DOI : https://doi.org/10.32628/CSEIT1953116

**Stability of **

## π³ β

**periodic equilibrium solutions of Navier-Stokes equations on **

**infinite strip **

S. Khabid

LaΓ’youne Higher School of Technology, Ibn Zohr University, Morocco E-mail : k.sidati@uiz.ac.ma

ABSTRACT

In this paper, we assume that a smooth equilibrium solution π0, π0 of Navier-Stokes equations is given on an infinite strip Ξ© = πΌπ Γ] β1

2, 1

2[, the problem of stability that arises in the infinite plate (Ξ© = πΌπ

2_{Γ] β}1
2,

1 2[) disappears in our case using the same tools in [7].

Keywords : Navier-Stokes equations, Fourier series, Stability, Regularity

I. Notations and Preliminaries

### For

### π³, π΄

### Banach spaces,

### β₯. β₯

_{π³}

### , β₯. β₯

_{π΄}

### are their

### respective norms.

### πΏ(π³, π΄)

### is the space of bounded

### operators from

### π³

### to

### π΄

### with

### β₯ π β₯

### the operator

### norm.

### For

### π΄

### a linear operator on

### π³

### and

### πΈ β π³

### a

### subspace,

### π΄|

_{πΈ}

### is the restriction of

### π΄

### to

### πΈ

### .

### For any

### Ξ©, π»

π_{(Ξ©)}

_{ is the Sobolev space of functions }

### having square integrable derivatives up to order

### π

### with

### (. , . )

_{π}

### and

### β₯. β₯

_{π»}π

_{(Ξ©)}

### the usual scalar product

### and norm on

### π»

π_{(Ξ©) }

_{.We set }

_{β}

2_{(Ξ©) = π»}

0_{(Ξ©) }

_{and }

### β₯. β₯

π»π### =β₯. β₯

_{π»}π

_{(Ξ©)}

### and extend this notation to vectors

### and set :

### β₯ π’ β₯

_{β}22

### =β₯ π’

_{1}

### β₯

_{β}22

### +β₯ π’

_{2}

### β₯

_{β}22

### where

### π’ = (π’

1### , π’

2### ) β (β

2### (Ξ©))

2### , Likewise with the

### Sobolev norms. The scalar product on

### (π»

π### (Ξ©))

2### is

### β©. , . βͺ

_{π}

### , with :

### β©π’, π£βͺ

π### = β

2π=1

### (π’

π### , π£

π### )

π### , π’

π### , π£

π### β π»

π### (Ξ©)

### where

### π’ = (π’

_{1}

### , π’

_{2}

### ), π£ = (π£

_{1}

### , π£

_{2}

### )

### and we set

### β©. , . βͺ = β©. , . βͺ

_{0}

### .

### πΆ

π### (Ξ©

### Μ )

### is the space of functions

### π

### times

### continously differentiable on

### Ξ©

### Μ

### and

### πΆ

_{0}π

### (Ξ©

### Μ )

### is the

### space of functions

### π β πΆ

π_{(Ξ©}

_{Μ
) }

_{with supp}

_{π }

_{compact. }

II. Introduction

Consider the Navier-Stokes equation on an infinite strip Ξ© = IR Γ] β1

2, 1 2[: βU

βt = Ξ½ΞU β (U. β)U + βp + f, div U = 0 (1) Where U = (u1, u2) satisfies Dirichlet boundary conditions at y = Β±1/2, p is the pressure, Ξ½ kinematic viscosity and f a time independent outer force.

We assume that a smooth equilibrium solution

We can study The stability of U0= (u1, u2), p0 against small perturbations under two aspects: (i) The perturbations are L -periodic in x. (ii)The perturbations are in (β2(Ξ©))2.

The relation between (i) and (ii) is the mathematical tools used by physicists in connection with SchrΓΆdinger equations with periodic potentials [4]. We will use the same tools used before like the direct integrals (see [1] , [4] , [7]) and the Ξ-Periodic functions (ie. generalisation of periodic function).

To discuss the stability of U0, p0 with respect to various classes of perturbations we replace U, p in ( 1) by U0+ v, p0+ Ο and we use that U0, p0 is an equilibrium solution of ( 1), Whereby it is supposed that v satisfy the same Dirichlet boundary conditions. By a straightforward computation we obtain:

{ βv

βt = Ξ½Ξv β (u0. β)v β (v. β)u0β (v. β)v β βΟ

div v = 0, v/ βΞ© = 0 (2)

Now letβs put this problem in a functional analytic frame.

Set V = {f β (H01(Ξ©))2, div f = 0}andE = V (β2(Ξ©))2

,

and π« the orthogonal projection onto E

We apply π« to the both sides of (2) we get:

βtv = (As+ π«T0)v + π«N(v) (3)

Where As is the Stokes operator defined by:

f β D(AS) iff {f β (H

2_{(Ξ©) β© H}
0

1_{(Ξ©))}2_{, div f = 0}
ASf = Ξ½π«Ξf, βf β D(AS)

T0 the linear operator defined by

T0f = β(u0. β)f β (f. β)u0, βf β (H1(Ξ©) β© H01(Ξ©))2. N is no-linear operator defined on E by N(v) = β(v. β)v and let B = As+ π«T0.

Proposition 2.1 [7]

E = {v β L2_{(Ξ©), div v = 0\v β₯ βp, βp β H}1_{(Ξ©)}}_{ }

Remark 1

Eβ₯β {βΟ, Ο β H1(Ξ©)}

We have Ο β H1(Ξ©) then according to the previous remark βΟ β Eβ₯, and on the other hand we have for all v β E, βtv β E. cause βt commute with v.

Proposition 2.2

AS is selfadjoint and β€ βΞ΅ for some Ξ΅ > 0, and exists
c > 0 such if ASv = f for v β D(AS) and f β E, then
βvβ_{H}2_{(Ξ©)}β€ cβfβ_{L}2_{(Ξ©)} (β)

Proof

We have β©ASv, vβͺ β€ βΞ½ c(Ξ©)βvβL2

2 _{ , where }_{c(Ξ©)}_{ is the }
PoincarΓ© constant, then it suffice to take Ξ΅ =_{c(Ξ©)}Ξ½ .
β©βASv, vβͺ β₯ Ξ΅βvβL2 so βA_{S}vβ_{L}2β₯ βvβ_{L}2 which

implies AS is injective .

Finaly, AS is bijective from D(AS) to Im(AS).

Corollary 1

T0 is relatively bounded with respect to AS ie : for all Ξ΄ > 0 there is KΞ΄> 0 such that:

βT0uβL2β€ Ξ΄βA_{S}uβ_{L}2+ K_{Ξ΄}βuβ_{L}2 , βu β D(A_{S})

Proof

Set A operator on L2_{(Ξ©)}_{ such that }_{(Au)}

i= Ξui for all
u in D(A) with D(A) = (H2_{(Ξ©) β© H}

0

1_{(Ξ©))}2_{, i = 1,2.}

Ξ© bounded in the direction ππ¦ββββ then according to the PoincarΓ© inequality A is bounded ie: there is a constant cβ² > 0 such that:

βAuβ_{L}2β€ cβ²βuβ_{H}2_{(Ξ©)} , for all u β D (A)

Since A β€ βΞ΅ for some Ξ΅ > 0 then (βA)12_{ is well }

For all Ξ΄ > 0 exists KΞ΄> 0 such that:

According to (*) of the proposition 2.2 there is c > 0

such that:

According to proposition 2.2 we have A is selfadjoint and β₯ Ξ΄ for some Ξ΄ > 0. example the quadratic forms theory in [4]) that:

D (A12) = H_{0}1(Ξ©), β₯ A
(****) with the proposition 2.2 to obtain:

β equivalents independently to Ξ», and defined by :

βfβΞ³ = βBΞ»

equilibrium solution of (3) stable in the sense of Ljapounov.

Theorem 1 [5]

1. If Ο(B) β {Ξ»/Re(Ξ») β€ βΞ΄} for some Ξ΄ > 0 then v = 0 is stable in the sense of Ljapounov of (3),

2. If Re(Ξ») > 0 for some Ξ» β Ο(B) then v = 0 is unstable in the sense of Ljapounov of (3)

Since the source of the difficulties is the singularity of the Stokes operator then we will focus on it using the same theory. This theory, well established in the case of SchrΓΆdinger equations with periodic potentials [4] extends to the stokes operators which occur in Navier-Stokes and related equations, but the corresponding theory is now more involved, see [7] where the three dimensional case (3d) is treated. The Stokes operators which appear in connection with (1), either 2d or 3d, are of the form:

AsU = Ξ½π«ΞU. (4)

U is the argument on which the operator acts, while

π« is the orthogonal projection onto the space of divergence free fields. Three cases are of interest: (a) U β (H2(Ξ©) β© H01(Ξ©))2, div U = 0.

(b) U is periodic in the unbounded space direction. (c) U is Floquet-periodic in the unbounded space direction.

Case (b) subsumes under case (c) [2]; case (a) is handled in [6]. There is a spectral formulas relation between the cases (a) and (c), well known in case of the SchrΓΆdinger equations with periodic potentials. In the 3d-case however, these spectral formulas associated with (2) are more complicated than in the SchrΓΆdinger case due to the appearance of singularities [7]. The purpose of this paper is to show that in the 2d-case these singularities are disappear and that the spectral formulas associated with (2) have the same results as in the SchrΓΆdinger equations case. To this effect we study first the most important special, ie. U = 0. We have to perform estimates

similar to those in [7]. In our estimates, which are considerably simpler, singularities do not appear.

How to exploit this fact so as to obtain the mentioned spectral formulas is the outlined in subsequent section.

III. π―-Periodic function

We fix a period πΏ > 0 , set ππΏ=]0, πΏ[ and π = ππΏΓ] β

1 2,

1

2[, for some small π > 0 we put ππ=] β π, 2π + π[ with π = [0,2π] (ππ is a neighbourhood of π) .

And let πΜπbe ππminus the numbers 0 and 2π. We define a Ξ-Periodic function (Floquet-periodic function):

For Ξ in ππ ; π β πΆΞ π

(π) if π β πΆπ_{(π)}_{ and }_{π(π₯ +}
ππΏ, π¦) = πππΞ_{π(π₯, π¦), π β ππ, (π₯, π¦) β Ξ©, with π}2_{= β1}
we define the functional spaces :

π»_{Ξ}π(π) is the set of π β β2(π) such that lim

πβ+ββ₯ ππβ π β₯π»π= 0 for some sequence (ππ)πβ πΌ π β πΆΞ

π (π)

We also let βπ2be the subspace of β2(π) containing the elements π such that π(π₯, βπ¦) = π(π₯, π¦) π. π.

Likewise with βπ’2and π(π₯, βπ¦) = βπ(π₯, π¦) π. π Finaly, we put

πΏ2_{= (β}2_{)}2_{, πΏ}
π
2 _{= β}

π2Γ βπ’2and βπ’2 = βπ’2Γ βπ2

Itβs easy to prove that :

πΏ2_{= πΏ}
π
2 _{β πΏ}

π’ 2

IV.Fourier series

We consider the eigenvalue problem: π¦" + ππ¦ = 0 on

] β1_{2},1_{2}[ with Neumann resp. Dirichlet boundary
conditions.

In the first case we have a complete orthonormal (C.O.N) system in β2(π):

π2π+1= (β1)πβ2 sin((2π + 1)ππ¦)) for π β₯ 0

Ξπ= π2π2 is an eigenvalue associated to ππ, π2π is even, π2π+1 odd and morever ππ(1/2) = β2 for π β₯ 1.

And in other case we have a (C.O.N) system given by

βΞπππ= πβ²π where πβ²π= ββΞπππ for π β₯ 1. Since parity in π¦ will be important we introduce notations:

ππ = π2π+1, ππ = π2π+1, ππ = Ξ2π+1, π β₯ 0, and ππ= π2π, ππ = π2π for π β₯ 1, π0= 1 and ππ = π2π And for Ξ β ππ we set: πΌΜ = (2ππΌ + Ξ)πΏβ1, πΌ β β€ and ππΌ= πππΌΜπ₯.

We have a characterization of spaces π»Ξ,01 , π»Ξ1, π»Ξ2 with the Fourier series :

Let π β β2_{(π) }_{have Fourier series : }
π = β ππΌ,πππ = β πΜπΌ,πππ

With respect to {ππΌππ} resp {ππΌππ}.

Proposition 4.1 (a) π β π»Ξ1iff

β (πΌΜ2_{+ Ξ}

π)|ππΌ,π|2< β (b) π β π»Ξ,01 iff

β (πΌΜ2_{+ Ξ}

π)|πΜπΌ,π|2< β.

For the proof see [7].

We have the characterization of space π»Ξ2too :

Proposition 4.2

Let π β β2(π) satisfy β (πΌΜ2_{+ Ξ}

π)2|ππΌ,π|2< β. then π β π»Ξ2and :

β₯ π β₯_{π»}22β€ πΆ(β (πΌΜ2+ Ξ_{π})2|π_{πΌ,π}|2)

for a πΆ independent of Ξ β ππ. Likewise with β (πΌΜ2+ Ξπ)2|πΜπΌ,π|2. For the proof see [7].

V. Main theorem

Our aim is to prove: Theorem 2

(π) There is πΆ > 0 as follows: If π β πππ(π΄π (Ξ)) β© πΈΞ

π

and π΄π (Ξ)π = π for some Ξ β ππ, π β πΈΞ

π

then π β (π»_{Ξ}2)2_{ and }
βπβ_{π»Β²}β€ πΆβπβ_{βΒ²}

β₯ π β₯_{π»}2β€ πΆ β₯ π β₯_{πΏ}2.

(π) Under the conditions π β πππ(π΄π (Ξ)) β© πΈΞπ’ or π β πππ(π΄π (Ξ)) β© πΈΞ the assertion (π) holds.

Proposition 5.1

If π β π»Ξ1 has Fourier series βπΌ,π ππΌ,πππΌππ then βπ|ππΌ,π| β€ β and π β π»Ξ,01 iff βπππΌ,π = 0, πΌ in β€

For the proof of the proposition 5.1 see [7].

We need also the proposition 6.12 used in [7] page 115:

We recall ππ=

(2π+1)2

π2 :

Proposition 5.2 There are π€0, π€1 such that for π β₯ 0:
(i) Ξ0(1 + π )β3β€ β (ππ+ π 2)β2β€ Ξ1(1 + π )β3
(ii) β (π_{π}+ π 2)β1β€ Ξ1(1 + π )β1

(iii)β π_{π}β1_{(π}

π+ π 2)β2β€ Ξ1(1 + π )β4 (iv)β ππ(ππ+ π 2)β2β€ Ξ1(1 + π )β1

VI.Proof of the main theorem

Since in the first part of the proof factor πΌΜβ1 appear which later cancel, it is advantagious to assume first

Ξ β πΜπ.

We take π = (π΄, π΅) β (π»Ξ,01 ) β© πΏπ2 such that dππ£ π = 0.

π is a weak solution of π΄π (Ξ)π = π for π β πΈΞ if and certain equations for the Fourier coefficients

|πΌπ|2β€ using Proposition 5.1 that there is a Ξ βindependent

πΆ2such that :

As indicated, due to the fact that the singularity Ξ = 0 resp. Ξ = 2π drops out in the computations presented in the previous section, the spectral theory, carried out for dimension π = 3 in [7] is simplified

for such π£ we set π΄π(pππ)π£ = ππ«pππΞπ£, π€βπππ π«pππ
is the orthogonal projection from β2_{(π)}2_{ onto }_{πΈ}

pππ. With this definition, π΄π (pππ) is selfadjoint on πΈpππ. Finally, we need corresponding objects defined on the whole strip Ξ© = πΌπ Γ] β1

2, 1

2[. Thus πΈ is the
βΒ² βclosure of π β π»1_{(Ξ©) Γ π»}

01(Ξ©) such that div π = 0,

π β dππ(π΄π) πππ π β (π»2(Ξ©) β© π»01(Ξ©))2 πππ πππ£ π = 0,

and for such π we set π΄ππ = ππ«Ξπ.

For elements π β dππ(π΄π), the operator π acts again via (18). Under these conditions, the operators

πΊ = π΄π+ π«π, πΊΞ = π΄π(Ξ) + π«Ξπ, πΊpππ = π΄π(pππ) + π«pπππ

all become holomorphic semigroup generators on

πΈ, πΈΞ, πΈpππ respectively. The spectral formulas, announced above now are:

(22)1 π(π΄π+ π«π) = (βΜ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Ξβ[0,2π] (π΄π(Ξ) + π«Ξπ))

(22)2 π(π΄π+ π«π) = β Ξβ[0,2π]

(π΄π(Ξ) + π«Ξπ).

These formulas correspond to formulas (*), (**) in [7], page. 169. While (22)1 looks the same as (*) in [7], (22)2 is definitely simpler; it implies in particular that if π β π(π΄π(pππ) + π«pπππ) then π β π(π΄π+ π«π), a statement which cannot be asserted in dimension

π = 3 as can be seen from formula (**) in [7]. The proof of (22)2 is based on the computations in the present section 5, which entails that the singularities which arise in dimension π = 3 in [7], drop out. The detailed verification of this claim is treated by a careful examination of the arguments in [7], a task within the scope of this paper.

VIII. REFERENCES

[1]. J. Dixmier, Les algebres d'operateurs dans l'espace hilbertien, Academic Press, New york, 1978.

[2]. K.KirchgΓ€ssner, H.KielhΓΆfer, Stability and bifurcation in fluid dynamics, Rocky mountain J. of Math. Vol. 3, no. 2, 275-318 (1973). [3]. Pazy, A., Semigroupes of linear operators and

applications to partial differential equations, Appl. Math. Sci. 44, Springer, New York, 1983. [4]. M.Reed and B.Simon, Methods of Math.

Physics IV. Analysis of operators. Gouthier-villars, Paris, 1957.1995.

[5]. D.Sattinger, Bifurcation and symmetry breaking in applied mathematics, Bull. Amer. Math. Soc., 3, 1980, pp. 779-819.

[6]. V.A.Romanov, Stability of plane parallel couette flows, Functional analysis and its applications 7, 137-146. (1973).

[7]. B.Scarpellini, Stability, Instability and direct integrals. Chapman & Hall/CRC, Boca Raton, London, 1999.

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