Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 41, 2004
REGULARITY PROPERTIES OF SOME STOKES OPERATORS ON AN INFINITE
STRIP
A. ALAMI-IDRISSI AND S. KHABID
U NIVERSITÉ M OHAMMED V-AGDAL FACULTÉ DES S CIENCES
D ÉPT DE M ATHÉMATIQUES & I NFORMATIQUE
AVENUE I BN BATOUTA BP 1014
R ABAT 10000 M ORROCO .
alidal@fsr.ac.ma
sidati@caramail.com
Received 08 May, 2002; accepted 19 December, 2003
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we try to solve the problem which arises in connection with the
stability
of a periodic equilibrium solution of Navier-Stokes equations on an infinite strip
theory
R × − 12 , 12 .
Key words and phrases: Navier-Stokes equations, Regularity, Fourier series.
2000 Mathematics Subject Classification. 35Q30, 76D05, 42B05.
1. I NTRODUCTION
This problem arises in connection with the
stability
theory of a periodic equilibrium solution
1 1
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 .
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
With all our thanks to Professor Doctor Bruno Scarpellini of Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel,
Switzerland who suggests this problem.
046-02
2
A. A LAMI -I DRISSI AND S. K HABID
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 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.
2. N OTATION
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:
kuk2L2 = ku1 k2L2 + ku2 k2L2 ,
where u = (u1 , u2 ) ∈ (L2 )2 , Likewise with the Sobolev norms. The scalar product on
(H p (Ω))2 is h·, ·ip , with:
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.
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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R EGULARITY P ROPERTIES OF S TOKES O PERATORS
3
3. Θ -P ERIODIC F UNCTION
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 .
4. F OURIER S ERIES
We consider the eigenvalue problem: y” + λy = 0 on − 21 , 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,
2 2
Λ
moreover ϕp (1/2) =
√p = p π is an eigenvalue associated to ϕp , ϕ2k is even, ϕ2k+1 odd andp
2 for p ≥ 1. For the other case we have a (C.O.N) system given by Λp ψp = ϕ0p , where
p
ψp0 = − 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
X
2
(α̂ + Λi ) f˜α,i < ∞.
2
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
kF k2H 2 ≤ C
(α̂2 + Λi )2 |fα,i |2
2
P 2
2˜ for a C independent of θ ∈ M . Likewise with (α̂ + Λi ) fα,i .
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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4
A. A LAMI -I DRISSI AND S. K HABID
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
1
Proposition 4.4. If f ∈ Hθ1 has Fourier series α,j aα,j eα σj then j |aα,j | ≤ ∞ and f ∈ Hθ,0
P
iff j aα,j = 0, α in Z.
Remark 4.5. Proposition 4.4 is a consequence of Propositions 6.1 and 6.3 in [8].
For the proof of this theorem we need the Proposition 6 used in [7]; we recall λk =
(2k+1)2
:
π2
Proposition 4.6. There are Γ0 , Γ1 such that for s ≥ 0:
P
−3
(i): ΓP
≤ (λk + s2 )−2 ≤ Γ1 (1 + s)−3 ;
0 (1 + s)
(ii): P(λk + s2 )−1 ≤ Γ1 (1 + s)−1 ;
2 −2
(iii): P λ−1
≤ Γ1 (1 + s)−4 ;
k (λk + s )
(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 characterization of
1
space Hθ,0
by Fourier series we have
X
X
A=
Ajα eα τj and B =
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:
2
X
(4.1)
h∇Uj , ∇Vj i + hf, V i = 0
j=1
1 2
for all V ∈ (Hθ,0
).
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
(4.2)
λj wj = −∂x uj and
p
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 deduce that since
√
√1 u0
λ0
λ
j
+ √1 uj is constant Θ-periodic, then uj = − √λ0 u0 .
λj
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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R EGULARITY P ROPERTIES OF S TOKES O PERATORS
5
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,
(B)j (α) = λˆj Bj,α − bj,α , j ≥ 0, α ∈ Z.
We obtain:
p
(4.3)
−√
λj
iα̂
iα̂
(A)j (α) + (A)0 (α) − √ (B)j (α) + √ (B)0 (α) = 0,
λ0
λ0
λ0
, j ≥ 0.
From the divergence condition for u, f we get:
(4.4)
iα̂
(B)j (α) = − p (A)j (α), j ≥ 0.
λj
From the condition θ ∈ Ṁ we get α̂ 6= 0 then:
p
i λj
(4.5)
(A)j (α) =
(B)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:
!


P
P


B0,α = k λˆ0 (λˆj )−2 b0,α − j≥1 (λˆj )−1 bj,α ,



j≥1

(4.7)
!−1



P


ˆ 2 (λˆj )−2

= k(α).
 k = 1 + (λ0 )
(4.6)
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:
λˆ
λˆ
p j (A)j = √ 0 (A)0
(4.8)
λ0
λj
Equation (4.5) gives us (for j = 0):
√
i λ0
(A)0 (α) =
(B)0 (α).
α̂
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
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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6
A. A LAMI -I DRISSI AND S. K HABID
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
1/2
−1
√ a0,α + λˆ0
(4.11)
(A)j (α) =
(λs λˆs ) as,α .
λ0
λˆj
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
p
− λj λˆ0 k
IIj =
a0,α .
√
λˆj λ0
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
X
λ−1
s
s≥1
!
X
|as,α |2
s≥1
0
C X
≤
|as,α |2 .
λj s≥1
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
λˆj λ0
2
then:
|IIj |2 ≤
λj (λ0 + s2 )2 Γ22 (1 + s)−2
|a0,α |2
(λj + s2 )2 λ0
and
X
|IIj |2 ≤ C 0 (1 + s)−2 |a0,α |2 .
j≥1
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 4.6.
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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R EGULARITY P ROPERTIES OF S TOKES O PERATORS
7
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
α
j
j
The proof of (b) is very similar.
Conclusion:
kU kH 2 ≤ C kF kL2 .
5. C OMMENTS
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 spaces
1
Hθp (Q), Hθ,0
(Q), θ ∈] − ε, 2π + ε[, the orthogonal projection Pθ from L2 (Q)2 onto Eθ , with Eθ
1
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
(5.2)
T u = −(v1 ∂x u1 + v3 ∂z u1 , v1 ∂x u3 + v3 ∂z u3 ).
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 or θ = 2π. Thus AS (per) =
p
p
AS (0) = AS (2π), Hper
(Q) = H0p (Q) = H2π
(Q), 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
f dxdz = 0
in Hper (Q) × Hper,0 (Q) such that div f = 0 and
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 × − 12 , 12 . Thus
(5.5)
E is the L2 −closure of f ∈ H 1 (Ω) × H01 (Ω)
such that div f = 0,
J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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8
(5.6)
A. A LAMI -I DRISSI AND S. K HABID
f ∈ dom(AS ) iff f ∈ (H 2 (Ω) ∩ H01 (Ω))2 and div f = 0,
and for such f we set AS f = νP ∆f.
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
all become holomorphic semigroup generators on E, Eθ , Eper respectively. The spectral formulas, announced above now are:


[
((22)1 )
σ(AS + P T ) = closure 
(AS (θ) + Pθ T ) ,
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.
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J. Inequal. Pure and Appl. Math., 5(2) Art. 41, 2004
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