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1 Advances in Differential Equations Volume 22, Numbers 9-27), LOCAL STABILIZATION OF COMPRESSIBLE NAVIER-STOKES EQUATIONS IN ONE DIMENSION AROUND NON-ZERO VELOCITY Debanjana Mitra Department of Mathematics, Virginia Tech, Blacksburg, VA 246 Mythily Ramaswamy T.I.F.R Centre for Applicable Mathematics Post Bag No. 653, GKVK Post Office, Bangalore-5665, India Jean-Pierre Raymond Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS 362 Toulouse Cedex, France Submitted by: Jean-Michel Coron) Abstract. In this paper, we study the local stabilization of one dimensional compressible Navier-Stokes equations around a constant steady solution ρ s, u s), where ρ s >, u s. In the case of periodic boundary conditions, we determine a distributed control acting only in the velocity equation, able to stabilize the system, locally around ρ s, u s), with an arbitrary exponential decay rate. In the case of Dirichlet boundary conditions, we determine boundary controls for the velocity and for the density at the inflow boundary, able to stabilize the system, locally around ρ s, u s), with an arbitrary exponential decay rate.. Introduction Stabilization of fluid flows around unstable stationary solutions is an important issue in many engineering applications see e.g. [5]). The case of the incompressible Navier-Stokes equations has been widely studied both in the mathematical and engineering literatures [2, 3, 4, 5, 6, 7, 2]. Similar issues for the compressible Navier-Stokes equations are much more recent. There are a few papers studying the controllability of such systems in the one dimensional case [, 2, ]. The null controllability of linearized systems []) or nonlinear systems [, 2]) implies their stabilization. But they do not give an explicit way for computing stabilizing controls. One of the AMS Subject Classifications: 93C2, 93D5, 76N25. Accepted for publication: January

2 694 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond goals of this paper is to fill this gap. We would like to determine stabilizing controls for the one dimensional Navier-Stokes equations, around a constant steady state ρ s >, u s. We first study the local stabilization of the one dimensional compressible Navier-Stokes system, with periodic boundary conditions, by a distributed control. Next, we shall see that the stabilization of the one dimensional compressible Navier-Stokes system by Dirichlet boundary controls may be deduced from this first result. We consider the following compressible isentropic Navier-Stokes equations in the interval, L) with periodic boundary conditions ρ t + ρu) y = in, L), ),.) ρu t + uu y ) + pρ)) y νu yy = ρfχ l,l 2 ) in, L), ), ρ, ) = ρl, ), u, ) = ul, ), u y, ) = u y L, ) in, ), ρ, ) = ρ ), u, ) = u ) in, L). Here, ρy, t) is the density, uy, t) is the fluid velocity, ν > is the fluid viscosity, and the pressure p is assumed to satisfy the constitutive law pρ) = aρ γ, for some constants a > and γ. Here, f is an interior control with support in l, l 2 ), a nonempty interval of, L). We set Ω y =, L), and Q y := Ω y, )..2) Let us first notice that any pair of constants ρ s, u s ), with ρ s >, is a steady state solution of.) for f =. By integrating the first equation in.) and using the periodic boundary conditions, we also observe that L ρy, t)dy = L ρ y)dy, t >. Thus, there is no effect of the control f on the mean value of the density. So, we start with an initial density ρ satisfying L L ρ y)dy = ρ s and min y Ω y ρ y) >..3) To study the local stabilization of.) around the pair of constants ρ s, u s ), where ρ s > and u s, we define σ = ρ ρ s, v = u u s. The system satisfied by σ, v) is σ t + ρ s v y + u s σ y + σ y v + σv y = in Q y,.4) v t + vv y + u Please s v y + aγσ + ρ DO s ) NOT γ 2 σ y ν distribute v yy = fχ σ + ρ l,l 2 ) in Q y, s σ, ) = σl, ), v, ) = vl, ), v y, ) = v y L, ) in, ),

3 Local Stabilization of Navier-Stokes equations 695 σ, ) = σ ) = ρ ) ρ s, v, ) = v ) = u ) u s in Ω y, L L σ y)dy =. Now, note that σ satisfies L σy, t)dy =, t >. To achieve the stabilization of.4) with exponential decay e ωt, for any ω >, it is convenient to introduce the new unknowns σ = e ωt σ, = e ωt v, f = e ωt f. We notice that σ,, f) satisfies the system σ t + u s σ y + ρ s y ω σ + e ωt { σ y + σ y } = in Q y,.5) t + u s y + e ωt y + aγe ωt σ + ρ s ) γ 2 σ y ν e ωt σ ω = + ρ fχ l,l 2 ) in Q y, s σ, ) = σl, ),, ) = L, ), y, ) = y L, ) in, ), σ, ) = σ ),, ) = v ) in Ω y, yy L L σ y)dy =. Thus, we have L σy, t)dy =, t >. L To study the associated stabilization problem, we need to introduce the one dimensional Sobolev spaces with periodic boundary conditions. For s N {}, we denote by Hper, s L), the space of L-periodic functions belonging to Hloc s R), and by Ḣs per, L) the subspace of the functions belonging to Hper, s L), with mean value zero. Our first main result is regarding stabilization of system.5). Theorem.. Let ω be any positive number. There exist positive constants µ and κ, depending on ω, ρ s, u s, l, l 2 and L, such that, for all < µ µ and all initial condition σ, v ) Ḣ perω y ) H perω y ) satisfying σ, v ) Ḣ per Ω y) HperΩ y) κ µ,

4 696 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond there exists a control f L 2, ; L 2 Ω y )) for which system.5) admits a unique solution σ, ) satisfying σ, ) L, ;Ḣ per Ωy)) L2, ;Ḣ per Ωy)) H, ;L 2 Ω y)) L 2, ;H 2 per Ωy)) Moreover, σ, ) C b [, ); Ḣ perω y ) H perω y )), for all y, t) Q y. σy, t) ρ s 2, µ. The above theorem leads us to the following stabilization result for system.). Theorem.2. Case of periodic boundary conditions.) Let ω be any positive number. There exist positive constants µ and κ, depending on ω, ρ s, u s, l, l 2 and L, such that, for all < µ µ and all initial condition ρ, u ) H perω y ) H perω y ), where ρ satisfies.3) and ρ, u ) obeys ρ ρ s, u u s ) Ḣ per Ω y) H per Ωy) κ µ, there exists a control f L 2, ; L 2 Ω y )) for which system.) admits a unique solution ρ, u) satisfying ρ, t) ρ s, u, t) u s ) Ḣ per Ω y) H perω y) C µ e ωt, for some positive constant C depending on ω, ρ s, u s, l, l 2 and L but independent of µ. Moreover, we have ρy, t) ρ s for all y, t) Q y. 2 The proofs of the above theorems appear in Section 4 and the details about the controls f and f obtained via a nonlinear control law are given in Section 4.4. It is well known that the main difficulty in studying the one dimensional compressible Navier-Stokes equations.) comes from the nonlinear term ρ y u. There are two classical ways to deal with that term. One way consists in using the Schauder fixed point theorem, to prove the existence of a solution to system.). This method is well adapted for finite time interval [, T ] and when there is no feedback control see e.g. [2, 9]). In our case, since we look for aplease solution to DO.) or NOT.5) over distribute the time interval, ), the Schauder fixed point method cannot be used.

5 Local Stabilization of Navier-Stokes equations 697 The second method consists in using a change of variables and in writing the nonlinear system in Lagrangian variables. We follow that way. Since we deal with the equations satisfied by σ = e ωt ρ ρ s ) and = e ωt u u s ), we do not use the classical Lagrangian change of variables, but a modified one, adapted to system.5). In our situation, the change of variables is defined through the solution to a transport equation and not to an ordinary differential equation. We shall refer to the transformed system 2.9) as the Lagrangian system and the transformed variable σ, ṽ), as the Lagrangian variables. Similarly, system.5) will be referred to as the Eulerian system. Finally, our method consists in finding a feedback control operator able to stabilize first the linearized Lagrangian system, and next the nonlinear one. This is done by using a fixed point method. Then, coming back to the Eulerian system, we prove the stabilization of system.5) and hence of system.). The transformed nonlinear system presents two new difficulties. One is that the control zone is also evolving with time. Thus, the control operator becomes time dependent. But there is no general stabilization method for finding a feedback control operator for nonautonomous systems. We manage this situation by choosing a fixed control domain, which lies inside each transformed control zone for every t > Lemma 2.7). The second difficulty is that the nonlinear term F appearing in the right hand side of the density equation of the Lagrangian system, is no longer with mean value zero. Hence, the associated density is also not with mean value zero, but the control has no effect on the mean value of density. To handle this difficulty, we split F and σ in a unique manner as, for all x, L), t >, F x, t) = F,m x, t) + F,Ω t), σx, t) = σ m x, t) + σ Ω t). Here, F,m and σ m are with mean value zero, and F,Ω and σ Ω which are the mean values of F and σ respectively) are functions only depending on time. We estimate the two components differently. In our fixed point argument, we will see that it is convenient to deal with a solution σ Ω and a right hand side F,Ω in weighted Lebesgue spaces. Proceeding in this way, we prove that σ Ω is bounded in the corresponding weighted Lebesgue space. However, we can deduce afterwards that σ Ω is indeed bounded see Theorem 4.4). Next, we consider the one dimensional compressible Navier-Stokes equations around ρ s, u s ), ρ s >, u s > with boundary controls ρ t + ρu) y = in, L), ),.6)

6 698 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond ρu t + uu y ) + pρ)) y νu yy = in, L), ), ρ, ) = q ), u, ) = q 2 ), ul, ) = q 3 ) in, ), ρ, ) = ρ ), u, ) = u ) in, L), pρ) = a ρ γ, for some constants a > and γ. We prove a local stabilization result for system.6) with initial conditions ρ, u ) close to ρ s, u s ). Since u s >, we prove that uy, t) u s 2 > for all y, t) Q y. Thus, the boundary condition for density has to be prescribed only at y =. For u s < the boundary condition for density should be prescribed at y = L and the result of local stabilization can be easily adapted from the case u s >. Our main theorem for this case, reads as follows. Theorem.3. Case of Dirichlet boundary controls.) Let ω be any positive number. There exists a positive constant µ d, depending on ω, ρ s, u s >, and L, such that for any initial condition satisfying min [,L] ρ, u ) H, L) H, L), ρ > and ρ ρ s, u u s ) H,L) H,L) µ d,.7) there exist controls q L 2, ) C b [, )) and q 2, q 3 H 3 4, ), for which system.6) admits a unique solution ρ, u) satisfying ρ, t) ρ s, u, t) u s ) H,L) H,L) C e ωt,.8) for some positive constant C depending on ω, ρ s, u s, and L but independent of µ d. Furthermore, ρy, t) ρ s 2 and uy, t) u s 2 for all y, t), L), ). We prove the above stabilization result by extending system.6) to L, L) with periodic boundary conditions and then using Theorem.2 with a control localized in L, ). Finally, the traces of the velocity and the density at boundary give the boundary controls for system.6). Chowdhury et. al [9]) prove the exponential stabilization of compressible Navier-StokesPlease system in, DO π) with NOT homogeneous distribute Dirichlet boundary conditions around ρ s, ), by using a localized control for velocity, for initial

7 Local Stabilization of Navier-Stokes equations 699 conditions in H, π) H, π). Our approach to prove the local stabilization of system.) around ρ s, u s ), ρ s >, u s, with periodic boundary conditions, is also using Lagrangian coordinate transformation, similar to [9]. However, there are crucial differences in the behavior of the two systems and hence in the techniques to handle them. While the transformation is given by an ODE in [9], it is given by a pde of transport type and hence the estimates require more intricate analysis. The linearized system around ρ, ) in [9]) is not null controllable by localized control because of the accumulation point ω in the spectrum of the linearized operator see [8]). That is why, in [9], the decay rate has to be chosen strictly less than ω. But, in our case, we are able to show that the system is locally stabilizable with exponential decay e ωt, for any ω >. When u s, even though the unstable subspace is of infinite dimension for ω arbitrarily large, the unstable eigenvalues are isolated and there is a uniform lower bound for the differences between any two eigenvalues. We are able to manage the infinite dimensional unstable spaces by using the null controllability of the linearized system associated with.) by a localized control see []). Furthermore, in [9], because the unstable subspace of the linearized system is of finite dimension, the feedback control operator turns out to be a Hilbert-Schmidt operator. In contrast, in our case, the infinite dimensional unstable subspace necessitates a totally different argument to get the structure of the feedback operator. To complete the references, we mention that Ervedoza et. al [2]) prove the local exact controllability of compressible Navier-Stokes system to constant states ρ s, u s ) with ρ s >, u s using boundary controls for density and velocity, when the initial conditions for density and velocity both belong to H 3, L). Our stabilization result, Theorem.3, is also with similar boundary controls but in less regular space. However, our approach is entirely different from that of [2]. In [], the authors consider the linearized compressible Navier-Stokes system around ρ s, u s ) with ρ s >, u s with periodic boundary conditions. By the moment method they prove the null controllability of this system in Ḣs+ per Hper, s for s > 6.5, using a localized L 2 -interior control only for the velocity equation. In [], the null controllability of that system is obtained in Ḣ per L 2 by proving an observability inequality. In [4], the authors study the stabilizability of the same linearized system with exponential decay e ωt, for any ω >, using L 2 -control acting only in velocity equation. It is proved that Ḣ per L 2 is the largest space in which that system is stabilizable with any arbitrary exponential decay rate.

8 7 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond The plan of the paper is as follows. In Section 2, we introduce the Lagrangian change of variables and study its properties in Section 2.. We explain how we can choose a fixed control zone in Section 2.2. In Section 3, we study the feedback stabilization of the linearized Lagrangian system. The stabilization of the nonlinear system is treated in Section 4. We state and prove the stabilization results for the Lagrangian system in Section 4.. The Lagrangian system and its equivalence with the initial one are studied in Section 4.2. Section 4.3 is devoted to the proofs of Theorems. and.2. In Section 4.4, we determine the nonlinear control law for the Eulerian system. The case of Dirichlet boundary controls is studied in Section 5. For the sake of completeness, some classical proofs and estimations are added in an appendix Section 6). 2. Rewriting system.5) The goal of this section is to explain how we can transform system.5) through a change of variables. A similar approach is used in [9] when u s =. In the case u s, the method is more complicated. For any s N {}, we equip the spaces H s per, L) = Ḣ s per, L) = { ϕ ϕ = 2πx ik c k e k Z { ϕ H s per, L) L, L k Z Please DO NOT distribute ) } k 2s c k 2 <, } ϕx)dx =, with the norms, ϕ H s per,l) = + k 2s ) c k 2) 2, ϕ Ḣs per,l) = k Z k Z\{} k 2s c k 2) 2. We mention that the Sobolev space Hper, s L) for s = corresponds to L 2, L). Let us also recall that for any bounded open interval L, L 2 ) R, a Sobolev constant s of the embedding H L, L 2 ) L L, L 2 ) can be chosen as see for example, Theorem 8.8 in [7]) s = 4 ) ) L 2 L In particular, we shall use the notation s for the interval, L) and in this case s = L. 2.2)

9 Local Stabilization of Navier-Stokes equations Lagrangian variables. To define properly the change of variables, in addition to Ω y =, L), we introduce the notation Ω x :=, L), and Q x := Ω x, ), 2.3) to consider functions depending on the x variable. Since we deal with periodic boundary conditions, it is convenient to identify Ω y as well as Ω x with the one dimensional torus R/LZ). For any smooth function, L-periodic in the space variable and bounded in L 2, ; H 2 perω y )), we consider the L-periodic mapping Y, t) from Ω x to Ω y satisfying the following equation Y x, t) Y x, t) + u s = u s + e ωty x, t), t), x, t) Q x, 2.4) t Y x, ) = Ix), x Ω x, Y x, ) = Y x + L, ), x Ω x, where Ix) is the identity mapping in R/LZ). By the method of characteristics, this is equivalent to the following ordinary differential equation, for all x, t) Q x, d [ ] Y x + u s τ t), τ) = u s + e ωτ Y x + u s τ t), τ), τ), τ >, dτ Y x + u s τ t), τ) τ= = Ix u s t), x, t) Q x, and hence to the following integral formulation, for all τ >, Y x + u s τ t), τ) = Ix + u s τ t)) + τ Please DO t NOT distribute e ωry x + u s r t), r), r) dr. Thus, to prove the existence of a solution to 2.4), it is enough to prove the existence of a solution to the above integral formulation. In order to do that, we introduce the spaces V = C b [, ); HperΩ y )) L 2, ; HperΩ 2 y )), 2.5) { } V ω = V ω ω L 2, ;Hper 2 Ωy)) min{ 2, }, 2s 2Ls where s is defined in 2.2). The following proposition gives the existence and uniqueness of a solution to 2.4) under some conditions of. Proposition 2.. If V ω, then there exists a unique function Y C b [, ); L 2 Ω x )) satisfying Y x, t) = Ix) + e ωτ Y x + u s τ t), τ), τ) dτ. 2.6)

10 72 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond For every t >, the periodic mapping x Y x, t) is bijective from the d-torus Ω x to the d-torus Ω y. Moreover, Y belongs to and it satisfies equation 2.4). C b [, ); H 2 perω x )) C b [, ); H perω x )) The proof of the above proposition follows from the Picard s iteration method and careful estimations of the integrals. For the sake of completeness, the proof is given in Section 6.. Lemma 2.2. Let V ω and let Y be the solution of equation 2.4). Then Y L Q x ) 2. The proof is given in Section 6.. Corollary 2.3. Let V ω and let Y be the solution of equation 2.4). Then, for every t >, the periodic map x Y x, t) is C diffeomorphism from the d-torus Ω x to the d-torus Ω y. Denoting by X, t) the L-periodic inverse of Y, t), we have Let us introduce the constants We set X Y x, t), t) = x, x, t) Q x, Y X y, t), t) = y, y, t) Q y. b := aγρ γ 2 s, ν := ν ρ s. 2.7) σx, t) = σy x, t), t), ṽx, t) = Y x, t), t), fx, t) = fy x, t), t), 2.8) for all x, t) Q x. Let us consider the system σ t + u s σ x + ρ s ṽ x ω σ = F σ, ṽ, t) in Q x, 2.9) ṽ t + u s ṽ x + b σ x ν ṽ xx ωṽ = fχ l,ṽ t), l 2,ṽ t)) + F 2 σ, ṽ, t) in Q x, σ, ) = σl, ), ṽ, ) = ṽl, ), ṽ x, ) = ṽ x L, ) in, ), σ, ) = σ ), ṽ, ) = v ) in Ω x, Y x, t) = Ix) + t L Please DO NOT distribute σ x)dx =, e ωτ ṽx + u s τ t), τ)dτ, x, t) Q x, XY x, t), t) = x, x, t) Q x, Y Xy, t), t) = y, y, t) Q y,

11 Local Stabilization of Navier-Stokes equations 73 l,ṽ t) = Xl, t), l2,ṽ t) = Xl 2, t), t >, where Y ) ) F σ, ṽ, t) = ρ s ṽ x e ωt σṽ Y ), x 2.) [ F 2 σ, ṽ, t) = σ x b aγe ωt σ + ρ s ) γ 2 Y ) ] νṽ x 2 Y Y e ωt σ + ρ s ) 2 Then, we have the following theorem: Theorem 2.4. Let ) 3 νṽxx [ ρ s σ L, ; Ḣ perω y )) L 2, ; Ḣ perω y )), L 2, ; H 2 perω y )) H, ; L 2 Ω y )), Y ) 2 ] e ωt σ. + ρ s ) be the solution of.5) with control f L 2, ; L 2 Ω y )). If in addition V, then σ, ṽ, f) defined by 2.8), together with Y, X) = Y, X ), satisfies system 2.9). Further, σ L, ; H perω x )) L 2, ; H perω x )), ṽ L 2, ; H 2 perω x )) H, ; L 2 Ω x )), f L 2, ; L 2 Ω x )), and there exists a constant M,ω, depending on ω, such that σ, ṽ) D M,ω σ, ) D, where σ, ṽ) D denotes the norm of σ, ṽ) in L, ; H perω x )) L 2, ; H perω x ))) L 2, ; H 2 perω x )) H, ; L 2 Ω x ))), and σ, ) D denotes the norm of σ, ) in L, ; Ḣ perω y )) L 2, ; Ḣ perω y ))) L 2, ; H 2 perω y )) H, ; L 2 Ω y ))). Proof. For σ,, f) with the L-periodic transformation Y defined in 2.6), by the chain rule differentiation formula, σ, ṽ, f) satisfies 2.9) in Q x, with L-periodic boundary conditions. From Lemma 2.2, it follows that Y 2 for V ω. The rest of the proof follows from this and the change of variables formula.

12 74 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond The converse of the above theorem will be handled in Section 4.2. We shall need the following spaces Ṽ = C b [, ); HperΩ x )) L 2, ; HperΩ 2 x )), 2.) { } ω Ṽ ω = ṽ Ṽ ṽ L 2, ;HperΩ 2 x)), 2s and, for ṽ Ṽ, the transformation Yṽx, t) = Ix) + t We have the following lemma. e ωτ ṽx + u s τ t), τ) dτ, x, t) Q x. 2.2) Lemma 2.5. Let ṽ Ṽ and let Y ṽ be defined by 2.2), then Y ṽ L ṽ L, ;L 2 Ω x)) 2ω 2, ;HperΩ 2 x)), 2.3) 2 Yṽ L 2 ṽ L, ;L 2 Ω x)) 2ω 2, ;Hper 2 Ωx)), 2.4) Y ṽ L ṽ, ;HperΩ x)) ω L 2, ;Hper 2 Ωx)). 2.5) Moreover, if ṽ Ṽω, we have Y ṽ L Q x ) ) Proof. By differentiating 2.2), we get Yṽ x, t) = t e ωs ṽ x x + u s s t), s) ds, x, t) Q x. 2.7) Thus, Y ṽ, t) 2 e 2ωt ṽ x 2 L 2 L Ω x) 2ω 2, ;L 2 Ω, x)) and 2.3) is proved. Estimate 2.4) follows from 2 Yṽ x, t) = 2 t e ωs ṽ xx x + u s s t), s) ds. The estimate 2.5) is a direct consequence of 2.3) and 2.4). If ṽ Ṽω, we have Please Y ṽ DO NOT distribute L s Q x ) Y ṽ L, ;Hper Ωx)) 2.

13 Local Stabilization of Navier-Stokes equations 75 Corollary 2.6. Let ṽ Ṽω and let Yṽ be defined by 2.2). Then, for each t >, the periodic mapping x Yṽx, t) is C -diffeomorphism from the d-torus Ω x to the d-torus Ω y. Denoting by Xṽ, t) the L-periodic inverse of Yṽ, t), we have XṽYṽx, t), t) = x, x, t) Q x, YṽXṽy, t), t) = y, y, t) Q y From a moving to a fixed control zone. As mentioned in the introduction, in the transformed system 2.9), the control zone depends on the time variable t. To handle this situation, we choose an open interval O Ω x such that O lies inside the control zone l,ṽ t), l 2,ṽ t)) for all t >. This is detailed in the following Lemma. Lemma 2.7. Let ṽ belong to C b [, ); HperΩ x )) L 2, ; HperΩ 2 x )) and let us also assume that { 2ωl 2 l ) ω } ṽ L 2, ;HperΩ 2 x)) min,. 2.8) 8s 2s Then, we have l j,ṽ t) l j l 2 l, t >, j =, ) 8 Furthermore, if we choose the open set O Ω x defined by 7l + l 2 ) O :=, 7l 2 + l ) ), 2.2) 8 8 then, we have O l,ṽ t), l 2,ṽ t)), t >. 2.2) Proof. For every t > the moving domain for the control is l,ṽ t), l 2,ṽ t)), where For we get l j = l j,ṽ t) + t e ωτ ṽ l j,ṽ t) + u s τ t), τ) dτ, j =, 2, t >. ṽ L 2, ;H 2 per Ωx)) 2ω l2 l 8s, l j,ṽ t) l j ṽ L 2ω 2, ;L Ω x)) l 2 l. 8 For δ = l 2 l 8 Please, we also have DOl + NOT δ < l 2 δ distribute. Therefore, the lemma follows by choosing O := l + δ, l 2 δ).

14 76 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond Let us notice that, with 2.2), we have χ l,ṽ t), l 2,ṽ t)) χ O = χ O, t >. Thus, to study the stabilizability of system 2.9), it is enough to study the stabilizability of the system σ t + u s σ x + ρ s ṽ x ω σ = F σ, ṽ, t) in Q x, 2.22) ṽ t + u s ṽ x + b σ x ν ṽ xx ωṽ = χ O f + F2 σ, ṽ, t) in Q x, σ, ) = σl, ), ṽ, ) = ṽl, ), ṽ x, ) = ṽ x L, ) in, ), σ, ) = σ ), ṽ, ) = v ) in Ω x, Y x, t) = Ix) + t L σ x)dx =, e ωτ ṽx + u s τ t), τ)dτ, x, t) Q x, XY x, t), t) = x, x, t) Q x, Y Xy, t), t) = y, y, t) Q y, l,ṽ t) = Xl, t), l2,ṽ t) = Xl 2, t), t >, where F and F 2 are defined in 2.). 3. Stabilization of the linearized Lagrangian system In this section, we will use the notation Ω and Q instead of Ω x and Q x, since we are going to study the Lagrangian system 2.22) where the unknowns are functions of x, t) only. Associated to the transformed system 2.22), with the control zone O, let us consider the following linearized system σ t + u s σ x + ρ s ṽ x = in Q, 3.) ṽ t ν ṽ xx + u s ṽ x + b σ x = f χ O in Q, σ, ) = σl, ), ṽ, ) = ṽl, ), ṽ x, ) = ṽ x L, ) in, ), σ, ) = σ, ṽ, ) = ṽ in Ω, σ x)dx =, where the control f belongs to L 2, ; L 2 Ω)). Let us introduce the complex Hilbert space Ω Z = Ḣ perω) L 2 Ω),

15 Local Stabilization of Navier-Stokes equations 77 endowed with the inner product ) ) ρ σ L L, u v := b ρ x x)σ x x) dx + ρ s ux)vx)dx. z We define the unbounded operator A, DA)) in Z by and DA) = Ḣ2 perω) H 2 perω), A = [ us d dx Setting zt) = σ, t), ṽ, t)) T written as ρ s d dx b d dx ν d 2 dx 2 u s d dx ]. 3.2) and B f =, fχ O ) T, system 3.) can be z t) = Azt) + B ft), z) = z Z. 3.3) Let us mention that A, DA)) generates a C semigroup in Z, denoted by {e ta } t, and the control operator B belongs to LL 2 Ω), Z). We recall Lemma 2.2 from [4] regarding the spectrum of A for L = 2π see also []). Lemma 3.. The spectrum of A consists of and two sequence of complex eigenvalues { λ h k, λp k } k Z with λh k = λh k, λp k = λp k for all k Z. Moreover, for k =, we denote λ h =. For k Z with k 2 < 4bρs ν 2, and, for k Z with k 2 4bρs ν 2, λ h k = [k2 ν ik 4bρ s k 2 ν 2 + 2u s )], 2 λ p k = [k2 ν + ik 4bρ s k 2 ν 2 2u s )], 2 λ h k = [k2 ν k k 2 ν 2 4bρ s ) 2iku s ], 2 λ p k = [k2 ν + k k 2 ν 2 4bρ s ) 2iku s ]. 2 Let us denote ω = bρs ν. We have the following asymptotic behaviors Reλ h k ω, Reλ p k k 2 ν as k, Im λh k u s Im λh k u s as k. k k

16 78 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond 2 5 imaginary part 5 5 ω= ω = real part Figure. Spectrum of A with u s = = ρ s = a = ν, γ = 25, ω = 25 and ω = In view of the above lemma, for ω > ω, A + ωi) has an infinite number of eigenvalues with positive real part see [4] and []). In spite of having an infinite dimensional unstable space, system 3.) is stabilizable in Z with exponential decay e ωt, for any ω >, by a L 2 -control acting everywhere in Ω see [4]). We know that system 3.) is null controllable in Z at T, by L 2 -localized control, for any T > L u s see Theorem.2 in []). Now, from this null controllability result, we obtain the complete stabilization of system 3.) see Theorem 3.3 in [2]). One can also get a feedback stabilization result as in the following theorem. Theorem 3.2. Let ω be any positive number. There exists K m LZ, L 2 Ω)) such that the semigroup e ta+ωi+bkm) ) t is exponentially stable. The solution σ, ṽ) of d dt σ ṽ ) σ = A+ωI +BK m ) ṽ ), σ ṽ belongs to C b [, ); Z) L 2, ; Z) and satisfies ) ) σ, ) = ṽ Please P LZ, Z DO ), P NOT = P >, distribute Z, 3.4) σ, ṽ), t) Z < Me δt σ, ṽ ) Z for all t >, 3.5) for some δ > and M >. Furthermore, K m can be chosen in the form K m = B P, where P is the solution of the following algebraic Riccati equation P A + ωi) + A + ωi)p P BB P + I =. 3.6)

17 Local Stabilization of Navier-Stokes equations 79 The above theorem follows from Theorem 3.3 in [2]. It is convenient to define the feedback operator K LH perω) L 2 Ω), L 2 Ω)), by K σ, ṽ) = K m σ m, ṽ), σ, ṽ) HperΩ) L 2 Ω), 3.7) where σ m x, t) = σx, t) σy, t) dy, L Ω and K m LḢ perω) L 2 Ω), L 2 Ω)). Now, we analyze further the structure of this feedback operator to see if it can be expressed by a kernel in a suitable Sobolev space. We have the following proposition in this direction. Proposition 3.3. Let the operator K LHperΩ) L 2 Ω), L 2 Ω)) be defined in 3.7). Then there exist two kernel operators k σ L 2 Ω; HperΩ)) and k v L 2 Ω Ω) such that for all σ, ṽ) HperΩ) L 2 Ω), K σ, ṽ)x) = k σ x, ), σ ) H per,h + k per v x, ξ)ṽξ) dξ. Ω Proof. The operator K LH perω) L 2 Ω), L 2 Ω)) can be decomposed in the form K σ, t), ṽ, t)) = K σ σ, t) + K v ṽ, t), with K σ σ L 2 Ω) C σ H per Ω) for all σ H perω), 3.8) and K v ṽ L 2 Ω) C ṽ L 2 Ω) for all ṽ L 2 Ω). 3.9) Let us denote by D per Ω) the set of functions which are the restrictions to Ω of L-periodic C functions. The space D per Ω Ω) is defined in an analogous manner. The dual of D per Ω) is denoted by D perω) and the dual of D per Ω Ω) is denoted by D perω Ω). From Schwartz s kernel Theorem see [8, Theorem II], [3]) adapted to D per Ω), it follows that there exist k σ D perω Ω) and k v D perω Ω) such that for all x Ω, and K σ σ)x) = k σ x, ), σ ) D per Ω),D perω) for all σ D per Ω), 3.) K v ṽ)x) = k v x, ), ṽ ) D per Ω),D perω) for all ṽ D per Ω). 3.) Due to 3.8), it follows that k σ is a distribution of order, and due to 3.9) that k v is a distribution of order.

18 7 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond Since D per Ω) is dense in H perω) as well as in L 2 Ω), using the calculations so far and the definitions of k σ and k v, we have a unique extension k σ x, ) H perω) and k v x, ) L 2 Ω) such that and K σ σ)x) = k σ x, ), σ ) H perω),h perω) for all σ H perω), K v ṽ)x) = k v x, ), ṽ ) L 2 Ω),L 2 Ω) for all ṽ L 2 Ω). Moreover, due to 3.8) and 3.9), k σ belongs to L 2 Ω; HperΩ)) and k v belongs to L 2 Ω Ω). Let us set V = Ḣ perω) H perω). To use a fixed point argument in later analysis, we need to consider initial condition σ, ṽ ) in V. We have the following regularity theorem for the solution σ, ṽ) of 3.4). Theorem 3.4. For σ, ṽ ) V, the solution σ, ṽ) of 3.4) satisfies σ L, ;Ḣ perω)) + σ L 2, ;Ḣ perω)) + ṽ L 2, ;H 2 per Ω)) + ṽ H, ;L 2 Ω)) C σ, ṽ ) V. Proof. The estimate of σ, ṽ) in L 2, ; Z) L, ; Z) follows from the exponential stability of the semigroup e ta+ωi+bkm) ) t, see 3.5). Next, the estimate of ṽ in L 2, ; H 2 perω)) H, ; L 2 Ω)) follows from regularity results for parabolic equations, with a right hand side in L 2, ; L 2 Ω)). To handle the nonlinear terms in 2.22), we need to consider the linearized system 3.4) with forcing terms, i.e., σ t + u s σ x + ρ s ṽ x ω σ = f in Q, 3.2) ṽ t + u s ṽ x + b σ x ν ṽ xx ωṽ = χ O K m σ, t) ) σξ, t) dξ, ṽ, t) + f 2 in Q, L Ω σ, ) = σl, ), ṽ, ) = ṽl, ), ṽ x, ) = ṽ x L, ) in, ), σ, ) = σ ), ṽ, ) = v ) in Ω. As explained in the Introduction, to study system 3.2) we decompose f and σ uniquely as follows f x, t) = f,m x, t) + f,ω t), σx, t) = σ m x, t) + σ Ω t), 3.3)

19 Local Stabilization of Navier-Stokes equations 7 for all x, t) Q x, where f,ω t) := f x, t)dx, f,m x, t)dx =, t >, 3.4) L Ω L Ω σ Ω t) := σx, t)dx, σ m x, t) dx =, t >. L Ω L Ω Setting z m t) = σ m, t), ṽ, t)) T, we easily check that σ, ṽ) is a solution to 3.2) if and only if z m, σ Ω ) is a solution to z mt) = A + ωi + BK m )z m t) + f,m, t), f 2, t)) T for all t >, z m ) = z, 3.5) σ Ωt) = ω σ Ω t) + f,ω t), t >, σ Ω ) =. This leads to t σ Ω t) = e ωt e ωs f,ω s) ds, t >. 3.6) Thus, we introduce the weighted Lebesgue spaces L, ; e ω ) ) = {h e ω ) h L, )}, L, ; e ω ) ) = {h e ω ) h L, )}. 3.7) We have the following results for the linearized closed loop system 3.2). Theorem 3.5. For σ, ṽ ) V and f,m L 2, ; Ḣ perω)), f,ω L, ; e ω ) ), and f 2 L 2, ; L 2 Ω)), the solution zt) = σ, t), ṽ, t)) T of system 3.2) satisfies σ m L 2, ;Ḣ per Ω)) + σ m L, ;Ḣ per Ω)) + σ Ω L, ;e ω ) ) 3.8) + ṽ L 2, ;H 2 per Ω)) + ṽ H, ;L 2 Ω)) C f,m L 2, ;Ḣ perω)) + f,ω L, ;e ω ) ) + f 2 L 2, ;L 2 Ω)) + σ, ṽ ) V ). Proof. The estimate of σ m, ṽ) in L 2, ; Z) L, ; Z) follows from the exponential stability of the semigroup e ta+ωi+bkm) ) t, see 3.5), the Duhamel formula and the Young inequality for convolution products. The estimate of σ Ω in L Please, ; e DO ω ) ) is obvious. Next, as in Theorem 3.4, the estimate of ṽ in L 2, ; HperΩ)) 2 NOT H, ; distribute L 2 Ω)) follows from regularity results for parabolic equations, with a right hand side in L 2, ; L 2 Ω)).

20 72 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond 4. Stabilization of the nonlinear system With the feedback operator K L H perω x ) L 2 Ω x ), L 2 Ω x ) ) defined in 3.7), the closed loop nonlinear system corresponding to 2.9) is σ t + u s σ x + ρ s ṽ x ω σ = F σ, ṽ, t) in Q x, 4.) ṽ t + u s ṽ x + b σ x ν ṽ xx ωṽ = χ O K σ, t), ṽ, t)) + F 2 σ, ṽ, t) in Q x, σ, ) = σl, ), ṽ, ) = ṽl, ), ṽ x, ) = ṽ x L, ) in, ), σ, ) = σ ), ṽ, ) = v ) in Ω x, σ x)dx =, Ω x Y x, t) = Ix) + t e ωτ ṽx + u s τ t), τ)dτ, x, t) Q x, XY x, t), t) = x, x, t) Q x, Y Xy, t), t) = y, y, t) Q y, l,ṽ t) = Xl, t), l2,ṽ t) = Xl 2, t), t >, where F and F 2 are defined in 2.). In this section, first, we show that 4.) admits a unique solution in a suitable ball defined by some estimates. Next, we show that it is possible to come back from system 4.) to the original system.5), since σ, ṽ, X, Y ) satisfies the required estimates. Using these results, we finally prove Theorems. and Stabilization of the Lagrangian system 4.). Recall the unique decomposition introduced in 3.3)-3.4). We first state the following lemma which will be useful in deriving several estimates. Lemma 4.. Let σx, t) = σ m x, t)+ σ Ω x, t) with σ m L, ; Ḣ perω x )) and σ Ω L, ; e ω ) ). If then σ m L, ;Ḣ perω x)) + σ Ω L, ;e ω ) ) ρ s 4s, 4.2) e ωt σx, t) ρ s 2, x, t) Q x. Proof. Using 4.2) and s >, for all x, t) Q x, we have e ωt σx, t) σ m L Q x ) + σ Ω L, ;e ω ) ) s σ m L, ;Ḣ per Ωx)) + σ Ω L, ;e ω ) ) ρ s 2.

21 Local Stabilization of Navier-Stokes equations 73 Let us consider the space { D = ζ, ϑ) ϑ H, ; L 2 Ω x )) L 2, ; HperΩ 2 x )), 4.3) ζ = ζ m + ζ Ω, ζ m L 2, ; Ḣ perω x )) L, ; Ḣ perω x )), } ζ Ω L, ; e ω ) ), equipped with the norm ζ, ϑ) D = ζ m L 2, ;Ḣ perω x)) + ζ m L, ;Ḣ perω x)) + For any µ >, we define We set ζ Ω L, ;e ω ) ) + ϑ L 2, ;H 2 perω x)) + ϑ H, ;L 2 Ω x)). F ζ, ϑ, t) = ρ s ϑ x D µ = { σ, ṽ) D σ, ṽ) D µ}. 4.4) Y ζ,ϑ) F 2 ζ, ϑ, t) = ζ x b aγe ωt ζ + ρ s ) γ 2 νϑ x 2 Y ζ,ϑ) e ωt ζ + ρ s 2 ) ) e ωt Y ζ,ϑ) ζϑ x ), 4.5) Y ζ,ϑ) Y ζ,ϑ) ) ) ) 3 νϑ xx Y ζ,ϑ) ρ s e ωt ζ + ρ s ) 2 ). As mentioned in the Introduction, we use the decomposition F ζ, ϑ, t) = F,m ζ, ϑ, t) + F,Ω ζ, ϑ, t), t >, where F,Ω ζ, ϑ, t) = F ζ, ϑ, t)dx. L Ω The next lemma gives some useful estimations of 4.5). Lemma 4.2. There exists a positive constant C 2 depending on ω, ρ s, u s, s, L, ν { such that for all ζ, ϑ), ζ, ϑ ), ζ 2, ϑ 2 ) belonging to D µ, with µ = min ρs 4s, ω 2s }, we have the following estimates F,m ζ, ϑ, ) L 2, ;Ḣ per Ωx)) C 2 ζ, ϑ) 2 D, 4.6) F,Ω ζ, ϑ, ) L, ;e ω ) ) C 2 ζ, ϑ) 2 D, 4.7) F 2 ζ, ϑ, ) L 2, ;L 2 Ω x)) C 2 ζ, ϑ) 2 D, 4.8)

22 74 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond F,m ζ, ϑ, ) F,m ζ 2, ϑ 2, ) L 2, ;Ḣ per Ωx)) 4.9) C 2 ζ, ϑ ) D + ζ 2, ϑ 2 ) D ) ζ, ϑ ) ζ 2, ϑ 2 ) D, F,Ω ζ, ϑ, ) F,Ω ζ 2, ϑ 2, ) L, ;e ω ) ) 4.) C 2 ζ, ϑ ) D + ζ 2, ϑ 2 ) D ) ζ, ϑ ) ζ 2, ϑ 2 ) D, F 2 ζ, ϑ, ) F 2 ζ 2, ϑ 2, ) L 2, ;L 2 Ω x)) 4.) C 2 ζ, ϑ ) D + ζ 2, ϑ 2 ) D ) ζ, ϑ ) ζ 2, ϑ 2 ) D. The proof of all these estimates 4.6)-4.) is given in the appendix Section 6.2). We have the following theorem. Theorem 4.3. Let K L H perω x ) L 2 Ω x ), L 2 Ω x ) ) be defined in 3.7) and O defined by 2.2). There exist constants µ > and κ >, depending on s, ω, L, l, l 2, u s, ρ s, such that, for < µ µ, and any initial conditions σ, v ) satisfying σ, v ) Ḣ per Ω x) H perω x) κ µ, 4.2) the closed loop system 4.) admits a unique solution σ, ṽ, X, Y ) such that σ, ṽ) belongs to D µ, X C b [, ); H 2 perω y )) C b [, ); H perω y )), and Y C b [, ); H 2 perω x )) C b [, ); H perω x )). Moreover, σ C[, ); H perω x )), ṽ belongs to Ṽω and satisfies 2.8). Proof. The proof is based on the Banach fixed point Theorem. Let us choose { ρs ω 2ω l2 l } µ := min,,,, κ =, 4.3) 4s 2s 8s 4C C 2 2C where C and C 2 are the constants appearing in Theorem 3.5 and Lemma 4.2 respectively. Let µ belong to, µ ]. For any ζ, ϑ) D µ, we denote by σ ζ,ϑ), ṽ ζ,ϑ)) the solution of the following linear system σ ζ,ϑ) t ṽ ζ,ϑ) t + u s σ ζ,ϑ) x + u s ṽ ζ,ϑ) x + ρ s ṽ ζ,ϑ) x ω σ ζ,ϑ) = F ζ, ϑ, t) in Q x, 4.4) + b σ ζ,ϑ) x ν ũ ζ,ϑ) xx ωṽ ζ,ϑ) = χ O x)k σ ζ,ϑ), t), ṽ ζ,ϑ), t)) + F 2 ζ, ϑ, t) in Q x, σ ζ,ϑ), ) = σ ζ,ϑ) L, ), ṽ ζ,ϑ), ) = ṽ ζ,ϑ) L, ),

23 ṽ ζ,ϑ) x Local Stabilization of Navier-Stokes equations 75, ) = ṽ x ζ,ϑ) L, ) in, ), σ ζ,ϑ), ) = σ ), ṽ ζ,ϑ), ) = v ) in Ω x, Y ζ,ϑ) x, t) = Ix) + t where F and F 2 are defined in 4.5). Let us prove that the mapping Ω x σ x)dx =, e ωτ ϑx + u s τ t), τ)dτ, x, t) Q x, ζ, ϑ) σ ζ,ϑ), ṽ ζ,ϑ) ), 4.5) is a contraction in D µ. Since µ is less than or equal to ρs 4s and ω 2s, from 3.5), 4.6), 4.7), 4.8) and 4.2), it follows that σ ζ,ϑ), ṽ ζ,ϑ)) D 4.6) C σ, v ) Ḣ per Ω x) Hper Ωx) + F,mζ, ϑ, ) L 2, ;Ḣ per Ωx)) ) + F,Ω ζ, ϑ, ) L, ;e ω ) ) + F 2ζ, ϑ, ) L 2, ;L 2 Ω x)) ) C σ, v ) Ḣ per Ω x) HperΩ + C x) 2 ζ, ϑ) 2 D µc κ + C C 2 µ) µ. Hence, if ζ, ϑ) D µ, then σ ζ,ϑ), ṽ ζ,ϑ)) belongs to D µ. We set The couple Σ, Ṽ ) satisfies Σ = σ ζ,ϑ ) σ ζ2,ϑ 2), Ṽ = ṽ ζ,ϑ ) ṽ ζ2,ϑ 2). Σ t + u s Σx + ρ s Ṽ x ω Σ = F ζ, ϑ, t) F ζ 2, ϑ 2, t) in Q x, 4.7) Ṽ t + u s Ṽ x + b Σ x ν Ṽ xx ωṽ = χ O K Σ, t), Ṽ, t)) + F 2ζ, ϑ, t) F 2 ζ 2, ϑ 2, t) in Q x, Σ, ) = ΣL, ), Ṽ, ) = Ṽ L, ), Ṽ x, ) = ṼxL, ) in, ), Σ, ) =, Ṽ, ) = in Ω x. If ζ, ϑ ) D µ and ζ 2, ϑ 2 ) D µ, with Theorem 3.5 and estimates 4.9), 4.), 4.) and 4.3), we have σ ζ,ϑ ) ), ṽ ζ,ϑ ) σ ζ2,ϑ 2) ), ṽ ζ2,ϑ 2 ) D 4.8) C F,m ζ, ϑ, ) F,m ζ 2, ϑ 2, ) L 2, ;Ḣ per Ωx)

24 76 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond + F,Ω ζ, ϑ, ) F,Ω ζ 2, ϑ 2, ) L, ;e ω ) ) + F 2 ζ, ϑ, ) F 2 ζ 2, ϑ 2, ) L 2, ;L 2 Ω x))) 2C C 2 µ ζ, ϑ ) ζ 2, ϑ 2 ) D 2 ζ, ϑ ) ζ 2, ϑ 2 ) D. Hence, the mapping defined in 4.5) is a contraction. Further, ṽ obtained from this fixed point argument belongs to Ṽω and satisfies 2.8) because of our choice of µ. The proof is complete Transformation to original system. Now, we want to prove the converse of Theorem 2.4. Theorem 4.4. Let µ belong to, µ ], where µ is defined by 4.3). Let σ, ṽ, X, Y ) be a solution to system 4.) such that σ, ṽ) belongs to D µ. Let us set f, t) = χ O K σ, t), ṽ, t)), σy, t) = σxy, t), t), y, t) = ṽxy, t), t), fy, t) = fxy, t), t). If ṽ Ṽω, then σ belongs to L 2, ; Ḣ perω y )) L, ; Ḣ perω y )), belongs to L 2, ; H 2 perω y )) H, ; L 2 Ω y )), f belongs to L 2, ; L 2 Ω y )) and there exists a constant M 2,ω, depending on ω, such that σ, ) D M 2,ω σ, ṽ) D. Moreover, σ,, f) satisfies system.5), and, for t >, Y, t) = Y, t) is the solution of 2.4) and X, t) = X, t) is the inverse of Y, t). In addition, σ Ω belongs to L 2, ) L, ). Proof. For ṽ Ṽω, using Lemma 2.5 and the change of variables formula, we get that σ,, f) satisfies.5) in Q y. Notice that all the solutions to the density equation.5) are with mean zero. Consequently, σ also satisfies this condition. Since we have if ṽ Ṽω. σ y y, t) = σ X Xy, t), t) y, t), y σ, t) Ḣ per Ω y) 2 σ m, t) Ḣ per Ω x), t >, Indeed, in that case we have X y y, t) 2. This inequality provides an estimate of σ in L, ; Ḣ perω y )) L 2, ; Ḣ perω y )), but also in L, ; HperΩ y )) L 2, ; HperΩ y )) because σ is with mean value zero.

25 Local Stabilization of Navier-Stokes equations 77 Let us now recall the identity σ Ω t) = L L σx, t)dx = L L Xy, t) σy, t) dy, t >. 4.9) y From 4.9) and the estimate of σ in L, ; HperΩ y )) L 2, ; HperΩ y )), it follows that σ Ω also belongs to L, ) L 2, ), because y, t) 2. X y Remark 4.5. The Lebesgue spaces L, ; e ω ) ) and L, ; e ω ) ) are well adapted to study the ordinary differential equation satisfied by σ Ω. We can prove the convergence of the fixed point method by using these spaces. But it is not possible to do it directly in L, ) L 2, ). We have finally deduced from 4.9) that σ Ω is actually bounded. However, it is not possible to use an identity similar to 4.9) for the different iterates of the fixed point method. The next remark will be useful to prove the stabilization of system.5) by using the stabilization of system 2.8). Remark 4.6. In Section 4., we first obtain the unique solution σ, ṽ, X, Y ) of system 2.9) satisfying ṽ Ṽω. Then applying Theorem 4.4, we get a solution σ,, X, Y ) for system.5) where Y is the solution of 2.4) and X its inverse. But the solution σ,, X, Y ), obtained in this way, is not necessarily unique. To prove the uniqueness, we first need to guarantee that the solutions to system.5) provide solutions to system 2.9) by a change of variables. That is obtained by imposing one more condition on the norm of ṽ L 2, ;HperΩ 2 x)), in addition to the fact that ṽ Ṽω. The details are given in the following lemma. The uniqueness of solution for system.5) will be obtained as a consequence of the uniqueness of solution for system 2.9) see the proof of Theorem.). Lemma 4.7. Let ṽ Ṽω and let us set y, t) = ṽxy, t), t) for all y, t) Q y. There exists a positive C ω such that if ṽ L 2, ;H 2 per Ωx)) C ω, then belongs to V ω. Proof. For each t >, we have y, t) = ṽxy, t), t) and Y Xy, t), t) = y for y Ω y. Now, differentiating these two terms with respect to y, we get Please DO X NOT Y X distribute y = ṽ y, y =. 4.2)

26 78 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond Using change of variables and the fact that X ) ) Please DO Y x, NOT t) distribute y 2, we have L 2 Q y ) 2 ṽ L 2 Q x ). 4.2) Using Y 2, from Lemma 2.5, we get y ṽ X L 2 Q y ) L 2 Q x ) y 2 ṽ L Q y ). 4.22) L 2 Q x ) By differentiating 4.2) with respect to y, we get 2 y 2 = 2 ṽ X ) 2 ṽ 2 X + 2 Y y 2, 2 Y X 2 y ) 2 + Y With Lemma 2.5, we have 2 X L y Y L, ;L 2 Ω y)) 2, ;L 2 Ω x)) 2 X =. 4.23) y2 8 2ω ṽ L 2, ;H 2 per Ωx)) 2 2 s. Thus, we get 2 L y ṽ L 2 Q y ) ṽ 2 Q x ). 4.24) L 2, ;HperΩ x)) Finally, by using 4.2), 4.22), 4.24), we obtain L 2, ;HperΩ 2 y)) = L 2 Q x ) L L 2 Q x ) 2 2 Q x ) Let us choose ) ṽ L 2, ;H 2 perω x)). { ω C ω := min 8, 2 + )s For C ω defined above, belongs to V ω. ω 4 L 2 + )s }. 4.25) 4.3. Stabilization of the original system. By making a change of variables, we can transform system 4.) to find a control law for system.5). For σ, v ) satisfying 4.2), let σ, ṽ, X, Y ) be the unique solution to system 4.) satisfying σ, ṽ) D µ. Associated to this solution, we consider the change of variables x t t, x, t) Q x. 4.26)

27 Local Stabilization of Navier-Stokes equations 79 Then, for each t >, the feedback control K σ, t), ṽ, t)), is transformed in the form K σt), ṽt)) X, t), 4.27) where X, t), the inverse of Y, t), is also one of the components of the solution to system 4.). As in Theorem 4.4, we can set σy, t) = σxy, t), t), y, t) = ṽxy, t), t), y, t) Q y, 4.28) and for each t, we have X, t) = X, t), where X, t) is the inverse of Y, t) and Y, t) is the solution to transport equation 2.4). Therefore, the feedback law K, transformed with the change of variables 4.26), depends not only on, t) but also on X, t). This is why we set K σt), t), X t))y) = K σ, t), ṽ, t)) Xy, t), y, t) Q y. 4.29) The feedback operator K is linear but, due to the change of variables, K is a nonlinear operator. With the change of variables 4.26), system 4.) is transformed into σ t + u s σ y + ρ s y ω σ + e ωt { σ y + σ y } = in Q y, 4.3) t + u s y + e ωt y + aγe ωt σ + ρ s ) γ 2 σ y ν e ωt σ ω + ρ s = χ l,l 2 ) K σt), t), X t)) in Q y, σ, ) = σl, ),, ) = L, ), y, ) = y L, ) σ, ) = σ ),, ) = v ) in Ω y, L L yy in, ), σ y)dy =, Y x, t) Y x, t) + u s = u s + e ωty x, t), t), x, t) Q x, t Y x, ) = Ix), x Ω x, Y x, ) = Y x + L, ), x Ω x, X Y x, t), t) = x, x, t) Q x. Theorem 4.8. Let ω be any positive number. There exist positive constants µ and κ, depending on ω, ρ s, u s, l, l 2 and L, such that for all < µ µ and all initial condition σ, v ) Ḣ perω y ) H perω y ) satisfying σ, v ) Ḣ per Ω y) H per Ωy) κ µ, 4.3) the nonlinear closed loop system 4.3) admits a unique solution σ,, Y, X ) satisfying σ, ) D µ. 4.32)

28 72 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond Moreover, σ, ) C b [, ); Ḣ perω y ) H perω y )), for all y, t) Q y. σy, t) ρ s 2, Proof. In view of Remark 4.6 and Lemma 4.7, µ in Theorem 4.3 can be reduced further, if necessary, so that µ C ω, where C ω is defined in Lemma 4.7. Then by Theorems 4.3 and 4.4, we deduce that there exist constants µ > and κ such that, for < µ µ and any initial condition σ, v ) Ḣ perω y ) H perω y ) satisfying σ, v ) Ḣ per Ω y) H per Ωy) κ µ, 4.33) the closed loop system 4.3) admits a solution σ,, X, Y ) such that σ, ) L, ;Ḣ perω y)) L 2, ;Ḣ perω y)) H, ;L 2 Ω y)) L 2, ;H 2 perω y)) µ. Moreover, X C b [, ); H 2 perω y )) C b [, ); H perω y )), Y C b [, ); H 2 perω x )) C b [, ); H perω x )), σ C[, ); Ḣ perω y )), and C b [, ); H perω y )). If the initial condition σ, v ) satisfies 4.33) and if σ,, X, Y ) is a solution to 4.3), we can define σ, ṽ) by change of variables, and σ, ṽ, X, Y ), with X, Y ) = X, Y ) is a solution to system 4.). From the decomposition σ = σ m + σ Ω and the definition of σ Ω, we deduce that σ Ω L, ;e ω ) ) L /2 σ L, ;H perω y)) L 2, ;H perω y)), σ Ω L, ) L 2, ) σ L, ;H perω y)) L 2, ;H perω y)), and σ m L, ;H perω y)) L 2, ;H perω y)) + σ Ω L, ;e ω ) ) σ L R + ;H perω y)) L 2 R + ;H perω y)) + σ Ω L R + ) L 2 R + ) L R + ;e ω ) ) 2 + L /2 ) σ L, ;H per Ωy)) L2, ;H per Ωy)). Therefore, due to Theorem 2.4 and to the definition of the norms D and D, we have σ, ṽ) D 2 + L /2 ) σ, ṽ) D 2 + L /2 ) M,ω σ, ) D. Thus, if we choose 2 + L Please DO /2 ) M,ω µ µ, we shall have σ, ṽ) D µ, and σ, ṽ, X, Y ) will be the unique NOT solutiondistribute of system 4.) in D µ. Thus, under the additional condition 2 + L /2 ) M,ω µ µ, 4.3) admits a

29 Local Stabilization of Navier-Stokes equations 72 unique solution σ,, X, Y ) such that σ, ) D µ. The proof is complete. From Theorem 4.8, the stabilization result of Theorem. is obtained with the control f defined by fy, t) = χ l,l 2 )y) K σ, t),, t), X, t)) y), y, t) Q y. 4.34) Proof of Theorem.2. The closed loop nonlinear system corresponding to system.) reads as follows ρ t + ρu) y = in Q y, 4.35) ρu t + uu y ) + pρ)) y νu yy = ρχ l,l 2 )e ωt Ke ωt ρt) ρ s ), e ωt ut) u s ), Xt)) in Q y, ρ, ) = ρl, ), u, ) = ul, ), u y, ) = u y L, ) in, ), ρ, ) = ρ ), u, ) = u ) in Ω y, Y x, t) Y x, t) + u s = uy x, t), t), x, t) Q x, t Y x, ) = Ix), x Ω x, Y x, ) = Y x + L, ), x Ω x, XY x, t), t) = x, x, t) Q x. From the proof of Theorem., it follows that system 4.35) admits at least a solution ρ, u, X, Y ) defined by ρ = e ωt σ + ρ s, u = e ωt + u s, X = X, Y = Y, where σ,, X, Y ) is the solution of system 4.3). The solution ρ, u, X, Y ) is unique in the set of functions satisfying e ωt ρ ρ s ), e ωt u u s )) D µ. Thus, the stabilization result of Theorem.2 is obtained with the control f defined by fy, t) = χ l,l 2 )y) e ωt K e ωt ρt) ρ s ), e ωt ut) u s ), Xt) ) y), 4.36) for all y, t) Q y Control law for the original system. In this section, we explain why the control defined in 4.27) or in 4.29) is a nonlinear control law for system 4.3). The state variable of system 4.3) is σ,, Y, X ). It is clear that σ,, Y ) are state variables, since the triplet satisfies an evolution equation. The Please last equation DOin 4.3) NOT is a nonlinear distribute equation characterizing X in terms of Y. Since we deal with one dimensional problems, X can be

30 722 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond easily deduced from Y. This is why we can consider X as an additional component of the state variable for system 4.3). Due to Proposition 3.3, the feedback K is expressed by two functions k σ L 2 Ω x ; HperΩ x )) and k v L 2 Ω x Ω x ). For all t >, we compose k σ, ) and k v, ) with X, t) and we set kσ y, ζ, t) = y X ζ, t) k σ X y, t), X ζ, t)), and kv y, ζ, t) = y X ζ, t) k v X y, t), X ζ, t)). Since X belongs to C b [, ); HperΩ 2 y )) Cb [, ); H perω y )), it follows that k σ C b [, ); L 2 Ω y ; HperΩ y ))) and k v C b [, ); L 2 Ω y Ω y )). Therefore, the control law K for the original system is defined by K σ, t),, t), X t))y) 4.37) ) = kσ y,, t), σ, t) kv + y,, t),, t). HperΩ y),hperω y) L 2 Ω y) Remark 4.9. The corresponding control law for system 4.35) is e ωt K e ωt ρt) ρ s ), e ωt ut) u s ), Xt) ) y) = kσ y,, t), ρ, t) ρ s ) H perω y),h per Ωy) + kv y,, t), u, t) u s ) ) L 2 Ω y). From Theorem 4.8, it follows that the first two components of the state of system 4.35), namely ρ, u), are stabilized towards the steady state ρ s, u s ), exponentially in the H perω y ) H perω y ) norm, even if Y, X) is not stabilized. 5. Dirichlet boundary control In this section, we study the local stabilization of the one dimensional compressible Navier-Stokes system around ρ s, u s ), ρ s >, u s > by boundary controls. We prove Theorem.3 by using Theorem.2. To do that, we need to extend the domain, L) to L, L) and to consider system.6) in L, L) with periodic boundary conditions. The next theorem gives the stabilization result for domain L, L), ), analogous to Theorem.2 and we skip its proof as it is similar to that of Theorem.2. Theorem 5.. Let L be any positive number and l, l 2 ) is an open subset of L, L). Let us consider system.) in L, L), ) with periodic boundary Please conditions. DO For any NOT positivedistribute number ω, there exist positive constants µ and κ, depending on ω, ρ s, u s, l, l 2 and L, such that, for

31 Local Stabilization of Navier-Stokes equations 723 < µ µ and any initial condition ρ, u ) Hper L, L) Hper L, L), where ρ satisfies 2L and ρ, u ) obeys L L ρ y)dy = ρ s and min y [ L,L] ρ ρ s, u u s ) Ḣ per L,L) H per L,L) ρ y) >, κ µ, there exists a control f L 2, ; L 2 L, L)) for which system.) admits a unique solution ρ, u) satisfying ρ, t) ρ s, u, t) u s ) Ḣ per L,L) H per L,L) C µe ωt, for some positive constant C depending on ω, ρ s, u s, l, l 2 and L but independent of µ. Moreover, we have ρy, t) ρ s for all y, t) L, L), ). 2 Furthermore, if u s >, we can choose µ in such a way that uy, t) us 2 for all y, t) L, L), ). To prove the stabilization result for system.6) using the above theorem, we extend the initial condition ρ, u ) as follows. Proposition 5.2. Let µ and κ be the positive constants defined in Theorem 5. corresponding to the domain L, L). There exists a positive constant µ d depending on ω, ρ s, u s, L and µ, such that any ρ, u ) H, L) H, L) satisfying.7), i.e., min [,L] ρ > and ρ ρ s, u u s ) H,L) H,L) µ d, admits an extension ρ e, ue ) in H per L, L) H per L, L) obeying and 2L L L ρ e x)dx = ρ s, min [ L,L] ρ e >, 5.) ρ e ρ s, u e u s ) Ḣ per L,L) H per L,L) κ µ. 5.2) Proof. We want to determine µ d and the extension ρ e, ue ) H per L, L) Hper L, L) of ρ, u ) depending on µ in such a way that.7) implies 5.) and 5.2). Recall that ρ ρ s H,L) µ d gives ρ ρ s L,L) s µ d, 5.3)

32 724 Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond where s is the Sobolev constant depending on the domain. To get 5.), we need the compatibility condition In view of 5.3), we have 2L 2L L L ρ x)dx < ρ s. 5.4) ρ x)dx 2 ρ s + s µ d ). We notice that 5.4) is satisfied if we choose µ d such that < µ d < ρ s s. 5.5) From now on, we assume that 5.5) is true. In order to determine the extension, we consider the minimization problem { } min ϱ x) 2 dx : ϱ H L, ) and satisfies 5.7), 5.6) with L ϱ) = ρ ), ϱ L) = ρ L), L ϱx)dx = ρ s + ˇρ)L, where ˇρ = ρ s L L { Please DO ρ NOT, on, distribute L), ρ x)dx. 5.7) By solving Euler-Lagrange equation associated with 5.6), we obtain the solution ϱ min x) = ax 2 + bx + σ, on L, ), with a = 3 ρ L) ρ s ) + ρ ) ρ s ) 2ˇρ L 2, b = 2 al 3 ρ s + ˇρ ρ ) ). 5.8) L Since ρ L) ρ s s µ d, ρ ) ρ s s µ d and ˇρ s µ d, we have a 2s µ d L 2, b 2s µ d L. 5.9) Thus, we get a positive constant M depending only on L, s such that 4a ϱ minx) 2 dx = 2ax + b 2 2 L 3 ) 2 + b 2 L M µ 2 d 3. 5.) L Now, define L ρ e = ϱ min, on L, ]. 5.)