Journal of Inequalities in Pure and
Applied Mathematics
REGULARITY PROPERTIES OF SOME STOKES OPERATORS
ON AN INFINITE STRIP
volume 5, issue 2, article 41,
2004.
A. ALAMI-IDRISSI AND S. KHABID
Université Mohammed V-Agdal Faculté des Sciences
Dépt de Mathématiques & Informatique
Avenue Ibn Batouta BP 1014
Rabat 10000 Morroco.
Received 08 May, 2002;
accepted 19 December, 2003.
Communicated by: S.S. Dragomir
EMail: alidal@fsr.ac.ma
EMail: sidati@caramail.com
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2000
Victoria University
ISSN (electronic): 1443-5756
046-02
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Abstract
In this paper, we try to solve the problem which arises in connection with the
stability theory of a periodic
equilibrium solution of Navier-Stokes equations on
an infinite strip R × − 12 , 12 .
2000 Mathematics Subject Classification: 35Q30, 76D05, 42B05.
Key words: Navier-Stokes equations, Regularity, Fourier series.
With all our thanks to Professor Doctor Bruno Scarpellini of Mathematisches Institut,
Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland who suggests this
problem.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Θ -Periodic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4
Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References
Regularity Properties of Some
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1.
Introduction
This problem arises in connection with the stability theory of
equili a1 periodic
1
birium solution of Navier-Stokes on infinite strip Ω = R × − 2 , 2 . Consider the Navier-Stokes equation on an infinite strip Ω = R × − 12 , 12 :
(1.1)
∂t U = ν∆U − (U · ∇)U + ∇p + f
with f = f (x, y) a smooth time independent outer force on Ω ,which is L
-periodic in x for some L.
Let a smooth equilibrium solution U0 = (u0 , v0 ), p0 of (1.1) be given, which
is L -periodic in x and U0 = 0 on ∂Ω. The stability of U0 = (u0 , v0 ), p0 can be
studied against small perturbations under two aspects:
(I) The perturbations are themselves L -periodic in x.
(II) The perturbations are in (L2 (Ω))2 .
The relation between (I) and (II) is the mathematical tools used by physicists
in connection with Schroedinger equations with periodic potentials [3]. The
main tool thereby is the notion of direct integrals (see [1] , [3] , [5], [8]). This
notion is based on Θ-Periodic functions (ie. generalisation of periodic functions).
In this paper we study the Stokes operators which arise in the so-called Bloch
space theory of equation (1.1). This theory, well established in the case of
Schroedinger equations with periodic potentials [3] extends to the Stokes operators which occur in Navier-Stokes and related equations, but the corresponding
theory is now more involved, see [8] where the three dimensional case (3d) is
Regularity Properties of Some
Stokes Operators on an Infinite
Strip
A. Alami-Idrissi and S. Khabid
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treated. The Stokes operators which appear in connection with (1.1), either 2d
or 3d, are of the form:
(1.2)
P ∆U − P (V · ∇) − P (U · ∇)V.
Here V is a fixed velocity field, periodic in the unbounded space directions (x
or x, y), U is the argument on which the operator acts, while P is the orthogonal
projection onto the space of divergence free fields. Three cases are of interest:
(a) U ∈ (H 2 (Ω) ∩ H01 (Ω))3 , divU = 0.
(b) U is periodic in the unbounded space directions.
(c) U is Floquet - periodic in the unbounded space directions.
Case (b) subsumes under case (c) [2]; case (a) is handled in [4]. Case
(a) and (c) are related by certain spectral formulas, well known in case of
the Schroedinger equations with periodic potentials. In the 3d-case however,
these spectral formulas associated with (1.2) are more complicated than in the
Schroedinger case due to the appearance of singularities ([8, Sect 9.4, 9.5]). The
purpose of the present paper is to show that in the 2d-case these singularities are
absent and that the spectral formulas associated with (1.2) have precisely the
same formula as in the Schroedinger case. To this effect we study first the most
important special, ie. V = 0. We have to perform estimates similar to those
in Sections 6.4–6.7 of [8]. In our estimates, which are considerably simpler,
singularities do not appear.
How this fact can be exploited so as to obtain the mentioned spectral formulas is outlined in subsequent sections.
Regularity Properties of Some
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A. Alami-Idrissi and S. Khabid
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2.
Notation
For X , Y Banach spaces, k·kX , k·kY are their respective norms. L(X , Y) is the
space of bounded operators from X to Y with kT k the operator norm.
For A a linear operator on X and E ⊆ X a subspace, A |E is the restriction
of A to E.
For any Ω, H p (Ω) is the Sobolev space of functions having square integrable
derivatives up to order p with (·, ·)p and k·kH p (Ω) the usual scalar product and
norm on H p (Ω).We set L2 (Ω) = H 0 (Ω) and k·kH p = k·kH p (Ω) and extend this
notation to vectors and set:
Regularity Properties of Some
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kuk2L2 = ku1 k2L2 + ku2 k2L2 ,
A. Alami-Idrissi and S. Khabid
where u = (u1 , u2 ) ∈ (L2 )2 , Likewise with the Sobolev norms. The scalar
product on (H p (Ω))2 is h·, ·ip , with:
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hu, vip =
2
X
(ui , vi )p ,
ui , vi ∈ H p (Ω),
i=1
where u = (u1 , u2 ), v = (v1 , v2 ) we set h·, ·i = h·, ·i0 .
C p (Ω̄) is the space of functions p times continuously differentiable on Ω̄
and C0p (Ω̄) is the space of functions f ∈ C p (Ω̄) with supp f compact.
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3.
Θ -Periodic Function
We fix a period L > 0, set QL =]0, L[ and Q = QL × − 12 , 12 , for some small
> 0 and put M =] − , 2π + [ with M = [0, 2π]. Also, let Ṁ be M minus
the numbers 0 and 2π.
We define a Θ-Periodic function: For Θ in M ; f ∈ CΘp (Q) if f ∈ C p (Q)
and
f (x + jL, y) = eijΘ f (x, y), j ∈ Z, (x, y) ∈ Ω.
We define the functional spaces: HΘp (Q) is the set of f ∈ L2 (Q) such that
limn kfn − f kH p = 0 for some sequence fn ∈ CΘp (Q).
We also let L2g be the subspace of L2 (Q) containing the elements f such
that f (x, −y) = f (x, y) a.e. Likewise with L2u and f (x, −y) = −f (x, y) a.e.
Finally, we put L2 = (L2 )2 , L2g = L2g × L2u and L2u = L2u × L2g .
It is easy to prove that:
L2 = L2g ⊕ L2u .
Regularity Properties of Some
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A. Alami-Idrissi and S. Khabid
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4.
Fourier Series
We consider the eigenvalue problem: y” + λy = 0 on − 12 , 12 with Neumann
resp. Dirichlet boundary conditions.
In the first case we have a complete orthonormal (C.O.N) system in L2 (Q):
√
ϕ2k = (−1)k 2 cos 2πky for k ≥ 1, ϕ0 = 1,
√
ϕ2k+1 = (−1)k 2 sin(2k + 1)πy for k ≥ 0,
Λp = p2 π 2 is an eigenvalue
associated to ϕp , ϕ2k is even, ϕ2k+1 odd and more√
over ϕp (1/2)
p ≥ 1. For the
p = 2 for
pother case we have a (C.O.N) system
0
0
given by Λp ψp = ϕp , where ψp = − Λp ϕp for p ≥ 1.
Since parity in y will be important we introduce notations: σk = ϕ2k+1 , τk =
ψ2k+1 , λk = Λ2k+1 , k ≥ 0, and ρk = ϕ2k , πk = ψ2k for k ≥ 1, ϕ0 = 1 and
µk = λ2k . For θ ∈ M we set: α̂ = (2πα + θ)L−1 , α ∈ Z and eα = eiα̂x .
1
We have a characterization of spaces Hθ,0
, Hθ1 , Hθ2 with the Fourier series:
Let f ∈ L2 (Q) have Fourier series:
X
X
f=
fα,i eα ϕi =
f˜α,i eα ψi .
With respect to {eα ϕi } resp {eα ψi }.
Proposition 4.1. (a) f ∈ Hθ1 iff
X
(α̂2 + Λi ) |fα,i |2 < ∞.
1
(b) f ∈ Hθ,0
iff
Regularity Properties of Some
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A. Alami-Idrissi and S. Khabid
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X
2
(α̂2 + Λi ) f˜α,i < ∞.
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For a proof see [6]. We have the characterization of space Hθ2 too:
P
Proposition 4.2. Let f ∈ L2 (Q) satisfy (α̂2 + Λi )2 |fα,i |2 < ∞. then f ∈
Hθ2 and
X
2
2
2
2
kF kH 2 ≤ C
(α̂ + Λi ) |fα,i |
2
P
for a C independent of θ ∈ M . Likewise with (α̂2 + Λi )2 f˜α,i .
For a proof see [6].
Our aim is to prove:
Theorem 4.3. (a) There is C > 0 as follows. If U ∈ dom(As (θ)) ∩ Eθg and
As (θ)U = f for some θ ∈ M , f ∈ Eθg then U ∈ (Hθ2 )2 and
kU kH 2 ≤ C kf kL2 .
(b) Under the conditions U ∈ dom(As (θ)) ∩ Eθu or U ∈ dom(As (θ)) ∩ Eθ the
assertion (a) holds.
P
P
Proposition 4.4. If f ∈ Hθ1 has Fourier series α,j aα,j eα σj then j |aα,j | ≤
P
1
∞ and f ∈ Hθ,0
iff j aα,j = 0, α in Z.
Regularity Properties of Some
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A. Alami-Idrissi and S. Khabid
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Remark 4.1. Proposition 4.4 is a consequence of Propositions 6.1 and 6.3 in
[8].
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For the proof of this theorem we need the Proposition 6 used in [7]; we recall
2
λk = (2k+1)
:
π2
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Proposition 4.5. There are Γ0 , Γ1 such that for s ≥ 0:
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P
(i) Γ0 (1 + s)−3 ≤ (λk + s2 )−2 ≤ Γ1 (1 + s)−3 ;
P
(ii) (λk + s2 )−1 ≤ Γ1 (1 + s)−1 ;
P −1
(iii)
λk (λk + s2 )−2 ≤ Γ1 (1 + s)−4 ;
P
(iv)
λk (λk + s2 )−2 ≤ Γ1 (1 + s)−1 .
Proof of Theorem 4.3. Since, in the first part of the proof, the factor α̂−1 appears
which is later cancelled, it is advantageous to assume first that θ ∈ Ṁ .
1
We take U = (A, B) ∈ (Hθ,0
) ∩ L2g such that divU = 0.
We know that if L2g = L2g × L2u then A ∈ L2g and B ∈ L2u and with the
1
characterization of space Hθ,0
by Fourier series we have
A=
X
Ajα eα τj and B =
X
Bjα eα σj
P
such that (λj + α̂2 )|Ajα |2 < ∞, likewise for B, the components of f = (a, b)
admit expansions too,
X
X
a=
ajα eα τj and b =
bjα eα σj .
U is a weak solution of As (θ)U = f for f ∈ Eθ if and only if:
Regularity Properties of Some
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A. Alami-Idrissi and S. Khabid
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(4.1)
2
X
j=1
1 2
for all V ∈ (Hθ,0
).
h∇Uj , ∇Vj i + hf, V i = 0
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As a test vector in (4.1) we take:
1 2
V = (u0 τ0 + uj τj , w0 σ0 + wj σj ) ∈ (Hθ,0
),
whereby divV = 0, thus:
p
p
(4.2)
λj wj = −∂x uj and λ0 w0 = −∂x u0 .
Here u0 ∈ Hθ2 (QL ) is arbitrarily fixed.
As in paper [7], we have w0 + wj = 0. From the divergence condition
√ we
λj
deduce that since √1λ0 u0 + √1 uj is constant Θ-periodic, then uj = − √λ0 u0 .
Regularity Properties of Some
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λj
By exploiting the arbitrariness of U0 , ψ we reach certain equations for the
Fourier coefficients Aj,α , Bj,α , aj,α , bj,α .
We note:
λˆj = λj + α̂2 , j ≥ 0, α ∈ Z,
(A)j (α) = λˆj Aj,α − aj,α , j ≥ 0, α ∈ Z,
From the divergence condition for u, f we get:
(4.4)
iα̂
(B)j (α) = − p (A)j (α), j ≥ 0.
λj
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(B)j (α) = λˆj Bj,α − bj,α , j ≥ 0, α ∈ Z.
We obtain:
p
λj
iα̂
iα̂
(4.3) − √ (A)j (α)+(A)0 (α)− √ (B)j (α)+ √ (B)0 (α) = 0,
λ0
λ0
λ0
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, j ≥ 0.
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From the condition θ ∈ Ṁ we get α̂ 6= 0 then:
p
i λj
(B)j (α).
(4.5)
(A)j (α) =
α̂
So according to (4.3) and (4.5) we have:
λˆj (B)j (α) = λˆ0 (B)0 (α).
P
By using Proposition 4.4 we have j Bjα = 0, and then:
(4.6)
(4.7)
!


P
P


B0,α = k λˆ0 (λˆj )−2 b0,α − j≥1 (λˆj )−1 bj,α ,



j≥1







 k=
= k(α).
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j≥1
Having B0,α , we can express Bj,α , j ≥ 1 via (4.7) and then Aj,α , j ≥ 0 via (4.5).
Then (4.3) becomes:
(4.8)
A. Alami-Idrissi and S. Khabid
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!−1
P
1 + (λˆ0 )2 (λˆj )−2
Regularity Properties of Some
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λˆ
λˆ
p j (A)j = √ 0 (A)0
λ0
λj
Equation (4.5) gives us (for j = 0):
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√
(A)0 (α) =
i λ0
(B)0 (α).
α̂
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Thus,
(4.9)
(A)j (α) =
i
λj λˆ0
(B)0 (α),
α̂λˆj
p
and from (4.7) we deduce:
(4.10)
(B)0 (α) = −k(b0,α + λˆ0
X
(λˆj )−1 bj,α ).
j≥1
By the divergence condition we replace bj,α by aj,α in (4.10). If we replace
(B)0 (α) in (4.9) by its value we obtain:
!
p
X
− λj λˆ0 k
1
ˆ −1
√ a0,α + λˆ0
(4.11)
(A)j (α) =
.
(λ1/2
s λs ) as,α
ˆ
λ0
λj
s≥1
As can be seen from (4.11), the expression for (A)j (α) does not contain any
factor α̂−1 , that is no singularity, we may therefore assume from now on that
θ ∈ M .
By (4.11) we have:
(A)j (α) = Ij + IIj ,
where
p
− λj λˆ0 k ˆ X 1/2 ˆ −1
Ij =
λ0
(λj λj ) aj,α
λˆj
j≥1
and
Regularity Properties of Some
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p
− λj λˆ0 k
IIj =
a0,α .
√
λˆj λ0
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We note that by Proposition 4.4 (i) a Γ2 is found such that,
k ≤ Γ2 (1 + s)−1 , (s = |α̂|),
then:
2
λj λˆ0 k 2
2
|Ij | ≤
2
λˆj
!
X
!
X
(λˆs )−2 (λˆ0 )2 (λs )−1
s≥1
|as,α |2
s≥1
!
Γ2 (1 + s)−2 (λ0 + s)2 λj
≤ 2
(λj + s2 )2
C0 X
≤
|as,α |2 .
λj s≥1
X
λ−1
s
s≥1
!
X
s≥1
Regularity Properties of Some
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Thus,
XX
α
|Ij |2 ≤ C
XX
α
j≥1
|as,α |2
s≥1
and for IIj we have:
2
λj λˆ0 k 2
|IIj | =
|a0,α |2
2
ˆ
λj λ0
2
then:
|as,α |2
λj (λ0 + s2 )2 Γ22 (1 + s)−2
|a0,α |2
|IIj |2 ≤
(λj + s2 )2 λ0
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and
X
j≥1
|IIj |2 ≤ C 0 (1 + s)−2 |a0,α |2 .
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Therefore
XX
α
j≥1
|IIj |2 ≤ C1
X
|a0,α |2 .
α
We still have to look at (A)0 (α). We recall (4.11) for j = 0 and we can
estimate k(α) by Proposition 4.5.
For (B)j (α): By (4.4) and (4.9) we can deduce by using Proposition 4.4 that
there is a θ−independent C2 such that:
XX
XX
|(B)j (α)|2 ≤ C2
|bj,α |2 .
α
j
α
The proof of (b) is very similar.
Conclusion:
kU kH 2 ≤ C kF kL2 .
j
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5.
Comments
As indicated, due to the fact that the singularity θ = 0 resp. θ = 2π drops out in
the computations presented in the previous sections, the spectral theory, carried
out for dimension d = 3 in [6], [8] simplifies considerably. Partly for this
reason and partly for reasons of space we concentrate here on briefly describing
the final result which emerges from this simplification. In order to describe the
manner in which the spectral formula (**) in [6] simplifies, we recall the objects
which appear in it. Following Sections 2 and 3, we have the θ-periodic Sobolev
1
spaces Hθp (Q), Hθ,0
(Q), θ ∈] − ε, 2π + ε[, the orthogonal projection Pθ from
2
2
1
L (Q) onto Eθ , with Eθ the L2 -closure of the set of f ∈ Hθ1 (Q) × Hθ,0
(Q)
such that div f = 0. The periodic Stokes operator AS (θ) is now defined as
follows:
(5.1)
f ∈ dom(AS (θ)) iff
1
f ∈ (Hθ2 (Q) ∩ Hθ,0
(Q))2
and div f = 0, and for such f, AS (θ)f = νPθ ∆f.
Next, we recall that, as stressed in the introduction, we are given a smooth
velocity field v = (v1 , v3 ) on R × [ 21 , 12 ] which is L-periodic in the unbounded
variable x, that gives rise to an operator T acting on elements u = (u1 , u3 ) ∈
dom(AS (θ)) according to
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T u = −(v1 ∂x u1 + v3 ∂z u1 , v1 ∂x u3 + v3 ∂z u3 ).
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We briefly digress on the periodic case which arises for θ = 0 of θ = 2π. In
accordance with [6] we stress this case by the label ‘per’ rather than by θ = 0
p
p
(Q) = H0p (Q) = H2π
or θ = 2π. Thus AS (per) = AS (0) = AS (2π), Hper
(Q),
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(5.2)
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etc. In order for the spectral formulas below to be valid, we have to define Eper ,
AS (per), Pper as follows:
(5.3)
Eper is the L2 −closure of all vector fields v = (f, h)
Z
1
1
in Hper (Q) × Hper,0 (Q) such that div f = 0 and
f dxdz = 0
Q
(5.4)
2
1
(Q))2 ,
v = (f, h) is in dom(AS (per)) if v ∈ (Hper
(Q) ∩ Hper,0
Z
div v = 0 and
f dxdz = 0; for such v we set
Q
AS (per)v = νPper ∆v, where Pper is the orthogonal projection
from L2 (Q)2 onto Eper .
With this definition, As (per) is selfadjoint on Eper .
Finally
we need corresponding objects defined on the whole strip Ω = R ×
1 1
− 2 , 2 . Thus
(5.5)
E is the L2 −closure of f ∈ H 1 (Ω) × H01 (Ω)
such that div f = 0,
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(5.6)
2
H01 (Ω))2
f ∈ dom(AS ) iff f ∈ (H (Ω) ∩
and for such f we set AS f = νP ∆f.
and div f = 0,
For elements f ∈ dom(AS ), the operator T acts again via (5.2). Under these
stipulations, the operators
G = AS + P T,
Gθ = AS (θ) + Pθ T,
Gper = AS (per) + Pper T
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all become holomorphic semigroup generators on E, Eθ , Eper respectively. The
spectral formulas, announced above now are:


[
((22)1 )
(AS (θ) + Pθ T ) ,
σ(AS + P T ) = closure 
θ∈(0,2π)
((22)2 )
σ(AS + P T ) =
[
(AS (θ) + Pθ T ).
θ∈[0,2π]
These formulas correspond to formulas (*), (**) in [6, p. 169]. While (22)1
looks the same as (*) in [6], (22)2 is definitely simpler; it implies in particular
that if λ ∈ σ(AS (per) + Pper T ) then λ ∈ σ(AS + P T ), a statement which
cannot be asserted in dimension d = 3 as can be seen from formula (**) in [6].
The proof of (22)2 is based on the computations in the present Section 4, which
entail that the singularities which arise in dimension d = 3 in [6], drop out. The
detailed verification of this claim is by a careful examination of the arguments
in [6], a task within the scope of this paper.
Regularity Properties of Some
Stokes Operators on an Infinite
Strip
A. Alami-Idrissi and S. Khabid
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References
[1] J. DIXMIER, Les Algebres d’Operateurs dans l’ESpace Hilbertien, Academic Press, New York, 1978.
[2] K. KIRCHGA̋SSNER AND H. KIELHŐFER, Stability and bifurcation in
fluid dynamics, Rocky Mountain J. of Math., 3(2) (1973), 275–318.
[3] M. REED AND B. SIMON, Methods of Math. Physics IV, Analysis of Operators, Gouthier-Villars, Paris, 1957.
[4] V.A. ROMANOV, Stability of plane parallel couette flows, Functional Analysis and its Applications, 7 (1973), 137–146.
[5] B. SCARPELLINI, Stability of space periodic equilibrium solutions of
Navier-Stokes on an infinite plate, C.R. Acad. Sci. Paris, Série I, 320 (1995),
1485–1488.
Regularity Properties of Some
Stokes Operators on an Infinite
Strip
A. Alami-Idrissi and S. Khabid
Title Page
Contents
2
[6] B. SCARPELLINI, L -perturbations of periodic equilibria solutions of
Navier-Stokes, Journal for Analysis and its Applications, 14(4) (1995),
779–828.
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[7] B. SCARPELLINI, Regularity properties of some Stokes operators on an
infinite plate, Pitman Res. Notes Math. Sci., 314 (1994), 221–231.
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[8] B. SCARPELLINI, Stability, Instability and Direct Integrals, Chapman &
Hall/CRC, Boca Raton, London, 1999.
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