ON INTERACTION OF CIRCULAR CYLINDRICAL SHELLS WITH A POISEUILLE TYPE FLOW

. We study dynamics of a coupled system consisting of the 3D Navier–Stokes equations which is linearized near a certain Poiseuille type ﬂow between two unbounded circular cylinders and nonlinear elasticity equations for the transversal displacements of the bounding cylindrical shells. We show that this problem generates an evolution semigroup S t possessing a compact ﬁnite-dimensional global attractor.


Introduction.
1.1. Description of the model. Let a domain between two concentric circular cylinders of radii R 1 and R 2 respectively and infinite length with the axes directed along the X-line be denoted by Ω ⊂ R 3 . We denote the surface of the inner cylinder by ∂Ω 1 and the surface of the outer cylinder by ∂Ω 2 . Let ∂Ω 1 = Γ 1 0 ∪ Γ 1 1 and ∂Ω 2 = Γ 2 0 ∪ Γ 2 1 where Γ i 0 = ∂Ω i \ Γ i 1 , i = 1, 2 are finite flexible parts of the inner and outer cylindrical surfaces respectively whose projections on the X-line are the intervals (l i , L i ). The projections of the rigid parts Γ 1 1 and Γ 2 1 of the inner and outer cylinders on the X-line are (−∞, l 1 ) ∪ (L 1 , +∞) and (−∞, l 2 ) ∪ (L 2 , +∞) respectively. We denote by n i (x, y, z) = (n i 1 (x, y, z), n i 2 (x, y, z), n i 3 (x, y, z)) the outward unit normals to ∂Ω i .
We consider the image Π = (R 1 , R 2 ) × (0, 2π) × (−∞, ∞) of the domain Ω after the coordinates transformation into cylindrical coordinates T (r, φ, x) = (x, y, z) = (x, r cos φ, r sin φ). We use the following notations Let z = (r, φ, x) be the point in R 3 in the cylindrical coordinates. We consider the following linear Navier-Stokes equations in Ω for the fluid velocity field V (z, t) = (v(z, t), u(z, t), w(z, t)) and for the pressure p(z, t): 606 IGOR CHUESHOV AND TAMARA FASTOVSKA div V = 0 in Π × (0, +∞), (2) where µ > 0 is the dynamical viscosity, G f (z, t) = (g 1 (z, t), g 2 (z, t), g 3 (z, t)) is a volume force (which may depend on t). The functions v(z, t) and u(z, t) denote the radial and angular velocities, while the function w(z, t) represents the longitudinal velocity of the fluid. We remind that the Laplace, divergence, and gradient operators in the cylindrical coordinates have the form The linear first order operator L 0 has the form L 0 V = (0, 0, a(r)w x ) T , where and k is a positive constant. We supplement 1 and 2 with the boundary conditions imposed on the velocity field V = V (z, t): V (r, 0, x, t) = V (r, 2π, x, t), V ≡ (v(R i ); u(R i ); w(R i )) = (η it ; 0; 0) on Υ i 0 , (0; 0; 0) on Υ i 1 , i = 1, 2.
Here G i b (t) are given external (non-autonomous) loads, the operators ∆ 2 Ri are given by the formulas and correspondingly ∆ Ri η i = η ixx + 1 The bracket ·, · of von Karman type contrary to the classical von Karman bracket for functions in the Cartesian coordinates x1x2 v, is defined in the cylindrical coordinates and has the form for i = 1, 2.. The functions F solve the boundary value problems 1 We supply 1-11 with initial data of the form v(z, 0) = v 0 (z), u(z, 0) = u 0 (z) w(z, 0) = w 0 (z), satisfying the condition v 0 (r, x) r + v 0r (r, x) + u 0φ (r, φ, x) r + w 0x (r, φ, x) = 0.
If we assume that the velocity field V decays sufficiently fast as |x| → +∞ and x ∈ Ω, then 2 and 3 imply the following compatibility conditions which can be interpreted as preservation of the volume of the fluid. Thus, our general model includes the case of interaction of the Poiseuille (laminar axisymmetric) flow in the unbounded circular cylindrical domain bounded by the (solid) cylindrical walls Γ i 1 and a circular cylindrical elastic shell Γ i 0 . The motion of the fluid is described by the 3D Navier-Stokes equations linearized around the Poiseuille flow.
In general the Poiseuille flow is a laminar flow in an incompressible and Newtonian fluid flowing through a long cylindrical domainof the form where B is a domain in R 2 , and the Poiseuille velocity field has the form a 0 = (a(x 2 ; x 3 ); 0; 0), where a(x 2 ; x 3 ) solves the elliptic problem where k is a positive parameter. Linearization of the nonlinear Navier-Stokes equations around the flow a 0 gives us the model with There are two important special cases of the choice of B: (i) B is a bounded domain in R 2 (the Poiseuille flow in cylindrical tubes) and (ii) a flow between two parallel planes.
To describe deformations of the shell we use the Donnell's nonlinear shallow shell model ( [17,18]) which accounts only for transversal displacements. Since we deal with linearized fluid equations the interaction model considered assumes that large deflections of the shell produce small effect on the corresponding underlying flow.
1.2. Previous work. We note that the mathematical studies of the problem of fluid-structure interaction in the case of viscous fluids and elastic plates/bodies have a long history.
The case of moving elastic bodies [16] and the case of elastic bodies with the fixed interface [2,4,6,22] were studied from the point of view of the well-posedness and stability of the problems.
We refer to [7,10,13,24,25,26,27,30] and the references therein for the case of shallow plates/membranes. Works [7,24,27] are devoted to the well-posedness of the problems of fluid-structure interaction, in papers [25,26] the stability of linear problems of interaction of a viscous incompressible fluid and (damped) plate equations accounting for the longitudinal displacements. In paper [30] the approximate controllability of a linear model of interaction between a viscous incompressible fluid and a thin elastic structure located on a part of the fluid domain boundary, when the rigid and the elastic parts of the boundary make a rectangular corner and if the control acts on the whole elastic structure. The existence of global attractor for fluid-structure interaction problems was investigated in [10,13]. While in the first paper a nonlinear system describing the interaction of a viscous incompressible fluid in a bounded vessel with a flat elastic part of the boundary moving in the in-plane directions only is considered, the second one deals with the transversal displacement on a flexible flat part of the boundary.
We also mention the paper [14] which deals with dynamical issues for a model taking into account both transversal and longitudinal deformations. All these sources deal with the case of bounded reservoirs Ω and a flat elastic shallow shell or plate.
Regarding infinite reservoirs Ω it can be mentioned, to the best of our knowledge, only work [15] which establishes the existence of a compact global attractor to a linearized around a Poiseuille type flow Navier-Stokes system in an unbounded domain coupled with a nonlinear equation for a bounded flat part of the boundary accounting for the transverse displacements only.
In our paper we consider for the first time, to the best of our knowledge, the interaction of the Newtonian fluid with nonlinear cylindrical (concentric) shells whose dynamics is described by a von Karman-type (Donnell's) model in the cylindrical coordinates. The peculiarity of the problem considered consists in the presence of the von Karman-type nonlinearity in the polar coordinates which looses the classical properties of the von Karman bracket after transition to the Cartesian coordinates. However, in the cylindrical coordinates the domain is not smooth and there exists no bounded domain with smooth boundary which lyes in Ω and whose boundary contains the flexible wall of the vessel. This renders impossible the direct construction (see [15]) of the extension operators of functions defined on the flexible part of the boundary to the solenoidal fields in the whole domain and the trace operators for the normal components of the fluid velocity lying in L 2 (see, e.g. [19,20,33]). To overcome this obstacle we resort to the construction of homeomorphisms between suitable Sobolev classes on the boundary and in the domain in Cartesian and cylindrical coordinates. This method allows also to define trace operators on Υ i 0 for functions defined in the Lipschitz domain Π in spaces of Sobolev type of order higher then 3/2. However, the approach used is not applicable in case of one cylindrical tube (R 1 = 0) since the above mentioned mappings between spaces in the Cartesian and in the cylindrical coordinates are not homeomorphisms on the axis of the cylinder. This case is the subject for further investigations.
We establish the well-posedness of the system considered and investigate the long-time dynamics of solutions to the coupled problem in 1-11.
In our argument we use the ideas and methods developed in papers [13,15]. Since we do not assume any kind of mechanical damping in the plate component, this means that dissipation of the energy in the fluid flow due to viscosity is sufficient to stabilize the system. We model considered does not takes into account rotational inertia of the cylindrical filaments. In case then rotational inertia terms are present in the shell equations under absence of mechanical damping it is impossible to establish results on dissipativity or asymptotical smoothness of the system, since the energy terms accounting for the rotational inertia cannot be estimated from above by the terms corresponding to the viscous damping of the fluid (cf. [13,15]). Though, there is a result on the exponential stability of a linear problem of such a type [3], the methods used in this work are not applicable in the nonlinear case.

Abstract results on attractors.
For the readers' convenience we recall some basic definitions and results from the theory of attractors. Definition 1.1 ([5, 8, 9, 11, 32]). A global attractor of a dynamical system (S t , H) with the evolution operator S t on a complete metric space H is defined as a bounded closed set A ⊂ H which is invariant (S t A = A for all t > 0) and uniformly attracts all other bounded sets: To establish the existence of attractor we use the concept of gradient systems. The main features of these systems are: (i) in the proof of the existence of a global attractor we can avoid a dissipativity property (existence of an absorbing ball) in the explicit form and (ii) the structure of the attractor can be described via unstable manifolds ( [8]). 8,9,11]). Let Y ⊆ H be a forward invariant set of a dynamical system (S t , H). A continuous functional Φ(y) defined on Y is said to be a Lyapunov function on Y for the dynamical system (S t , H) if t → Φ(S t y) is a nonincreasing function for any y ∈ Y .
The Lyapunov function is said to be strict on Y if the equation Φ(S t y) = Φ(y) for all t > 0 and for some y ∈ Y implies that S t y = y for all t > 0; that is, y is a stationary point of (S t , H).
The dynamical system is said to be gradient if there exists a strict Lyapunov function on the whole phase space H.
We single out a class of the so-called quasi-stable?systems that enjoy some kind of stabilizability inequalities written in some general form. These inequalities, although often difficult to establish (most often they are obtained by means of multipliers technic), once they are proved provide a number of consequences that describe various properties of attractors [8,11]. Definition 1.4 ( [8,11]). A seminorm n(x) on a Banach space H is said to be compact if any bounded sequence {x m } ⊂ H contains a subsequence {x m k } which is Cauchy with respect to n, i.e., n(x m k − x m l ) → 0 as k, l → ∞.
The dynamical system (S t , H) is said to be quasi-stable on a set B ⊂ H (at time t * ) if there exist (a) time t * > 0, (b) a Banach space Z, (c) a globally Lipschitz mapping K : B → Z, and (d) a compact seminorm n Z (·) on the space Z, such that S t * y 1 − S t * y 2 H ≤ q · y 1 − y 2 H + n Z (Ky 1 − Ky 2 ) (16) for every y 1 , y 2 ∈ B with 0 ≤ q < 1. The space Z, the operator K, the seminorm n Z and the time moment t * may depend on B.
The following statement collects criteria on existence, finite dimensionality and properties of attractors to gradient systems. and that is, any trajectory stabilizes to the set N of stationary points.
The paper is organized as follows. In Section 2 we introduce Sobolev type spaces we need and prove the result on the well-posedness of the system. In Sections 3 and 4 we deal with extension operators and characterization of the phase spaces for the problem in Cartesian and in cylindrical coordinates respectively. In Section 5 we prove the well-posedness of the problem. Section 6 is devoted to the existence of a compact finite-dimensional global attractor.
Remark 2.1. We note that the domain Ω is unbounded, Π is unbounded and possesses non-smooth boundary, the domains Υ i 0 are bounded but their boundaries are non-smooth. However, all these domains satisfy the cone property, therefore standard results on the equivalent norms and interpolation theorems used below are applicable (see, e.g. [35] Section 4.2).
Remark 2.2. The boundaries of Π and Υ i 0 are piecewise-smooth, however, one can define the standard traces on each smooth part of the boundary. Definition 2.3. Let B be a sufficiently smooth or Lipschitz domain in R d , d = 2, 3 equipped with Cartesian coordinates and H s (B) be the Sobolev space of order s ∈ R on B which we define (see [35]) as restriction (in the sense of distributions) of the space H s (R d ) (introduced via Fourier transform). We define the norm in H s (B) by . We also introduce the spaces For fractional Sobolev spaces on manifolds with boundary there exists a bounded surjective trace operator γ ∂D for s > 1/2 from H s (D) to H s−1/2 (∂D) in case both manifold and its boundary are of bounded geometry (see e.g. [21]). We also denote by H s 0 (D) the closure of C ∞ 0 (D) with respect to the norm in H s (D) for s > 0.  [31]). where and, consequently, do not depend on the choice of M , and It is easy to see from 20, 22 and from the well-known property of dual spaces that for 0 < s < 1/2 , assertion 21 holds also for −1/2 < s < 0.
Definition 2.13. We introduce the space C p (Π) of [C ∞ ] 3 vector-valued solenoidal (i.e., divergence-free), 2π-periodic functions in the cylindrical coordinates vanishing in a neighborhood of ∂Π and also for |x| large enough Remark 2.14. Due to Remark 1.1 the following properties hold: (i) The norms in the spaces H k (Π) for k ∈ N are equivalent to (ii) The following interpolation result holds true for 0 < θ < 1 [35].
We will also need the following spaces: Definition 2. 16.
3. Extension operator and characterization of spaces on Ω. Let Ω ⊆ Ω be a smooth bounded domain such that Γ i 0 ⊂ ∂Ω for i = 1, 2. For instance, it can be constructed by joining bounded parts of ∂Ω 1 and ∂Ω 2 containing Γ 1 0 and Γ 2 0 respectively by two halves of a torus. We define the space consider the following Stokes problem where g ∈ [L 2 (Ω)] 3 and ψ = (ψ 1 , ψ 2 ) ∈L are given. This type of boundary value problems for the Stokes equation was studied by many authors (see, e.g., [28] and [33] and the references therein). We collect some properties of solutions to 24 in the following assertion.
In particular, we can define a linear operator N 0 : for ψ ∈L (N 0 ψ solves 24 with g ≡ 0). It follows from 25 and 26 that continuously for σ ≥ 0. Here X 0 (Ω ) is the closure in the L 2 -norm of the class of C ∞ solenoidal functions, vanishing at infinity and in a neiborhood of For the proof see the Appendix. Now we adjust the result from [15] to our situation: There exists a linear bounded operator Ext : L → L 2 (Ω) 3 such that for i = 1, 2 div and the relations Ext The proof of Proposition 3.2 is presented in the Appendix. Remark 3.3. We excluded the cases of semi-integer σ to guarantee the independence of the spaces H σ * (Γ i 0 ) from the choice of Ω . It follows from Propositions 3.1, 3.2 (see [19,33] for similar arguments) that and Using the extension operator constructed above we introduce the set and denote byX s (Ω) the closure of M(Ω) with respect to the H s -norm. One can see that For details concerning this type of spaces see, e.g., [19,28,33].
By the localization argument one can show that the inequality in 32 implies a similar property for any f ∈ {g ∈ H 1 (Ω) : g| Γ 1 4. Extension operator and characterization of spaces on Π.

4.1.
Trace operators for Π. Our aim in this subsection is to construct a operators mapping traces from Γ i 0 to Υ i 0 , i = 1, 2 and their inverse operators. For this purpose we give the following definition: where y 2 + z 2 = R 2 i . Now we extend these operators to the Sobolev-type spaces. We denote by H s (Π) the closure of C(Π) in the H s (Π)-norm and by H s (Υ i 0 ) the closure of C(Υ i 0 ) in the H s (Υ i 0 )-norm. Let q(x, y, z) ∈ H s (Γ i 0 ), there exists a sequence q n (x, y, z) ∈ C(Υ i 0 ) such that q n − q H s (Γ i 0 ) → 0 as n → ∞ and we define operators j s i : Lemma 4.2. The operators j i are one-to-one and onto mappings.
It is easy to see that j i are injections. Consequently, there exist operators j −1 i :  Proof. We can rewrite the function g(y, z) in 36 in local coordinates on the surfaces and q(x, y, z) = q(x, y, R 2 i − y 2 ) =q 1 (x, y) = b(g 1 (y), x), for 0 ≤ φ ≤ π or φ = g(y, z) = g 2 (y) = 2π − arccos y R i .
The unit normal field on the surfaces of the outer and inner cylinders have the form n i (x, y, z) = 0, ± y √ y 2 +z 2 , ± z √ y 2 +z 2 . The tangential gradient ∇ Γ i 0 on the surfaces of the cylinders are defined by The Laplace-Beltrami operator on the cylinders in the Cartesian coordinates then has the form and in the cylindrical coordinates ∆ p Using the definition of Sobolev spaces on manifolds 36, and 37 we obtain that for any q ∈ C ∞ (Γ i 0 ) and k ∈ N Arguing in the same way we get the reverse estimate and by the interpolation argument for any s ≥ 0 We can analogously infer the inverse estimate and interpolating 4.1 with 43 and arguing as above for we get the statement of the lemma.
in H s (Υ i 0 ). Consequently, the operators j s i are surjective. We introduce the spaces By H s l (Υ i 0 ) we denote the closures of C l (Υ 0 ) in H s (Υ i 0 ).In particular, by the trace theorems for functions in Lipschitz domains [1], The following assertion is a corollary of Lemma 4.4.
For any f ∈ C(Ω) J s f = Jf . Indeed, we can choose in definition 47 f n = f for any n ∈ N. For any f ∈ C(Ω) Relying on the interpolation theorem we get that for fractional s ≥ 0 For any V ∈ C(Π) where P k (r cos φ, r sin φ) are polynomials of the kth degree. Relying on the interpolation theorem we get that for fractional s ≥ 0 Arguing as in Lemma 4.4 we obtain from 53, 54 the statement of the lemma.

Extension operator and characterization of spaces. Let
Proposition 4.8. The following properties of operators Ext s Υ i 0 hold true for any s ≥ 0 such that s = 1/2 + Z: By Proposition 3.2, Lemmas 4.4, 4.6 we infer the estimate Then (ii) follows from 57 and 58.
Obviously, Ext s For s > 1 we can replace b i m with b i in 56 with the same computations, for 0 ≤ s ≤ 1 properties in (i) are satisfied owing to the trace theorem for Lipschitz domains (see, [1]).
Using the extension operator constructed above we introduce the set Then we denote byX s (Π) the closure of M(Π) with respect to the H s -norm.
Proposition 4.9. The following characterization of spaces holds true: (60) Proof. It follows from the definition of the operators Ext s We consider the domain Π being the image of Ω constructed in Proposition 3.1 after coordinates transformation T . Since for functions from H 1 in a Lipschitz domain one can define the continuous trace operator acting onto H 1/2 on the boundary with the right inverse lifting operator (see, e.g. [1]), then the conditions of Remark 1.3 in [33] are satisfied and the generalized Stokes formula is valid for Π . We note that ∂Π = (Υ 1 0 ∪ Υ 1 0 ) ∪ (Υ 1 1 ∪Υ 2 1 ) ∪ (Σ 1 ∪ Σ 2 ) ∪ Λ, whereΥ i 1 are bounded parts of Υ i 1 , Σ 1 and Σ 2 are bounded and belong to planes φ = 0 and φ = 2π (and have the same shape) and Λ = T (Ω \ Ω ). For any V ∈ M(Π) Consequently,by Proposition 4.9 for any Σ 1 belonging to the strip R 1 < r < R 2 Consequently, u(0, r, x) − u(2π, r, x) = 0 on any bounded smooth subdomain of the strip R 1 < r < R 2 .By the generalized Stokes formula one can show (analogously to [33]) that the same equality holds for V ∈ X 0 (Π) in the sense of distributions in H −1/2 . The proposition is proved.

5.
Well-posedness. To define weak (variational) solutions to problem 1-13 we need to define the class of test functions. Let We denote by P the projection on H in H 2 l (Υ 1 0 ) × H 2 l (Υ 2 0 ) which is orthogonal with respect to the inner product (∆ R1 ·, ∆ R1 ·) Similarly to [13] we consider the following class of test functions , where η(t) = (η 1 (t), η 2 (t)) is said to be a weak solution to 1-13 on a time interval [0, T ] if • for every Φ ∈ L T the following equality holds: and F i (η i ) are solutions to 9-11. • the compatibility condition V (t)| Υ i 0 = (η it (t); 0; 0) holds for almost all t and i = 1, 2.
Proof. To justify the subsequent computations we use the approximation procedure by the prolongations by zero to Υ i of functions from C l (Υ i 0 ). Then, Using the properties of functions from C l (Υ i 0 ) it easy to see that and Consequently, In further considerations we take into account the fact that the domain considered is bounded in two directions. Consequently, compactness properties of the fluid velocity variable are valid and we can use eigenfunctions of the Stokes operator [33].
6. Asymptotical behavior. In this section we are interested in global asymptotic behavior of the dynamical system generated in the space H. Our main result states the existence of a compact global attractor of finite fractal dimension.
Using the ideas presented in [12] one can prove the following result.
Proof. To justify the subsequent arguments we prolong the functions η i by zero to a smooth bounded domain Σ i 0 containing Υ i 0 by adding to Υ i 0 two "semicircles" on each side where Dirichlet boundary conditions hold.
on Σ i 0 . Let η i =η i −η i , then it is easy to see that we obtain we get . Taking into account that the solution to 70-72 equals to zero for Σ i 0 \Υ i 0 and belongs to H 2 , we get that the restrictions of solutions of 70-72 to Υ i 0 are solutions to 70 with boundary conditions 11-10 with the same estimates.
(1) First we show that the dynamical system (S t , H) is gradient. Indeed, it follows from energy inequality 68 that the set It is easy to see that there exist positive constants C i , i = 1, 4 such that Consequently, Φ(U n ) → +∞ if and only if U n H → +∞. Therefore, the set W R is bounded and any bounded set belongs to W R for some R. Moreover, it follows from energy inequality 68 that the continuous functional Φ on H possesses the properties: (1) Φ S t U ≤ Φ(U ) for all t ≥ 0 and U ∈ H; (2) the equality Φ(U ) = Φ(S t U ) holds for all t > 0 only if U is a stationary point of S t . This means that Φ(U ) is a strict Lyapunov function and (S t , H) is a gradient dynamical system.
(3) To prove the existence and finite dimensionality of the global attractor it is necessary due to Theorem 1.5 to show that the dynamical system (S t , H) is quasi-stable. Having in hand Lemma 6.1 the proof is analogous to that given in [13,15]. (4) To obtain the result on regularity we apply Theorem 7.9.8 [12]. (5) The results on the structure of the global attractor follow from Theorem 1.5.

Appendix A.
Proof of Proposition 3.1. We to keep to the scheme of the proof of the similar assertion given in [13]. Since the extension of elements from H σ * (Γ 1 0 ) × H σ * (Γ 2 0 ) by zero to the whole boundary ∂Ω do not change the smoothness in Sobolev class, i.e., leads to elements from H σ (∂Ω ), we can use the regularity results available for the Stokes problem with the Dirichlet type boundary conditions imposed on the whole ∂Ω (see, e.g., [28,33] and also the paper [20] and the references therein). This observation leads to the following arguments.
1. The existence and uniqueness of solutions along with the bound in 25 follow from Proposition 2.3 and Remark 2.6 on Sobolev norm's interpolation in [33, Chapter 1].
3. We first represent f in the form f =f + f * , wheref solves 24 with ψ ≡ 0 and f * satisfies 24 with g ≡ 0. Letp and p * be the corresponding representatives of the pressure (which are identified with an element in a factor-space). By the first statement we have thatp ∈ H 1/2+σ (Ω ) and thus by the standard trace theorem there existsp| ∂Ω ∈ H σ (∂Ω ). This implies thatp| In the case g ≡ 0 the pressure p * is a harmonic function in Ω which belongs to H −1/2+σ (Ω ). This allows us to assign a meaning to p * | Γ i 0 in H −1+σ (Γ i 0 ). Indeed, let φ ∈ C ∞ 0 (Γ i 0 ) andφ ∈ C ∞ 0 (∂Ω ) be the extension of φ by zero. Then by the trace theorem there exists a smooth function w i φ on Ω such that