) dx = In (2.2), dx is the one dimensional Lebesgue measure on Γc , τ (x) is the unitary vector tangent to Cx , oriented in the sense of increasing arc length coordinates, and n(x) is the unitary normal to Γc exterior to Ω Let us notice that condition (2.2) is not restrictive Indeed, if there exists a tube T (δ; C1 , C2 ; S1 , S2 ) satisfying (2.1), we can always modify T (δ; C1 , C2 ; S1 , S2 ) in a neighborhood of S1 in such a way that (2.2) is satisfied Figure is an example of a domain with a tube satisfying (H6 ) 2.2 Some function spaces In order to write (1.5) as a controlled system, we have to introduce the Leray projector associated with the boundary conditions of our problem We shall define the Oseen operator in section 2.4 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3011 STABILIZATION OF THE NAVIER–STOKES EQUATIONS Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Γo Γi Γc Γn C2 S1 S2 C1 Γo Fig Domain with a tube satisfying (H6 ) In the case of mixed Dirichlet/Neumann boundary conditions, we introduce the space (Ω) = z ∈ L2 (Ω; R2 ) | div z = in Ω, z · n = on Γd Vn,Γ d Lemma 2.2 We have the following orthogonal decomposition L2 (Ω; R2 ) = Vn,Γ (Ω) ⊕ grad HΓ1n (Ω), d HΓ1n (Ω) = {p ∈ H (Ω) | p = on Γn } (Ω) will be denoted by Π, and called The orthogonal projector in L2 (Ω; Rd ) onto Vn,Γ d the Leray projector for the above decomposition Proof This type of result may be deduced from results in [26] in the case when Γd and Γn have no junction point In the case of the present paper, let us give a short proof of that result Let us notice that, due to the divergence formula, Vn,Γ (Ω) and grad HΓ1n (Ω) are d d orthogonal closed subspaces in L (Ω; R ) For any z ∈ L (Ω; Rd ), let us denote by pz and qz the solutions to the following elliptic equations pz ∈ H01 (Ω), qz ∈ HΓ1n (Ω), Δpz = div z ∈ H −1 (Ω), Δqz = 0, ∂qz = (z − ∇pz ) · n on Γd , ∂n qz = on Γn (Ω) Thus It is clear that ∇(pz + qz ) ∈ grad HΓ1n (Ω) and that z − ∇(pz + qz ) ∈ Vn,Γ d Πz = z − ∇pz − ∇qz for all z ∈ L2 (Ω; R2 ), and the proof is complete To define the Oseen operator, we introduce the space (Ω) | z = on Γd } VΓ1d (Ω) = {z ∈ H (Ω; R2 ) ∩ Vn,Γ d Remark 2.3 Due to assumption (H2 ), we can extend Ω to a bounded Lipschitz domain Ωe ⊂ R2 in such a way that Γn ⊂ Ωe and Γd ⊂ ∂Ωe Thus, we have VΓ1d (Ω) = {z|Ω | z ∈ V01 (Ωe )} and Vn,Γ (Ω) = {z|Ω | z ∈ Vn0 (Ωe )}, d where V01 (Ωe ) = {z ∈ H (Ωe ; R2 ) | div z = 0, z|∂Ωe = 0} and Vn0 (Ωe ) = {z ∈ L2 (Ωe ; R2 ) | div z = 0, z · n|∂Ωe = 0} Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3012 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND If we identify Vn,Γ (Ω) with its dual, and if VΓ−1 (Ω) denotes the dual of VΓ1d (Ω), d d we have (Ω) → VΓ−1 (Ω) VΓ1d (Ω) → Vn,Γ d d with dense and continuous embeddings The density follows from Remark 2.3 and from the density of V(Ωe ) = {z ∈ D(Ωe ; R2 ) | div z = 0, z|∂Ωe = 0} in Vn0 (Ωe ) for the L2 -topology (see, e.g., [23]) We also introduce the intermediate spaces ε (Ω) = [Vn,Γ (Ω), VΓ1d (Ω)]ε Vn,Γ d d for < ε < 1/2 (See [12] or [33] for the definition of spaces defined by the so-called complex interpolation.) In the following, we shall also need the space HΓ1d (Ω) = {z ∈ H (Ω; R2 ) | z = on Γd } 2.3 The Stokes equation with homogeneous boundary conditions Before defining the Oseen operator in the next section, we first consider the equation −div σ(z, p) = f, (2.3) z=0 div z = on Γd , σ(z, p)n = in Ω, on Γn We assume that f belongs to L2 (Ω; R2 ) We shall say that (z, p) ∈ VΓ1d (Ω) × L2 (Ω) is a weak solution to (2.3) if and only if it satisfies the following mixed variational formulation find (z, p) ∈ HΓ1d (Ω; R2 ) × L2 (Ω) such that a0 (z, φ) − b(φ, p) = f φ dx Ω ∀φ ∈ HΓ1d (Ω; R2 ), ∀ψ ∈ L (Ω), b(z, ψ) = where a0 (z, ζ) = (2.4) Ω ν ∇z + (∇z)T : ∇ζ + (∇ζ)T dx and b(z, ψ) = div z ψ dx Ω The following theorem is deduced from [38, Theorem 9.1.5] Theorem 2.4 Let us assume that f ∈ L2 (Ω; R2 ) Equation (2.3) admits a unique solution (z, p) ∈ VΓ1d (Ω) × L2 (Ω) and z VΓ1 (Ω) d + p L2 (Ω) ≤C f L2 (Ω;R2 ) According to [25, page 174] (see also [39]), since there is no reentrant corner at a junction between two Dirichlet boundary conditions, the restriction to Ωδ,Jd,n of the solution (z, p) to (2.3) belongs to H (Ωδ,Jd,n ; R2 ) × H (Ωδ,Jd,n ), where Ωδ,Jd,n = {x ∈ Ω | dist(x, Jd,n ) > δ} and Jd,n is the set of junction points between Dirichlet and Neumann boundary conditions Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3013 Thus, to study the regularity of solutions to (2.3), we introduce weighted Sobolev spaces as in [39, 38, 36] For β > 0, we introduce the norms z Wβ2,2 (Ω;R2 ) |k|=0 i=1 (2.5) p Wβ1,2 (Ω) = Ω j∈J d,n = |k|=0 Ω j∈J d,n rj2β |∂k zi |2 dx and rj2β |∂k p|2 dx, where rj stands for the distance to the junction point Jj ∈ Jd,n , k = (k1 , k2 ) ∈ N2 denotes a two-index, |k| = k1 + k2 is its length, ∂k denotes the corresponding partial differential operator, and z = (z1 , z2 ) We denote by Wβ2,2 (Ω; R2 ) (respectively, Wβ1,2 (Ω)) the closure of C0∞ (Ω \ Jd,n ; R2 ) (respectively, C0∞ (Ω \ Jd,n )) in the norm · W 2,2 (Ω;R2 ) (respectively, · W 1,2 (Ω) ) According to [39, 38, 36], in order to determine β β the exponent β of weighted Sobolev spaces in which the solution to (2.3) will belong when f ∈ L2 (Ω; R2 ), we have to consider the complex roots λ to the equation (2.6) λ2 sin2 (π/2) − cos2 (λπ/2) = λ2 − cos2 (λπ/2) = Let us notice that, if ≤ Re λ < 1, (2.6) is equivalent to cos(λθ)(λ2 sin2 (θ) − cos2 (λθ)) = when θ = π/2, which is the equation that we have to consider in the case of a junction point between a Dirichlet and a Neumann condition with angle θ (see, e.g., [36, page 761]) We have the following regularity result Theorem 2.5 Let us assume that f ∈ L2 (Ω; R2 ) The unique solution (z, p) ∈ VΓd (Ω) × L2 (Ω) to (2.3) belongs to Wβ2,2 (Ω; R2 ) × Wβ1,2 (Ω) with z Wβ2,2 (Ω;R2 ) + p Wβ1,2 (Ω) ≤C f L2 (Ω;R2 ) for some < β < 1/2 In particular, there exists ε0 ∈ (0, 1/2) such that z H 3/2+ε0 (Ω;R2 ) + p H 1/2+ε0 (Ω) ≤C f L2 (Ω;R2 ) Proof If there is no solution λ to (2.6) in the strip ≤ Re λ ≤ − β, the estimate in Wβ2,2 (Ω; R2 ) × Wβ1,2 (Ω) follows from [38, Theorem 9.4.5] The solution λc to (2.6)1 with the smallest positive real part is such that Re λc ≈ 0.59 and Re λc > 0.58; see [10, page 71] Thus, the solution (z, p) to (2.3) belongs to Wβ2,2 (Ω; R2 ) × Wβ1,2 (Ω), if − β ≤ 0.58 In particular, we can choose β = 0.42 According to [10, Proposition A.1], z belongs to H 3/2+0.08 The proof is complete Remark 2.6 Analogous results to those in Theorem 2.5 are stated in three dimensions in the case of polyhedral cones; see [36] and [37, Theorem 4.2] Similar results in two dimensions are also stated in [36, Lemma 2.9], but for higher regularity conditions, and in [35, Lemma 2.6] with the same regularity as in Theorem 2.5 for second order elliptic systems With Theorem 2.4, we know that, for a given f , the pressure p corresponding to (2.3) is uniquely defined To find an equation for p, in terms of z, we first notice that (z, p) = (z, p0 ) + (0, p1 ), where (z, p0 ) is the solution to equation (2.7) −νΔz + ∇p0 = Πf, z = on Γd , div z = σ(z, p0 )n = in Ω, on Γn , Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3014 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND and p1 ∈ HΓ1n (Ω) is characterized by ∇p1 = (I − Π)f Since z belongs to H 3/2+ε0 (Ω; R2 ) and p0 belongs to H 1/2+ε0 (Ω), p0 |Γ and ν(∇z + (∇z)T )n · n|Γ belong to H ε0 (Γ) Now, since Πf belongs to Vn,Γ (Ω), by taking d the divergence of the first equation in (2.7), we obtain Δp0 = and div(−νΔz + ∇p0 ) = in Ω Thus the trace (∇p0 − νΔz) · n|Γ = Πf · n|Γ is well defined in H −1/2 (Γ) Using the fact that p0 ∈ L2 (Ω) and Δp0 ∈ L2 (Ω), with [25, Theorem −3/2 (Γd ), 1.5.2], it is also possible to give a meaning to ∂p ∂n |Γd in the trace space H 3/2 3/2 the dual of H (Γd ), where H (Γd ) is the trace space, defined in [25, Page 26], as a subspace of functions belonging to H 3/2 (Γd ) and satisfying some compatibility conditions at each junction points of Jd,n Therefore the equation ∂p0 = νΔz · n ∂n on Γd makes sense Thus we can say that if (z, p0 ) is a solution to (2.7), then p0 obeys (2.8) −Δp0 = in Ω, ∂p0 = νΔz · n on Γd , ∂n p0 = ν(∇z + (∇z)T )n · n on Γn But we not give the precise definition of solutions to (2.8) in the sense of transposition, because we not need to define p0 precisely, and we not want to introduce the space of functions belonging to H (Ω) and whose trace belongs to H 3/2 (Γd ) (the trace space introduced in [25], already mentioned above) 2.4 The Oseen operator We first define the Stokes operator (A0 , D(A0 )) corresponding to the boundary conditions in system (1.5) We set D(A0 ) = z ∈ VΓ1d (Ω) ∩ H 3/2+ε0 (Ω; R2 ) | ∃p ∈ H 1/2+ε0 (Ω) such that div σ(z, p) ∈ L2 (Ω; R2 ) and σ(z, p)n = on Γn and A0 z = Π div σ(z, p), where ε0 is the exponent appearing in Theorem 2.5 The condition div σ(z, p) ∈ L2 (Ω; R2 ) allows us to define σ(z, p)n in H −1/2 (Γ; R2 ) Thus σ(z, p)n|Γn = is meaningful in H −1/2 (Γn ; R2 ), where H −1/2 (Γn ; R2 ) is the 1/2 dual of the space H00 (Γn ; R2 ) defined in [33] Since we know that (z, p) belongs to 3/2+ε0 H (Ω; R ) × H 1/2+ε0 (Ω), σ(z, p)n|Γn is also defined as a function belonging to ε0 H (Γn ; R2 ) Remark 2.7 If f ∈ L2 (Ω; R2 ) and if (z, p) is the solution to (2.3), then we easily see that z belongs to D(A0 ) and (f, p) is a pair of functions for which −div σ(z, p) = f ∈ L2 (Ω; R2 ) and σ(z, p)n = on Γn Conversely, if z ∈ D(A0 ) and if p ∈ H 1/2+ε0 (Ω) are such that −div σ(z, p) = f ∈ L2 (Ω; R2 ) and σ(z, p)n = on Γn , then we can verify that (z, p) is the solution to (2.3) We can write p in the form p = p0 + p1 , where (z, p0 ) is the solution to (2.7) and p1 ∈ HΓ1n (Ω) is defined by ∇p1 = (I − Π)f And we obviously have A0 z = Π div σ(z, p) = div σ(z, p0 ) We can directly see that p is uniquely defined by z and f ∈ L2 (Ω; R2 ) for which divσ(z, p) = f Indeed, if p ∈ H 1/2+ε0 (Ω) and q ∈ H 1/2+ε0 (Ω) obey divσ(z, p) = divσ(z, q) = f and σ(z, p)n = = σ(z, q)n on Γn , we have ∇p − ∇q = in Ω and (p − q)n = on Γn Thus p = q Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3015 STABILIZATION OF THE NAVIER–STOKES EQUATIONS Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php The Oseen operator (A, D(A)) is defined by D(A) = D(A0 ) and Az = A0 z − Π ((ws · ∇)z + (z · ∇)ws ) Theorem 2.8 The operator (A, D(A)) is the infinitesimal generator of an ana0 (Ω) Its resolvent is compact lytic semigroup on Vn,Γ d Proof The proof follows from the following inequality ν z 2V (Ω) ∀z ∈ D(A), (2.9) (λ0 I − A)z, z L2 (Ω;R2 ) ≥ Γd with λ0 > large enough (see the proof in [41] in the case of Dirichlet boundary conditions We rewrite it in the case of mixed boundary conditions for the convenience of the reader) For all z ∈ D(A), we have − Az, z L2 (Ω;R2 ) = Ω ν ∇z + (∇z)T : ∇z + (∇z)T + (ws · ∇)z · z + (z · ∇)ws · z dx We notice that the boundary terms are zero because z = on Γd and σ(z, p)n = on Γn Using the classical inequality (in two dimensions) z L4 (Ω) ≤ 21/4 z 1/2 L2 (Ω) ∇z 1/2 L2 (Ω) , we have the following estimates Ω (ws · ∇)z · z dx ≤ ws L4 (Ω) ∇z L2 (Ω) z L4 (Ω) ≤ 21/4 ws L4 (Ω) z 1/2 L2 (Ω) ∇z 3/2 L2 (Ω) and Ω (z · ∇)ws · z dx ≤ ∇ws L2 (Ω) z L4 (Ω) ≤ 21/2 ∇ws L2 (Ω) z L2 (Ω) ∇z L2 (Ω) Thus (2.9) follows from the above estimates and from Young’s and Korn’s inequalities Let us set (2.10) ν ∇z + (∇z)T : ∇ζ + (∇ζ)T + (ws · ∇)z · ζ + (z · ∇)ws · ζ dx, a(z, ζ) = Ω for all z ∈ VΓ1d (Ω) and all ζ ∈ VΓ1d (Ω) We can verify that the bilinear form a is continuous in VΓ1d (Ω) × VΓ1d (Ω) Thus, from Theorem 2.12 in [12, Chapter 1], it follows that A is the infinitesimal generator of an analytic semigroup on Vn,Γ (Ω) d Let us prove that its resolvent is compact We first verify that, for λ0 > for which (2.9) is true, the solution to the equation (λ0 I − A)z = f belongs to VΓ1d (Ω) if f ∈ Vn,Γ (Ω) This follows from the Lax–Milgram lemma Finally, we use the fact that d (Ω) is compact The proof is complete the imbedding from VΓ1d (Ω) into Vn,Γ d Remark 2.9 Due to assumption (H3 ), the restriction of functions in D(A) to Ωδ,Γn = {x ∈ Ω | dist(x, Γn ) > δ} belongs to H (Ωδ,Γn ; R2 ) for all δ > Before defining the adjoint of (A, D(A)), we introduce the equation λ0 φ − div σ(φ, ψ) − (ws · ∇)φ + (∇ws )T φ = f, (2.11) div φ = in Ω, φ = on Γd , σ(φ, ψ)n + ws · n φ = on Γn , Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS for all φ ∈ D(A∗ ) Since (2.29), we have Ω 0= Ω 3025 (I − Π)w (λ0 φ − A∗ φ) = 0, using the second equality in w (λ0 φ − A∗ φ) dx + M u σ(φ, ψ) dx Γ for all φ ∈ D(A∗ ) and all ψ ∈ L2 (Ω) such that div σ(φ, ψ) ∈ L2 (Ω; R2 ) This implies that w is the solution to (2.20) To prove the regularity result stated in (2.30), we introduce the solution q to the following elliptic equation q ∈ HΓ1n (Ω), Δq = 0, ∂q = Mu · n ∂n on Γd , q=0 on Γn Let us recall that (I − Π)w = ∇q Moreover M u ·n = on Γd \ Γc and dist(Γc , J ) > Let us denote by ρ a function in H (Ω), with compact support in a neighborhood of ∂ρ Γc included in Ω ∪ Γc and such that ∂n = M u on Γd Since ρ is with compact support in Ω ∪ Γc , we necessarily have ρ = on Γn If we set p = q − ρ, the equation for p is Δp = −Δρ, ∂p = on Γd , ∂n p = on Γn Due to assumptions (H1 ) − (H4 ) and to elliptic regularity results, p ∈ H (Ω) Indeed, since the junction points between Γd and Γn correspond to right angles, the H regularity can also be directly obtained by using a symmetry argument as in [30, Exercise 3, Page 147] The estimate (2.30) can be easily obtained from the estimates for p and ρ, and from (2.26) The proof is complete Stabilization of the linearized Navier–Stokes equations 3.1 Criterion for stabilizability of parabolic-type systems From Theorem 2.8, it follows that spec(A), the spectrum of A, is contained in a sector (We have used the notation “spec” rather than the classical notation σ, because σ already denotes the Cauchy stress tensor of the fluid flow.) The eigenvalues are isolated, pairwise conjugate when they are not real, and of finite multiplicity In order to obtain a prescribed exponential decay rate e−ωt , we can always choose ω > to have (3.1) · · · < ReλNω,u +1 < −ω < ReλNω,u ≤ ReλNω,u −1 ≤ · · · ≤ Reλ1 , where (λj )j∈N∗ are the eigenvalues of A, repeated according to their multiplicity Indeed if ω is equal to −ReλNω,u for some λNω,u , we may choose ω > ω such that (3.1) holds true for ω in place of ω We denote by GR (λj ) the real generalized eigenspace for A, that is, the space generated by ReGC (λj ) ∪ ImGC (λj ) (where GC (λj ) is the complex generalized eigenspace for A), and G∗R (λj ) is the real generalized eigenspace for A∗ We set N ω,u GR (λj ) Zω,u = ⊕j=1 and ω,u ∗ Zω,u = ⊕j=1 G∗R (λj ) N ∗ the invariant We denote by Zω,s the invariant subspace under (etA )t≥0 and by Zω,s ∗ tA subspace under (e )t≥0 such that Z = Vn,Γ (Ω) = Zω,u ⊕ Zω,s d ∗ ∗ and Z ∗ = Vn,Γ (Ω) = Zω,u ⊕ Zω,s d Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3026 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND ∗ We have identified Z with Z ∗ , but Zω,u and Zω,u are not identified and neither are ∗ ∗ Zω,s and Zω,s We notice that Zω,u is the unstable space of Aω = A + ωI, while Zω,u ∗ ∗ is the unstable space of Aω = A + ωI, and Zω,s is the stable space of Aω = A + ωI, ∗ while Zω,s is the stable space of A∗ω = A∗ + ωI Let πω,u be the projection onto Zω,u along Zω,s and set πω,s = I − πω,u We set Aω,u = πω,u (A + ωI) and Bω,u = πω,u B Since Zω,u is invariant under A, we have Aω,u = Aω πω,u = πω,u Aω πω,u Similarly, ∗ ∗ ∗ ∗ ∗ we denote by πω,u the projection onto Zω,u along Zω,s and we set πω,s = I − πω,u ∗ These projectors can be easily defined by determining a basis of Zω,u and of Zω,u (see [20, 45]) Proposition 3.1 There exist a basis (e1 , · · · , edω,u ) of Zω,u , constituted of ∗ elements belonging to D(A), and a basis (ξ1 , · · · , ξdω,u ) of Zω,u , constituted of elements ∗ , such that belonging to D(A∗ ) with dω,u = dimZω,u = dimZω,u dω,u πω,u f = (3.2) dω,u (f, ξi )Ω ei ∗ πω,u f and = (f, ei )Ω ξi i=1 (ei , ξj )Ω = δi,j ∀1 ≤ i ≤ dω,u , ≤ j ≤ dω,u , where δi,j is the Kroenecker symbol and (f, g)Ω = ∗ may be extended to L2 (Ω; R2 ) by setting πω,u dω,u πω,u f = ∀f ∈ Z, i=1 Ω f g dx The operators πω,u and dω,u (f, ξi )Ω ei and ∗ πω,u f= i=1 (f, ei )Ω ξi ∀f ∈ L2 (Ω; R2 ) i=1 We notice that πω,u f = πω,u Πf ∀f ∈ L2 (Ω; R2 ) ∗ Moreover, the operators πω,u and πω,u may be extended, respectively, to (D(A∗ )) and (D(A)) by setting πω,u f = dω,u i=1 f, ξi (D(A∗ )) ,D(A∗ ) ei dω,u i=1 f, ei (D(A)) ,D(A) ξi ∀f ∈ (D(A∗ )) and ∗ πω,u f= ∀f ∈ (D(A)) In particular, we have dω,u πω,u Bu = dω,u Bu, ξi i=1 (D(A∗ )) (u, B ∗ ξi )L2 (Γc ;R2 ) ei ,D(A) ei = ∀u ∈ L2 (Γc ; R2 ) i=1 Due to the characterization of B ∗ (see Lemma 2.20 above), we have B ∗ ξi = M σ(ξi , ρi )n, ∗ where ρi is the pressure associated with ξi Since (ξ1 , · · · , ξdω,u ) is a basis of Zω,u = Nω,u ∗ ∗ ⊕j=1 GR (λj ), then ξi belongs to GR (λj ) for some λj , with j ∈ {1, · · · , Nω,u } We have to consider the different cases (see [20, 45]): Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3027 – If λj is real, then ξi is an eigenvector or a generalized eigenvector associated with λj and ρi is the corresponding pressure satisfying the corresponding partial differential equation (see below) √ √ is an eigenvector – If Im λj = 0, then ξi = Re φi or ξi = Im φi , where φi √ or a generalized eigenvector associated with λ In that case ρ = Re ψi or ρi = j i √ Im ψi , where ψi is the pressure associated with φi If φ is an eigenvector associated with λj , then (φ, ψ) obeys the equation φ ∈ H (Ω; C2 ), (3.3) ψ ∈ L2 (Ω; C), λj φ − div σ(φ, ψ) − (ws · ∇)φ + (∇ws )T φ = div φ = in Ω, φ = on Γd , in Ω, σ(φ, ψ)n + ws · n φ = on Γn The equations satisfied by a generalized eigenvector have to be modified accordingly In the next section, we are going to state a stabilizability result with controls of finite dimension For that we consider the real control space ω,u U0 = vect ∪j=1 (ReB ∗ E ∗ (λj ) ∪ ImB ∗ E ∗ (λj )), N where E ∗ (λj ) = Ker(A∗ −λj I) Let {g1 , · · · , gNc } be a basis of U0 Since B ∗ E ∗ (λj ) is generated by all the functions of the form M (σ(φ, ψ)n), where (φ, ψ) are the solutions to system (3.3), the functions g1 , · · · , gNc are regular Indeed, since Γc is of class C , it can be shown that ws and the solution φ to (3.3) are in H in a neighborhood of + 3/2 Γ+ (Γc ; R2 ) Since c (Γc is relatively open in Γc ; see (H4 )) Thus, we have gi ∈ H we are going to look for stabilizing controls u ∈ L (0, ∞; U0 ), we write Nc u(x, t) = vi (t)gi (x), i=1 and we consider v = (v1 , · · · , vNc )T ∈ L2 (0, ∞; RNc ) as the new control variable Thus, it is interesting to introduce the control operator B ∈ L(RNc , (D(A∗ )) ) defined by Nc Bv = vi B gi i=1 With such a definition, the controlled system z = (A + ωI)z + Bu, z(0) = z0 , z = (A + ωI)z + Bv, z(0) = z0 is rewritten as 3.2 Stabilizability of the Oseen equations Let us recall that the pair 0 (Ω) if and only if, for all z0 ∈ Vn,Γ (Ω), there ((A + ωI), B) is stabilizable in Vn,Γ d d Nc exists a control v ∈ L (0, ∞; R ) such that the solution to the controlled system z = (A + ωI)z + Bv, z(0) = z0 , obeys ∞ z(t) Vn,Γ (Ω) dt < ∞ d Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3028 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND Theorem 3.2 Let ω > be given, and assume that the spectrum of A and ω obeys the condition (3.1) The pair (A + ωI, B) is stabilizable in Vn,Γ (Ω) d α−1 B is bounded from the control space U0 into Vn,Γ (Ω) Proof Since (λ0 I − A) d α−1 Nc for < α < 1/4, (λ0 I − A) B is bounded from the control space R into Vn,Γd (Ω), and in that case proving the stabilizability of the pair ((A + ωI), B) in Vn,Γ (Ω) is d equivalent to verifying the Hautus criterion; see, e.g., [6] Therefore, we have to prove that if φ belongs to Ker(λj I − A∗ ) with j ∈ {1, · · · , Nω,u }, and if B ∗ φ = 0, then φ = If ψ is the pressure associated with φ, then (φ, ψ) is a solution to (3.3), and we can easily check that B∗φ = − =− Γc Γc gi M σ(φ, ψ)n dx 1≤i≤Nc −i gi M Re σ(φ, ψ)n dx 1≤i≤Nc Γc gi M Im σ(φ, ψ)n dx 1≤i≤Nc Let us notice that M Re σ(φ, ψ)n ∈ U0 and M Im σ(φ, ψ)n ∈ U0 Thus, if B ∗ φ = 0, since {g1 , · · · , gNc } is a basis of U0 , we have M (σ(φ, ψ)n) Γc = Since M > on Γ+ c , we have (3.4) (σ(φ, ψ)n) Γ+ c = (See (H4 ) for the assumptions on Γ+ c ) We have to show that if (3.3) and (3.4) are fulfilled, then necessarily φ = and ∇ψ = For that we can use an extension procedure and a unique continuation property due to Fabre and Lebeau [15] See also [16] and [48] 3.3 Stabilization by feedback of finite dimension In this section we are going to determine a feedback control law of finite dimension We assume that ω > is given and that the spectrum of A and ω obeys the condition (3.1) We consider the controlled system Nc (3.5) zω,u = Aω,u zω,u + vi Bω,u gi , zω,u (0) = πω,u z0 i=1 We set Aω,s = πω,s (A + ωI), Bω,s = πω,s B, and we define the operators Bω,u ∈ L(RNc , Zω,u ), Bω,s ∈ L(RNc , (D(A∗ )) ), and B ∈ L(RNc , (D(A∗ )) ) by setting Nc Bω,u v = Nc vi Bω,u gi , Bω,s v = i=1 Nc vi Bω,s gi , i=1 and Bv = vi B gi i=1 ∀v = (v1 , · · · , vNc )T ∈ RNc Notice that Bω,u is bounded from RNc into Zω,u , because πω,u is bounded from (D(A∗ )) into Zω,u , while Bω,s is not bounded from RNc into Zω,s Equation (3.5) is now (3.6) zω,u = Aω,u zω,u + Bω,u v Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3029 STABILIZATION OF THE NAVIER–STOKES EQUATIONS Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php To stabilize (3.5), we can choose the feedback law ∗ v = −Bω,u Pω,u zω,u , (3.7) ∗ ∗ where Bω,u ∈ L(Zω,u , RNc ) is the adjoint of Bω,u ∈ L(RNc , Zω,u ), and Pω,u is the solution to the algebraic Riccati equation ∗ ), Pω,u ∈ L(Zω,u , Zω,u ∗ Pω,u = Pω,u ≥ 0, ∗ Pω,u Aω,u + A∗ω,u Pω,u − Pω,u Bω,u Bω,u Pω,u = 0, Pω,u is invertible This result follows from [29] Moreover (see [29, Theorem 3]), the spectrum of Aω,u − ∗ Bω,u Bω,u Pω,u is determined in terms of the spectrum of Aω,u by ∗ spec(Aω,u − Bω,u Bω,u Pω,u )= {λj ∈ spec(Aω,u ) | λj = −2ω − Reλj + i Imλj , ≤ j ≤ Nω,u } ∗ Pω,u πω,u ∈ Proposition 3.3 (i) The operator K = −B ∗P, with P = πω,u L(Z), provides a stabilizing feedback for (A + ωI, B) More precisely, the operator A + ωI − BB ∗P, with domain D(A + ωI − BB ∗P) = {z ∈ Z | (A + ωI − BB ∗P)z ∈ Z} is the infinitesimal generator of an analytic semigroup exponentially stable on Z (ii) Moreover, it can be verified that P is the unique solution to the algebraic Riccati equation P ∈ L(Z), P = P ∗ ≥ 0, A + ωI − BB ∗P P(A + ωI) + (A∗ + ωI)P − PBB ∗P = 0, is exponentially stable on Z (iii) The domain of A∗ + ωI − PBB ∗ in Z, the adjoint of A + ωI − BB ∗P, is D(A∗ ) Proof From the definition of P, it follows that the equation (A + ωI − BB ∗P)z = λz, is equivalent to the system πω,s ((A + ωI − BB ∗P)z) πω,u ((A + ωI − BB ∗P)z) = =λ Aω,s πω,s z πω,u z ∗ −Bω,s Bω,u Pω,u πω,s z ∗ Aω,u − Bω,u Bω,u Pω,u πω,u z ∗ Pω,u zω,u = B ∗ Pzω,u , where zω,u = πω,u z (this is a consequence We have used that Bω,u of the definition of P) The solutions to this eigenvalue problem are λ = λj with ≤ j ≤ Nω,u , ∗ πω,u z is an eigenvector of Aω,u − Bω,u Bω,u Pω,u , or λ = λj with j > Nω,u , πω,u z = 0, and πω,s z is an eigenvector of Aω,s Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3030 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND ∗ When ≤ j ≤ Nω,u and when πω,u z is an eigenvector of Aω,u − Bω,u Bω,u Pω,u , then either πω,s z = or πω,s z is an eigenvector of Aω,s This last case happens if λj is also an eigenvalue of Aω,s , which could be excluded by adding additional conditions on the choice of ω Thus, it is clear that A + ωI − BB ∗P is the infinitesimal generator of an exponen∗ tially stable semigroup on Z Since the semigroup generated by Aω,u − Bω,u Bω,u Pω,u is analytic on Zω,u , because it is a finite dimensional system, and the semigroup generated by Aω,s is analytic on Zω,s , then the semigroup generated by Aω,s ∗ −Bω,s Bω,u Pω,u ∗ Aω,u − Bω,u Bω,u Pω,u is also analytic on Zω,u × Zω,s , because ∗ −Bω,s Bω,u Pω,u 0 is a bounded perturbation with relative bound equal to zero of the operator Aω,s 0 ∗ Aω,u − Bω,u Bω,u Pω,u ; see [28] Thus, the first statement of the theorem is proved The second and third statements can be easily verified 3.4 Regularity of solutions to the closed loop nonhomogeneous linearized system In this section, we want to prove regularity results for the solutions to the closed loop nonhomogeneous linear system (3.8) ∂z + (ws · ∇)z + (z · ∇)ws − ωz − div σ(z, p) = f, div z = in Q∞ = Ω × (0, ∞), ∂t Nc ∗ Bω,u Pω,u πω,u z i M gi z=− on Σ∞ d , σ(z, p)n = on Σ∞ n , i=1 Πz(0) = z0 on Ω, ∗ ∗ Pω,u πω,u z i denotes the ith component of the vector Bω,u Pω,u πω,u z ∈ where Bω,u Nc R Let us notice that we have written the initial condition in the form Πz(0) = z0 ; see Remark 2.19 In order to take into account the nonhomogeneous boundary condition on Σ∞ d , we denote by (ζi , πi ) the solution to the following stationary equation (3.9) λ0 ζi − div σ(ζi , πi ) + (ws · ∇)ζi + (ζi · ∇)ws = 0, ζi = M gi on Γd , div ζi = in Ω, σ(ζi , πi )n = on Γn We also introduce the solution wi to the equation (λ0 I − A)wi − Bgi = We have the identity wi = Πζi From Theorem 2.17 we know that ζi ∈ H 3/2+ε0 (Ω; R2 ), and from Theorem 2.21 that wi ∈ H (Ω; R2 ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3031 Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php To study the regularity of solutions to system (3.8), we rewrite it in the form ∗ zω,u = (Aω,u − Bω,u Bω,u Pω,u )zω,u + πω,u f, (3.10) zω,s = Aω,s zω,s − zω,u = πω,u z, ∗ Pω,u zω,u i Bω,u Nc i=1 zω,s = πω,s z, Nc i=1 (I − Π)z = −(I − Π) zω,u (0) = πω,u z0 , Bω,s gi + πω,s f, zω,s (0) = πω,s z0 , Πz = zω,u + zω,s , ∗ Bω,u Pω,u zω,u i DM gi Lemma 3.4 For all ≤ i ≤ Nc , DM gi belongs to H 3/2+ε0 (Ω; R2 ) Moreover, for all ≤ i ≤ Nc , the solution wi,s to the equation Aω,s wi,s + Bω,s gi = belongs to H (Ω; R2 ) Proof The equation Aω,s wi,s + Bω,s gi = admits a unique solution in Zω,s , since Aω,s is an isomorphism from Zω,s into (D(A∗ω,s )) In particular, we know that wi,s ∈ Zω,s The equation can be rewritten in the form (λ0 I − A)wi,s − Bω,s gi = (λ0 + dω,u M gi , B ∗ ξk L2 (Γc ;R2 ) ek ω)wi,s Since Bω,s = πω,s B, we have Bω,s gi = Bgi − k=1 Thus wi,s = Πζi + ηi , where ζi is the solution to (3.9), and ηi is the solution of dω,u (λ0 I − A)ηi = (λ0 + ω)wi,s − k=1 M gi , B ∗ ξk L2 (Γc ;R2 ) ek (wi,s intervenes in the right-hand side of the equation, but we already know that wi,s belongs to Zω,s ) It is clear that ηi ∈ D(A) and D(A) → H 3/2+ε0 (Ω; R2 ) (see Theorem 2.13) From Theorem 2.21, it follows that Πζi belongs to H (Ω; R2 ) The proof is complete ε Theorem 3.5 Let ε belong to (0, 1/2) Assume that z0 belongs to Vn,Γ (Ω) and d −1+ε f ∈ L (0, ∞; HΓd (Ω)) Then the solution z to system (3.8) satisfies z (3.11) L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) d ≤ C3 f −1+ε L2 (0,∞;HΓ (Ω)) d + z0 H ε (Ω;R2 ) Proof We first prove that z belongs to (Ω)) + H (0, ∞; H (Ω; R2 )) L (0, ∞; D((λ0 I − A)1/2+ε/2 )) ∩ H 1/2+ε/2 (0, ∞; Vn,Γ d We will improve that result in Step Step Since the equation satisfied by zω,u is a finite dimensional system and ∗ Aω,u − Bω,u Bω,u Pω,u is stable, it is easy to verify that zω,u ∈ H (0, ∞; Zω,u ) Using that Zω,u ⊂ D(A), we also have zω,u ∈ H (0, ∞; D(A)) and (3.12) zω,u H (0,∞;D(A)) ≤ C( πω,u z0 Z + πω,u f Next, we look for zω,s in the form zω,s = ξ + ζ = − where wi,s is the solution to L2 (0,∞;Zω,u ) ) Nc i=1 ∗ Bω,u Pω,u zω,u i wi,s + ζ, Aω,s wi,s + Bω,s gi = 0, and ζ is the solution to Nc ζ = Aω,s ζ + ∗ Bω,u Pω,u zω,u i wi,s + πω,s f, i=1 Nc ζ(0) = πω,s z0 + ∗ Bω,u Pω,u zω,u (0) i Πwi,s i=1 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3032 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php From Lemma 3.4, we know that wi,s belongs to H (Ω; R2 ) Thus, we have Nc ∗ Bω,u Pω,u zω,u i wi,s i=1 H (0,∞;H (Ω;R2 )) ≤ C zω,u H (0,∞;D(A)) , and ξ belongs to H (0, ∞; H (Ω; R2 )) ∗ Let us now estimate ζ We know that f belongs to L2 (0, ∞; HΓ−1+ε (Ω)), [Bω,u d Nc ∗ Pω,u zω,s ]i wi,s belongs to L2 (0, ∞; H (Ω; R2 )), and that i=1 Pω,u zω,u (0) i Πwi,s Bω,u ε and z0 belong to Vn,Γd (Ω) Due to Theorem 2.13, πω,s z0 belongs to D((λ0 I − A)ε/2 ) and πω,s f belongs to L2 (0, ∞; (D((λ0 I − A∗ )1/2−ε/2 )) ) Thus, we have (see, e.g., [12]) ζ L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω)) d ≤C f + z0 −1+ε L2 (0,∞;HΓ (Ω)) d H ε (Ω;R2 ) Collecting together all the previous estimates, we obtain Πz L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H (Ω;R2 )) d (3.13) ≤C f −1+ε L2 (0,∞;HΓ (Ω;R2 )) d + z0 H ε (Ω;R2 ) The following estimate for (I − Π)z, (I − Π)z H (0,∞;H (Ω;R2 )) ≤C f −1+ε L2 (0,∞;HΓ (Ω)) d + z0 H ε (Ω;R2 ) , is a consequence of Lemma 3.4 and of (3.12) Step We now improve the estimates obtained in Step For that, we look for the solution (z, p) to system (3.8) in the form Nc (z, p) = ∗ Bω,u Pω,u zω,u i ζi + ζ0 , − − i=1 Nc ∗ Bω,u Pω,u zω,u i πi + π0 , i=1 where (ζi , πi ) is the solution to (3.9) and (ζ0 , π0 ) is the solution to N c ∂ζ0 ∗ + (ws · ∇)ζ0 + (z · ∇)ws − ωζ0 − div σ(ζ0 , π0 ) = f + Bω,u Pω,u zω,u i ζi , ∂t i=1 div ζ0 = in ζ0 = Q∞ = Ω × (0, ∞), on Σ∞ d , ζ0 (0) = z0 + Nc i=1 σ(ζ0 , π0 )n = on Σ∞ n , ∗ Bω,u Pω,u zω,u (0) i Πζi on Ω Since ζi belongs to H 3/2+ε0 (Ω; R2 ), we have Nc i=1 ∗ Pω,u zω,u i ζi Bω,u ≤ C( πω,u z0 Z + πω,u f H (0,∞;H 3/2+ε0 (Ω;R2 )) ≤ C zω,u H (0,∞;D(A)) L2 (0,∞;Zω,u ) ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS Nc i=1 Let us now estimate ζ0 We know that ζ0 = z + L2 (0, ∞; L2 (Ω; R2 )) and that Nc ζ0 ≤ z L2 (0,∞;L2 (Ω;R2 )) L2 (0,∞;L2 (Ω;R2 )) ∗ Bω,u Pω,u zω,u i ζi belongs to ∗ Bω,u Pω,u zω,u i ζi + 3033 L2 (0,∞;L2 (Ω;R2 )) i=1 ≤C f + z0 −1+ε L2 (0,∞;HΓ (Ω;R2 )) d H ε (Ω;R2 ) We rewrite the system satisfied by ζ0 in the form ∂ζ0 + (ws · ∇)ζ0 + (z · ∇)ws + λ0 ζ0 − div σ(ζ0 , π0 ) ∂t Nc ∗ Bω,u = f + i=1 Pω,u zω,u i ζi + (λ0 + ω)ζ0 , Q∞ = Ω × (0, ∞), div ζ0 = in ζ0 = Σ∞ d , on Nc i=1 ζ0 (0) = z0 + σ(ζ0 , π0 )n = on Σ∞ n , ∗ Bω,u Pω,u zω,u (0) i Πζi on Ω From the estimates in Step 1, it follows that Nc ∗ Bω,u Pω,u zω,u i ζi + (λ0 + ω)ζ0 f+ −1+ε L2 (0,∞;HΓ (Ω;R2 )) i=1 ≤C d f −1+ε L2 (0,∞;HΓ (Ω;R2 )) d + z0 H ε (Ω;R2 ) and Nc z0 + ∗ Pω,u zω,u (0) i Πζi Bω,u i=1 ≤C f −1+ε L2 (0,∞;HΓ (Ω;R2 )) d H ε (Ω;R2 ) + z0 H ε (Ω;R2 ) Since (A − λ0 I) is stable, from the above estimates it follows that ζ0 L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω)) d ≤C f −1+ε L2 (0,∞;HΓ (Ω;R2 )) d + z0 ε Vn,Γ (Ω) d The proof is complete Stabilization of the Navier–Stokes equations When we apply the previous feedback control law (3.5) to system (2.16), we obtain the following closed loop system: ∂z + (ws · ∇)z + (z · ∇)ws + (z · ∇)z − div σ(z, p) = 0, ∂t div z = in Q∞ , Nc (4.1) z=− ∗ Bω,u Pω,u zω,u i M gi on Σ∞ d , i=1 σ(z, p)n = on Σ∞ n , Πz(0) = z0 on Ω Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3034 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND In order to stabilize z(t) in H ε (Ω; R2 ) with the prescribed exponential decay rate e−ωt , we make the change of unknowns zˆ = eωt z and pˆ = eωt p The system satisfied by zˆ and pˆ is ∂ zˆ + ω zˆ + (ws · ∇)ˆ z + (ˆ z · ∇)ws + e−ωt (ˆ z · ∇)ˆ z − div σ(ˆ z , pˆ) = 0, ∂t div zˆ = (4.2) zˆ = − Nc i=1 in Q∞ , ∗ Pω,u zˆω,u i M gi Bω,u on Σ∞ d , σ(ˆ z , pˆ)n = on Σ∞ n , Πˆ z (0) = z0 on Ω In the proof of the main result (Theorem 4.2 below), we have to introduce the following space H 1+η,1/2+η/2 (Ω × (0, ∞)) = L2 (0, ∞; H 1+η (Ω)) ∩ H 1/2+η/2 (0, ∞; L2 (Ω)); see [34] It is equipped with the norm ζ H 1+η,1/2+η/2 (Ω×(0,∞)) = ζ L2 (0,∞;H 1+η (Ω)) + ζ H 1/2+η/2 (0,∞;L2 (Ω)) 1/2 Lemma 4.1 We have the following continuous imbedding H 1+η,1/2+η/2 (Ω × (0, ∞)) → L∞ (0, ∞; H η (Ω)) Proof This lemma is a direct consequence of a trace theorem by Grisvard [24, Lemma 4.1] See [14] for a detailed proof Theorem 4.2 Let ε belong to (0, 1/2) There exist a constant C0 > 0, depending on ε, and a nondecreasing function θ from R+ into itself, such that if z0 belongs to ε (Ω), C ∈ (0, C0 ), and Vn,Γ d z0 H ε (Ω;R2 ) ≤ θ(C), then the system (4.1) admits a unique solution in the space VC = z ∈ L2 (0, ∞; D((λ0 I − A)1/2+ε/2 )) (Ω)) + H (0, ∞; H 3/2+ε0 (Ω; R2 )), ∩ H 1/2+ε/2 (0, ∞; Vn,Γ d eω· z L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) d ≤C In particular the solution to system (4.1) obeys (4.3) z(t) H ε (Ω;R2 ) ≤ Ce−ωt , where C depends on z0 H ε (Ω;R2 ) Proof The proof is based on the Banach fixed point theorem Let ζ belong to (Ω))+H (0, ∞; H 3/2+ε0 (Ω; R2 )) L2 (0, ∞; D((λ0 I −A)1/2+ε/2 ))∩H 1/2+ε/2 (0, ∞; Vn,Γ d Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3035 Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 0) ≤ 12 + 2ε , where η(ε, ε0 ) = ε2 + εε0 Thus, due to TheoNotice that ≤ 12 + η(ε,ε rem 2.13, ζ belongs to L2 (0, ∞; H 1+η (Ω; R2 )) ∩ H 1/2+η/2 (0, ∞; L2 (Ω; R2 )), with η = (Ω; R2 )) Since η(ε, ε0 ) We want to estimate F (ζ) = − ((ζ · ∇) · ζ) in L2 (0, ∞; HΓ−1+ε d (Ω; R2 )) 2η = ε+2εε0 > ε, it is enough to prove that F (ζ) belongs to L (0, ∞; HΓ−1+2η d We have ((ζ · ∇) · ζ) −1+2η L2 (0,∞;HΓ (Ω;R2 )) ≤ C (ζ · ∇) · ζ d ≤ C ∇ζ L2 (0,∞;L2/(1−η) (Ω;R2×2 )) ≤ C ∇ζ L2 (0,∞;H η (Ω;R2×2 )) ζ ζ L2 (0,∞;L1/(1−η) (Ω;R2 )) L∞ (0,∞;L2/(1−η) (Ω;R2 )) L∞ (0,∞;H η (Ω;R2 )) ≤C ζ L2 (0,∞;H 1+η (Ω;R2 )) ≤C ζ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ ζ (H 1+η,1/2+η/2 (Ω×(0,∞)))2 d (Ω) → L1/η (Ω) and, The first inequality follows from the Sobolev imbedding HΓ1−2η d (Ω) The third one follows from the imbedding by duality, L1/(1−η) (Ω) → HΓ−1+2η d H η (Ω) → L2/(1−η) (Ω), and the fourth one from the imbedding H 1+η,1/2+η/2 (Ω × (0, ∞)) → L∞ (0, ∞; H η (Ω)) (see Lemma 4.1) The last inequality follows from The(Ω)) orem 2.13 Thus e−ωt F (ζ) belongs to L2 (0, ∞; HΓ−1+ε d Let us denote by (zζ , pζ ) the solution to the system ∂zζ + (ws · ∇)zζ + (zζ · ∇)ws + e−ωt (ζ · ∇)ζ − div σ(zζ , pζ ) = 0, ∂t div zζ = in Q∞ , (4.4) zζ = − Nc i=1 ∗ Bω,u Pω,u πω,u zζ i M gi on Σ∞ d , σ(zζ , pζ )n = on Σ∞ n , Πzζ (0) = z0 on Ω From Theorem 3.5, it follows that zζ L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) d ≤ C3 ( e−ωt F (ζ) ≤ C3 ( ζ −1+ε L2 (0,∞;HΓ (Ω;R2 )) d + z0 H ε (Ω;R2 ) ) L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) d + z0 H ε (Ω;R2 ) ) As in [41], we can determine C0 > and θ such that, for all < C ≤ C0 , the nonlinear mapping ζ −→ zζ is a strict contraction in VC = zˆ ∈ L2 (0, ∞; D((λ0 I − A)1/2+ε/2 )) (Ω)) + H (0, ∞; H 3/2+ε0 (Ω; R2 )), ∩H 1/2+ε/2 (0, ∞; Vn,Γ d zˆ L2 (0,∞;D((λ0 I−A)1/2+ε/2 ))∩H 1/2+ε/2 (0,∞;Vn,Γ (Ω))+H (0,∞;H 3/2+ε0 (Ω;R2 )) d ≤C , Copyright © by SIAM Unauthorized reproduction of this article is prohibited 3036 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php provided that z0 H ε (Ω;R2 ) ≤ θ(C) Thus system (4.2) admits a unique solution in the metric space VC The existence of a unique solution to system (4.1) follows from the fact that zˆ is a solution to system (4.2) if and only if z = e−ωt zˆ is a solution to system (4.1) The proof is complete Remark 4.3 In the case when only Dirichlet boundary conditions are present, we can choose ε = in the statement of Theorem 4.2 Let us recall why, and why we cannot obtain the same estimates in the case of mixed boundary conditions Indeed, if ζ belongs to L2 (0, ∞; H (Ω; R2 )) ∩ H 1/2 (0, ∞; L2 (Ω; R2 )), we can identify F (ζ) = −Π ((ζ · ∇) · ζ) = −Π div (ζ ⊗ ζ) with an element in L2 (0, ∞; V −1 (Ω)) if Γd = Γ, and we can use a fixed point method in L2 (0, ∞; H (Ω; R2 )) ∩ H 1/2 (0, ∞; L2 (Ω; R2 )) to prove the existence of solutions to the nonlinear closed loop system; see, e.g., [45, 4] This is no longer true with mixed boundary conditions Indeed we have F (ζ), φ = Ω (ζ · ∇)φ · ζ dx − Γn (ζ · φ) (ζ · n) dx The mapping φ −→ Ω (ζ · ∇)φ · ζ dx can be identified with an element in HΓ−1 (Ω; R2 ), d but not the mapping φ −→ Γn (ζ · φ) (ζ · n) dx This is why a better regularity condition is needed for ζ To obtain the existence of a solution in a class smaller ε (Ω)), we need that z0 belongs to Vn,Γ (Ω) than L2 (0, ∞; VΓ1d (Ω)) ∩ H 1/2 (0, ∞; Vn,Γ n d for some ε > Conclusion To conclude let us recall that, by using a feedback control depending only on the unstable component zω,u of the state variable, we have been able to write the nonhomogeneous closed loop linear system (3.8) in the triangular form stated in (3.10) The regularity of zω,u follows from the fact that zω,u (t) belongs to the finite dimensional space Zω,u ⊂ D(A) ⊂ H 3/2+ε0 (Ω; R2 ) We see that the limitation in regularity for zω,u comes from the presence of junctions between Dirichlet and Neumann boundary conditions This is why D(A) is imbedded in H 3/2+ε0 (Ω; R2 ) and not H (Ω; R2 ) We first obtain the estimate for z by studying separately the equations for zω,s and (I − Π)z In order to improve the regularity result for z, we have to come back to the partial differential equation satisfied by z, because by using the projector Π we lose some regularity properties Appendix In this appendix, we would like to explain when we can expect to have the existence of a solution (ws , qs ) to (1.1) in H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω) For that, we only consider the geometrical configuration corresponding to Figure First of all, let us recall that we not claim that (1.1) admits a solution (ws , qs ) in H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω), for any regular data fs , us , and gs satisfying some compatibility conditions But we claim that there exist a pair (ws , qs ) ∈ H 3/2+εo (Ω; R2 )×H 1/2+εo (Ω), and data fs , us , and gs for which (1.1) is satisfied Thus, for us given and gs = (which is the case of interest in several applications), proceeding as in the proof of Theorem 2.17, we can define a solution ws ∈ H 3/2+εo (Ω; R2 ) to the Stokes equation with nonhomogeneous Dirichlet boundary conditions on Γi , homogeneous Dirichlet boundary conditions on Γ0 ∪ Γc , and homogeneous Neumann boundary conditions on Γn Therefore, setting fs = −(ws · ∇ws ), ws = ws will be a solution to (1.1) This is a little bit artificial in the sense that fs is defined afterwards Let us explain what can be justified when the data are a priori given Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILIZATION OF THE NAVIER–STOKES EQUATIONS 3037 As is well known, contrary to the stationary Navier–Stokes equation with nonhomogeneous Dirichlet boundary conditions, in the case of mixed or Neumann boundary conditions, the existence of a weak solution is guarenteed only under smallness conditions on the data (see, e.g., [37] or [21]) Also notice that the problem studied in [21] corresponds to the geometrical configuration of Figure Therefore, it seems that there is a contradiction between what is claimed in the first paragraph of this appendix and the results from the literature that we have mentioned We consider the configuration corresponding to Figure 1, when fs = 0, us is a parabolic profile of Poiseuille type given on Γi , and gs = To find a configuration for which a solution (ws , qs ) to (1.1) exists in H (Ω; R2 ) × L2 (Ω), following [13, Chapter V], the idea is to look for a solution in the form (ws , qs ) = (w, q) + (wr , qr ), where (wr , qr ) is an a priori known reference stationary solution to a Stokes equation in which the boundary condition σ(ws , qs )n = on Γn is replaced by a nonlinear boundary condition of the form (A.1) σ(w + wr , q + qr )n = σ(ws , qs )n = − (ws · n)− ws + σ(wr , qr )n on Γn , where (ws · n)− = − min((ws · n), 0) There are several ways for choosing wr but it is convenient to choose it as the solution to the equation −div σ(wr , qr ) = and div wr = in Ω, wr = us on Γi ∪ Γn , wr = on Γ0 Since the function g defined on Γ by g = us on Γi ∪ Γn and g = on Γ0 obeys g · n dx = and is the trace of a function belonging to H (Ω; R2 ), it can be easily Γ verified that (wr , qr ) ∈ H (Ω; R2 ) × H (Ω) The existence of (ws , qs ) ∈ H (Ω; R2 ) × L2 (Ω) satisfying (1.1)1 , (1.1)2 , and (A.1) can be shown by using a Galerkin method as in [46, Chapter II] Next we notice that if the solution (ws , qs ) to (1.1)1 , (1.1)2 , and (A.1) obeys (A.2) (ws · n)− = on Γn , then (ws , qs ) is also a solution to (1.1) We claim that if fs = 0, us is a parabolic profile of Poiseuille type, gs = 0, and if the channel corresponding to Figure is long enough, then the solution (ws , qs ) to (1.1)1 , (1.1)2 , and (A.1) will satisfy (A.2) We have no mathematical proof of that, but there are enough numerical simulations for this type of problem to be convinced that it is true (see the numerical simulations on this type of problem reported in [1, 2, 27, 32]) If (ws , qs ) is a solution to (1.1) in H (Ω; R2 ) × L2 (Ω), far from Γn , the solution is of class H Since ws ∈ H (Ω; R2 ), (ws · ∇)ws belongs to Lp (Ω; R2 ) for all < p < In particular it belongs to HΓ−ε (Ω; R2 ) for d 1+η all < ε < 1/2 Thus ws belongs to HΓd (Ω; R ), for some η > That follows by interpolation from Theorems 2.4 and 2.5 Thus, using this improved regularity result, we verify that (ws · ∇)ws belongs to L2 (Ω; R2 ) Therefore, from Theorem 2.5, it follows that (ws , qs ) belongs to H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω) Let us notice that, instead of replacing the homogeneous Neumann boundary condition of (1.1) by the nonlinear boundary condition (A.1), we could also replace it by a variational inequality as in [31] We could follow the same steps as above to prove the existence of a solution (ws , qs ) in H 3/2+εo (Ω; R2 ) × H 1/2+εo (Ω) to (1.1) Indeed following [31], if the solution (ws , qs ) of the variational inequality obeys (A.2), then it is also a solution to (1.1) This type of result is stated in [31] only for solutions of the variational inequality regular enough, but we think that the proof can be adapted to Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3038 PHUONG ANH NGUYEN AND JEAN-PIERRE RAYMOND obtain the same result under the weaker regularity (ws , qs ) ∈ H (Ω; R2 )× L2 (Ω) The only difficulty is giving a correct meaning to the boundary inequality of the variational inequality Acknowledgments The authors would like to thank the referees for their comments and questions which have been at the origin of several improvements in writing the paper The second author would like to thank Mehdi Badra for stimulating discussions during the preparation of this paper REFERENCES [1] L Amodei and J.-M Buchot, An invariant subspace method for large-scale algebraic Riccati equation, Appl Numer Math., 60 (2010), pp 1067–1082 [2] L Amodei and J.-M Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numer Linear Algebra Appl., 19 (2012), pp 700–727 [3] M Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim Calc Var., 15 (2009), pp 934–968 [4] M Badra, Lyapunov function and local feedback boundary stabilization of the Navier–Stokes equations, SIAM J Control Optim., 48 (2009), pp 1797–1830 [5] M Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccatibased strategy Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin Dyn Syst., 32 (2012), pp 1169–1208 [6] M Badra and T Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier–Stokes system, SIAM J Control Optim., 49 (2011), pp 420–463 [7] V Barbu, Stabilization of the Navier-Stokes Equations, Springer-Verlag, Berlin, 2011 [8] V Barbu, I Lasiecka, and R Triggiani, Tangential Boundary Stabilization of the NavierStokes Equations, Mem Amer Math Soc 852, Providence, RI, 2006 [9] V Barbu, S S Rodrigues, and A Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier–Stokes equations, SIAM J Control Optim., 49 (2011), pp 1454–1478 [10] L Baffico, C Grandmont, and B Maury, Multiscale modeling of the respiratory tract, Math Models Methods Appl Sci., 20 (2010), pp 59–93 [11] P Benner, J Saak, F Schieweck, P Skrzypacz, and H K Weichelt, A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations, J Numer Math., 22 (2014), pp 191–219 [12] A Bensoussan, G Da Prato, M C Delfour, and S K Mitter, Representation and Control of Infinite Dimensional Systems, Vol 1, Birkhă auser, Boston, 1992 ements danalyse pour l [13] F Boyer and P Fabrie, El´ etude de quelques mod` eles d’´ ecoulements de fluides visqueux incompressibles, Springer, Berlin, 2006 [14] J.-M Buchot, J.-P Raymond, and J Tiago, Coupling estimation and control for a two dimensional Burgers type equation, ESAIM Control Optim Calc Var 21 (2015), pp 535– 560 [15] C Fabre and G Lebeau, Prolongement unique des solutions de l’´ equation de Stokes, Comm Partial Differential Equations, 21 (1996), pp 573–596 [16] C Fabre and G Lebeau, R´ egularit´ e et unicit´ e pour le probl` eme de Stokes, Comm Partial Differential Equations, 27 (2002), pp 437–475 [17] E Fernandez-Cara, S Guerrero, O Yu Imanuvilov, and J.-P Puel, Local exact controllability of the Navier-Stokes system, J Math Pures Appl (9), 83 (2004), pp 1501–1542 [18] A V Fursikov, Optimal Control of Distributed Systems, Theory and Applications, AMS, Providence, RI, 2000 [19] A V Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J Math Fluid Mech., (2001), pp 259–301 [20] A V Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete Contin Dyn Systems, 10 (2004), pp 289–314 [21] A V Fursikov and R Rannacher, Optimal Neumann control for the two-dimensional steady-state Navier-Stokes equations, in New Directions in Mathematical Fluid Mechanics, Adv Math Fluid Mech., Birkhă auser Verlag, Basel, 2010, pp 193–221 [22] G P Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol 1, Springer-Verlag, New York, 1994 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 09/23/15 to 132.203.227.61 Redistribution subject to SIAM license or copyright; 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