On the interaction of an elasticwall with a poiseuille-type flow
We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first...
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Інститут математики НАН України
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irk-123456789-1649302020-02-12T01:26:42Z On the interaction of an elasticwall with a poiseuille-type flow Chueshov, I. Ryzhkova, I. Статті We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup St in an appropriate phase space. Then, under some conditions imposed on the underlying (Poiseuille-type) flow, we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that St is an exponentially stable C₀ -semigroup of linear operators in the completely linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that the dissipation of energy in the flow of fluid caused by viscosity is sufficient to stabilize the system. Вивчається динамiка зв’язної системи, що складається з тривимiрних рiвнянь Нав’є – Стокса, якi лiнеаризованi в околi деякої течiї пуазейлiвського типу в (необмеженiй) областi, та класичного (можливо, нелiнiйного) рiвняння для поперечного вiдхилення пружної пластини на гнучкiй частинi межi. Показано, що задача породжує еволюцiйну пiвгрупу St у придатному фазовому просторi. При деяких умовах щодо основної течiї встановлено iснування компактного скiнченновимiрного глобального атрактора цiєї пiвгрупи, а також показано, що St є екпоненцiально стiйкою C₀-пiвгрупою лiнiйних операторiв у повнiстю лiнiйному випадку. Оскiльки не припускається наявнiсть механiчного демпфiрування у пластинi, отриманi результати означають, що дисипацiї енергiї в потоцi рiдини через в’язкiсть достатньо для стабiлiзацiї системи. 2013 Article On the interaction of an elasticwall with a poiseuille-type flow / I. Chueshov, I. Ryzhkova // Український математичний журнал. — 2013. — Т. 65, № 1. — С. 143-160. — Бібліогр.: 35 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164930 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Chueshov, I. Ryzhkova, I. On the interaction of an elasticwall with a poiseuille-type flow Український математичний журнал |
| description |
We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup St in an appropriate phase space. Then, under some conditions imposed on the underlying (Poiseuille-type) flow, we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that St is an exponentially stable C₀ -semigroup of linear operators in the completely linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that the dissipation of energy in the flow of fluid caused by viscosity is sufficient to stabilize the system. |
| format |
Article |
| author |
Chueshov, I. Ryzhkova, I. |
| author_facet |
Chueshov, I. Ryzhkova, I. |
| author_sort |
Chueshov, I. |
| title |
On the interaction of an elasticwall with a poiseuille-type flow |
| title_short |
On the interaction of an elasticwall with a poiseuille-type flow |
| title_full |
On the interaction of an elasticwall with a poiseuille-type flow |
| title_fullStr |
On the interaction of an elasticwall with a poiseuille-type flow |
| title_full_unstemmed |
On the interaction of an elasticwall with a poiseuille-type flow |
| title_sort |
on the interaction of an elasticwall with a poiseuille-type flow |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164930 |
| citation_txt |
On the interaction of an elasticwall with a poiseuille-type flow / I. Chueshov, I. Ryzhkova // Український математичний журнал. — 2013. — Т. 65, № 1. — С. 143-160. — Бібліогр.: 35 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT chueshovi ontheinteractionofanelasticwallwithapoiseuilletypeflow AT ryzhkovai ontheinteractionofanelasticwallwithapoiseuilletypeflow |
| first_indexed |
2025-07-14T17:41:04Z |
| last_indexed |
2025-07-14T17:41:04Z |
| _version_ |
1837645027515301888 |
| fulltext |
UDC 517.9
I. Chueshov, I. Ryzhkova (Kharkov Nat. Univ.)
ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW
ПРО ВЗАЄМОДIЮ ПРУЖНОЇ СТIНКИ З ПОТОКОМ
ПУАЗЕЙЛIВСЬКОГО ТИПУ
We study dynamics of a coupled system consisting of the 3D Navier – Stokes equations which is linearized near a certain
Poiseuille type flow in an (unbounded) domain and a classical (possibly nonlinear) elastic plate equation for transversal
displacement on a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup St
on an appropriate phase space. Then under some conditions concerning the underlying (Poiseuille type) flow we prove the
existence of a compact finite-dimensional global attractor for this semigroup and also show that St is an exponentially
stable C0-semigroup of linear operators in the fully linear case. 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.
Вивчається динамiка зв’язної системи, що складається з тривимiрних рiвнянь Нав’є – Стокса, якi лiнеаризованi в
околi деякої течiї пуазейлiвського типу в (необмеженiй) областi, та класичного (можливо, нелiнiйного) рiвняння
для поперечного вiдхилення пружної пластини на гнучкiй частинi межi. Показано, що задача породжує еволюцiйну
пiвгрупу St у придатному фазовому просторi. При деяких умовах щодо основної течiї встановлено iснування
компактного скiнченновимiрного глобального атрактора цiєї пiвгрупи, а також показано, що St є екпоненцiально
стiйкою C0-пiвгрупою лiнiйних операторiв у повнiстю лiнiйному випадку. Оскiльки не припускається наявнiсть
механiчного демпфiрування у пластинi, отриманi результати означають, що дисипацiї енергiї в потоцi рiдини через
в’язкiсть достатньо для стабiлiзацiї системи.
1. Introduction. Let O ⊂ R3 be a (possibly unbounded) domain with a sufficiently smooth boundary
∂O. We assume that ∂O = Ω ∪ S, where Ω ∩ S = ∅,
Ω ⊂
{
x = (x1;x2; 0) : x′ ≡ (x1;x2) ∈ R2
}
is bounded in R2 and has the smooth contour Γ = ∂Ω. We refer to Assumption 2.1 below for further
hypotheses concerning the domain O.
Let a0(x) =
(
a1
0(x); a2
0(x); a3
0(x)
)
be a smooth bounded field defined on O such that div a0 = 0,
(n, a0) = 0 on ∂O (n is the exterior normal to ∂O, n = (0; 0; 1) on Ω) and A = A(x) be a bounded
measurable 3× 3 matrix, x ∈ O. We introduce a linear first order operator L0 of the form
L0v = (a0,∇)v +Av (1)
and consider the following linear Navier – Stokes equations in O for the fluid velocity field v =
= v(x, t) =
(
v1(x, t); v2(x, t); v3(x, t)
)
and for the pressure p(x, t):
vt − ν∆v + L0v +∇p = Gf (t) in O × (0,+∞), (2)
div v = 0 in O × (0,+∞), (3)
where ν > 0 is the dynamical viscosity, Gf (t) is a volume force (which may depend on t). We
supplement (2) and (3) with the (non-slip) boundary conditions imposed on the velocity field v =
= v(x, t):
v = 0 on S, v ≡ (v1; v2; v3) = (0; 0;ut) on Ω. (4)
c© I. CHUESHOV, I. RYZHKOVA, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1 143
144 I. CHUESHOV, I. RYZHKOVA
Here, as in [14], u = u(x, t) is the transversal displacement of the plate occupying Ω and satisfying
the following equation:
utt + ∆2u+ F(u) = Gpl(t) + p|Ω in Ω× (0,∞), (5)
where F(u) is a nonlinear (feedback) force (see Assumption 4.1 below), p is the pressure from (2),
Gpl(t) is a given external (non-autonomous) load. We refer to [14] and to the references therein for
some discussion of this plate model and for an explanation of the structure of the force exerted by
the fluid on the plate.
We impose clamped boundary conditions on the plate deflection
u|∂Ω =
∂u
∂n
∣∣∣∣
∂Ω
= 0 (6)
and supply (2) – (6) with initial data of the form
v(0) = v0, u(0) = u0, ut(0) = u1. (7)
If we assume that the velocity field v decays sufficiently fast as |x| → +∞ and x ∈ O, then (3) and
(4) imply the following compatibility condition:∫
Ω
ut(x
′, t)dx′ = 0 for all t ≥ 0, (8)
which can be interpreted as preservation of the volume of the fluid.
Below (see Definitions 3.1 and 4.1) we define a solution to (2) – (8) as a pair (v;u) satisfying
some variational type relation. If the pair (v;u) is already determined, then (at least formally) we can
find ∇p in O and the trace of p on Ω from (2) and (5). Thus the pressure p is uniquely defined by
(v;u).
The main example which we have in mind is the Poiseuille flow (see, e.g., [6] for some details).
In this case we deal with the domain
O =
{
(x1;x2;x3) : (x2;x3) ∈ B ⊂ R2, x1 ∈ R
}
, (9)
where B is a domain in R2, and the Poiseuille velocity field has the form a0 = (a(x2;x3); 0; 0),
where a(x2;x3) solves the elliptic problem
ν∆a = −k in B, a = 0 on ∂B, (10)
where k is a positive parameter. Linearization of the nonlinear Navier – Stokes equations around the
flow a0 gives us the model with
L0v = (a0,∇)v + (v,∇)a0. (11)
There are two important special cases of the choice of B in (9): (i) B is a bounded domain in R2 (the
Poiseuille flow in a cylindrical tube) and (ii) a flow between two parallel planes. In the latter case
B =
{
(x2;x3) : x2 ∈ R, x3 ∈ (−h, 0)
}
, a(x2;x3) = −kx3
2ν
(h+ x3). (12)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 145
Another possibility included in the framework above is the Oseen modification of (2) (see, e.g.,
[6]). In this case L0v = U∂x1v, where U is the parameter which has the sense of the speed of the
unperturbed flow moving along the x1-axis. This corresponds to the case when a0 = (U ; 0; 0) and
A(x) ≡ 0 in (1). We can also consider the situation when a0 ≡ 0 and A(x) 6≡ 0 in (1). In this case we
note that if A(x) is a symmetric strictly positive matrix (e.g., A(x) = σI, σ > 0), then L0v = A(x)v
can be interpreted as a drag/friction term which models the resistance offered by the fluid against its
flow (see, e.g., [29] for some discussion).
Thus, our general model includes the case of interaction of the Poiseuille (or Oseen type)
flow (with a possible drag/friction) in the domain O bounded by the (solid) wall S and a hor-
izontal boundary Ω on which a thin (nonlinear) elastic plate is placed. The motion of the fluid
is described by the 3D Navier – Stokes equations linearized around the Poiseuille (or Oseen) flow
a0(x). To describe deformations of the plate we consider a generalized plate model which ac-
counts only for transversal displacements and covers a general large deflection Karman type model
and can be also applied to nonlinear Berger and Kirchhoff plates (see the discussion in [14] and
also in Section 4). Since we deal with linearized fluid equations the interaction model considered
assumes that large deflections of the plate produce small effect on the corresponding underlying
flow.
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. We refer to [5, 8, 14, 20 – 24, 30] and the
references therein for the case of plates/membranes. The case of moving elastic bodies [17] and the
case of elastic bodies with the fixed interface [1, 2, 4, 19] were studied; see also the literature cited
in these references. We also mention the recent short survey [16] and the paper [15] which deals with
dynamical issues for a model taking into account both transversal and longitudinal deformations. All
these sources deals with the case of bounded reservoirs O.
In this paper our main point of interest is well-posedness and long-time dynamics of solutions to
the coupled problem in (2) – (7) for the velocity v and the displacement u in the case of unbounded
domains O.
In our argument we use the ideas and methods developed in our previous paper [14]. Our main
difficulties in comparison with [14] are related to the facts that (i) we deal with the (possibly) un-
bounded domain O (hence, we loose some compactness properties of the fluid velocity variable and
cannot use eigenfunctions of the Stokes operator) and (ii) the fluid equation (2) is perturbed by non-
conservative and nondissipative term (hence, we can loose the energy monotonicity and need some
additional argument for non-monotone parts). To overcome these difficulties we are enforced to use a
general basis in the fluid component and a specially constructed extension operator Ext of functions
on Ω into solenoidal functions on O.
The paper is organized as follows. In Section 2 we introduce Sobolev type spaces we need
and provide with some results concerning the extension operator Ext. In Section 3 we prove Theo-
rem 3.1 on well-posedness in the case of linear model and study stability properties of solutions in
Theorem 3.2. Section 4 is devoted to nonlinear problem. We prove here that the problem generates a
dynamical system (see Theorem 4.1) which, under some additional conditions, possesses a compact
finite-dimensional global attractor (Theorem 4.2).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
146 I. CHUESHOV, I. RYZHKOVA
2. Preliminaries. Let D be a sufficiently smooth domain in Rd and Hs(D) be the Sobolev
space of order s ∈ R on D which we define (see [35]) as restriction (in the sense of distributions) of
the space Hs(Rd) (introduced via Fourier transform). We define the norm in Hs(D) by the relation
‖u‖2s,D = inf
{
‖w‖2s,Rd : w ∈ Hs(Rd), w = u on D
}
.
We also use the notation ‖ ·‖D = ‖ ·‖0,D for the corresponding L2 norm and, similarly, (·, ·)D for the
L2 inner product. We denote by Hs
0(D) the closure of C∞0 (D) in Hs(D) (with respect to ‖ · ‖s,D)
and introduce the spaces
Hs
∗(D) :=
{
f
∣∣
D
: f ∈ Hs(Rd), supp f ⊂ D
}
, s ∈ R,
to describe boundary traces on Ω ⊂ ∂O. Since the extension by zero of elements from Hs
∗(D) give
us elements of Hs(Rd), these spaces Hs
∗(D) can be treated not only as functional classes defined
on D (and contained in Hs(D)) but also as (closed) subspaces of Hs(Rd). We endow the classes
Hs
∗(D) with the induced norms ‖f‖∗s,D = ‖f‖s,Rd for f ∈ Hs
∗(D). It is clear that
‖f‖s,D ≤ ‖f‖∗s,D, f ∈ Hs
∗(D).
It is known (see [35], Theorem 4.3.2/1) that C∞0 (D) is dense in Hs
∗(D) and
Hs
∗(D) = Hs
0(D) for − 1/2 < s <∞, s− 1/2 6∈ {0, 1, 2, . . .}.
The norms ‖ · ‖∗s,D and ‖ · ‖s,D are equivalent for these s. Note that in the notations of [27] the space
H
m+1/2
∗ (D) is the same as Hm+1/2
00 (D) for every m = 0, 1, 2, . . . . Below we also use the factor-
spaces Hs(D)/R with the naturally induced norm. To describe fluid velocity fields we first introduce
the class C0(O) of C∞ vector-valued solenoidal (i.e., divergence-free) functions v = (v1; v2; v3) on
O which vanish in a neighborhood of ∂O and also for |x| large enough. Then we denote by X̃ the
closure of C0(O) with respect to the L2-norm and by Ṽ the closure of C0(O) with respect to the
H1-norm. One can see that
X̃ =
{
v = (v1; v2; v3) ∈ [L2(O)]3 : div v = 0; γnv ≡ (v, n) = 0 on ∂O
}
and
Ṽ ⊆ Ṽ � ≡
{
v = (v1; v2; v3) ∈ [H1(O)]3 : div v = 0; v = 0 on ∂O
}
. (13)
For some details concerning this type of spaces see, e.g., [25, 34] and [18].
The following (geometry type) hypothesis plays an important role in our further considerations.
Assumption 2.1 (Domain Hypothesis). We assume that
(i) there exists a smooth bounded domain O′ ⊆ O such that Ω ⊂ ∂O′;
(ii) we have the equality in (13), i.e., Ṽ = Ṽ �.
The sense of the first requirement in Assumption 2.1 is obvious. As for the second one we refer to
[26] for a discussion of conditions on the domain which guarantee the equality Ṽ = Ṽ � (see also [18]
(Sect. 4.3) and the references therein). Here, as examples, we only note that this property holds in
the following cases: (i) O is a smooth domain with the compact boundary; (ii) O = R3
− = {x3 ≤ 0};
(iii) O is given by (9) with smooth bounded B or with B as in (12) (infinitely long pipes and tubes
of possibly varying cross section are also admissible).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 147
We also need the Sobolev spaces consisting of functions with zero average on the domain Ω,
namely we consider the subspace
L̂2(Ω) =
u ∈ L2(Ω):
∫
Ω
u(x′)dx′ = 0
in L2(Ω) and also the subspaces Ĥs(Ω) = Hs(Ω) ∩ L̂2(Ω) in Hs(Ω) for s > 0 with the standard
Hs(Ω)-norm. The notations Ĥs
∗(Ω) and Ĥs
0(Ω) have a similar meaning. We denote by P̂ the projec-
tion on Ĥ2
0 (Ω) in H2
0 (Ω) which is orthogonal with respect to the inner product (∆·,∆·)Ω. As it was
already mentioned in [14] the subspace (I − P̂ )H2
0 (Ω) consists of functions u ∈ H2
0 (Ω) such that
∆2u = const and thus has dimension one.
In further considerations we need the following assertion concerning extension of functions de-
fined on Ω.
Proposition 2.1. Let Assumption 2.1 (i) be in force. Then there exists a linear bounded operator
Ext: L̂2(Ω) 7→
[
L2(O)
]3
such that
div Ext[ψ] = 0 in O, (Ext[ψ], n)
∣∣
Ω
= ψ, (Ext[ψ], n)
∣∣
S
= 0,
and
‖Ext[ψ]‖[
H1/2−δ(O)
]3 ≤ C‖ψ‖Ω ∀δ > 0 ∀ψ ∈ L̂2(Ω).
Moreover,
if ψ ∈ Hs
∗(Ω) for some 0 < s < 1, then Ext[ψ] ∈
[
Hs+1/2(O)
]3
with the estimate
‖Ext[ψ]‖[
Hs+1/2(O)
]3 ≤ C‖ψ‖Hs
∗(Ω), (14)
and the relations Ext[ψ]
∣∣
S
= (0; 0; 0) and Ext[ψ]
∣∣
Ω
= (0; 0;ψ) on the boundary of ∂O;
there exists a smooth bounded subdomain O′ in O such that (i) Ω ⊂ ∂O′, (ii) Ext[ψ]
∣∣
O\O′ = 0,
and (iii) Ext[ψ]
∣∣
O′ ∈
[
H2(O′)
]3
provided ψ ∈ H3/2+δ
0 (Ω) for some δ > 0.
Proof. On a smooth bounded subdomain O′ in O such that Ω ⊂ ∂O′ we consider the following
Stokes problem:
−ν∆v +∇p = 0, div v = 0 in O′;
v = 0 on ∂O′ \ Ω; v = (0; 0;ψ) on Ω, (15)
where ψ ∈ L̂2(Ω) is given. This type of boundary-value problems in bounded domains was studied
by many authors (see, e.g., [25, 34] and also the recent monograph [18] and the references therein).
To construct an extension operator we need the following properties of solutions to (15) (for some
discussion and references concerning the assertion below we refer to [14]).
Proposition 2.2. Let ψ ∈ Hs
∗(Ω) with −1/2 ≤ s ≤ 3/2 and
∫
Ω
ψ(x′)dx′ = 0. Then prob-
lem (15) has a unique solution
{v; p} ∈ [Hs+1/2(O′)]3 × [Hs−1/2(O′)/R]
such that
‖v‖[Hs+1/2(O′)]3 + ‖p‖Hs−1/2(O′)/R ≤ c0‖ψ‖Hs
∗(Ω).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
148 I. CHUESHOV, I. RYZHKOVA
Now we can take a solution v to (15) and define Ext[ψ] as the zero extension of v on the domain
O. One can see that for this operator Ext all statements of Proposition 2.1 are in force.
Remark 2.1. We could not find in the literature an appropriate statement of Proposition 2.2 for
unbounded domains. On the other hand we do not know an extension result in the class solenoidal
functions with estimate (14) for some range of the parameter s. This is why we use this way for
a construction of the operator Ext. We also note that in the case when O is bounded we can take
O′ = O. In this case Ext is a Green type operator which maps ψ into v according to (15). Exactly
this extension operator was used in [14].
Using the extension operator constructed above we introduce the spaces which we need to de-
scribe the interaction between fluid and plate.
Let Assumption 2.1 be valid and
M(O) =
{
v = v0 + Ext[ψ] : v0 ∈ C0(O), ψ ∈ Ĥ2
0 (Ω)
}
.
Then we denote by X the closure of M(O) with respect to the L2-norm and by V the closure of
M(O) with respect to the H1-norm. One can see that
X =
{
v = (v1; v2; v3) ∈ [L2(O)]3 : div v = 0; γnv ≡ (v, n) = 0 on S
}
and
V = V � ≡
{
v = (v1; v2; v3) ∈ [H1(O)]3
∣∣∣∣∣ div v = 0, v = 0 on S,
v1 = v2 = 0 on Ω.
}
.
We equip X with L2-type norm ‖ · ‖O and denote by (·, ·)O the corresponding inner product. The
space V is endowed with the standard H1-norm.
In conclusion of this section we mention that in the the case of the Poiseuille flow in the tube
or between two planes described above we deal with a domain satisfying the Friedrichs – Póincare
property1:
∃dO > 0 :
∫
O
|v(x)|2dx ≤ d2
O
∫
O
|∇v(x)|2dx ∀v ∈ H1
0 (O). (16)
By the localization argument one can show that the inequality in (16) implies a similar property for
any v ∈ {g ∈ H1(O) : g|S = 0} and thus
∃cO > 0: ‖v‖O ≤ cO‖∇v‖O ∀v ∈ V, (17)
for the Friedrichs – Póincare domains.
3. Linear problem. In this section we consider a linear version of (2) – (8) which is obtained by
replacing equation (5) with its linear counterpart. Thus we deal with the following problem:
vt − ν∆v + L0v +∇p = Gf (t) and div v = 0 in O × (0,+∞), (18)
v = 0 on S and v ≡ (v1; v2; v3) = (0; 0;ut) on Ω, (19)
utt + ∆2u = Gpl(t) + p|Ω on Ω, (20)
1This property is valid in the case when the domain O is bounded at least in one direction.
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 149
u =
∂u
∂n
= 0 on ∂Ω,
∫
Ω
ut(x
′, t)dx′ = 0 for all t ≥ 0, (21)
which we supply with initial data of the form
v(0) = v0, u(0) = u0, ut(0) = u1. (22)
Similarly to [14] we consider the following class of test functions
LT =
φ
∣∣∣∣∣∣∣∣
φ ∈ L2(0, T ;
[
H1(O)
]3
), φt ∈ L2(0, T ; [L2(O)]3),
divφ = 0, φ|S = 0, φ|Ω = (0; 0; b), φ(T ) = 0,
b ∈ L2(0, T ; Ĥ2
0 (Ω)), bt ∈ L2(0, T ; L̂2(Ω)).
and introduce the following definition.
Definition 3.1. A pair of functions (v(t);u(t)) is said to be a weak solution to the problem in
(18) – (22) on a time interval [0, T ] if
v ∈ L∞(0, T ;X)
⋂
L2(0, T ;V );
u ∈ L∞(0, T ;H2
0 (Ω)), ut ∈ L∞(0, T ; L̂2(Ω)) and u(0) = u0;
for every φ ∈ LT the following equality holds:
−
T∫
0
(v, φt)Odt+ ν
T∫
0
(∇v,∇φ)Odt+
T∫
0
(L0v, φ)Odt−
−
T∫
0
(ut, bt)Ωdt+
T∫
0
(∆u,∆b)Ωdt =
=
T∫
0
(Gf , φ)Odt+
T∫
0
(Gpl, b)Ωdt+ (v0, φ(0))O + (u1, b(0))Ω; (23)
the compatibility condition v(t)|Ω = (0; 0;ut(t)) holds for almost all t.
The same argument as in [14] shows that a weak solution (v(t);u(t)) satisfies the relation
(v(t), ψ)O + (ut(t), β)Ω = (v0, ψ)O + (u1, β)Ω−
−
t∫
0
[
ν(∇v,∇ψ)O + (L0v, ψ)O + (∆u,∆β)Ω − (Gf , ψ)O − (Gpl, β)Ω
]
dτ
for almost all t ∈ [0, T ] and for all ψ = (ψ1;ψ2;ψ3) ∈W, where β = ψ3
∣∣
Ω
and
W =
{
ψ ∈ V
∣∣∣ ψ|Ω = (0; 0;β), β ∈ Ĥ2
0 (Ω)
}
. (24)
It also follows from the compatibility condition and the standard trace theorem that the plate velocity
ut possesses an additional spatial regularity, namely we have that ut ∈ L2(0, T ;H
1/2
∗ (Ω)).
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150 I. CHUESHOV, I. RYZHKOVA
Below as phase spaces we use
H =
{
(v0;u0;u1) ∈ X ×H2
0 (Ω)× L̂2(Ω): (v0, n) ≡ v3
0 = u1 on Ω
}
(25)
and
Ĥ =
{
(v0;u0;u1) ∈ H : u0 ∈ Ĥ2
0 (Ω)
}
⊂ H (26)
with the norm ‖(u0;u0;u1)‖2H = ‖v0‖2O + ‖∆u0‖2Ω + ‖u1‖2Ω.
Our main result in this section is the following well-posedness theorem concerning the linear
problem.
Theorem 3.1. Let Assumption 2.1 be in force. Assume that
U0 = (v0;u0;u1) ∈ H, Gf (t) ∈ L2(0, T ;V ′), Gpl(t) ∈ L2(0, T ;H−1/2(Ω)).
Then for any interval [0, T ] there exists a unique weak solution (v(t);u(t)) to (18) – (22) with the
initial data U0. This solution possesses the property
U(t;U0) ≡ U(t) ≡ (v(t);u(t);ut(t)) ∈ C(0, T ;X ×H2
0 (Ω)× L̂2(Ω)),
and satisfies the energy balance equality
E0(v(t), u(t), ut(t)) +
t∫
0
[
ν‖∇v‖2O + (Av, v)O
]
dτ =
= E0(v0, u0, u1) +
t∫
0
(Gf , v)Odτ +
t∫
0
(Gpl, uτ )Ωdτ (27)
for every t > 0, where the energy functional E0 is defined by the relation
E0(v(t), u(t), ut(t)) =
1
2
(
‖v(t)‖2O + ‖ut(t)‖2Ω + ‖∆u(t)‖2Ω
)
.
If Gf ≡ 0 and Gpl ≡ 0, then Theorem 3.1 implies that the problem in (18) – (22) generates a
strongly continuous semigroup. In order to state our result on asymptotic stability of this semigroup
we need additional assumptions.
Assumption 3.1 (Stability Hypothesis). Assume that one of the following conditions is valid:
either the matrix A(x) in (1) is uniformly strictly positive, i.e.,
∃σ > 0: (A(x)ξ, ξ)R3 ≥ σ|ξ|2R3 ∀ξ ∈ R3, x ∈ O;
or the domain O satisfies the Friedrichs – Póincare property (16) and
∃δ > 0: (A(x)ξ, ξ)R3 ≥ −
(
ν
c2
O
− δ
)
|ξ|2R3 ∀ξ ∈ R3, x ∈ O, (28)
where cO is the constant from the Friedrichs – Póincare inequality in (17).
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 151
Thus in the case of a general domain O satisfying Assumption 2.1 to obtain a result on long-time
dynamics we need to assume the presence of some additional damping mechanism (drag/friction
terms). If the domain satisfies the Friedrichs – Póincare property, then the result can be achieved
without any damping (e.g., we can take A(x) ≡ 0). Moreover, we note that the condition in (28) is
true when supx∈O |A(x)| < νc−2
O , where |A(x)| is the operator (Euclidian) norm in R3. In the case
of the Poiseuille type flow (see (11)) this means that |∇x1,x2a| is small enough. Since the profile
a can be written in the form a = kν−1a∗, where a∗ solves (10) with ν = 1 and k = 1, the latter
condition is satisfied when kν−2 ≤ c(B). Here k is the Poiseuille velocity parameter and c(B) is a
constant depending on the cross-section B of the tube O. In the case of the Oseen model we have
a0 = (U ; 0; 0) in (11) and thus there are no restrictions on the velocity U of the underlining flow for
Friedrichs – Póincare domains.
Theorem 3.2. In addition to the hypotheses of Theorem 3.1 we assume that Assumption 3.1 is
in force. Then there exist positive constants M and γ such that for every initial data U0 = (v0;u0;u1)
from Ĥ we have
‖U(t)‖2H ≤Me−γt‖U0‖2H +M
t∫
0
e−γ(t−τ)
[
‖Gf (τ)‖2V ′ + ‖Gpl(τ)‖2−1/2,Ω
]
dτ. (29)
In particular, if Gf ≡ 0 and Gpl ≡ 0, then the C0-semigroup generated by (18) – (22) is exponentially
stable in Ĥ.
In the case of a general operator L0 we need to add the term(
M1 +M2 sup
x∈O
|A(x)|
) t∫
0
e−γ(t−τ)‖v(τ)‖2dτ
in the right-hand side of (29). Here |A(x)| denotes the operator (Euclidian) norm in R3 and M1 = 0
when the domain O satisfies the Friedrichs – Póincare property in (16).
Proof of Theorem 3.1. In the case when L0 ≡ 0 and O is bounded this theorem was proved
in [14] (see also [30] for a similar result). We use the same idea as in [14]. The main difficulty which
we are faced is that we loose several compactness properties of the model (e.g., we cannot use the
basis of eigenfunctions of the Stokes operator).
Step 1. Existence of an approximate solution. Let {ψi}i∈N be an (orthonormal) basis in the
space Ṽ consisting of the smooth finite in O functions. Denote by {ξi}i∈N the basis in Ĥ2
0 (Ω) which
consists of eigenfunctions of the following problem:
(∆ξi,∆w)Ω = κi(ξi, w)Ω ∀w ∈ Ĥ2
0 (Ω),
with the eigenvalues 0 < κ1 ≤ κ2 ≤ . . . and ‖ξi‖Ω = 1. Let φi = Ext[ξi], where the operator Ext
is defined in Proposition 2.1. This proposition also yields φi is H2 in some vicinity of Ω and thus as
in [14] one can conclude that ∂x3φ
3
i = 0 on Ω.
We define an approximate solution as a pair of functions
vn,m(t) =
m∑
i=1
αi(t)ψi +
n∑
j=1
β̇j(t)φj , un(t) =
n∑
j=1
βj(t)ξj + (I − P̂ )u0, (30)
satisfying the relations
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152 I. CHUESHOV, I. RYZHKOVA
(v̇n,m(t), χ)O + (ün(t), h)Ω + ν(∇vn,m(t),∇χ)O + (∆un(t),∆h)Ω =
= −(L0vn,m, χ)O + (Gf (t), χ)O + (Gpl(t), h)Ω (31)
for t ∈ [0, T ] and for every χ and h of the form
χ =
m′∑
k=1
χkψk + Ext[h] with h =
n′∑
k=1
hkξk, (32)
where m′ ≤ m and n′ ≤ n. It is clear that χ ∈ W and χ
∣∣
Ω
= (0; 0;h). The system in (31) is
endowed with the initial data
vn,m(0) = Πm(v0 − Ext[u1]) + Ext[Pnu1],
un(0) = PnP̂ u0 + (I − P̂ )u0, u̇n(0) = Pnu1,
where Πm is the orthoprojector on Lin{ψj : j = 1, . . . ,m} in X̃ and Pn is orthoprojector on
Lin{ξi : i = 1, . . . , n} in L̂2(Ω). Since Ext: L̂2(Ω) 7→ X, it is clear that
(vn,m(0);un(0); u̇n(0))→ (v0;u0;u1) strongly in H as m,n→∞.
As in [14] one can show that (31) can be reduced to some ODE in Rm+n and with given initial
data has a unique solution on any time interval [0, T ].
It follows from (30) that
vn,m(t) =
m∑
i=1
αi(t)ψi + Ext[∂tun(t)].
This implies the boundary compatibility condition:
vn,m(t) = (0; 0; ∂tun(t)) on Ω. (33)
Step 2. Energy relation and a priori estimate for an approximate solution. Taking χ = vn,m
and h = ∂tun(t) in (31) we obtain the following energy balance relation for approximate solutions:
E0(vn,m(t), un(t), ∂tun(t)) + ν
t∫
0
∫
O
|∇vn,m|2dxdτ +
t∫
0
(Avn,m, vn,m)Odτ =
= E0(vn,m(0), un(0), ∂tun(0)) +
t∫
0
(Gf , vn,m)Odτ +
t∫
0
(Gpl, ∂tun)Ωdτ. (34)
We use here the structure of L0 which after simple calculations (see, e.g., Lemma 1.3 [34], Ch. 2)
yields the equality (L0vn,m, vn,m)O = (Avn,m, vn,m)O. The relation in (34) and Gronwall’s lemma
implies the following a priori estimate:
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 153
sup
t∈[0,T ]
{
‖vn,m(t)‖2O + ‖∆un(t)‖2Ω + ‖∂tun(t))‖2Ω
}
+
+
T∫
0
(
‖∇vn,m‖2O + ‖vn,m‖2O
)
dτ ≤ CT . (35)
By the trace theorem from (33) and (35) we also have that
T∫
0
‖∂tun(τ))‖2
H
1/2
∗ (Ω)
dτ =
T∫
0
‖vn,m(τ)‖21/2,∂Odτ ≤ CT . (36)
Step 3. Limit transition. By (35) the sequence {(vn,m;un; ∂tun)} contains a subsequence such
that
(vn,m;un; ∂tun) ⇀ (v;u; ∂tu) ∗-weakly in L∞(0, T ;H), (37)
un → u strongly in C(0, T ;H2−ε
0 (Ω)) ∀ε > 0, (38)
vn,m ⇀ v weakly in L2(0, T ;V ). (39)
To obtain (38) we use the Aubin – Dubinsky theorem (see, e.g., [32], Corollary 4). By (36) we can
also suppose that
∂tun ⇀ ∂tu weakly in L2(0, T ;H
1/2
∗ (Ω)), (40)
vn,m ⇀ v weakly in L2(0, T ;H1/2(∂O)). (41)
Applying the same argument as in [14] and using relations (37) – (41) we conclude the proof of the
existence of weak solutions which satisfy the corresponding energy balance inequality. At this point
we use Assumption 2.1(ii) to approximate elements from W by elements of the form (32). We need
this to establish (23) for φ ∈ LT .
Step 4. Uniqueness. We first consider the case when L0 ≡ 0 and use Lions’ idea (see [28]),
with the same test function as [14] in the case of a bounded domain. After establishing properties of
solutions in this case we consider the term L0v as a perturbation.
Let U j(t) = (vj(t);uj(t);ujt (t)), j = 1, 2, be two different solutions to the problem in ques-
tion with the same initial data and L0 ≡ 0. Then their difference U(t) = U1(t) − U2(t) =
= (v(t);u(t);ut(t)) satisfies the variational equality
−
T∫
0
(v, φt)O + ν
T∫
0
(∇v,∇φ)O −
T∫
0
(ut, bt)Ω +
T∫
0
(∆u,∆b)Ω = 0
for all φ ∈ LT , b = (φ|Ω)3. Now for every 0 < s < T we take
φ(t) ≡ φs(t) =
−
s∫
t
dτ
τ∫
0
dζv(ζ), t < s,
0, t ≥ s,
as a test function. The same calculation as in [14] yields the uniqueness in the case L0 ≡ 0.
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154 I. CHUESHOV, I. RYZHKOVA
Step 5. Continuity with respect to t and the energy equality. Using the Lions lemma (see [27],
Lemma 8.1) by the same argument as in [14] we first prove any weak solution (v(t);u(t);ut(t)) is
weakly continuous in X ×H2
0 (Ω)× L̂2(Ω).
To prove the energy equality (in the case L0 = 0), we follow the scheme of [28] (Ch. 1), see
also [27] (Ch. 3), in the form presented in [14]. Thus as in [14] we can conclude that the solution is
strongly continuous in t. Moreover, the energy relation in the case L0 = 0 with Gf = 0 and Gpl = 0
implies that the corresponding solutions generates strongly continuous semigroup.
Step 6. Case L0 6= 0. Using the energy relation for the problem with L0 = 0 and Gf (t) :=
:= Gf (t) − L0v(t) we can establish the uniqueness of solutions via the Gronwall’s type argument
and also the smoothness properties in the general case.
Theorem 3.1 is proved.
Proof of Theorem 3.2. To prove the estimate in (29), we construct a Lyapunov function using
an idea from [8] (see also [14]). Let
V (v0, u0, u1) = E0(v0, u0, u1) + εΨ(v0, u0, u1),
where Ψ(v0, u0, u1) = (u0, u1)Ω + (v0,Ext[u0])O and ε > 0 is a small parameter which will be
chosen later. We consider these functionals on approximate solutions (vn,m;un) for which P̂ u0 = u0
and thus P̂ un(t) = un(t) for all t > 0. This allow us to substitute in (31) Ext[un] instead of χ and
obtain that
d
dt
Ψn,m(t) ≡ d
dt
Ψ(vn,m(t), un(t), ∂tun(t)) =
= ‖∂tun‖2Ω + (vn,m,Ext[∂tun])O − (L0vn,m,Ext[un])O−
−ν(∇vn,m,∇Ext[un])O − ‖∆un‖2Ω + (Gf ,Ext[un])O + (Gpl, un)Ω. (42)
By Proposition 2.1, using the compatibility condition in (33) and the trace theorem we have that∣∣(vn,m,Ext[∂tun])O
∣∣ ≤ C‖vn,m‖O‖∂tun‖Ω ≤ C [‖∇vn,m‖2O + ‖vn,m‖2O
]
.
Similarly, for every η > 0 we have∣∣(∇vn,m,∇Ext[un])O
∣∣ ≤ η‖∆un‖2Ω + Cη
[
‖∇vn,m‖2O + ‖vn,m‖2O
]
and ∣∣(Gf ,Ext[un])O + (Gpl, un)Ω
∣∣ ≤ η‖∆un‖2Ω + Cη
[
‖Gf‖2V ′ + ‖Gpl‖2−1/2,Ω
]
.
It is also clear that∣∣(L0vn,m,Ext[un])O
∣∣ ≤ η‖∆un‖2Ω + Cη
[
‖∇vn,m‖2O + ‖vn,m‖2O
]
.
Therefore it follows from (42) that
d
dt
Ψn,m(t) ≤ −1
2
‖∆un‖2Ω + C
[
‖∇vn,m‖2O + ‖vn,m‖2O
]
+ C
[
‖Gf‖2V ′ + ‖Gpl‖2−1/2,Ω
]
.
Using the energy relation in (34) we also have that
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 155
d
dt
E0(vn,m(t), un(t), ∂tun(t)) ≤ −(ν − η)‖∇vn,m‖2O + η‖vn,m‖2O+
+Cη
[
‖Gf‖2V ′ + ‖Gpl‖2−1/2,Ω
]
− (Avn,m, vn,m)O ∀η > 0.
One can see that the function Vn,m(t) ≡ V (vn,m(t), un(t), ∂tun(t)) satisfies the relations
a0E0(vn,m(t), un(t), ∂tun(t)) ≤ Vn,m(t) ≤ a1E0(vn,m(t), un(t), ∂tun(t))
for sufficiently small ε > 0. Using the stability hypothesis in Assumption 3.1 we can choose η > 0
and σ > 0 such that
(ν − η)‖∇vn,m‖2O − η‖vn,m‖2O + (Avn,m, vn,m)O ≥ σ
[
‖∇vn,m‖2O + ‖vn,m‖2O
]
.
Therefore we have that
d
dt
Vn,m(t) + a2Vn,m(t) ≤ a3
[
‖Gf‖2V ′ + ‖Gpl‖2−1/2,Ω
]
with positive constants ai. This implies relation (29) for approximate solutions. The limit transition
yields (29) for every weak solution.
In the general case we can apply (29) with L0 := (a0,∇)v + µv and Gf := Gf − Av + µv,
where µ > 0 (in the case of the Friedrichs – Póincare domains we can take µ = 0). This implies the
desired conclusion and completes the proof of Theorem 3.2.
4. Nonlinear problem. In this section we deal with problem (2)–(8) with a nonlinear feedback
force. First we impose the following hypotheses concerning the force F(u) in the plate equation (5).
Assumption 4.1. There exists ε > 0 such that F(u) is locally Lipschitz from H2−ε
0 (Ω) into2
H−1/2(Ω) in the sense that∥∥F(u1)−F2(u2)
∥∥
−1/2,Ω
≤ CR‖u1 − u2‖2−ε,Ω (43)
for any ui ∈ H2
0 (Ω) such that ‖ui‖2,Ω ≤ R.
There exists a C1-functional Π(u) on H2
0 (Ω) such that F(u) = Π′(u), where Π′ denotes the
Fréchet derivative of Π.
The plate force potential Π is bounded on bounded sets from H2
0 (Ω) and there exist η < 1/2 and
C ≥ 0 such that
η‖∆u‖2Ω + Π(u) + C ≥ 0 ∀u ∈ H2
0 (Ω). (44)
Examples of nonlinear feedback (elastic) forces F(u) satisfying Assumption 4.1 are described
in [9] and [14], see also [13]. They represent different plate models and include Kirchhoff, von
Karman, and Berger models.
2We recall [35] that H−1/2(Ω) =
[
H
1/2
∗ (Ω)
]′ ' [
H
1/2
0 (Ω)
]′
.
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156 I. CHUESHOV, I. RYZHKOVA
4.1. Well-possedness.
Definition 4.1. A pair of functions (v(t);u(t)) is said to be a weak solution to (2) – (8) on a
time interval [0, T ] if
v ∈ L∞(0, T ;X)
⋂
L2(0, T ;V );
u ∈ L∞(0, T ;H2
0 (Ω)), ut ∈ L∞(0, T ; L̂2(Ω)), u(0) = u0;
the equality in (23) holds with Gpl(t) := −F(u(t)) +Gpl(t);
the compatibility condition v(t)|Ω = (0; 0;ut(t)) holds for almost all t.
Theorem 4.1. Let Assumptions 2.1 and 4.1 be in force. Assume that U0 = (v0;u0;u1) ∈ H,
Gf (t) ∈ L2(0, T ;V ′) and Gpl(t) ∈ L2(0, T ;H−1/2(Ω)). Then for any interval [0, T ] there exists
a unique weak solution (v(t);u(t)) to (2) – (8) with the initial data U0. This solution possesses the
property
U(t) ≡ (v(t);u(t);ut(t)) ∈ C(0, T ;H), (45)
where H is given by (25), and satisfies the energy balance equality
E(v(t), u(t), ut(t)) +
t∫
0
[
ν‖∇v‖2O + (Av, v)O
]
dτ =
= E(v0, u0, u1) +
t∫
0
(Gf , v)Odτ +
t∫
0
(Gpl, uτ )Ωdτ (46)
for every t > 0, where the energy functional E is defined by the relation
E(v, u, ut) =
1
2
(
‖v‖2O + ‖ut‖2Ω + ‖∆u‖2Ω
)
+ Π(u).
Moreover, there exists a constant aR,T > 0 such that for any couple of weak solutions U(t) =
=
(
v(t);u(t);ut(t)
)
and Û(t) =
(
v̂(t); û(t); ût(t)
)
with the initial data possessing the property
‖U0‖H, ‖Û0‖H ≤ R we have
‖U(t)− Û(t)‖2H +
t∫
0
‖∇(v − v̂)‖2Odτ ≤ aR,T ‖U0 − Û0‖2H, ∈ [0, T ]. (47)
The spatial average of u(t) is preserved. In particular, if U0 ∈ Ĥ, then U(t) ∈ Ĥ for every t > 0.
We recall that Ĥ is defined by (26).
Proof. The proof of the local existence of an approximate solution is almost the same, as in the
linear case (see Theorem 3.1). We use approximate solutions of the same structure which satisfy (31)
with −F(un(t))+Gpl(t) instead of Gpl(t). Then using the standard argument we establish the energy
relation in (46) for these approximate solutions. Now the positivity type estimate in (44) allow us to
obtain the same a priori estimates as in (35) and (36). Therefore we can prove the global existence
of approximate solutions and establish the existence of a weak solution U(t) = (v(t);u(t);ut(t)) by
the same argument as in the linear case. To make limit transition in the nonlinear term we use (43).
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 157
Next we consider the pair (v(t);u(t)) as a solution to the linear problem with Gpl(t) :=
:= −F(u(t)) + Gpl(t). This allow us to obtain (45) and also derive energy balance relation (46)
from (27) using the potential structure of the force F : F(u) = Π′(u).
Since the difference of two weak solution can be treated as a solution to the linear problem with
Gf ≡ 0 and Gpl(t) := F(û(t)) − F(u(t)), we can obtain (47) from the energy equality (27). The
uniqueness follows from (47).
Preservation of the spatial average of u(t) follows from the same property for approximate
solutions.
Theorem 4.1 is proved.
We can derive from Theorem 4.1 the following assertion.
Corollary 4.1. In addition to the hypotheses of Theorem 4.1 we assume that Gf (t) ≡ G0 ∈ V ′
is independent of t and Gpl(t) ≡ 0. Then problem (2) – (8) generates dynamical systems (St,H) and
(St, Ĥ) with the evolution operator defined by the formula StU0 = (v(t);u(t);ut(t)), where (v;u)
is a weak solution to (2) – (8) with the initial data U0 = (v0;u0;u1).
If we assume in addition that G0 = 0 and Assumption 3.1 holds, then these systems are gradient
with the full energy E(v0, u0, u1) as a Lyapunov function. This means that (a) U 7→ E(U) is contin-
uous on H, (b) E(StU0) is not increasing in t, and (c) if E(StU0) = E(U0) for some t > 0, then U0
is a stationary point of St (i.e., StU0 = U0 for all t ≥ 0). Moreover, the set ER = {U0 : E(U0) ≤ R}
is a bounded closed forward invariant set for every R > 0.
Proof. The argument is the same as in [14]. We only note that under Assumption 3.1 from (46)
(with Gf = 0 and Gpl = 0) we have that every stationary point U∗ for St has the form U∗ = (0;u; 0),
where u ∈ H2
0 (Ω).
4.2. Stationary solutions. As above we assume that Gpl ≡ 0 and Gf ≡ 0 and Assumptions 2.1
and 3.1 holds. It follows from Definition 4.1 that a stationary (time-independent) solution is a pair
(v;u) from Ṽ ×H2
0 (Ω) satisfying the relation
ν(∇v,∇ψ)O + (L0v, ψ)O + (∆u,∆β)Ω + (F(u), β)Ω = 0 (48)
for any ψ ∈ W with ψ3
∣∣
Ω
= β, where W is given by (24). Taking ψ = v we conclude that
ν‖∇v‖2O + (Av, v)O = 0 and hence from Assumption 3.1 we have v = 0. Therefore we obtain the
following variational problem for u ∈ H2
0 (Ω):
(∆u,∆β)Ω + (F(u), β)Ω = 0 ∀β ∈ Ĥ2
0 (Ω). (49)
As in [14] we can show the existence of a family of solutions to (49) parameterized by a real
parameter. In the case of the zero average of u we can fix this parameter and obtain the following
assertion (see [14] for details).
Proposition 4.1 [14]. In addition to Assumption 4.1 we assume that there exist η < 1/2 and
c ≥ 0 such that
η‖∆u‖2Ω + (u,F(u))Ω ≥ −c ∀u ∈ H2
0 (Ω). (50)
Then the set N0 of solutions u to problem (49) with the property
∫
Ω
udx = 0 is nonempty compact
set in Ĥ2
0 (Ω). This implies that the set N of all stationary points of St in the space Ĥ is nonempty
compact set and has the form
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
158 I. CHUESHOV, I. RYZHKOVA
N =
{
(0;u; 0) : u ∈ Ĥ2
0 (Ω) solves (49)
}
. (51)
4.3. Asymptotical behavior. In this section we are interested in global asymptotic behavior of
the dynamical system (St, Ĥ). Our main result states the existence of a compact global attractor of
finite fractal dimension.
We recall (see, e.g., [3, 7, 33]) that a global attractor of the dynamical system (St, Ĥ) is defined
as a bounded closed set A ⊂ Ĥ which is invariant (S(t)A = A for all t > 0) and uniformly attracts
all other bounded sets:
lim
t→∞
sup{distH(S(t)y,A) : y ∈ B} = 0 for any bounded set B in Ĥ.
Theorem 4.2. Let Assumptions 2.1, 3.1, and 4.1 be in force. Assume that Gpl ≡ 0, Gf ≡ 0
and (50) hold. Then the dynamical system (St, Ĥ) possesses a compact global attractor A of finite
fractal dimension3.
Moreover,
(1) Any trajectory γ = {(v(t);u(t);ut(t)) : t ∈ R} from the attractor A possesses the properties
(vt;ut;utt) ∈ L∞(R;X × Ĥ2
0 (Ω)× L̂2(Ω)) (52)
and there is R > 0 such that
sup
γ⊂A
sup
t∈R
(
‖vt‖2O + ‖ut‖22,Ω + ‖utt‖2Ω
)
≤ R2. (53)
(2) The global attractor A consists of full trajectories {(v(t);u(t);ut(t)) : t ∈ R} which are
homoclinic to the set N , i.e.,
lim
t→±∞
inf
u∗∈N0
(
‖v(t)‖2O + ‖u− u∗‖22,Ω + ‖ut‖2Ω
)
= 0,
where N0 =
{
u ∈ Ĥ2
0 (Ω) solves (49)
}
. In addition we have
lim
t→+∞
distĤ(Sty,N ) = 0 for any initial data y ∈ Ĥ. (54)
We emphasize that Theorem 4.2 deals with dynamics in the space Ĥ (the case of the zero spatial
average of the deflection). For a possible approach to description of the system long-time behavior
in the space H we refer to [14] (Remark 4.9).
To obtain the result stated in Theorem 4.2 it is sufficient to show that the system is quasi-stable
in the sense of Definition 7.9.2 [11] (see also Section 4.4 in [12]). For this we can use the stability
properties of linear problem (18) – (22) established in Theorem 3.2 and the same argument as in [14]
which yields the following assertion.
Lemma 4.1 (Quasi-stability). Let the hypotheses of Theorem 4.2 be in force and U i(t) =
= (vi(t);ui(t);uit(t)), i = 1, 2, be two weak solutions with initial data U i0 = (vi0;ui0;ui1) from
Ĥ such that ‖U i0‖H ≤ R, i = 1, 2. Then the difference
Z(t) = U1(t)− U2(t) ≡ (v(t);u(t);ut(t))
satisfies the relation
3For the definition and some properties of the fractal dimension, see, e.g., [7] or [33].
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ON INTERACTION OF AN ELASTIC WALL WITH A POISEUILLE TYPE FLOW 159
‖Z(t)‖2H ≤MRe
−γ∗t‖Z0‖2H +MR
t∫
0
e−γ∗(t−τ)‖u(τ)‖2Ωdτ (55)
for some positive constant MR and γ∗.
Proof. See [14] for some details.
To complete the proof of Theorem 4.2 we note that by Proposition 7.9.4 [11] (St, Ĥ) is asymptot-
ically smooth, i.e., for any bounded set B in Ĥ such that StB ⊂ B for t > 0 there exists a compact
set K in the closure B of B, such that StB converges uniformly to K. By Corollary 4.1 the system
is gradient. Proposition 4.1 yields that the set N of the stationary points (see (51)) is bounded in Ĥ.
Therefore to prove the existence of a global attractor we can use well-known criteria for gradient
systems (see, e.g., [31] (Theorem 4.6) or Corollary 2.29 in [10]).
The standard results on gradient systems with compact attractors (see, e.g., [3, 7, 33]) imply
(54). Since (St, Ĥ) is quasi-stable, the finiteness of fractal dimension dimfA follows from Theo-
rem 7.9.6 [11]. To obtain the result on regularity stated in (52) and (53) we apply Theorem 7.9.8 [11].
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Received 25.11.12
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