九州大学学術情報リポジトリKyushu University Institutional Repository

Instability of plane Poiseuille flow in viscouscompressible gas

Kagei, YoshiyukiFaculty of Mathematics, Kyushu University

Nishida, TakaakiDepartment of Applied Complex System, Kyoto University

http://hdl.handle.net/2324/1470206

出版情報：MI Preprint Series. 2014-11, 2014-09-12. Faculty of Mathematics, Kyushu Universityバージョン：権利関係：

MI Preprint SeriesMathematics for Industry

Kyushu University

Instability of plane Poiseuille

flow in viscous compressible gas

Yoshiyuki Kagei

& Takaaki Nishida

MI 2014-11

( Received September 12, 2014 )

Institute of Mathematics for IndustryGraduate School of Mathematics

Kyushu UniversityFukuoka, JAPAN

Instability of plane Poiseuille flowin viscous compressible gas

Yoshiyuki Kagei1 and Takaaki Nishida2

1 Faculty of Mathematics,Kyushu University,

Nishi-ku, Motooka 744,Fukuoka 819-0395, Japan

2 Department of Applied Complex System,Kyoto University,

Yoshida Honmachi, Sakyo-ku,Kyoto, 606-8317, Japan

Abstract

Instability of plane Poiseuille flow in viscous compressible gas is investigated.A condition for the Reynolds and Mach numbers is given in order for planePoiseuille flow to be unstable. It turns out that plane Poiseuille flow is un-stable for Reynolds numbers much less than the critical Reynolds number forthe incompressible flow when the Mach number is suitably large. It is provedby the analytic perturbation theory that the linearized operator around planePoiseuille flow has eigenvalues with positive real part when the instabilitycondition for the Reynolds and Mach numbers is satisfied.

Mathematics Subject Classification (2000). 35Q30, 76N15.

Keywords. Compressible Navier-Stokes equation, Poiseuille flow, instability.

1 Introduction

This paper is concerned with the stability of plane Poiseuille flow of the compressibleNavier-Stokes equation. We consider the following system of equations

∂tρ+ div (ρv) = 0, (1.1)

ρ(∂tv + v · ∇v)− µ∆v − (µ+ µ′)∇div v +∇P (ρ) = ρg (1.2)

in a 3-dimensional infinite layer Ωℓ = R2 × (0, ℓ):

Ωℓ = x = (x′, x3) : x′ = (x1, x2) ∈ R2, 0 < x3 < ℓ.

1

Here ρ = ρ(x, t) and v = ⊤(v1(x, t), v2(x, t), v3(x, t)) denote the density and velocityat time t ≥ 0 and position x ∈ Ωℓ, respectively; P = P (ρ) is the pressure that isassumed to be a smooth function of ρ satisfying

P ′(ρ∗) > 0

for a given constant ρ∗ > 0; µ and µ′ are the viscosity and the second viscositycoefficients, respectively, that are assumed to be constants and satisfy

µ > 0,2

3µ+ µ′ ≥ 0;

div , ∇ and ∆ denote the usual divergence, gradient and Laplacian with respectto x; and g is a given external force. Here and in what follows ⊤· stands for thetransposition.

We assume that the external force g takes the form

g = ge1,

where g is a positive constant and e1 =⊤(1, 0, 0) ∈ R3.

The system (1.1)–(1.2) is considered under the boundary condition

v|x3=0,ℓ = 0. (1.3)

It is easily seen that (1.1)–(1.3) has a stationary solution us =⊤(ϕs, vs) satisfying

ϕs = ρ∗, vs =ρ∗g

2µx3(ℓ− x3)e1,

that is the so-called plane Poiseuille flow.The aim of this paper is to give a condition for the Reynolds and Mach numbers

in order for plane Poiseuille flow to be unstable.The function us is also a stationary solution of the incompressible Navier-Stokes

equationdiv v = 0, (1.4)

ρ∗(∂tv + v · ∇v)− µ∆v +∇p = 0, (1.5)

v|x3=0,ℓ = 0 (1.6)

with p = ρ∗gx1.It is well known that stationary parallel flow of the incompressible Navier-Stokes

equation is in general stable under arbitrary size of initial perturbations in L2 ifthe Reynolds number R is sufficiently small. Furthermore, plane Poiseuille flow isstable under sufficiently small initial perturbations if R < Rc for a critical numberRc ∼ 5772, and unstable if R > Rc.

In the case of the compressible Navier-Stokes equation, Iooss-Padula [1] investi-gated the linearized stability of stationary parallel flow in a cylindrical domain underperturbations that are periodic in the unbounded direction of the domain. It was

2

shown in [1] that stationary parallel flow is linearly stable for suitably small Reynoldsnumber. In [2] (cf., [3]), nonlinear stability of parallel flow in the infinite layer Ωℓ

was studied; and it was proved that parallel flow is asymptotically stable under per-turbations sufficiently small in some Sobolev space over Ωℓ if the Reynolds and Machnumbers are sufficiently small. In this paper we will show that plane Poiseuille flow of(1.1)–(1.2) is linearly unstable if (3/Re+1/Re′)/Re ≤ 30(1/280−1/Ma2)/Ma2, pro-vided that Ma2 > 280. Here Re, Re′ and Ma are the numbers given by Re = 16R,Re′ = 16R′ and Ma = M/8 with the Reynolds number R, second Reynolds numberR′ and Mach number M defined by

R =ρ∗ℓV0

2µ, R′ =

ρ∗ℓV0

2µ′ , M =

√P ′(ρ∗)

V0

.

Here V0 is the maximum velocity ρ∗gℓ2

8µof plane Poiseuille flow. In particular, this

result shows that there appears an instability even when R ≪ Rc in the case ofcompressible flows.

To prove our result, we consider the spectrum of the linearized operator underperiodic boundary condition in x′ = (x1, x2) to find eigenvalues with positive realpart. As in the case of cylindrical domain analyzed in [1], the linearized operatorgenerates a C0-semigroup on L2

per(Pα1,α2 × (0, 1)). Here Pα1,α2 denotes the basicperiod cell [− π

α1, πα1)× [− π

α2, πα2) with α1, α2 > 0. We will investigate the spectrum

of the linearized operator on L2per(Pα1,α2 × (0, 1)) for sufficiently small α1 and α2 by

using the analytic perturbation theory to obtain our instability criterion mentionedabove.

This paper is organized as follows. In section 2 we deduce a non-dimensional formof system (1.1)–(1.2) and rewrite it into the system of equations for perturbations.We also introduce notations used in this paper. In section 3 we state the main resultof this paper precisely. Sections 4–6 are devoted to the proof of the main result. Insection 4 we consider the Fourier series expansion in x′ = (x1, x2) ∈ Pα1,α2 and reducethe spectral analysis of the linearized operator to the one for the Fourier coefficientsthat are functions of x3. Section 5 is devoted to the study of the spectrum of the zerofrequency part of the linearized operator. In section 6 we investigate the spectrumof the low frequency part of the linearized operator by the analytic perturbationtheory and complete the proof of our instability result.

2 Preliminaries

In this section we first deduce a non-dimensional form of system (1.1)–(1.2) andthen give the system of equations for perturbations. In the end of this section weintroduce function spaces used in this paper.

We introduce the following non-dimensional variables:

x = ℓx, t =ℓ

Vt, v = V v, ρ = ρ∗ρ, P = ρ∗V

2P

3

with

V =ρ∗gℓ

2

µ.

Under this transformation, Ωℓ is transformed into Ω = Ω1:

Ω = x = (x′, x3) : x′ = (x1, x2) ∈ R2, 0 < x3 < 1.

Using the relations ∂x = 1ℓ∂x, ∂t =

Vℓ∂t, we see that (1.1) and (1.2) are trans-

formed into∂tρ+ div (ρv) = 0, (2.1)

ρ(∂tv + v · ∇v)− ν∆v − (ν + ν ′)∇div v +∇P (ρ) = νρe1. (2.2)

Here, div , ∇ and ∆ denote the divergence, gradient and Laplacian with respect tox; and ν and ν ′ are the non-dimensional parameters given by

ν =µ

ρ∗ℓV, ν ′ =

µ′

ρ∗ℓV.

To derive (2.2) we have used the relation ℓgV 2 = ν. The assumption P ′(ρ∗) > 0 is

restated asP ′(1) > 0.

We next introduce plane Poiseuille flow. Let us consider the stationary problem

div (ρv) = 0, (2.3)

ρv · ∇v − ν∆v − (ν + ν ′)∇div v +∇P (ρ) = νρe1 (2.4)

in Ω under the boundary condition

v|x3=0,1 = 0. (2.5)

Proposition 2.1. Problem (2.3)–(2.5) has a stationary solution (plane Poiseuilleflow) us =

⊤(ρs, vs), where

ρs = 1, vs =⊤(v1s(x3), 0, 0), v1s(x3) =

1

2(−x2

3 + x3).

Proof. Set ρ = 1 and v = ⊤(v1(x3), 0, 0) in (2.3) and (2.4). Then, since

div v = ∂x1v1(x3) = 0, v · ∇v1 = v1∂x1v

1(x3) = 0,

together with (2.5), we have −∂2x3v1 = 1 and v1|x3=0,1 = 0, from which we obtain

v1(x3) =12(−x2

3 + x3). This completes the proof.

We next derive the system of equations for perturbations. We substitute u(t) =⊤(ϕ(t), w(t)) ≡ ⊤(γ2(ρ(t) − ρs), v(t) − vs) into (2.1) and (2.2), where γ is the non-dimensional number given by

γ =√

P ′(1) =

√P ′(ρ∗)

V.

4

Noting that ρs = 1, vs = ⊤(v1s(x3), 0, 0) and −∆vs = e1, we obtain the followingsystem of equations

∂tϕ+ v1s∂x1ϕ+ γ2divw = f 0, (2.6)

∂tw − ν∆w − ν∇divw +∇ϕ− ν

γ2ϕe1 + v1s∂x1w + (∂x3v

1s)w

3e1 = f . (2.7)

Here e1 =⊤(1, 0, 0) ∈ R3; and f 0 and f = ⊤(f ′, f 3) with f ′ = ⊤(f 1, f 2) denote the

nonlinearities:f 0 = −div (ϕw),

f = −w · ∇w − ϕγ2+ϕ

(ν∆w + ν

γ2ϕe1 + ν∇divw)

+ ϕγ4∇

(P (1)(γ−2ϕ)ϕ

)+ ϕ2

γ2(γ2+ϕ)∇ (P (1 + γ−2ϕ)) + 1

γ4∇(P (2)(γ−2ϕ)ϕ2)

),

where

P (1)(γ−2ϕ) =

∫ 1

0

P ′(1 + θγ−2ϕ) dθ

and

P (2)(γ−2ϕ) =

∫ 1

0

(1− θ)P ′′(1 + θγ−2ϕ) dθ.

We consider (2.6)–(2.7) under the boundary conditions

w|x3=0,1 = 0, ϕ, w: 2παj-periodic in xj (j = 1, 2), (2.8)

and the initial conditionu|t=0 = u0 =

⊤(ϕ0, w0). (2.9)

Here α1 and α2 are given positive numbers.

We are interested in the instability of plane Poiseuille flow. We will thus considerthe linearized problem for problem (2.6)–(2.9), i.e., with f 0 = 0 and f = 0.

In the remaining of this section we introduce some notations used in this paper.For given α1, α2 > 0, we denote the basic period cell by

Pα1,α2 =[− π

α1, πα1

)×[− π

α2, πα2

).

We setΩα1,α2 = Pα1,α2 × (0, 1).

We denote by C∞0,per(Ωα1,α2) the space of restrictions of functions in C∞(Ω) which

are Pα1,α2-periodic in x′ = (x1, x2) and vanish near x3 = 0, 1. We set

L2per(Ωα1,α2) = the L2(Ωα1,α2)-closure of C∞

0,per(Ωα1,α2),

H10,per(Ωα1,α2) = the H1(Ωα1,α2)-closure of C∞

0,per(Ωα1,α2).

We note that if f ∈ H10,per(Ωα1,α2), then f |xj=−π/αj

= f |xj=π/αj(j = 1, 2) and

f |x3=0,1 = 0.

5

For simplicity the set of all vector fields whose components are in L2per(Ωα1,α2)

(resp. H10,per(Ωα1,α2)) is also denoted by L2

per(Ωα1,α2) (resp. H10,per(Ωα1,α2)) if no

confusion will occur.We also use notation L2

per(Ωα1,α2) for the set of all u = ⊤(ϕ,w) with ϕ ∈L2

per(Ωα1,α2) and w = ⊤(w1, w2, w3) ∈ L2per(Ωα1,α2) if no confusion will occur. The

inner product of uj =⊤(ϕj, wj) ∈ L2

per(Ωα1,α2) (j = 1, 2) is defined by

(u1, u2) =1

γ2

∫Ωα1,α2

ϕ1(x)ϕ2(x) dx+

∫Ωα1,α2

w1(x) · w2(x) dx,

where z denotes the complex conjugate of z.We denote by L2(0, 1) the usual L2 space on (0, 1) with norm | · |L2 , and, likewise,

by Hk(0, 1) the k th order L2-Sobolev space on (0, 1) with norm | · |Hk . The H1-closure of C∞

0 (0, 1) is denoted by H10 (0, 1). As in the case of functions on Ωα1,α2 ,

function spaces of vector fields w = ⊤(w1, w2, w3) and, also, those of u = ⊤(ϕ,w),are simply denoted by L2(0, 1), H1

0 (0, 1), and so on, if no confusion will occur. Wedefine an inner product ⟨u1, u2⟩ of uj =

⊤(ϕj, wj) ∈ L2(0, 1) (j = 1, 2), by

⟨u1, u2⟩ =1

γ2

∫ 1

0

ϕ1(x3)ϕ2(x3) dx3 +

∫ 1

0

w1(x3) · w2(x3) dx3.

The mean value of a function ϕ(x3) over (0, 1) is denoted by ⟨ϕ⟩:

⟨ϕ⟩ =∫ 1

0

ϕ(x3) dx3.

The set of all ϕ ∈ L2(0, 1) with ⟨ϕ⟩ = 0 is denoted by L2∗(0, 1), i.e.,

L2∗(0, 1) = ϕ ∈ L2(0, 1) : ⟨ϕ⟩ = 0.

We define 4× 4 diagonal matrices Q0 and Q by

Q0 = diag (1, 0, 0, 0), Q = diag (0, 1, 1, 1).

Note thatQ0u = ⊤(ϕ, 0), Qu = ⊤(0, w) for u = ⊤(ϕ,w).

We denote the resolvent set of a closed operator A by ρ(A) and the spectrumof A by σ(A). The kernel and the range of A are denoted by KerA and R(A),respectively.

3 Main result

In this section we state our main result of this paper.The linearized problem is written as

∂tϕ+ v1s∂x1ϕ+ γ2divw = 0, (3.1)

6

∂tw − ν∆w − ν∇divw +∇ϕ− ν

γ2ϕe1 + v1s∂x1w + (∂x3v

1s)w

3e1 = 0, (3.2)

w|x3=0,1 = 0, ϕ, w : Pα1,α2-periodic in x′, (3.3)

u|t=0 = u0 =⊤(ϕ0, w0). (3.4)

We define the operator L on L2per(Ωα1,α2) by

D(L) =u = ⊤(ϕ,w) ∈ L2

per(Ωα1,α2) : w ∈ H10,per(Ωα1,α2), Lu ∈ L2

per(Ωα1,α2),

L =

(v1s∂x1 γ2div

∇ −ν∆− ν∇div

)+

0 0

− νγ2 v1s∂x1 + (∂x3v

1s)e1

⊤e3

.

As in [1] one can see that −L generates a C0-semigroup in L2per(Ωα1,α2).

We now state our main result of this paper. For α′ = (α1, α2) with α1, α2 > 0and (m1,m2) ∈ Z2, we introduce the notations |α′| and α′

m1,m2by

|α′| = (α21 + α2

2)12 and α′

m1,m2= (α1m1, α2m2).

Theorem 3.1. There exist constants r0 > 0 and η0 > 0 such that if |α′| ≤ r0, then

σ(−L) ∩λ ∈ C : |λ| ≤ η0

= λm1,m2 : |m1| = 0, 1, · · · , k1, |m2| = 0, 1, · · · , k2

for some k1, k2 ∈ N, where λm1,m2 are eigenvalues of −L that satisfies

λm1,m2 = − i

6(α1m1) + κ0(α1m1)

2 − γ2

12ν(α2m2)

2 +O(|α′m1,m2

|3)

as |α′m1,m2

| → 0. Here κ0 is the number given by

κ0 =1

12ν

(1

280− γ2 − ν2

15γ2− νν

30γ2

).

As a consequence, if γ2 < 1280

and 2ν2 + νν ≤ 30γ2(

1280

− γ2), then κ0 > 0 and

plane Poiseuille flow us =⊤(ϕs, vs) is linearly unstable.

Remark 3.2. Let, for example, γ = 0.05, ν = 1/173 and ν ′ = −2ν/3. Thenκ0 > 0 and thus plane Poiseuille flow is unstable. In this case, the Reynolds numberR = 1/(16ν) ∼ 10.81 and the Mach number M = 8/γ = 160.

We will prove Theorem 3.1 in the subsequent sections. In section 4 we considerthe Fourier series expansion in x′ = (x1, x2) ∈ Pα1,α2 and reduce the problem tothe ones for the Fourier coefficients u(α′

m1,m2, x3). In section 5 we investigate the

spectrum for the case α′m1,m2

= 0. In section 6 we complete the proof of Theorem 3.1by applying the analytic perturbation theory for small α′

m1,m2based on the analysis

in section 5.

7

4 Fourier series expansion in x′ ∈ Pα1,α2

In this section we consider the Fourier series expansion in x′ = (x1, x2) ∈ Pα1,α2 andreduce the problem to the ones for the Fourier coefficients u(α′

m1,m2).

To investigate the spectrum of −L, we consider the Fourier series expansion of(3.1)–(3.4) in x′ ∈ Pα1,α2 :

∂tϕ+ iξ1v1s ϕ+ iγ2ξ′ · w′ + γ2∂x3w

3 = 0, (4.1)

∂tw′ + ν(|ξ′|2 − ∂2

xn)w′ − iνξ′(iξ′ · w′ + ∂x3w

3) + iξ′ϕ

− νγ2 ϕe

′1 + iξ1v

1sw

′ + (∂x3v1s)w

3e′1 = 0,

(4.2)

∂tw3 + ν(|ξ′|2 − ∂2

xn)w3 − ν∂x3(iξ

′ · w′ + ∂x3w3) + ∂x3ϕ+ iξ1v

1sw

3 = 0, (4.3)

w|xn=0,1 = 0, (4.4)

u|t=0 = u0 =⊤(ϕ0, w0). (4.5)

Here and in what follows we simply write α′m1,m2

= (α1m1, α2m2) ((m1,m2) ∈ Z2)

as ξ′ = (ξ1, ξ2); ϕ = ϕ(ξ′, x3, t) and wj = wj(ξ′, x3, t) (j = 1, 2, 3) are the Fouriercoefficients of ϕ = ϕ(x′, x3, t) and wj = wj(x′, x3, t) (j = 1, 2, 3) with respect tox′ = (x1, x2) ∈ Pα1,α2 , respectively, with w′ = ⊤(w1, w2); and e′

1 =⊤(1, 0) ∈ R2.

We thus arrive at the following problem

∂tu+ Lξ′u = 0, u|t=0 = u0 (4.6)

with a parameter ξ′ = (ξ1, ξ2) ∈ R2, where Lξ′ is the operator on L2(0, 1) of theform

Lξ′ = Aξ′ + Bξ′ + C0

with domain

D(Lξ′) = u = ⊤(ϕ,w) ∈ L2(0, 1) : w ∈ H10 (0, 1), Lξ′u ∈ L2(0, 1).

Here

Aξ′ =

0 0 0

0 ν(|ξ′|2 − ∂2x3)I2 + νξ′⊤ξ′ −iνξ′∂x3

0 −iν⊤ξ′∂x3 ν(|ξ′|2 − ∂2x3)− ν∂2

x3

,

where I2 denotes the 2× 2 identity matrix,

Bξ′ =

iξ1v

1s iγ2 ⊤ξ′ γ2∂x3

iξ′ iξ1v1sI2 0

∂x3 0 iξ1v1s

,

8

and

C0 =

0 0 0

− νγ2e

′1 0 (∂x3v

1s)e

′1

0 0 0

.

Note that D(Lξ′) = D(L0) for all ξ′ ∈ R2.

5 Spectrum of −L0

To prove Theorem 3.1, we first consider the spectrum of −L0, i.e., −Lξ′ with ξ′ = 0:

L0 =

0 0 γ2∂x3

− νγ2e

′1 −ν∂2

x3I2 (∂x3v

1s)e

′1

∂x3 0 −(ν + ν)∂2x3

.

Let us introduce the adjoint operator L∗ξ′ of Lξ′ with respect to the inner product

⟨·, ·⟩:L∗

ξ′ = Aξ′ − Bξ′ + C∗0 ,

where

C∗0 =

0 −ν⊤e′

1 0

0 0 0

0 (∂x3v1s)

⊤e′1 0

.

We consider L∗ξ′ as an operator on L2(0, 1) with domain

D(L∗ξ′) = u = ⊤(ϕ,w) ∈ L2(0, 1) : w ∈ H1

0 (0, 1), L∗ξ′u ∈ L2(0, 1).

Note that

L∗0 =

0 −ν⊤e′

1 −γ2∂x3

0 −ν∂2x3I2 0

−∂x3 (∂x3v1s)

⊤e′1 −(ν + ν)∂2

x3

.

In this paper we only consider the spectrum near the origin since we focus onthe instability of plane Poiseuille flow.

Lemma 5.1. The following assertions hold true.

(i) There is a positive number η1 = η1(ν, ν, γ) such that λ ∈ C : |λ| < η1\0 ⊂ρ(−L0). Furthermore, the following estimate holds uniformly for λ ∈ λ ∈ C : |λ| ≤η1/2 \ 0: ∣∣(λ+ L0)

−1f∣∣L2 +

∣∣∂x3Q(λ+ L0)−1f∣∣2≤ C

|λ||f |L2 .

9

The same assertion holds with L0 replaced by L∗0.

(ii) λ = 0 is a simple eigenvalue of −L0, i.e., R(L0) is closed and

L2(0, 1) = Ker L0 ⊕R(L0) with dimKer L0 = 1.

The same assertion holds with L0 replaced by L∗0.

(iii) The eigenspaces for λ = 0 of L0 and L∗0 are spanned by u(0) and u(0)∗

respectively, where

u(0) = ⊤(ϕ(0), w(0)), w(0) = ⊤(w(0),1, 0, 0)

andu(0)∗ = ⊤(ϕ(0)∗, w(0)∗), w(0)∗ = ⊤(0, 0, 0)

with

ϕ(0)(x3) = 1, w(0),1(x3) =1

2γ2(−x2

3 + x3), ϕ(0)∗(x3) = γ2.

(iv) The eigenprojections Π (0) and Π (0)∗ for λ = 0 of −L0 and −L∗0 are given by

Π (0)u = ⟨u, u(0)∗⟩u(0) = ⟨ϕ⟩u(0),

andΠ (0)∗u = ⟨u, u(0)⟩u(0)∗

for u = ⊤(ϕ,w), respectively. In particular, it holds that

u = ⊤(ϕ,w) ∈ R(I − Π (0)) if and only if ⟨ϕ⟩ = ⟨u, u(0)∗⟩ = 0.

To prove Lemma 5.1, we introduce some operators. We define 2 × 2 matrixoperators L0, L

∗0 on L2(0, 1)2 = L2(0, 1)× L2(0, 1), and A, C0, C

∗0 on L2(0, 1)2 by

L0 =

(0 γ2∂x3

∂x3 −(ν + ν)∂2x3

),

L∗0 =

(0 −γ2∂x3

−∂x3 −(ν + ν)∂2x3

)with domain

D(L0) = u = ⊤(ϕ,w3) ∈ L2(0, 1)2 : w ∈ H10 (0, 1), L0u ∈ L2(0, 1)2,

D(L∗0) = u = ⊤(ϕ,w3) ∈ L2(0, 1)2 : w ∈ H1

0 (0, 1), L∗0u ∈ L2(0, 1)2,

and

A =

( −ν∂2x3

0

0 −ν∂2x3

)

10

with domain D(A) = [H2(0, 1) ∩H10 (0, 1)]

2, and

C0 =

( − νγ2 (∂x3v

1s)

0 0

),

C∗0 =

( −ν 0

(∂x3v1s) 0

)with domain D(C0) = D(C∗

0) = L2(0, 1)2.

Lemma 5.2. (i) It holds that

σ(−L0) ∩ λ ∈ C : |λ| < η0 = 0, λ ∈ C : |λ| < νπ ⊂ ρ(−A),

for some constant η0 = η0(ν, ν, γ2) > 0. Furthermore,

L2(0, 1)2 = Ker L0 ⊕R(L0)

withKer L0 = span u(0), u(0) = ⊤(1, 0),

R(L0) = L2∗(0, 1)× L2(0, 1).

In particular, 0 is a simple eigenvalue of −L0 with eigenprojection Π0 given byΠ0u = ⟨ϕ⟩u(0) (u = ⊤(ϕ,w3)).

(ii) There hold the estimates∣∣∣(λ+ L0)−1g∣∣∣L2

+∣∣∣∂x3Q2(λ+ L0)

−1g∣∣∣L2

≤ C( 1

|λ|+

1

η0 − |λ|)|g|L2

uniformly for λ ∈ λ ∈ C : |λ| < η0 \ 0 and g = ⊤(f 0, f 3) ∈ L2(0, 1)2, and∣∣∂lx3(λ+ A)−1f ′∣∣

L2 ≤1

νπ2 − |λ||f ′|L2 , l = 0, 1.

uniformly for λ ∈ λ ∈ C : |λ| < νπ2 and f ′ ∈ L2(0, 1)2. Here Q2 = diag (0, 1).

(iii) The assertions in (i) and (ii) also hold with L0 replaced by L∗0.

Proof. The assertions for A is well-known, and so we here omit the proof for A.As for L0, let us consider the problem to find u satisfying

L0u = g, u = ⊤(ϕ,w3) ∈ D(L0) (5.1)

for a given g = ⊤(f 0, f 3) ∈ L2(0, 1)2.To solve this problem, we expand ϕ and w3 into the Fourier cosine and sine series

respectively:

ϕ =∞∑n=0

ϕn cosnπx3, w3 =∞∑n=1

w3n sinnπx3,

11

and likewise,

f 0 =∞∑n=0

f 0n cosnπx3, f 3 =

∞∑n=1

f 3n sinnπx3.

It then follows from (5.1) thatf 00 = 0,

and, for n ≥ 1,

w3n =

1

γ2

1

nπf 0n,

ϕn =ν + ν

γ2f 0n − 1

nπf 3n.

We thus see that problem (5.1) is uniquely solvable if and only if ⟨f 0⟩ = f 00 = 0 and

⟨ϕ⟩ = ϕ0 = 0; and in this case, the unique solution is given by

ϕ =ν + ν

γ2f 0 + F 3, w3 =

1

γ2

∫ x3

0

f 0(y) dy, (5.2)

where F 3 = −∑∞

n=11nπf 3n cosnπx3. Furthermore, it holds that

|u|L2 ≤ η−10 |g|L2 , |∂x3w

3|L2 ≤ 1

γ2|f 0|L2 , (5.3)

where η0 =[maxν+ν

γ2 , 1γ2π

, 1π]−1

. Therefore, we see that R(L0) = L2∗(0, 1) ×

L2(0, 1). Moreover, we find that Ker L0 = span u(0) with u(0) = ⊤(1, 0) andL2(0, 1)2 = Ker L0 ⊕ R(L0). It then follows that 0 is a simple eigenvalue of −L0

and the eigenspace for 0 is spanned by u(0). The eigenprojection Π0 for 0 is givenby Π0u = ⟨ϕ⟩u(0) for u = ⊤(ϕ,w3) ∈ L2(0, 1).

We decompose L2(0, 1)2 as L2(0, 1)2 = X0 ⊕ X1, where X0 = Π0L2(0, 1)2 =

Ker L0 and X1 = Π1L2(0, 1)2 = R(L0) with Π1 = I − Π0.

Let us consider the resolvent problem

λu+ L0u = g, u ∈ D(L0). (5.4)

This is equivalent toλu = g, u ∈ X0 (5.5)

for g ∈ X0 andλu+ L0u = g, u ∈ X1 ∩D(L0) (5.6)

for g ∈ X1.We see from (5.3) that if |λ| < η0, then λ ∈ ρ(−L0|X1) and

|((λ+ L0)|X1)−1g|L2 ≤ 1

η0 − |λ||g|L2 ,

|∂x3Q2((λ+ L0)|X1)−1g|L2 ≤ 1

γ2

η0η0 − |λ|

|g|L2

12

for |λ| < η0 and g ∈ X1. On the other hand, it follows from (5.5) that if λ = 0, then

Π0u =1

λΠ0g.

As a result, we see that

(λ+ L0)−1g =

1

λΠ0g + ((λ+ L0)|X1)

−1Π1g

for λ = 0 with |λ| < η0 and g ∈ L2(0, 1)2. The desired estimate for L0 now follows.The assertion for L∗

0 can be shown in a similar manner. This completes the proof.

We now give a proof of Lemma 5.1.

Proof of Lemma 5.1. For u = ⊤(ϕ,w), w = ⊤(w1, w2, w3), we write

u = ⊤(ϕ,w3), w′ = ⊤(w1, w2).

Then the resolvent problem(λ+ L0)u = f (5.7)

is written as

(λ+ L0)u = g, (5.8)

(λ+ A)w′ = f ′ − C0u. (5.9)

Here f = ⊤(f 0, f 1, f 2, f 3), g = ⊤(f 0, f 3) and f ′ = ⊤(f 1, f 2).Set η1 = minη0, νπ2. Since |C0u|L2 ≤ C|u|L2 , we see from Lemma 5.2 that

λ ∈ C : |λ| < η1\0 ⊂ ρ(−L0). Furthermore, if λ ∈ λ ∈ C : |λ| < η1\0, thenthe u- and w′-components of u = (λ+L0)

−1f are given by u = ⊤(ϕ,w3) = (λ+L0)−1g

and w′ = ⊤(w1, w2) = (λ+ A)−1[f ′ − C0u], respectively, and it holds that

|u|L2 +∣∣∂x3w

3∣∣L2 ≤ C

( 1

|λ|+

1

η0 − |λ|)|g|L2

and ∣∣∂lx3w′∣∣

L2 ≤C

(νπ2 − |λ|)

(|f ′|L2 +

( 1

|λ|+

1

η0 − |λ|)|g|L2

), l = 0, 1.

We thus find that if |λ| ≤ η12, then∣∣∣(λ+ L0)

−1f∣∣∣L2

+∣∣∣∂x3Q(λ+ L0)

−1f∣∣∣L2

≤ C

|λ||f |L2 .

This proves (i) for L0.We next prove assertions (ii)–(iv) for L0. We first prove Ker L0 = span u0.

Let L0u = 0. Then L0u = 0 and Aw′ = −C0u. It follows from Lemma 5.2 thatu = αu(0) (α: constant) and, hence, Ker L0 = span u0.

13

Let us show that R(L0) is closed and L2(0, 1) = Ker L0 ⊕ R(L0). Set Π (0)u =⟨u, u(0)∗⟩u(0) = ⟨ϕ⟩u(0) for u = ⊤(ϕ,w) ∈ L2(0, 1). It then follows that Π (0) is aprojection onto Ker L0 with property Π (0)L0 ⊂ L0Π

(0). Furthermore, it holds thatf = ⊤(f 0, f ′, f 3) ∈ R(I − Π (0)) if and only if ⟨f, u(0)∗⟩ = ⟨f 0⟩ = 0.

Let ⟨f, u(0)∗⟩ = ⟨f 0⟩ = 0. Then (5.7) with λ = 0 is written in the form of(5.8)–(5.9) with λ = 0 and g = ⊤(f 0, f 3) ∈ L2

∗(0, 1)× L2(0, 1). It then follows fromLemma 5.2 that (5.8) with λ = 0 has a unique solution u ∈ D(L0) and, hence, (5.9)with λ = 0 has a unique solution w′ ∈ D(A). As a result, (5.7) with λ = 0 has aunique solution u = ⊤(ϕ,w′, w3) that is given by

u = ⊤(ϕ,w3) = (L0|X1)−1g, g = ⊤(f 0, f 3),

w′ = A−1[f ′ − C0u].

We thus find that R(I − Π (0)) ⊂ R(L0). On the other hand, if f = ⊤(f 0, f ′, f 3) ∈R(L0), then it is easy to see that ⟨f, u(0)∗⟩ = ⟨f 0⟩ = 0, and, hence, f ∈ R(I −Π (0)).We thus find that R(I − Π (0)) = R(L0). Consequently, we see that R(L0) is closedand L2(0, 1) = Ker L0 ⊕R(L0). This proves (ii)–(iv) for L0.

The assertions for L∗0 can be obtained similarly and we omit the details. This

completes the proof.

6 Perturbation argument

In this section we investigate σ(−Lξ′) ∩ |λ| ≤ η1/2 for |ξ′| ≪ 1.

Let Lξ′ be denoted by

Lξ = L0 +2∑

j=1

ξjL(1)j +

2∑j,k=1

ξjξkL(2)jk ,

where

L0 =

0 0 γ2∂x3

− νγ2e

′1 −ν∂2

x3I2 (∂x3v

1s)e

′1

∂x3 0 −(ν + ν)∂2x3

,

L(1)j =

iδ1jv

1s iγ2⊤e′

j 0

ie′j iv1sδ1jI2 −iνe′

j∂x3

0 −iν⊤e′j∂x3 iv1sδ1j

, j = 1, 2,

L(2)jk =

0 0 0

0 νδjkI2 + νe′j⊤e′

k 0

0 0 νδjk

, j, k = 1, 2.

14

Here e′1 =

⊤(1, 0) and e′2 =

⊤(0, 1).

We will apply the analytic perturbation theory to prove Theorem 3.1. To do so,we prepare the following estimates.

Lemma 6.1. There hold the following estimates uniformly for λ with |λ| = η12and

f ∈ L2(0, 1): ∣∣∣L(1)j (λ+ L0)

−1f∣∣∣L2

≤ C|f |L2 , j = 1, 2,∣∣∣L(2)jk (λ+ L0)

−1f∣∣∣L2

≤ C|f |L2 , j, k = 1, 2.

Proof. Let λ satisfy |λ| = η12. It then follows from Lemma 5.1 that∣∣∣L(1)

j (λ+ L0)−1f∣∣∣L2

≤ C∣∣∣(λ+ L0)

−1f∣∣∣L2×H1

≤ C|f |L2

and ∣∣∣L(2)jk (λ+ L0)

−1f∣∣∣L2

≤ C∣∣∣(λ+ L0)

−1f∣∣∣L2

≤ C|f |L2 .

This completes the proof.

Theorem 3.1 follows from the following result on the spectrum of −Lξ′ .

Theorem 6.2. There exists a positive number r1 = r1(ν, ν, γ) such that if |ξ′| ≤ r1,then it holds

σ(−Lξ′) ∩λ ∈ C : |λ| ≤ η0

2

= λξ′,

where λξ′ is a simple eigenvalue of −Lξ′ and it satisfies

λξ′ = − i

6ξ1 + κ0ξ

21 −

γ2

12νξ22 +O(|ξ′|3)

as ξ′ → 0. Here

κ0 =1

12ν

(1

280− γ2 − ν2

15γ2− νν

30γ2

).

Therefore, if γ2 < 1280

and 2ν2 + νν ≤ 30γ2(

1280

− γ2), then κ0 > 0.

Proof of Theorem 6.2. Based on Lemma 5.1 and Lemma 6.1 we can apply theanalytic perturbation theory (see, e.g., [4, Chap VII], [5, Chap. XII]) to see that if|ξ′| ≪ 1, then

σ(−Lξ′) ∩λ ∈ C : |λ| ≤ η1

2

= λξ′,

where λξ′ is a simple eigenvalue. Furthermore, λξ′ is given by

λξ′ = λ0 +2∑

j=1

ξjλ(1)j +

2∑j,k=1

ξjξkλ(2)jk +O(|ξ′|3).

15

Here λ(2)jk = λ

(2)kj ,

λ0 = 0,

λ(1)j = −⟨L(1)

j u(0), u(0)∗⟩,

λ(2)jk = −⟨1

2(L

(2)jk + L

(2)kj )u

(0), u(0)∗⟩+ ⟨12(L

(1)j SL

(1)k + L

(1)k SL

(1)j )u(0), u(0)∗⟩,

where S = [(I − Π (0))L0(I − Π (0))]−1.

The proof of Theorem 6.2 will be completed if we compute λ(1)j and λ

(2)jk . We will

compute them in the following propositions.

Proposition 6.3. λ(1)j = − i

6δ1j, j = 1, 2.

Proof. We have

L(1)1 u(0) = i

v1s + γ2w(0),1

(1 + v1sw(0),1)e′

1

−ν∂x3w(0),1

= i

−x23 + x3

1 + 14γ2 (−x2

3 + x3)2

0

− ν2γ2 (−2x3 + 1)

, (6.1)

L(1)2 u(0) = i

0

e′2

0

= i

0

0

1

0

. (6.2)

It then follows that

λ(1)1 = −⟨L(1)

1 u(0), u(0)∗⟩ = −i⟨v1s + γ2w(0),1⟩ = −i

∫ 1

0

(−y2 + y) dy = − i

6

and λ(1)2 = 0. This completes the proof.

Proposition 6.4. λ(2)22 = − γ2

12ν.

Proof. Since L(2)jk u

(0) = ⊤(0, ∗, ∗, ∗), we have

⟨L(2)jk u

(0), u(0)∗⟩ = 0 for j, k = 1, 2. (6.3)

Let us compute L(1)j SL

(1)2 u(0). We see from (6.2) that ⟨L(1)

2 u(0), u(0)∗⟩ = 0. There-

fore, L(1)2 u(0) ∈ R(I − Π (0)), and so, SL

(1)2 u(0) is a unique solution u = ⊤(ϕ,w) of

L0u = L(1)2 u(0), ⟨ϕ⟩ = 0.

16

By (6.2), we see that the solution u of this problem is given by ϕ = w1 = w3 = 0and w2 that satisfies −ν∂2

x3w2 = i and w2|x3=0,1 = 0. We thus obtain

SL(1)2 u(0) = ⊤(0, 0,

i

2ν(−x2

3 + x3), 0).

This implies thatL(1)1 SL

(1)2 u(0) = 0 (6.4)

and

L(1)2 SL

(1)2 u(0) = ⊤(−γ2

2ν(−x2

3 + x3), ∗, ∗, ∗). (6.5)

It then follows from (6.3) and (6.5) that

λ(2)22 = −γ2

2ν⟨ − x2

3 + x3⟩ = − γ2

12ν.

This completes the proof.

To obtain λ(2)j1 (= λ

(2)1j ), we compute SL

(1)1 u(0).

Proposition 6.5. SL(1)1 u(0) is given by

SL(1)1 u(0) = u

(1)1 ,

where u(1)1 = ⊤(ϕ

(1)1 , w

(1),11 , w

(1),21 , w

(1),31 ) with

ϕ(1)1 (x3) = i

(νγ2 +

ν2γ2

) (−x2

3 + x3 − 16

),

w(1),11 (x3) = i

(νγ4 +

ν2γ4

) (112x43 − 1

6x33 +

112x23

)+ i

12νγ2

(130x63 − 1

10x53 +

112x43 − 1

60x3

)+ i

2ν(−x2

3 + x3),

w(1),21 (x3) = 0,

w(1),31 (x3) = i

γ2

(−1

3x33 +

12x23 − 1

6x3

).

Proof. We set f = ⊤(f 0, f 1, f 2, f 3) = (I− Π (0))L(1)1 u(0). Then SL

(1)1 u(0) is a unique

solution u ofL0u = f, ⟨ϕ⟩ = 0,

namely, u = ⊤(ϕ,w1, w2, w3) is a solution of

γ2∂x3w3 = f 0, (6.6)

− ν

γ2ϕ− ν∂2

x3w1 + (∂x3v

1s)w

3 = f 1, (6.7)

−ν∂2x3w2 = f 2, (6.8)

17

∂x3ϕ− (ν + ν)∂2x3w3 = f 3, (6.9)

w|x3=0,1 = 0, (6.10)

⟨ϕ⟩ = 0. (6.11)

To solve (6.6)–(6.11), let us first compute f . Since

Π (0)L(1)1 u(0) = ⟨L(1)

1 u(0), u(0)∗⟩u(0) = −λ(1)1 u(0) =

i

6u(0),

we havef = L

(1)1 u(0) − i

6u(0)

= i

v1s + γ2w(0),1 − 16ϕ(0)

1 + v1sw(0),1 − 1

6w(0),1

0

−ν∂x3w(0),1

= i

−x23 + x3 − 1

6

112γ2 (3x

43 − 6x3

3 + 4x23 − x3) + 1

0

ν2γ2 (2x3 − 1)

.

(6.12)

Computation of w2: It follows from (6.8), (6.10) and (6.12) that w2 = 0.

Computation of w3: Integrating (6.6), we have

w3(x3) =

∫ x3

0

f 0(y) dy =i

γ2

(−1

3x33 +

1

2x23 −

1

6x3

). (6.13)

Note that this w3 also satisfies (6.10) since ⟨f 0⟩ = 0.

Computation of ϕ: We see from (6.9)–(6.11), (6.12) and (6.13) that

ϕ(x3) = i

(ν

γ2+

ν

2γ2

)(−x2

3 + x3 −1

6

). (6.14)

Computation of w1: From (6.7), we have

∂2x3w1 = − 1

γ2ϕ+

1

ν(∂x3v

1s)w

3 − 1

νf 1,

which, together with (6.12)–(6.14), gives

w1(x3) = c0 + c1x3 − i(

νγ4 +

ν2γ4

) (− 1

12x43 +

16x33 − 1

12x23

)+ i

12νγ2

(130x63 − 1

10x53 +

112x43

)− i

2νx23.

18

Here c0 and c1 are some constants. Since w1(0) = w1(1) = 0, we see that c0 = 0 andc1 = − i

12νγ2 · 160

+ i2ν. We thus find that

w1(x3) = −i(

νγ4 +

ν2γ4

) (− 1

12x43 +

16x33 − 1

12x23

)+ i

12νγ2

(130x63 − 1

10x53 +

112x43 − 1

60x3

)+ i

2ν(−x2

3 + x3).(6.15)

This completes the proof.

Proposition 6.6. λ(2)j1 = λ

(2)1j = κ0δj1, j = 1, 2. Here κ0 is given by

κ0 =1

12ν

(1

280− γ2 − ν2

15γ2− νν

30γ2

).

Proof. Since w(1),21 = 0 by Proposition 6.5, we have L

(1)2 SL

(1)1 u(0) = ⊤(0, ∗, ∗, ∗).

It then follows that ⟨L(1)2 SL

(1)1 u(0), u(0)∗⟩ = 0. This, together with (6.3) and (6.4),

implies λ(2)21 = λ

(2)12 = 0.

We next compute λ(2)11 . Since

v1sϕ(1)1 + γ2w

(1),11 = i

12

(νγ2 +

ν2γ2

)(7x4

3 − 14x33 + 8x2

3 − x3)

+ i12ν

(130x63 − 1

10x53 +

112x43 − 1

60x3

)+ iγ2

2ν(−x2

3 + x3),

we haveλ(2)11 = ⟨L(1)

1 SL(1)1 u(0), u(0)∗⟩

= i⟨v1sϕ(1)1 + γ2w

(1),11 ⟩

= − 112

(νγ2 +

ν2γ2

)· 115

+ 112ν

· 1280

− γ2

2ν· 16= κ0.

This completes the proof.

The proof of Theorem 6.2 is now completed in view of Propositions 6.3, 6.4 and6.6.

Acknowledgements. Y. Kagei was partly supported by JSPS KAKENHI GrantNumber 24340028, 22244009, 24224003.

References

[1] G. Iooss and M. Padula, Structure of the linearized problem for compressibleparallel fluid flows, Ann. Univ. Ferrara, Sez. VII, 43 (1998), pp. 157–171.

[2] Y. Kagei, Asymptotic behavior of solutions to the compressible Navier-Stokesequation around a parallel flow, Arch. Rational Mech. Anal., 205 (2012), pp.585–650.

19

[3] Y. Kagei, Y. Nagafuchi and T. Sudo, Decay estimates on solutions of the lin-earized compressible Navier-Stokes equation around a Poiseuille type flow, J.Math-for-Ind., 2A (2010), pp. 39–56. Correction to ”Decay estimates on solu-tions of the linearized compressible Navier-Stokes equation around a Poiseuilletype flow” in J. Math-for-Ind. 2A (2010), pp. 39–56 J. Math-for-Ind., 2B (2010),pp. 235.

[4] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin,Heidelberg, New York (1980).

[5] M. Reed and B. Simon, Methods of modern mathematical physics IV, AcademicPress (1979).

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MI2009-18 Me Me NAING & Yasuhide FUKUMOTOLocal Instability of an Elliptical Flow Subjected to a Coriolis Force

MI2009-19 Mitsunori KAYANO & Sadanori KONISHISparse functional principal component analysis via regularized basis expansions andits application

MI2009-20 Shuichi KAWANO & Sadanori KONISHISemi-supervised logistic discrimination via regularized Gaussian basis expansions

MI2009-21 Hiroshi YOSHIDA, Yoshihiro MIWA & Masanobu KANEKOElliptic curves and Fibonacci numbers arising from Lindenmayer system with sym-bolic computations

MI2009-22 Eiji ONODERAA remark on the global existence of a third order dispersive flow into locally Hermi-tian symmetric spaces

MI2009-23 Stjepan LUGOMER & Yasuhide FUKUMOTOGeneration of ribbons, helicoids and complex scherk surface in laser-matter Interac-tions

MI2009-24 Yu KAWAKAMIRecent progress in value distribution of the hyperbolic Gauss map

MI2009-25 Takehiko KINOSHITA & Mitsuhiro T. NAKAOOn very accurate enclosure of the optimal constant in the a priori error estimatesfor H2

0 -projection

MI2009-26 Manabu YOSHIDARamification of local fields and Fontaine’s property (Pm)

MI2009-27 Yu KAWAKAMIValue distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

MI2009-28 Masahisa TABATANumerical simulation of fluid movement in an hourglass by an energy-stable finiteelement scheme

MI2009-29 Yoshiyuki KAGEI & Yasunori MAEKAWAAsymptotic behaviors of solutions to evolution equations in the presence of transla-tion and scaling invariance

MI2009-30 Yoshiyuki KAGEI & Yasunori MAEKAWAOn asymptotic behaviors of solutions to parabolic systems modelling chemotaxis

MI2009-31 Masato WAKAYAMA & Yoshinori YAMASAKIHecke’s zeros and higher depth determinants

MI2009-32 Olivier PIRONNEAU & Masahisa TABATAStability and convergence of a Galerkin-characteristics finite element scheme oflumped mass type

MI2009-33 Chikashi ARITAQueueing process with excluded-volume effect

MI2009-34 Kenji KAJIWARA, Nobutaka NAKAZONO & Teruhisa TSUDAProjective reduction of the discrete Painleve system of type(A2 +A1)

(1)

MI2009-35 Yosuke MIZUYAMA, Takamasa SHINDE, Masahisa TABATA & Daisuke TAGAMIFinite element computation for scattering problems of micro-hologram using DtNmap

MI2009-36 Reiichiro KAWAI & Hiroki MASUDAExact simulation of finite variation tempered stable Ornstein-Uhlenbeck processes

MI2009-37 Hiroki MASUDAOn statistical aspects in calibrating a geometric skewed stable asset price model

MI2010-1 Hiroki MASUDAApproximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

MI2010-2 Reiichiro KAWAI & Hiroki MASUDAInfinite variation tempered stable Ornstein-Uhlenbeck processes with discrete obser-vations

MI2010-3 Kei HIROSE, Shuichi KAWANO, Daisuke MIIKE & Sadanori KONISHIHyper-parameter selection in Bayesian structural equation models

MI2010-4 Nobuyuki IKEDA & Setsuo TANIGUCHIThe Ito-Nisio theorem, quadratic Wiener functionals, and 1-solitons

MI2010-5 Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling and detecting change point via the relevance vectormachine

MI2010-6 Shuichi KAWANO, Toshihiro MISUMI & Sadanori KONISHISemi-supervised logistic discrimination via graph-based regularization

MI2010-7 Teruhisa TSUDAUC hierarchy and monodromy preserving deformation

MI2010-8 Takahiro ITOAbstract collision systems on groups

MI2010-9 Hiroshi YOSHIDA, Kinji KIMURA, Naoki YOSHIDA, Junko TANAKA & YoshihiroMIWAAn algebraic approach to underdetermined experiments

MI2010-10 Kei HIROSE & Sadanori KONISHIVariable selection via the grouped weighted lasso for factor analysis models

MI2010-11 Katsusuke NABESHIMA & Hiroshi YOSHIDADerivation of specific conditions with Comprehensive Groebner Systems

MI2010-12 Yoshiyuki KAGEI, Yu NAGAFUCHI & Takeshi SUDOUDecay estimates on solutions of the linearized compressible Navier-Stokes equationaround a Poiseuille type flow

MI2010-13 Reiichiro KAWAI & Hiroki MASUDAOn simulation of tempered stable random variates

MI2010-14 Yoshiyasu OZEKINon-existence of certain Galois representations with a uniform tame inertia weight

MI2010-15 Me Me NAING & Yasuhide FUKUMOTOLocal Instability of a Rotating Flow Driven by Precession of Arbitrary Frequency

MI2010-16 Yu KAWAKAMI & Daisuke NAKAJOThe value distribution of the Gauss map of improper affine spheres

MI2010-17 Kazunori YASUTAKEOn the classification of rank 2 almost Fano bundles on projective space

MI2010-18 Toshimitsu TAKAESUScaling limits for the system of semi-relativistic particles coupled to a scalar bosefield

MI2010-19 Reiichiro KAWAI & Hiroki MASUDALocal asymptotic normality for normal inverse Gaussian Levy processes with high-frequency sampling

MI2010-20 Yasuhide FUKUMOTO, Makoto HIROTA & Youichi MIELagrangian approach to weakly nonlinear stability of an elliptical flow

MI2010-21 Hiroki MASUDAApproximate quadratic estimating function for discretely observed Levy driven SDEswith application to a noise normality test

MI2010-22 Toshimitsu TAKAESUA Generalized Scaling Limit and its Application to the Semi-Relativistic ParticlesSystem Coupled to a Bose Field with Removing Ultraviolet Cutoffs

MI2010-23 Takahiro ITO, Mitsuhiko FUJIO, Shuichi INOKUCHI & Yoshihiro MIZOGUCHIComposition, union and division of cellular automata on groups

MI2010-24 Toshimitsu TAKAESUA Hardy’s Uncertainty Principle Lemma inWeak Commutation Relations of Heisenberg-Lie Algebra

MI2010-25 Toshimitsu TAKAESUOn the Essential Self-Adjointness of Anti-Commutative Operators

MI2010-26 Reiichiro KAWAI & Hiroki MASUDAOn the local asymptotic behavior of the likelihood function for Meixner Levy pro-cesses under high-frequency sampling

MI2010-27 Chikashi ARITA & Daichi YANAGISAWAExclusive Queueing Process with Discrete Time

MI2010-28 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAMotion and Backlund transformations of discrete plane curves

MI2010-29 Takanori YASUDA, Masaya YASUDA, Takeshi SHIMOYAMA & Jun KOGUREOn the Number of the Pairing-friendly Curves

MI2010-30 Chikashi ARITA & Kohei MOTEGISpin-spin correlation functions of the q-VBS state of an integer spin model

MI2010-31 Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling and spike detection via Gaussian basis expansions

MI2010-32 Nobutaka NAKAZONOHypergeometric τ functions of the q-Painleve systems of type (A2 +A1)

(1)

MI2010-33 Yoshiyuki KAGEIGlobal existence of solutions to the compressible Navier-Stokes equation aroundparallel flows

MI2010-34 Nobushige KUROKAWA, Masato WAKAYAMA & Yoshinori YAMASAKIMilnor-Selberg zeta functions and zeta regularizations

MI2010-35 Kissani PERERA & Yoshihiro MIZOGUCHILaplacian energy of directed graphs and minimizing maximum outdegree algorithms

MI2010-36 Takanori YASUDACAP representations of inner forms of Sp(4) with respect to Klingen parabolic sub-group

MI2010-37 Chikashi ARITA & Andreas SCHADSCHNEIDERDynamical analysis of the exclusive queueing process

MI2011-1 Yasuhide FUKUMOTO& Alexander B. SAMOKHINSingular electromagnetic modes in an anisotropic medium

MI2011-2 Hiroki KONDO, Shingo SAITO & Setsuo TANIGUCHIAsymptotic tail dependence of the normal copula

MI2011-3 Takehiro HIROTSU, Hiroki KONDO, Shingo SAITO, Takuya SATO, Tatsushi TANAKA& Setsuo TANIGUCHIAnderson-Darling test and the Malliavin calculus

MI2011-4 Hiroshi INOUE, Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling via Compressed Sensing

MI2011-5 Hiroshi INOUEImplications in Compressed Sensing and the Restricted Isometry Property

MI2011-6 Daeju KIM & Sadanori KONISHIPredictive information criterion for nonlinear regression model based on basis ex-pansion methods

MI2011-7 Shohei TATEISHI, Chiaki KINJYO & Sadanori KONISHIGroup variable selection via relevance vector machine

MI2011-8 Jan BREZINA & Yoshiyuki KAGEIDecay properties of solutions to the linearized compressible Navier-Stokes equationaround time-periodic parallel flowGroup variable selection via relevance vector machine

MI2011-9 Chikashi ARITA, Arvind AYYER, Kirone MALLICK & Sylvain PROLHACRecursive structures in the multispecies TASEP

MI2011-10 Kazunori YASUTAKEOn projective space bundle with nef normalized tautological line bundle

MI2011-11 Hisashi ANDO, Mike HAY, Kenji KAJIWARA & Tetsu MASUDAAn explicit formula for the discrete power function associated with circle patternsof Schramm type

MI2011-12 Yoshiyuki KAGEIAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda parallel flow

MI2011-13 Vladimır CHALUPECKY & Adrian MUNTEANSemi-discrete finite difference multiscale scheme for a concrete corrosion model: ap-proximation estimates and convergence

MI2011-14 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAExplicit solutions to the semi-discrete modified KdV equation and motion of discreteplane curves

MI2011-15 Hiroshi INOUEA generalization of restricted isometry property and applications to compressed sens-ing

MI2011-16 Yu KAWAKAMIA ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolicthree-space

MI2011-17 Naoyuki KAMIYAMAMatroid intersection with priority constraints

MI2012-1 Kazufumi KIMOTO & Masato WAKAYAMASpectrum of non-commutative harmonic oscillators and residual modular forms

MI2012-2 Hiroki MASUDAMighty convergence of the Gaussian quasi-likelihood random fields for ergodic Levydriven SDE observed at high frequency

MI2012-3 Hiroshi INOUEA Weak RIP of theory of compressed sensing and LASSO

MI2012-4 Yasuhide FUKUMOTO & Youich MIEHamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

MI2012-5 Yu KAWAKAMIOn the maximal number of exceptional values of Gauss maps for various classes ofsurfaces

MI2012-6 Marcio GAMEIRO, Yasuaki HIRAOKA, Shunsuke IZUMI, Miroslav KRAMAR,Konstantin MISCHAIKOW & Vidit NANDATopological Measurement of Protein Compressibility via Persistence Diagrams

MI2012-7 Nobutaka NAKAZONO & Seiji NISHIOKA

Solutions to a q-analog of Painleve III equation of type D(1)7

MI2012-8 Naoyuki KAMIYAMAA new approach to the Pareto stable matching problem

MI2012-9 Jan BREZINA & Yoshiyuki KAGEISpectral properties of the linearized compressible Navier-Stokes equation aroundtime-periodic parallel flow

MI2012-10 Jan BREZINAAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda time-periodic parallel flow

MI2012-11 Daeju KIM, Shuichi KAWANO & Yoshiyuki NINOMIYAAdaptive basis expansion via the extended fused lasso

MI2012-12 Masato WAKAYAMAOn simplicity of the lowest eigenvalue of non-commutative harmonic oscillators

MI2012-13 Masatoshi OKITAOn the convergence rates for the compressibleNavier- Stokes equations with potential force

MI2013-1 Abuduwaili PAERHATI & Yasuhide FUKUMOTOA Counter-example to Thomson-Tait-Chetayev’s Theorem

MI2013-2 Yasuhide FUKUMOTO & Hirofumi SAKUMAA unified view of topological invariants of barotropic and baroclinic fluids and theirapplication to formal stability analysis of three-dimensional ideal gas flows

MI2013-3 Hiroki MASUDAAsymptotics for functionals of self-normalized residuals of discretely observed stochas-tic processes

MI2013-4 Naoyuki KAMIYAMAOn Counting Output Patterns of Logic Circuits

MI2013-5 Hiroshi INOUERIPless Theory for Compressed Sensing

MI2013-6 Hiroshi INOUEImproved bounds on Restricted isometry for compressed sensing

MI2013-7 Hidetoshi MATSUIVariable and boundary selection for functional data via multiclass logistic regressionmodeling

MI2013-8 Hidetoshi MATSUIVariable selection for varying coefficient models with the sparse regularization

MI2013-9 Naoyuki KAMIYAMAPacking Arborescences in Acyclic Temporal Networks

MI2013-10 Masato WAKAYAMAEquivalence between the eigenvalue problem of non-commutative harmonic oscilla-tors and existence of holomorphic solutions of Heun’s differential equations, eigen-states degeneration, and Rabi’s model

MI2013-11 Masatoshi　OKITAOptimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations

MI2013-12 Shuichi KAWANO, Ibuki HOSHINA, Kazuki MATSUDA & Sadanori KONISHIPredictive model selection criteria for Bayesian lasso

MI2013-13 Hayato CHIBAThe First Painleve Equation on the Weighted Projective Space

MI2013-14 Hidetoshi MATSUIVariable selection for functional linear models with functional predictors and a func-tional response

MI2013-15 Naoyuki KAMIYAMAThe Fault-Tolerant Facility Location Problem with Submodular Penalties

MI2013-16 Hidetoshi MATSUISelection of classification boundaries using the logistic regression

MI2014-1 Naoyuki KAMIYAMAPopular Matchings under Matroid Constraints

MI2014-2 Yasuhide FUKUMOTO & Youichi MIELagrangian approach to weakly nonlinear interaction of Kelvin waves and a symmetry-breaking bifurcation of a rotating flow

MI2014-3 Reika AOYAMADecay estimates on solutions of the linearized compressible Navier-Stokes equationaround a Parallel flow in a cylindrical domain

MI2014-4 Naoyuki KAMIYAMAThe Popular Condensation Problem under Matroid Constraints

MI2014-5 Yoshiyuki KAGEI & Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokesequation for time periodic external force with symmetry

MI2014-6 This paper was withdrawn by the authors.

MI2014-7 Masatoshi OKITAOn decay estimate of strong solutions in critical spaces for the compressible Navier-Stokes equations

MI2014-8 Rong ZOU & Yasuhide FUKUMOTOLocal stability analysis of azimuthal magnetorotational instability of ideal MHDflows

MI2014-9 Yoshiyuki KAGEI & Naoki MAKIOSpectral properties of the linearized semigroup of the compressible Navier-Stokesequation on a periodic layer

MI2014-10 Kazuyuki TSUDAOn the existence and stability of time periodic solution to the compressible Navier-Stokes equation on the whole space

MI2014-11 Yoshiyuki KAGEI & Takaaki NISHIDAInstability of plane Poiseuille flow in viscous compressible gas