MI Preprint SeriesMathematics for Industry

Kyushu University

On Chorin’s method for

stationary solutions of the

Oberbeck-Boussinesq equation

Yoshiyuki Kagei

& Takaaki Nishida

MI 2016-5

( Received April 22, 2016 )

Institute of Mathematics for IndustryGraduate School of Mathematics

Kyushu UniversityFukuoka, JAPAN

On Chorin’s method for stationary solutions of

the Oberbeck-Boussinesq equation

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

Stability of stationary solutions of the Oberbeck-Boussinesq system (OB) andthe corresponding artificial compressible system is considered. The lattersystem is obtained by adding the time derivative of the pressure with smallparameter ε > 0 to the continuity equation of (OB), which was proposed byA. Chorin to find stationary solutions of (OB) numerically. Both systemshave the same sets of stationary solutions and the system (OB) is obtainedfrom the artificial compressible one as the limit ε → 0 which is a singularlimit. It is proved that if a stationary solution of the artificial compressiblesystem is stable for sufficiently small ε > 0, then it is also stable as a solution of(OB). The converse is proved provided that the velocity field of the stationarysolution satisfies some smallness condition.

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

Keywords. Oberbeck-Boussinesq equation, artificial compressible system,Chorin’s method, singular perturbation, stability, cellular convection patterns.

1 Introduction

This paper is concerned with the Oberbeck-Boussinesq equation

divv = 0, (1.1)

Pr−1 (∂tv + v · ∇v)−∆v +∇p− Raθe3 = 0, (1.2)

∂tθ + v · ∇θ −∆θ − Rav · e3 = 0, (1.3)

1

and the artificial compressible system for (1.1)–(1.3):

ε2∂tp+ divv = 0, (1.4)

Pr−1 (∂tv + v · ∇v)−∆v +∇p− Raθe3 = 0, (1.5)

∂tθ + v · ∇θ −∆θ − Rav · e3 = 0. (1.6)

Here v = ⊤(v1(x, t), v2(x, t), v3(x, t)), p = p(x, t) and θ = θ(x, t) denote the unknownvelocity field, pressure and temperature deviation from the heat conductive state,respectively, at time t > 0 and position x ∈ R3; e3 = ⊤(0, 0, 1) ∈ R3; Pr > 0and Ra > 0 are non-dimensional parameters, called Prandtl and Rayleigh numbers,respectively; and ε > 0 is a small parameter, called artificial Mach number. Hereand in what follows, the superscript ⊤ · stands for the transposition. The systems(1.1)–(1.3) and (1.4)–(1.6) are considered in the infinite layer Ω:

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

The Oberbeck-Boussinesq equation (1.1)–(1.3) is a system of equations whichdescribes convection phenomena of viscous fluid occupying Ω heated from below(heated at x3 = 0) under the gravitational force. It is well known ([1, 6, 7, 10]) thatunder the boundary condition

v|x3=0,1 = 0, θ|x3=0,1 = 0, (1.7)

there exists a critical number Rac > 0 such that when Ra < Rac, the heat con-ductive state v = 0, θ = 0 is stable, while, when Ra > Rac, the heat conductivestate is unstable and convective cellular stationary solutions bifurcate from the heatconductive state.

A. Chorin ([2, 3, 4]) proposed the artificial compressible system such as (1.4)–(1.6) to find stationary solutions of equations for viscous incompressible fluid numer-ically. In the context of the Oberbeck-Boussinesq equation (1.1)–(1.3), the idea isstated as follows. Obviously, the sets of stationary solutions of (1.1)–(1.3) and (1.4)–(1.6) are the same ones. If solutions of the artificial compressible system (1.4)–(1.6)converge to a function us =

⊤(ps,vs, θs) as t → ∞, then the limit us is a stationarysolution of (1.4)–(1.6) which is thus a stationary solution of (1.1)–(1.3). By usingthis method, Chorin numerically obtained stationary cellular convection solutionsof (1.1)–(1.3).

Since the limit us in Chorin’s method described above is a large time limit ofsolutions of (1.4)–(1.6), us is stable as a solution of (1.4)–(1.6). It is of interest toconsider whether us is stable as a solution of (1.1)–(1.3), in other words, whether us

represents an observable stationary flow in the real world, and, conversely, what kindof stationary flows can be computed by Chorin’s method. These questions are to beformulated as stability problem for stationary solutions of the systems (1.1)–(1.3)and (1.4)–(1.6). Since the system (1.1)–(1.3) is obtained from (1.4)–(1.6) as thelimit ε → 0, one could expect that solutions of (1.1)–(1.3) would be approximatedby solutions of (1.4)–(1.6) when ε ≪ 1. However, this limiting process is a singularlimit, and hence, it is not straightforward to conclude that stability properties of us

2

as a solution of (1.1)–(1.3) are the same as those as a solution of (1.4)–(1.6) evenwhen 0 < ε ≪ 1.

The purpose of this paper is to investigate stability relations of stationary solu-tions between the systems (1.1)–(1.3) and (1.4)–(1.6) when ε is sufficiently small.We investigate the spectra of the linearized operators around a stationary solutionof (1.1)–(1.3) and (1.4)–(1.6) for 0 < ε ≪ 1.

We first show that if a stationary solution us = ⊤(ps,vs, θs) of (1.4)–(1.6) isasymptotically stable for sufficiently small ε, then so is us as a stationary solutionof (1.1)–(1.3). More precisely, we consider (1.1)–(1.3) and (1.4)–(1.6) under theboundary condition (1.7) and the periodicity condition

p, v and θ are Q-periodic in (x1, x2), (1.8)

where Q = [−π/α1, π/α1)×[−π/α2, π/α2). Here αj, j = 1, 2, are positive constants.We denote the basic period domain by Ωper = Q×(0, 1). We introduce the linearizedoperators around us associated with (1.1)–(1.3) and (1.4)–(1.6). Let L : L2

per,σ ×L2

per → L2per,σ × L2

per be the operator acting on U = ⊤(w, θ) ∈ D(L) defined by

L =

(−PrP∆+ P(vs · ∇+ ⊤(∇vs)) −PrRaPe3

⊤(∇θs)− Ra⊤e3 −∆+ vs · ∇

)with domain D(L) = [(H2

per ∩ H10,per)

3 ∩ L2per,σ] × [H2

per ∩ H10,per]. Here L2

per, Hkper,

· · · , denote L2, Hk, · · · spaces over Ωper with periodicity condition in x′, P denotesthe Helmholtz projection from (L2

per)3 to L2

per,σ, where L2per,σ denotes the set of

all solenoidal vector fields w = ⊤(w1, w2, w3) in (L2per)

3 with w3|x3=0,1 = 0. Wedefine the operator Lε : H

1per,∗ × (L2

per)3 × L2

per → H1per,∗ × (L2

per)3 × L2

per, acting onu = ⊤(p,w, θ) ∈ D(Lε), by

Lε =

0 1ε2div 0

Pr∇ −Pr∆ + vs · ∇+ ⊤(∇vs) −PrRae3

0 ⊤(∇θs)− Ra⊤e3 −∆+ vs · ∇

with domain D(Lε) = H1

per,∗ × [H2per ∩H1

0,per]3 × [H2

per ∩H10,per]. Here H1

per,∗ = p ∈H1

per;∫Ωper

p(x) dx = 0.We prove that if there exists a positive number b0 such that ρ(−Lεn) ⊃ λ ∈

C; Reλ ≥ −b0 for some sequence εn → 0 as n → ∞, then there exists a con-stant b1 > 0 such that ρ(−L) ⊃ λ ∈ C; Reλ ≥ −b1. Therefore, a stationarysolution obtained by Chorin’s method with 0 < ε ≪ 1 is stable as a solution ofthe Oberbeck-Boussinesq system (1.1)–(1.3). Furthermore, we prove an instabilityresult: if σ(−L) ∩ λ ∈ C; Reλ > 0 = ∅, then σ(−Lε) ∩ λ ∈ C; Reλ > 0 = ∅ for0 < ε ≪ 1. This shows that unstable stationary solutions of (1.1)–(1.3) cannot beobtained by Chorin’s method with 0 < ε ≪ 1.

As a converse of the above result, we prove that if

ρ(−L) ⊃ λ ∈ C; Reλ ≥ −b0 (1.9)

for some constant b0 > 0, then there exist constants δ0 > 0 and b1 > 0 such that

ρ(−Lε) ⊃ λ ∈ C; Reλ ≥ −b1 (1.10)

3

for 0 < ε ≪ 1, provided that

infw∈H10,per,w =0

Re (w · ∇vs,w)L2

∥∇w∥2L2

≥ −δ0. (1.11)

This gives a sufficient condition for us to be computed by Chorin’s method with0 < ε ≪ 1. We note that no smallness condition for the temperature θs of us isrequired.

This result is applicable to stable bifurcating cellular convective patterns such asroll pattern, hexagonal pattern and etc., when Ra ∼ Rac. In fact, the velocity fieldsof bifurcating convective patterns are small when Ra ∼ Rac since they bifurcatefrom v = 0, θ = 0 when Ra crosses Rac, and hence, condition (1.11) is satisfied.

To prove the first result, we show that an eigenvalue λ of −Lε with Imλ = O(1)can be obtained as a perturbation of an eigenvalue of −L as ε → 0. In fact, it isproved that if λ0 is an eigenvalue of −L, then, for any neighborhood of λ0, thereexists an eigenvalue of −Lε for 0 < ε ≪ 1 and the total projection Pε on theneighborhood satisfies Pε = P0 + O(ε), where P0 is a projection whose velocity-temperature part is the eigenprojection for the eigenvalue λ0 of −L (Theorem 5.2(ii)). Based on this observation we show the first result by a contradiction argument.In particular, we find that if −L has an eigenvalue with positive real part, then sodoes −Lε with 0 < ε ≪ 1, which gives the instability result.

To prove the converse, we investigate the resolvent (λ + Lε)−1 according to the

cases for λ near the imaginary axis with |Imλ| ≥ O(ε−1), |Imλ| ≤ O(ε−1) and|Imλ| = O(ε−1) under the condition (1.9). By a standard energy method one canshow that −Lε is a sectorial operator and λ; Reλ ≥ −b, |Imλ| ≥ O(ε−1) ⊂ ρ(−Lε)with some b > 0 for 0 < ε ≪ 1. We show that the spectrum of −Lε near theimaginary axis with |Imλ| ≤ O(ε−1) can be treated as a kind of regular perturbationof that of −L, which concludes that λ; Reλ ≥ −b, |Imλ| ≤ O(ε−1) ⊂ ρ(−Lε).The spectrum of −Lε in the region |Imλ| = O(ε−1) stems from the “compressible”aspect of −Lε and does not appear in the spectrum of −L. We prove λ; Reλ ≥−b, |Imλ| = O(ε−1) ⊂ ρ(−Lε) by an energy method provided that the condition(1.11) holds.

One more remark is added in order. Due to the translation invariance in x1 andx2 variables, 0 is an eigenvalue of −Lε whenever ∂x1us = 0 or ∂x2us = 0. In thiscase nonzero ∂xj

us are eigenfunctions for the eigenvalue 0. Correspondingly, 0 is aneigenvalue of −L whenever ∂x1U s = 0 or ∂x2U s = 0. We formulate the main resultsof this paper by taking this translation invariance into account. We note that underthe translation invariance one can show the orbital stability of U s and us for thecorresponding nonlinear problems.

This paper is organized as follows. In section 2 we introduce notations used inthis paper. In section 3 we state the main theorems of this paper. Section 4 isdevoted to the study of the null spaces of L and Lε for 0 < ε ≪ 1. We then provethe first results in section 5. The converse is proved in section 6; and we give theestimates for the derivatives of the pressure component of (λ+ Lε)

−1 in section 7.

4

2 Preliminaries

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

Q =[− π

α1, πα1

)×[− π

α2, πα2

).

We denote the basic period domain by Ωper:

Ωper = Q× (0, 1).

We denote by C∞per the space of restrictions of functions in C∞(Ω) which are

Q-periodic in x′ = (x1, x2). We also denote by C∞0,per the space of restrictions of

functions in C∞ which are Q-periodic in x′ = (x1, x2) and vanish near x3 = 0, 1.For 1 ≤ r ≤ ∞ we denote by Lr(Ωper) the usual Lebesgue space over Ωper, and itsnorm is denoted by ∥ · ∥r. The k th order L2 Sobolev space over Ωper is denoted byHk(Ωper), and its norm is denoted by ∥ · ∥Hk .

We setL2

per = the L2(Ωper)-closure of C∞0,per,

Hkper = the Hk(Ωper)-closure of C∞

per,

H10,per = the H1(Ωper)-closure of C∞

0,per.

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

= f |xj=π/αjand f |x3=0,1 = 0. The inner

product of fj ∈ L2per (j = 1, 2) is denoted by

(f1, f2) =

∫Ωper

f1(x)f2(x) dx,

where z denotes the complex conjugate of z.The mean value of a function ϕ(x) over Ωper is denoted by ⟨ϕ⟩:

⟨ϕ⟩ = 1

|Ωper|

∫Ωper

ϕ(x) dx.

The set of all ϕ ∈ L2per with ⟨ϕ⟩ = 0 is denoted by L2

per,∗, i.e.,

L2per,∗ = ϕ ∈ L2

per : ⟨ϕ⟩ = 0.

Furthermore, we setHk

per,∗ = Hkper ∩ L2

per,∗.

We denote by C∞0,per,σ the set of all vector fields v in (C∞

0,per)3 with divv = 0. We

setL2per,σ = the L2(Ωper)

3-closure of C∞0,per,σ.

It is known that (L2per)

3 = L2per,σ ⊕ G2

per, where G2per = ∇p; p ∈ H1

per is theorthogonal complement of L2

per,σ. The orthogonal projection P on L2per,σ is called

5

the Helmholtz projection. We define the projection P from (L2per)

3 × L2per onto

L2per,σ × L2

per by

P =

(P 00 I

).

For simplicity the set of all vector fields in (L2per)

3 (resp. (H10,per)

3, (Hkper)

3) arefrequently denoted by L2

per (resp. H10,per, H

kper) if no confusion will occur.

We also use notation L2per for the set of all u = ⊤(p,w, θ) with p ∈ L2

per, w =⊤(w1, w2, w3) ∈ L2

per and θ ∈ L2per if no confusion will occur.

Let ε be a positive number. We introduce an inner product ⟨⟨u1, u2⟩⟩ε for uj =⊤(pj,wj, θj) (j = 1, 2) defined by

⟨⟨u1, u2⟩⟩ε = ε2(p1, p2) + Pr−1(w1,w2) + (θ1, θ2).

We also define the inner product ⟨U 1,U 2⟩ for U j =⊤(wj, θj) (j = 1, 2) by

⟨U 1,U 2⟩ = Pr−1(w1,w2) + (θ1, θ2).

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 Ker (A) and R(A),respectively.

3 Main Results

In this section we state the main results of this paper. Let us = ⊤(ps,vs, θs) bea stationary solution of (1.1)–(1.3) under the boundary conditions (1.7) and (1.8)satisfying

∫Ωper

ps(x) dx = 0. We consider the linearized problem for the Oberbeck-

Boussinesq system (1.1)–(1.3) around us =⊤(ps,vs, θs):

divw = 0, (3.1)

Pr−1∂tw −∆w + Pr−1(vs · ∇w +w · ∇vs) +∇p− Raθe3 = 0, (3.2)

∂tθ −∆θ + vs · ∇θ +w · ∇θs − Raw · e3 = 0 (3.3)

under the boundary condition

w|x3=0,1 = 0, θ|x3=0,1 = 0, (3.4)

and the periodicity condition

p, w and θ are Q-periodic in (x1, x2). (3.5)

The linearized problem for the artificial compressible system is written as

ε2∂tp+ divw = 0, (3.6)

Pr−1∂tw −∆w + Pr−1(vs · ∇w +w · ∇vs) +∇p− Raθe3 = 0, (3.7)

∂tθ −∆θ + vs · ∇θ +w · ∇θs − Raw · e3 = 0 (3.8)

6

with the boundary conditions (3.4) and (3.5).By applying the projection P , problem (3.1)–(3.5) is written as

Pr−1∂tw − P∆w + Pr−1P(vs · ∇w +w · ∇vs)− RaPθe3 = 0, (3.9)

∂tθ −∆θ + vs · ∇θ +w · ∇θs − Raw · e3 = 0 (3.10)

with the boundary conditions (3.4) and (3.5). We introduce the linearized operatoraround U s =

⊤(vs, θs) associated with problem (3.9)–(3.10) under (3.4) and (3.5).We define the operator L : L2

per,σ × L2per → L2

per,σ × L2per by

L =

(−PrP∆+ P(vs · ∇+ ⊤(∇vs)) −PrRaPe3

⊤(∇θs)− Ra⊤e3 −∆+ vs · ∇

)with domain D(L) = [(H2

per ∩H10,per)

3 ∩ L2per,σ]× [H2

per ∩H10,per].

We also introduce the linearized operator around us = ⊤(ps,ws, θs) associatedwith (3.6)–(3.8) under (3.4) and (3.5). We define the operator Lε : H

1per,∗× (L2

per)3×

L2per → H1

per,∗ × (L2per)

3 × L2per by

Lε =

0 1ε2div 0

Pr∇ −Pr∆ + vs · ∇+ ⊤(∇vs) −PrRae3

0 ⊤(∇θs)− Ra⊤e3 −∆+ vs · ∇

with domain D(Lε) = H1

per,∗ × [H2per ∩H1

0,per]3 × [H2

per ∩H10,per].

Before going further we make one observation. Due to the translation invariancein x1 and x2 variables, 0 is an eigenvalue of −Lε whenever ∂x1us = 0 or ∂x2us = 0.In this case nonzero ∂xj

us are eigenfunctions for the eigenvalue 0. Correspondingly,0 is an eigenvalue of −L whenever ∂x1U s = 0 or ∂x2U s = 0. For simplicity weconsider the case ∂xj

us = 0 (and hence ∂xjU s = 0) for j = 1, 2.

Theorem 3.1. Let ∂xjus = 0 for j = 1, 2. If there exists a positive number b0 such

that ρ(−Lεn) ⊃ λ ∈ C; Reλ ≥ −b0\0 for some sequence εn → 0 as n → ∞ and0 is a semisimple eigenvalue of −Lεn with Ker (−Lεn) = span ∂x1us, ∂x2us, thenthere exists a constant b1 > 0 such that ρ(−L) ⊃ λ ∈ C; Reλ ≥ −b1 \ 0 and 0is a semisimple eigenvalue of −L with Ker (−L) = span ∂x1U s, ∂x2U s.

Theorem 3.1 shows that if us is obtained by Chorin’s method with 0 < ε ≪ 1,then it is stable as a solution of the Oberbeck-Boussinesq system. In particular, wehave the following instability result.

Theorem 3.2. Let ∂xjus = 0 for j = 1, 2. If σ(−L) ∩ λ ∈ C; Reλ > 0 = ∅, then

σ(−Lε) ∩ λ ∈ C; Reλ > 0 = ∅ for sufficiently small ε.

By Theorem 3.2, we see that unstable stationary solutions of the Oberbeck-Boussinesq system cannot be obtained by Chorin’s method with 0 < ε ≪ 1.

We next give a sufficient condition for us to be computed by Chorin’s methodwith 0 < ε ≪ 1.

7

Theorem 3.3. Let ∂xjU s = 0 for j = 1, 2. Suppose that ρ(−L) ⊃ λ ∈ C; Reλ ≥

−b0 \ 0 for some constant b0 > 0 and 0 is a semisimple eigenvalue of −L withKer (−L) = span ∂x1U s, ∂x2U s. Then there exist constants ε0 > 0, δ0 > 0 andb1 > 0 such that if

infw∈H1per,0,w =0

Re (w · ∇vs,w)

∥∇w∥22≥ −δ0, (3.11)

then ρ(−Lε) ⊃ λ ∈ C; Reλ ≥ −b1 \ 0 for all 0 < ε ≤ ε0 and 0 is a semisimpleeigenvalue of −Lε with Ker (−Lε) = span ∂x1us, ∂x2us.

In Theorem 3.3 we require smallness condition for the velocity field vs only butnot for the temperature θs.

Remark 3.4. It is known that −L is sectorial. Furthermore, −Lε is also sectorialfor each ε > 0. (See Proposition 6.1 below.) Consequently, under the assumptionsof Theorems 3.1 and 3.3, us =

⊤(ps,vs, θs) is asymptotically (orbitally) stable as asolution of both problems. More precisely, a solution of the nonlinear problem nearus converges to a stationary solution us(·+ a′, ·) for some a′ ∈ R2 as t → ∞.

Remark 3.5. Theorems 3.1 and 3.3 also hold with slight modifications when ∂xjus =

0 (and hence ∂xjU s = 0) for both j = 1, 2 or for one of j = 1, 2. For ex-

ample, if ∂x1U s = 0 and ∂x2U s = 0, then Theorem 3.3 holds with Ker (−L) =span ∂x1U s, ∂x2U s and Ker (−Lε) = span ∂x1us, ∂x2us replaced by Ker (−L) =span ∂x1U s and Ker (−Lε) = span ∂x1us, respectively. Likewise, the theoremshold for other cases with similar modifications.

Since the velocity fields of cellular stationary convective patterns bifurcatingfrom the heat conductive state are small when Ra ∼ Rac, we apply Theorem 3.3and Remark 3.5 to obtain the following corollary.

Corollary 3.6. If us is a stable convective pattern of (1.1)–(1.3) bifurcating from theheat conductive state, then it is also stable as a solution of (1.4)–(1.6) for 0 < ε ≪ 1when Ra ∼ Rac.

To prove Theorems 3.1–3.3, we will first investigate the null spaces of L and Lε

in section 4. Theorems 3.1 and 3.2 will be proved in section 5. The proof of Theorem3.3 will be given in sections 6 and 7.

4 The null spaces of L and Lε

In this section we investigate the relation of the null spaces of L and Lε. We introducethe adjoint operator L∗ : L2

per,σ × L2per → L2

per,σ × L2per of L:

L∗ =

(−PrP∆+ P(−vs · ∇+ (∇vs)) PrP((∇θs)− Rae3)

−Ra⊤e3 −∆− vs · ∇

)with domain D(L∗) = [(H2

per ∩H10,per)

3 ∩ L2per,σ]× [H2

per ∩H10,per].

8

In what follows we assume that 0 is a semisimple eigenvalue of−L with Ker (−L) =span ∂x1U s, ∂x2U s.

Since −L is a sectorial operator with compact resolvent and 0 is a semisimpleeigenvalue of −L, we have the following resolvent estimate for −L.

Proposition 4.1. Set U(0)j = ∂xj

U s for j = 1, 2. Then the following assertionshold.

(i) There exist U ∗j = ⊤(w∗

j , θ∗j ) ∈ D(L∗) such that L∗U ∗

j = 0 and ⟨U (0)j ,U ∗

j⟩ = δjkfor j, k = 1, 2. Furthermore,

L2per,σ × L2

per = X0 ⊕X1,

where X0 = Ker (−L) and

X1 = R(−L) = U ∈ L2per,σ × L2

per; ⟨U ,U ∗j⟩ = 0, j = 1, 2.

The eigenprojection Π0 for the eigenvalue 0 of −L is given by

Π0U = ⟨U ,U ∗1⟩U

(0)1 + ⟨U ,U ∗

2⟩U(0)2 .

(ii) Set Πc0 = I −Π0. There exist constants a0 > 0 and c0 ∈ R such that

Σ \ 0 ⊂ ρ(−L),

whereΣ := λ ∈ C; Reλ ≥ −a0|Imλ|2 + c0,

and the estimates

∥(λ+ L)−1F ∥2 ≤ C

1

|λ|∥Π0F ∥2 +

1

|λ|+ 1∥Πc

0F ∥2,

∥∂2x(λ+ L)−1F ∥2 ≤ C

1

|λ|∥Π0F ∥2 + ∥Πc

0F ∥2

hold uniformly for λ ∈ Σ \ 0. Furthermore, if Π0F = 0, then Π0(λ+ L)−1F = 0and the above estimates hold for any λ ∈ Σ.

We next introduce an operator L ε,λ : H1per,∗× (L2

per)3×L2

per → H1per,∗× (L2

per)3×

L2per defined by

D(L ε,λ) = H1per,∗ × [H2

per ∩H10,per]

3 × [H2per ∩H1

0,per],

L ε,λ =

0 1ε2div 0

Pr∇ λ− Pr∆ + vs · ∇+ ⊤(∇vs) −PrRae3

0 ⊤(∇θs)− Ra⊤e3 λ−∆+ vs · ∇

,

and its adjoint L ∗ε,λ : H1

per,∗ × (L2per)

3 × L2per → H1

per,∗ × (L2per)

3 × L2per by

D(L ∗ε,λ) = H1

per,∗ × [H2per ∩H1

0,per]3 × [H2

per ∩H10,per],

9

L ∗ε,λ =

0 − 1ε2div 0

−Pr∇ λ− Pr∆− vs · ∇+ (∇vs) Pr(∇θs)− PrRae3

0 −Ra⊤e3 λ−∆− vs · ∇

.

Note thatL ε,0 = Lε.

We also introduce the operators Aλ and A λ by

Aλw =

(λ− Pr∆ + vs · ∇+ ⊤(∇vs)

⊤(∇θs)− Ra⊤e3

)w,

and

A λw =

0λ− Pr∆ + vs · ∇+ ⊤(∇vs)

⊤(∇θs)− Ra⊤e3

w.

We first prepare the following lemma.

Lemma 4.2. There exists a bounded linear operator V : H1per,∗ → [H2

per ∩ H10,per]

3

such thatdivV f = f, ∥V f∥H2 ≤ C∥f∥H1

for f ∈ H1per,∗.

This lemma follows from the solvability for the nonhomogeneous Stokes problemdivv = f , −∆v + ∇p = 0 under the boundary conditions v|x3=0,1 = 0, and p, vbeing Q-periodic in x′. See, e.g., [5].

In what follows we set

Y := H1per,∗ × (L2

per)3 × L2

per.

Proposition 4.3. Let ε > 0.

(i) Let u(0)j = ∂xj

us (j = 1, 2). Then

Ker (L ε,0) = Ker (Lε) = span u(0)1 , u

(0)2 .

(ii) For each j = 1, 2, there exists a unique p∗j ∈ H1per,∗ such that L ∗

ε,0u∗j = 0,

u∗j =

⊤(p∗j ,w∗j , θ

∗j ), where

⊤(w∗j , θ

∗j ) (j = 1, 2) are the functions given in Proposition

4.1 (i). Furthermore,

R(L ε,0) = R(Lε) = u ∈ Y ; ⟨⟨u, u∗j⟩⟩ε = 0, j = 1, 2.

(iii) There exists a positive number ε1 such that if |ε| ≤ ε1, then

Y = Ker (Lε)⊕R(Lε).

Therefore, 0 is a semisimple eigenvalue of Lε. Furthermore, if F = ⊤(f, g, h) ∈R(Lε), then there exists a unique solution u = ⊤(p,w, θ) ∈ D(Lε)∩R(Lε) of Lεu = Fand u satisfies

∥u∥H1×H2×H2 ≤ Cε2∥f∥H1 + ∥F ∥2,where F = ⊤(g, h).

10

Proof. Let u = ⊤(p,w, θ) ∈ D(L ε,0) satisfy L ε,0u = 0. Then U = ⊤(w, θ)satisfies U ∈ D(L) and LU = 0. By assumption, there exist aj ∈ C such thatU = a1∂x1U s+ a2∂x2U s. On the other hand, L ε,0(a1∂x1us+ a2∂x2us) = 0. We thusobtain

∇p = ∇(a1∂x1ps + a2∂x2ps).

Since p and a1∂x1ps+a2∂x2ps are in H1per,∗, we have p = a1∂x1ps+a2∂x2ps, and hence,

u = a1∂x1us + a2∂x2us. This shows Ker (L ε,0) = span u(0)1 , u

(0)2 and (i) is proved.

As for (ii), since L∗U ∗j = 0, there exists a unique p∗j ∈ H1

per,∗ such that u∗j =

⊤(p∗j ,U∗j) satisfies uj ∈ D(L ∗

ε,0) and L ∗ε,0u

∗j = 0.

Let us prove R(L ε,0) = u ∈ Y ; ⟨⟨u, u∗j⟩⟩ε = 0, j = 1, 2. Set w = w − ε2V f .

ThenL ε,0u = F − ε2A 0V f, (4.1)

where u = ⊤(p, w, θ) and F = ⊤(0, g, h). Therefore, we have

LU = P [F − ε2A0V f ], (4.2)

where U = ⊤(w, θ) and F = ⊤(g, h). Since ⟨⟨F, u∗j⟩⟩ε = 0 for j = 1, 2, we have

⟨P [F − ε2A0V f ],U ∗j⟩

= ⟨F ,U ∗j⟩ − ε2Pr−1(−Pr∆V f + vs · ∇V f + V f · ∇vs,w

∗j)

−ε2(V f · ∇θs − RaV f · e3, θ∗j )

= ⟨F ,U ∗j⟩

−ε2Pr−1(V f,−Pr∆w∗j − vs · ∇w∗

j + (∇vs)w∗j + Pr(∇θs)θ

∗j − PrRaθ∗je3)

= ⟨F ,U ∗j⟩ − ε2Pr−1(V f,Pr∇p∗j)

= ⟨F ,U ∗j⟩+ ε2(divV f, p∗j)

= ⟨⟨F, u∗j⟩⟩ε = 0.

Applying Proposition 4.1, we see that there exists a unique solution U = ⊤(w, θ) ∈D(L) ∩X1 of (4.2) with estimate

∥U∥H2 ≤ Cε2∥f∥H1 + ∥F ∥2.

It then follows that there exists a unique p ∈ H1per,∗ such that u = ⊤(p, w, θ) ∈

D(L ε,0) is a solution of (4.1). Setting u = ⊤(p, w + ε2V f, θ), we see that u ∈D(L ε,0) and L ε,0u = F with estimate

∥u∥H1×H2×H2 ≤ Cε2∥f∥H1 + ∥F ∥2. (4.3)

We thus conclude that R(L ε,0) = u ∈ Y ; ⟨⟨u, u∗j⟩⟩ε = 0, j = 1, 2. Note that R(Lε)

is closed.

11

As for (iii), we first show that there exists u(0)j,ε ∈ D(Lε) (j = 1, 2) such that

Ker (Lε) = span u(0)1,ε, u

(0)2,ε, ⟨⟨u(0)

j,ε , u∗k⟩⟩ε = δjk (j, k = 1, 2).

We look for u(0)j,ε in the form u

(0)j,ε = a1ju

(0)1 + a2ju

(0)2 (j = 1, 2). By the conditions

⟨⟨u(0)j,ε , u

∗k⟩⟩ = δjk, we have

Bε

(a11 a12a21 a22

)=

(1 00 1

), (4.4)

where

Bε =

(⟨⟨u(0)

1 , u∗1⟩⟩ε ⟨⟨u(0)

2 , u∗1⟩⟩ε

⟨⟨u(0)1 , u∗

2⟩⟩ε ⟨⟨u(0)2 , u∗

2⟩⟩ε

).

Since ⟨U (0)j ,U ∗

k⟩ = δjk, we see that

Bε =

(1 + ε2(p

(0)1 , p∗1) ε2(p

(0)2 , p∗1)

ε2(p(0)1 , p∗2) 1 + ε2(p

(0)2 , p∗2)

).

Therefore,

Bε →(1 00 1

)as ε → 0, and hence, Bε has its inverse B−1

ε for 0 < ε ≪ 1. We thus obtain the

desired u(0)j,ε (j = 1, 2).

Set

P0,εu =2∑

j=1

⟨⟨u, u∗j⟩⟩εu

(0)j,ε .

Then P0,ε is the projection on Ker (Lε), i.e., the eigenprojection for the eigenvalue0 of −Lε. It is easy to see that R(I −P0,ε) = u ∈ Y ; ⟨⟨u, u∗

j⟩⟩ε = 0, j = 1, 2. This,together with (ii), implies that R(Lε) = R(I − P0,ε), and hence, Y = Ker (Lε) ⊕R(Lε). This decomposition and (ii), together with (4.3), yield the unique existenceof solution u ∈ D(Lε)∩R(Lε) of Lεu = F for F ∈ R(Lε) with the desired estimate.This completes the proof

5 Proof of Theorems 3.1 and 3.2

In this section we prove Theorems 3.1 and 3.2. We introduce some operators. Wedefine Lλ : (H2

per)3 × L2

per → (L2per)

3 by

LλU = λw − Pr∆w + vs · ∇w +w · ∇vs − PrRaθe3

for U = ⊤(w, θ) ∈ (H2per)

3 × L2per.

Let λ ∈ ρ(−L). For any F = ⊤(g, h) ∈ L2per,σ × L2

per, there exists a uniquep ∈ H1

per,∗ satisfying

∇p = IF − Lλ(λ+ L)−1F , (5.1)

12

and∥p∥H1 ≤ C|λ|C0,λ + C2,λ + 1∥F ∥2,

where I is the n× (n+1) matrix given by I = (I,0) with I being the n×n identitymatrix, i.e., IF = g, and

Ck,λ = sup

∥(λ+ L)−1F ∥Hk×Hk

∥F ∥2; F ∈ L2

per,σ × L2per, F = 0

(k = 0, 2).

We denote this p by pλ[F ]. The correspondence F ∈ L2per,σ × L2

per 7→ pλ[F ] ∈ H1per,∗

is linear, and hence, defines a bounded linear operator.We first give an expression of L −1

ε,λ in terms of (λ+ L)−1.

Proposition 5.1. Let λ ∈ ρ(−L). Then

L −1ε,λF =

(pλ[P (F − ε2AλV f)]

(λ+ L)−1P (F − ε2AλV f) + ε2V f

)for F = ⊤(f, g, h) ∈ H1

per,∗ × L2per × L2

per.

Proof. For a given F = ⊤(f, g, h) ∈ H1per,∗ × L2

per × L2per we consider the problem

L ε,λu = F . As in the proof of Proposition 4.3, we set u = u− ⊤(0, ε2V f, 0). Thenwe have L ε,λu = F − ε2A λV f with F = ⊤(0, g, h). We thus arrive at

(λ+ L)U = P (F − ε2AλV f),

where F = ⊤(g, h). Since λ ∈ ρ(−L) we have

⊤(w, θ) := U = (λ+ L)−1P (F − ε2AλV f).

It then follows that u = ⊤(p, w, θ) with

p = pλ[P (F − ε2AλV f)]

satisfies L ε,λu = F −ε2A λV f . Setting u = ⊤(p,w, θ) with w = w+ε2V f we havethe desired expression. This completes the proof.

We next give a perturbation result of eigenvalues of −L.

Theorem 5.2. Let λ0 ∈ σ(−L). Then λ0 is an eigenvalue of −L with finite multi-plicity and the following assertions hold.

(i) There exist C > 0, m ∈ N and r1 > 0 such that

λ; 0 < |λ− λ0| ≤ r1 ⊂ ρ(−L)

and

∥(λ+ L)−1g∥H2 ≤ C

|λ− λ0|m∥g∥2

for λ with 0 < |λ− λ0| ≤ r1.

13

(ii) For any r0 ∈ (0, r1], there exists a positive number ε0 = ε0(r0, r1,m, |λ0|)such that σ(−Lε) ∩ λ; |λ− λ0| < r0 = ∅ for 0 < ε ≤ ε0, and if |λ− λ0| = r0, thenλ ∈ ρ(−Lε) for 0 < ε ≤ ε0 and the total projection Pε, the sum of eigenprojectionsfor eigenvalues of −Lε lying inside the circle |λ− λ0| = r0, satisfies

PεF =1

2πi

∫|λ−λ0|=r0

(λ+ Lε)−1F dλ = P0F +O(ε2)F

in H1per,∗ × L2 × L2 as ε → 0. Here P0 is the projection defined by

P0F =

(Πλ0PF

Πλ0PF

)

for F = ⊤(f, g, h) ∈ H1per,∗×L2×L2, where F = ⊤(g, h), Πλ0 is the eigenprojection

for the eigenvalue λ0 of −L given by

Πλ0PF =1

2πi

∫|λ−λ0|=r0

(λ+ L)−1PF dλ

and

Πλ0PF = − 1

2πi

∫|λ−λ0|=r0

pλ[PF ] dλ,

i.e., Πλ0F ∈ H1per,∗ is the function satisfying

∇Πλ0PF = − 1

2πi

∫|λ−λ0|=r0

(I− Lλ(λ+ L)−1)PF dλ.

(iii) It holds that dimR(Pε) = dimR(P0) = dimR(Πλ0) for 0 < ε ≤ ε0.

Remark 5.3. Let m be the algebraic multiplicity of λ0. Then one can take a basisof R(Πλ0) consisting of functions U j, (λ0+L)U j, · · · , (λ0+L)mj−1U j, j = 1, · · · , r,with (λ0 + L)mjU j = 0 and m1 + · · · + mr = m. We denote by pjℓ ∈ H1

per,∗ thepressure satisfying

Lλ0(λ0 + L)mj−ℓ−1U j = I(λ0 + L)mj−ℓU j −∇pjℓ

for 1 ≤ j ≤ r, 0 ≤ ℓ ≤ mj − 1. It then follows that a set of functions ⊤(pjℓ, (λ0 +L)mj−ℓU j), 1 ≤ j ≤ r, 0 ≤ ℓ ≤ mj − 1, forms a basis of R(P0).

Proof. Since−L has compact resolvent, its spectrum consists of discrete eigenvalueswith finite multiplicities. Let the algebraic multiplicity of λ0 be m. Then we seethat there exist a m ∈ N, 1 ≤ m ≤ m, and a constant r1 > 0 such that λ; 0 <|λ− λ0| ≤ r1 ⊂ ρ(−L) and

∥(λ+ L)−1g∥H2 ≤ C

|λ− λ0|m∥g∥2

14

for λ with 0 < |λ− λ0| ≤ r1. This proves (i).As for (ii), we set

J =

1 0 00 0 00 0 0

and write λ+ Lε as

λ+ Lε = L ε,λ + λJ = L ε,λ(I + λL −1ε,λJ).

Since JF = ⊤(f,0, 0) for F = ⊤(f, g, h), we see from (i) and Proposition 5.1 that

∥L −1ε,λJF∥H1×H2×H2 ≤ Cε2

(|λ0|+ rm0 + 1)2

rm0∥f∥H1

for |λ− λ0| = r0. Therefore, if

ε2 ≤ rm04C(|λ0|+ rm0 + 1)2

1

|λ0|+ r0,

then

∥λL −1ε,λJF∥H1×H2×H2 ≤ Cε2(|λ0|+ r0)

(|λ0|+ rm0 + 1)2

rm0∥f∥H1 ≤ 1

2∥F∥H1×L2×L2

for |λ− λ0| = r0. It then follows that (I + λL −1ε,λJ) is boundedly invertible both on

Y and D(Lε) with estimates

∥(I + λL −1ε,λJ)

−1F∥H1×L2×L2 ≤ 2∥F∥H1×L2×L2

for F ∈ Y and

∥(I + λL −1ε,λJ)

−1F∥H1×H2×H2 ≤ 2∥F∥H1×H2×H2

for F ∈ D(Lε). We thus find that λ + Lε = L ε,λ + λJ has a bounded inverse(λ+ Lε)

−1 = (L ε,λ + ε2λJ)−1 on Y which satisfies

(λ+ Lε)−1 = (I + λL −1

ε,λJ)−1L −1

ε,λ

=∞∑

N=0

(−λ)N(L −1ε,λJ)

NL −1ε,λ

= L −1ε,λ − λL −1

ε,λJ

∞∑N=0

(−λ)N(L −1ε,λJ)

NL −1ε,λ

with estimates∥(λ+ Lε)

−1F∥H1×L2×L2 ≤ C∥F∥H1×L2×L2

and ∥∥∥∥∥λL −1ε,λJ

∞∑N=0

(−λ)N(L −1ε,λJ)

NL −1ε,λF

∥∥∥∥∥H1×L2×L2

≤ Cε2∥F∥H1×L2×L2 .

15

Using these estimates with Proposition 5.1 and the assertion (i), we obtain thedesired result (ii).

As for (iii), we first observe that P0 is a projection, i.e., P 20 = P0. By (ii) we

have Pε → P0 in the operator norm as ε → 0. It then follows from [8, Chap. I,Lemma 4.10] that dimR(Pε) = dimR(P0) for 0 < ε ≪ 1. On the other hand,dimR(P0) = dimR(Πλ0). We thus obtain the desired result. This completes theproof. Proof of Theorems 3.1 and 3.2. Theorem 3.2 is an immediate consequenceof Theorem 5.2 (ii). To prove Theorems 3.1, we first observe that dimKer (L) =dimKer (L ε,0). Suppose that 0 is not a semisimple eigenvalue of L. Then

dimR(Π0) > dimKer (L) = dimKer (L ε,0) = dimKer (Lε). (5.2)

Since 0 is a semisimple eigenvalue of Lε, we have dimKer (Lε) = dimR(Pε). This,together with (5.2), implies that dimR(Π0) > dimR(Pε), which contradicts toTheorem 5.2 (iii) with λ0 = 0. We thus conclude that 0 is a semisimple eigenvalueof −L.

Suppose that there exists λ0 ∈ σ(−L) with λ0 = 0 and Reλ0 ≥ 0. Then, byProposition 4.1 (ii), we have Imλ0 = O(1). Let r0 = (Reλ0+b0)/2 > 0. We see fromTheorem 5.2 (ii) that there exist ε0 > 0 such that λ; |λ − λ0| < r0 ∩ σ(−Lε) = ∅for all 0 < ε ≤ ε0. But if |λ− λ0| < r0, then

Reλ = Re (λ− λ0) + (Reλ0 + b0)− b0 ≥ r0 − b0 > −b0.

This contradicts to the assumption λ; Reλ ≥ −b0 \ 0 ⊂ ρ(−Lεn) with εn → 0.The proof of Theorem 3.1 is complete.

6 Proof of Theorem 3.3

In this section we give a proof of Theorem 3.3. We consider the resolvent problemfor −Lε:

λu+ Lεu = F, (6.1)

where u = ⊤(p,w, θ) ∈ D(Lε) and F = ⊤(f, g, h) ∈ Y . Problem (6.1) is written as

ε2λp+ divw = ε2f, (6.2)

Pr−1λw −∆w + Pr−1(vs · ∇w +w · ∇vs) +∇p− Raθe3 = Pr−1g, (6.3)

λθ −∆θ + vs · ∇θ +w · ∇θs − Raw · e3 = h, (6.4)

and u = ⊤(p,w, θ) satisfies the boundary conditions (3.4) and (3.5).

Proposition 6.1. There exist constants aε = O(1) > 0 and bε = O(1) > 0 as ε → 0such that λ ∈ C; Reλ ≥ −aεε

2|Imλ|2 + bε ⊂ ρ(−Lε) for all 0 < ε ≤ 1.

16

Proof. We set|||u|||2 =

(ε2∥p∥22 + Pr−1∥w∥22 + ∥θ∥22

) 12

for u = ⊤(p,w, θ). Taking the inner product ⟨⟨·, ·⟩⟩ε of (6.1) with u, we have

λ|||u|||22 + ∥∇w∥22 + ∥∇θ∥22

= 2RaRe (θ, w3) + 2iIm (p, divw)− Pr−1(vs · ∇w,w)− (vs · ∇θ, θ)

−Pr−1(w · ∇vs,w)− (w · ∇θs, θ) + ⟨⟨F, u⟩⟩ε.

(6.5)

Noting that Re (vs · ∇w,w) = Re (vs · ∇θ, θ) = 0, we see from the real part of (6.5)that

(Reλ− b2)|||u|||22 + ∥∇w∥22 + ∥∇θ∥22 ≤ |||F |||2|||u|||2, (6.6)

where b2 = 3 ∥∇vs∥∞ + (Pr + 1)(2Ra + ∥∇θs∥∞). The imaginary part of (6.5)yields

|Imλ||||u|||22 ≤ 2∥p∥2∥divw∥2 + Pr−1∥vs∥∞∥∇w∥2∥w∥2

+∥vs∥∞∥∇θ∥2∥θ∥2 + Pr−1∥∇vs∥∞∥w∥22 + ∥∇θs∥∞∥w∥2∥θ∥2

+|||F |||2|||u|||2

≤(

2ε−1 + Pr−12∥vs∥∞

)∥∇w∥2 + ∥vs∥∞∥∇θ∥2

+Pr−12∥∇vs∥∞∥w∥2 + Pr

12∥∇θs∥∞∥θ∥2 + |||F |||2

|||u|||2,

and hence,

|Imλ|2|||u|||22 ≤ 36(

4ε−2 + Pr−1∥vs∥2∞)∥∇w∥22 + ∥vs∥2∞∥∇θ∥22

+Pr−1∥∇vs∥2∞∥w∥22 + Pr∥∇θs∥2∞∥θ∥22 + |||F |||22.

(6.7)

It then follows from (6.6) and (6.7) that

(Reλ+ aεε2|Imλ|2 − bε)|||u|||22 + 1

2(∥∇w∥22 + ∥∇θ∥22)

≤ (δ−1 + 36aεε2)|||F |||22

(6.8)

for all δ > 0, where

aε =1

72(4 + ε2(Pr−1 + 1)∥vs∥2∞)

andbε = δ + b2 + 36aεε

2(∥∇vs∥2∞ + Pr∥∇θs∥2∞

).

It remains to estimate ∥∂xp∥2. To this end we use the estimates in section 7. Weintroduce quantities D(w, θ) and M(w, θ) defined by

D(w, θ) = ∥∇w∥22 + ∥∇θ∥22,

M(w, θ) = ∥vs∥2∞(Pr−2∥∇w∥22 + ∥∇θ∥22) + (Pr−2∥∇ws∥2∞ + ∥∇θs∥2∞)∥w∥22

+Ra(1 + Ra)(∥w∥22 + ∥θ∥22).

17

We setΣ1 = λ ∈ ; Reλ+ aεε

2|Imλ| − bε ≥ 0

andΣ2 = λ ∈ ; dist (λ,Σ1) ≥ 1.

In what follows we assume that λ ∈ Σ2. It then follows that

M(w, θ) ≤ C|||F |||22.

Here C is a positive constant depending on Ra and Pr but not on ε.We consider (1 + ε−2) × (6.8) + d0 × (7.8). Then taking d0 > 0 suitably small,

we see that

(Reλ+ aεε2|Imλ|2 − bε)|||u|||22 + (Reλ+ |Imλ|)D(w, θ)

+(Reλ+ ε2|Imλ|2) (|||∂x′u|||22 + ε2∥∂x3p∥22)

+b0((1 + 1

ε2

)D(w, θ) + ∥ε2λp∥2H1 +

12|λ|2|||u|||22

)+b0 (∥∂2

xw∥22 + ∥∂2xθ∥22 + ∥∂xp∥22)

≤ C(1 + ε2)ε2∥∂xf∥22 +

(1 + 1

ε2

)|||F |||22

(6.9)

with some positive constants b0 and C uniformly for 0 < ε ≤ 1. This completes theproof.

We next show that the spectrum of −Lε in a disc with radius O(ε−1) can beviewed as a perturbation of the one of −L. From the assumption of Theorem 3.3,we see that one can take the constant c0 in Proposition 4.1 (ii) so that c0 < 0 bychanging a0 > 0 suitably. In what follows we fix these a0 > 0 and c0 < 0. UsingProposition 4.1 (ii), Lemma 4.2, (5.1) and Proposition 5.1, we have the followingestimates for L −1

ε,λ.

Proposition 6.2. Let ε > 0. If λ ∈ Σ \ 0, then L ε,λ has a bounded inverse L −1ε,λ

and ⊤(p,v, θ) = L −1ε,λF for F = ⊤(f, g, h) ∈ Y satisfies

∥U∥2 ≤C

|λ|

2∑j=1

∣∣⟨⟨F, u∗j⟩⟩ε∣∣+ C

ε2∥f∥H1 +

1

|λ|+ 1∥F ∥2

,

∥∂2xU∥2 + ∥∂xp∥2 ≤

C

|λ|

2∑j=1

∣∣⟨⟨F, u∗j⟩⟩ε∣∣+ C

ε2(|λ|+ 1)∥f∥H1 + ∥F ∥2

,

where U = ⊤(w, θ) and F = ⊤(g, h). Furthermore, if F ∈ Y1,ε, then⊤(p,v, θ) =

L −1ε,λF exists uniquely in Y1,ε and the above estimates hold with λ = 0 and ⟨⟨F, u∗

j⟩⟩ε =0 (j = 1, 2).

18

Proof. As in the proof of Proposition 4.3 (ii), we see that

Π0P [F − ε2AλV f ] =2∑

j=1

⟨⟨F, u∗

j⟩⟩ε −ε2λ

Pr(V f,w∗

j)

U

(0)j .

Therefore, applying Proposition 4.1 (ii), Lemma 4.2 and Proposition 5.1 togetherwith (5.1), we have the desired result. This completes the proof.

Proposition 6.3. There exist positive numbers ε1 and a1 such that

Σ ∩ λ ∈ C; Reλ ≥ −b0, |λ| ≤ a1ε−1 ⊂ ρ(−Lε|Y1,ε)

for all 0 < ε ≤ ε1. Here

Y1,ε = R(Lε) = u ∈ Y ; ⟨⟨u, u∗j⟩⟩ε = 0, j = 1, 2.

Proof. As in the proof of Theorem 5.2, we write the resolvent problem

(λ+ Lε)u = F

on Y1,ε asL ε,λu+ λJu = F, (6.10)

where F = ⊤(f, g, h) ∈ Y1,ε. If λ ∈ Σ, then it follows from Proposition 6.2 that(6.10) is written as

L ε,λ(I + λL −1ε,λJ)u = F,

and, furthermore, we have

∥L −1ε,λJF∥H1×H2×H2 ≤ ε2C1(|λ|+ 1)∥f∥H1

for all F = ⊤(f, g, h) ∈ Y1,ε. It then follows that there exists ε1 > 0 such that if λ ∈ Σand |λ| ≤ 1/(4

√C1ε), then L −1

ε,λJF ∈ D(L ε,λ) = D(Lε) and ∥λL −1ε,λJF∥H1×H2×H2 ≤

12∥F∥H1×L2×L2 for 0 < ε ≤ ε1. Therefore, (I +λL −1

ε,λJ) is boundedly invertible bothon Y1,ε and Y1,ε ∩D(Lε) with estimates

∥(I + λL −1ε,λJ)

−1F∥H1×L2×L2 ≤ 2∥F∥H1×L2×L2

for F ∈ Y1,ε and

∥(I + λL −1ε,λJ)

−1F∥H1×H2×H2 ≤ 2∥F∥H1×H2×H2

for F ∈ Y1,ε ∩D(Lε). We thus find that λ+ Lε = L ε,λ + λJ has a bounded inverse(λ+ Lε)

−1 = (L ε,λ + ε2λJ)−1 on Y1,ε which satisfies

(λ+ Lε)−1 = L −1

ε,λ − λL −1ε,λJ

∞∑N=0

(−λ)N(L −1ε,λJ)

NL −1ε,λ

19

and

∥(λ+ Lε)−1F∥H1×L2×L2 ≤ 2C1

ε2(|λ|+ 1)∥f∥H1 + ∥F ∥2

≤ 2C1

ε

(1

4√C1

+ ε

)∥f∥H1 + ∥F ∥2

with F = ⊤(g, h). This completes the proof.

Theorem 3.3 follows from Propositions 4.3, 6.1 and 6.3 if√

bε/aε < a1 for 0 <

ε ≪ 1. In the case√

bε/aε ≥ a1, there is some range of λ near the imaginary axiswith |Imλ| = O(ε−1) to be proved that it belongs to ρ(−Lε).

To prove Theorem 3.3 when√

bε/aε ≥ a1, we first prepare an estimate for theθ-component. We recall that the Poincare inequality

∥∇θ∥2 ≥ β∥θ∥2holds for θ ∈ H1

0,per with some positive constant β.

Proposition 6.4. Let ⊤(p,w, θ) be a solution of (6.2)–(6.4) under boundary condi-

tions (3.4) and (3.5). Then if Reλ ≥ −β2

2, the following estimates hold:

∥θ∥2 ≤1

|Imλ|

(1 +

2∥vs∥∞β

)(∥∇θs∥∞ +Ra)∥w∥2 + ∥h∥2 ,

∥∇θ∥2 ≤2

β(∥∇θs∥∞ +Ra)∥w∥2 + ∥h∥2 .

Proof. We take the inner product of (6.4) with θ to obtain

λ∥θ∥22 + ∥∇θ∥22

= − (iIm (vs · ∇θ, θ) + (w · ∇θs, θ)− Ra(w3, θ)) + (h, θ).(6.11)

Using the Poincare inequality we see from the real part of (6.11) that(Reλ+

β2

2

)∥θ∥22 +

1

2∥∇θ∥22 ≤ (∥∇θs∥∞ +Ra) ∥w∥2 + ∥h∥2 ∥θ∥2.

This implies that if Reλ ≥ −β2

2, then

∥∇θ∥2 ≤2

β(∥∇θs∥∞ +Ra)∥w∥2 + ∥h∥2 . (6.12)

Furthermore, the imaginary part of (6.11), together with (6.12), implies that

|Imλ|∥θ∥22 = |−Im ((vs · ∇θ, θ) + (w · ∇θs, θ)− Ra(w3, θ)) + Im (h, θ)|

≤ ∥vs∥∞∥∇θ∥2 + (∥∇θs∥∞ +Ra)∥w∥2 + ∥h∥2 ∥θ∥2

≤(1 + 2∥vs∥∞

β

)(∥∇θs∥∞ +Ra)∥w∥2 + ∥h∥2 ∥θ∥2,

from which we have the desired estimate for ∥θ∥2. This completes the proof. We are now in a position to complete the proof of Theorem 3.3.

20

Proposition 6.5. For given µ∗ > 0 and η∗ > 0 there exist constants ε1 > 0 andc2 > 0 such that if

infw∈H10,per,w =0

Re (w · ∇vs,w)

∥∇w∥22≥ −Pr

32,

then λ = µ+ i

η

ε; −c2 ≤ µ ≤ µ∗, |η| ≥ η∗

⊂ ρ(−Lε)

for all 0 < ε ≤ ε1. Here ε1 and c2 are positive constants depending only on Pr, Ra,∥vs∥C1, ∥∇θs∥∞, β, µ∗ and η∗

Proof. We see from (6.2) that

p = − 1

ε2λdivw +

1

λf. (6.13)

Substituting (6.13) into (6.3), we have

ε2λ2

Prw − ε2λ∆w −∇divw +

ε2λ

Pr(vs · ∇w +w · ∇vs)− ε2λRaθe3 = ε2Gλ, (6.14)

where Gλ = λPrg −∇f .

Let λ = µ + iεη with |η| ≥ η∗(> 0). Without loss of generality we may assume

η ≥ η∗. Taking the inner product of (6.14) with w, we have

ε2λ2

Pr∥w∥22 + ε2λ∥∇w∥22 + ∥divw∥22

= −ε2λ(Pr−1(vs · ∇w,w) + Pr−1(w · ∇vs,w)− Ra(θ, w3)

)+ ε2(Gλ,w).

(6.15)Since λ2 = (µ2 − ε−2η2) + 2iε−1µη and Re (vs · ∇w,w) = 0, the real and imaginaryparts of (6.15) yield

1

Pr(ε2µ2 − η2)∥w∥22 + ε2µ∥∇w∥22 + ∥divw∥22

= −ε2µRe(Pr−1(w · ∇vs,w)− Ra(θ, w3)

)+εηIm

(Pr−1(vs · ∇w,w) + Pr−1(w · ∇vs,w)− Ra(θ, w3)

)+ε2Re (Gλ,w)

(6.16)

and

2εµη

Pr∥w∥22 + εη∥∇w∥22

= −ε2µIm(Pr−1(vs · ∇w,w) + Pr−1(w · ∇vs,w)− Ra(θ, w3)

)−εηRe

(Pr−1(w · ∇vs,w)− Ra(θ, w3)

)+ε2Im (Gλ,w).

(6.17)

21

Since |Imλ| = ηε, by Proposition 6.4, we have

|Ra(θ, w3)| ≤ Raε

η

(1 +

2∥vs∥∞β

)(∥∇θs∥∞ +Ra)∥w∥22 + ∥h∥2∥w∥2

,

and hence, we see from (6.16) that

1

Pr(η2 − ε2µ2)∥w∥22

≤(ε2µ+

η

η∗+ εη

)∥∇w∥22 +

((ε2|µ|+ εη)Pr−1∥∇vs∥∞ + εηPr−2∥vs∥2∞

)∥w∥22

+(ε2|µ|+ εη)Raε

η

(1 +

2∥vs∥∞β

)(∥∇θs∥∞ +Ra)∥w∥22 + ∥h∥2∥w∥2

+ε2∥Gλ∥2∥w∥2.

(6.18)Similarly, we see from (6.17) that

2µη

Pr∥w∥22 +

3η

4∥∇w∥22

≤ −ηPr−1Re (w · ∇vs,w) +

(ε2|µ|2Pr−2∥vs∥2∞

η+ ε|µ|Pr−1∥∇vs∥∞

)∥w∥22

+(ε|µ|+ η)Raε

η

(1 +

2∥vs∥∞β

)(∥∇θs∥∞ +Ra)∥w∥22 + ∥h∥2∥w∥2

+ε∥Gλ∥2∥w∥2.

(6.19)

We set M = infw∈H10,per,w =0

Re (w·∇vs,w)

∥∇w∥22. It then follows from η∗

4× (6.18) +

(6.19) that (Iε,µ,η +

η∗4Pr

η2)∥w∥22 +

(η

2− ε2η∗µ

4− εη∗η

4

)∥∇w∥22

≤ −ηPr−1M∥∇w∥22 + Cε.λ(∥F∥H1×(L2)2)∥w∥2,(6.20)

where

Iε,µ,η =2µη

Pr− ε2µ2

Pr

(η∗4

+∥vs∥2∞ηPr

)− ε|µ|

Pr

(1 +

εη∗4

)∥∇vs∥∞

−εη∗η

4Pr

(∥∇vs∥∞ +

∥vs∥2∞Pr

)

−εRa(1 +

εη∗4

)(ε|µ|η

+ 1

)(1 +

2∥vs∥∞β

)(∥∇vs∥∞ +Ra),

Cε,λ(∥F∥H1×(L2)2) = ε(1 +

εη∗4

)[Ra

(ε|µ|η

+ 1

)(1 +

2∥vs∥∞β

)∥h∥2 + ∥Gλ∥2

].

22

We take ε > 0 so small that ε2µ∗ ≤ 18and εη∗ ≤ 1

8in (6.20). Then by using the

Poincare inequality, we have(Iε,µ,η +

η∗4Pr

η2 +β2

8η

)∥w∥22+

η

16∥∇w∥22 ≤ −ηPr−1M∥∇w∥22+Cε,λ(∥F∥H1×(L2)2)∥w∥2.

(6.21)and hence, if

M ≥ −Pr

32,

we find from (6.21) that(Iε,µ,η +

β2

8η

)∥w∥22 +

η

32∥∇w∥22 ≤ Cε,λ(∥F∥H1×(L2)2)∥w∥2. (6.22)

Therefore, if we show Iε,µ,η +β2

8η ≥ β2

16η for −c2 ≤ µ ≤ µ∗ and η ≥ η∗ with some

c2 > 0 when 0 < ε ≪ 1, then the proof of Proposition 6.6 is complete.Let us prove Iε,µ,η +

β2

8η ≥ β2

16η. When 0 ≤ µ ≤ µ∗, we have

Iε,µ,η ≥ η

µ

(2

Pr−Kε

)− εη∗

4Pr

(∥∇vs∥∞ +

∥vs∥2∞Pr

)−εRa

η∗

(1 +

εη∗4

)(1 +

2∥vs∥∞β

)(∥∇vs∥∞ +Ra)

,

where

Kε =ε2µ∗

Pr

(1

4+

∥vs∥2∞η2∗Pr

)+

ε

Pr

(ε

4+

1

η∗

)∥∇vs∥∞

+ε2Ra

η2∗

(1 +

εη∗4

)(1 +

2∥vs∥∞β

)(∥∇vs∥∞ +Ra).

Therefore, if we take ε > 0 so that

2

Pr−Kε ≥ 0

andβ2

16− εη∗

4Pr

(∥∇vs∥∞ +

∥vs∥2∞Pr

)−εRa

η∗

(1 +

εη∗4

)(1 +

2∥vs∥∞β

)(∥∇vs∥∞ +Ra) ≥ 0,

then we have Iε,µ,η +β2

8η ≥ β2

16η for 0 ≤ µ ≤ µ∗ and η ≥ η∗, and hence,

β2

16η∥w∥22 +

η

32∥∇w∥22 ≤ Cε,λ(∥F∥H1×(L2)2)∥w∥2 (6.23)

for 0 ≤ µ ≤ µ∗ and η ≥ η∗.When µ < 0, we assume that |µ| ≤ Pr

64β2. We write Iε,µ,η as Iε,µ,η = ηIε(µ, η).

Then we see that Iε(µ, η) ≥ Iε(−Pr

64β2, η∗

)= −β2

32+ O(ε), and so we take ε > 0 in

23

such a way that Iε(−Pr

64β2, η∗

)+ β2

16≥ 0. It then follows that Iε,µ,η +

β2

8η ≥ β2

16η,

and hence, we have (6.23) for −c2 ≤ µ < 0 and η ≥ η∗ with c2 = Pr128

β2. Similarlyto the proof of Proposition 6.1, we can derive the necessary estimate for ∥∂xp∥22 bycombining estimate (6.23), Propositions 6.4 and 7.5. This completes the proof.

Theorem 3.3 now follows by taking η∗ =a12, µ∗ = 2bε|ε=1,δ=1 and ε > 0 sufficiently

small.

7 Estimate for ∥∂xp∥2In this section we give basic energy estimates which yield the estimate for ∥∂xp∥2 inthe resolvent estimate. The following Propositions 7.1–7.4 can be obtained by theMatsumura-Nishida energy method [9]. We here give an outline of the proof.

Proposition 7.1. Let u = ⊤(p,w, θ) be a solution of (6.2)–(6.4) satisfying boundaryconditions (3.4) and (3.5). Then

(Reλ+ ε2|Imλ|2)|||∂x′u|||22 + 14D(∂x′w, ∂x′θ) + 1

4∥ε2λ∂x′p∥22

≤ C(1 + ε2) δε2∥∂x′p∥22 + δ−1ε2∥∂x′f∥22 + |||F |||22 +M(w, θ)(7.1)

for any δ > 0, where C is a positive constant independent of ε and δ.

Proposition 7.1 can be proved as in the proof of Proposition 6.1 by taking theinner product ⟨⟨∂x′(6.1), ∂x′u⟩⟩ε and using the relation ε2λ∂x′p = −div ∂x′w+ε2∂x′fwhich follows from (6.2).

We next compute the inner product ⟨⟨(6.1), λu⟩⟩ε to obtain the following esti-mate.

Proposition 7.2. Let u = ⊤(p,w, θ) be a solution of (6.2)–(6.4) satisfying boundaryconditions (3.4) and (3.5). Then

(Reλ+ |Imλ|)D(w, θ) +1

2|λ|2|||u|||22 ≤ C

1

ε2∥divw∥22 +M(w, θ)

, (7.2)

where C is a positive constant independent of ε.

As for the normal derivative of p, we have the following estimate.

Proposition 7.3. Let u = ⊤(p,w, θ) be a solution of (6.2)–(6.4) satisfying boundaryconditions (3.4) and (3.5). Then

(Reλ+ ε2|Imλ|2)ε2∥∂x3p∥22 + 14∥∂x3p∥22 + 1

4∥ε2λ∂x3p∥22

≤ C ε2∥∂x3f∥22 + |||F |||22 +M(w, θ) + |λ|2∥w∥22 + ∥∇∂x′w∥22 ,(7.3)

where C is a positive constant depending on Pr but independent of ε.

24

Outline of Proof. We denote w′ = ⊤(w1, w2), ∇′ = ⊤(∂x1 , ∂x2) and ∆′ = ∂2x1+∂2

x2.

Applying ∂x3 to (6.2), we have

ε2λ∂x3p+∇′ · ∂x3w′ + ∂2

x3w3 = ε2∂x3f. (7.4)

The third equation of (6.3) is written as

Pr−1λw3 −∆′w3 − ∂2x3w3 + ∂x3p = g3, (7.5)

whereg3 = Pr−1g3 − Pr−1(vs · ∇w3 +w · ∇v3s) + Raθ.

We compute (7.4)+(7.5) to obtain

ε2λ∂x3p+ ∂x3p = H, (7.6)

whereH = ε2∂x3f + g3 − (Pr−1λw3 −∆′w3 +∇′ · ∂x3w

′).

Taking the L2 inner product of (7.6) with ∂x3p, we have

(Reλ+ ε2|Imλ|2)ε2∥∂x3p∥22 + 14∥∂x3p∥22 + 1

4∥ε2λ∂x3p∥22

≤ C ε2∥∂x3f∥22 + |||F |||22 +M(w, θ) + |λ|2∥w∥22 + ∥∇∂x′w∥22 .

This, together with the relation ε2λ∂x3p = −∂x3p + H, gives the desired estimate.

The elliptic estimates yield the following estimate for the second order derivativesof w and θ and the first order derivatives of p.

Proposition 7.4. Let u = ⊤(p,w, θ) be a solution of (6.2)–(6.4) satisfying boundaryconditions (3.4) and (3.5). Then

∥∂2xw∥22 + ∥∂2

xθ∥22 + ∥∂xp∥22

≤ Cε4∥f∥2H1 + |||F |||22 + ∥ε2λp∥2H1 + |λ|2(∥w∥22 + ∥θ∥22) +M(w, θ)

,

(7.7)

where C is a positive constant depending on Pr but independent of ε.

Outline of Proof. We rewrite (6.1) asdivw = ε2f − ε2λp,

−∆w +∇p = Pr−1g − (Pr−1λw + Pr−1(vs · ∇w +w · ∇vs)− Raθe3),

−∆θ = h− (λθ + vs · ∇θ +w · ∇θs − Raw · e3).

Applying the estimates for the Stokes system and the elliptic equation (see, e.g.,[5, 11]), we obtain the desired estimate.

Combining the estimates in Propositions 7.1–7.4, we have the following estimatefor ∥∂xp∥2 and ∥∂2

xw∥2.

25

Proposition 7.5. Let 0 < ε ≤ 1 and let u = ⊤(p,w, θ) be a solution of (6.2)–(6.4)satisfying boundary conditions (3.4) and (3.5). Then

(Reλ+ |Imλ|)D(w, θ) + (Reλ+ ε2|Imλ|2) (|||∂x′u|||22 + ε2∥∂x3p∥22)

+b(∥ε2λ∂xp∥2H1 +

12|λ|2|||u|||22 + ∥∂2

xw∥22 + ∥∂2xθ∥22 + ∥∂xp∥22

)≤ C(1 + ε2)

ε2∥∂xf∥22 +

(1 + 1

ε2

)(∥divw∥22 +M(w, θ) + |||F |||22)

(7.8)

Proof of Proposition 7.5. Let 0 ≤ ε ≤ 1. We see from (7.1) + (7.2) + d2 × (7.3)with suitably small d2 > 0 that

(Reλ+ |Imλ|)D(w, θ) + (Reλ+ ε2|Imλ|2) (|||∂x′u|||22 + ε2∥∂x3p∥22)

+b(D(∂x′w, ∂x′θ) + ∥∂x3p∥22 + ∥ε2λ∂xp∥2 + 1

2|λ|2|||u|||22

)≤ C(1 + ε2) δε2∥∂x′p∥22 + δ−1ε2∥∂xf∥22

+(1 + 1

ε2

)(∥divw∥22 +M(w, θ) + |||F |||22)

(7.9)

for any δ > 0 with positive constants b and C independent of δ and ϵ.Adding d3 × (7.3) to (7.9) and taking d3 > 0 suitably small, we have

(Reλ+ |Imλ|)D(w, θ) + (Reλ+ ε2|Imλ|2) (|||∂x′u|||22 + ε2∥∂x3p∥22)

+b(∥ε2λ∂xp∥22 + 1

2|λ|2|||u|||22 + ∥∂2

xw∥22 + ∥∂2xθ∥22 + ∥∂xp∥22

)≤ C(1 + ε2) δε2∥∂x′p∥22 + δ−1ε2∥∂xf∥22

+(1 + 1

ε2

)((∥divw∥22 +M(w, θ) + |||F |||22)

(7.10)

for any δ > 0 with positive constants b and C independent of δ and 0 < ϵ ≤ 1.Taking δ > 0 suitably small, we arrive at

(Reλ+ |Imλ|)D(w, θ) + (Reλ+ ε2|Imλ|2) (|||∂x′u|||22 + ε2∥∂x3p∥22)

+b(∥ε2λ∂xp∥2H1 +

12|λ|2|||u|||22 + ∥∂2

xw∥22 + ∥∂2xθ∥22 + ∥∂xp∥22

)≤ C(1 + ε2)

ε2∥∂xf∥22 +

(1 + 1

ε2

)((∥divw∥22 +M(w, θ) + |||F |||22)

(7.11)

This completes the proof.

Acknowledgements. Y. Kagei was partly supported by JSPS KAKENHI GrantNumber 24340028, 15K13449, 24224003, 16H03947; T. Nishida was partly supportedby JSPS KAKENHI Grant Number 26400163.

References

[1] F. H. Busse, Non-linear properties of thermal convection, Rep. Prog. Phys., 41(1978), pp. 1929–1967.

26

[2] A. Chorin, The numerical solution of the Navier-Stokes equations for an incom-pressible fluid, Bull. Amer. Math. Soc., 73 (1967), pp. 928–931.

[3] A. Chorin, A numerical method for solving incompressible viscous flow prob-lems, J. Comput. Phys., 2 (1967), pp. 12–26.

[4] A. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp.,22 (1968), pp. 745–762.

[5] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-StokesEquations, Vol. 1, Springer-Verlag New York (1994).

[6] V. I. Iudovich, On the origin of convection, PMM. J. appl. Math. Mech., 30(1966), pp. 1193–1199.

[7] D. D. Joseph, Stability of fluid motions, I, II, Springer-Verlag, Berlin, Heidel-berg, New York (1973).

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

[9] A. Matsumura and T. Nishida, Initial boundary value problems for the equa-tions of motion of compressible viscous and heat-conductive fluids. Comm.Math. Phys., 89 (1983) , pp. 445–464.

[10] P. H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of theBenard problem, Arch. Rational Mech. Anal., 29 (1968), pp. 32–57.

[11] H. Sohr, The Navier-Stokes equations : an elementary functional analytic ap-proach, Birkhauser, Basel (2001).

27

List of MI Preprint Series, Kyushu University

The Global COE ProgramMath-for-Industry Education & Research Hub

MI

MI2008-1 Takahiro ITO, Shuichi INOKUCHI & Yoshihiro MIZOGUCHIAbstract collision systems simulated by cellular automata

MI2008-2 Eiji ONODERAThe intial value problem for a third-order dispersive flow into compact almost Her-mitian manifolds

MI2008-3 Hiroaki KIDOOn isosceles sets in the 4-dimensional Euclidean space

MI2008-4 Hirofumi NOTSUNumerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme

MI2008-5 Yoshiyasu OZEKITorsion points of abelian varieties with values in nfinite extensions over a p-adic field

MI2008-6 Yoshiyuki TOMIYAMALifting Galois representations over arbitrary number fields

MI2008-7 Takehiro HIROTSU & Setsuo TANIGUCHIThe random walk model revisited

MI2008-8 Silvia GANDY, Masaaki KANNO, Hirokazu ANAI & Kazuhiro YOKOYAMAOptimizing a particular real root of a polynomial by a special cylindrical algebraicdecomposition

MI2008-9 Kazufumi KIMOTO, Sho MATSUMOTO & Masato WAKAYAMAAlpha-determinant cyclic modules and Jacobi polynomials

MI2008-10 Sangyeol LEE & Hiroki MASUDAJarque-Bera Normality Test for the Driving Levy Process of a Discretely ObservedUnivariate SDE

MI2008-11 Hiroyuki CHIHARA & Eiji ONODERAA third order dispersive flow for closed curves into almost Hermitian manifolds

MI2008-12 Takehiko KINOSHITA, Kouji HASHIMOTO and Mitsuhiro T. NAKAOOn the L2 a priori error estimates to the finite element solution of elliptic problemswith singular adjoint operator

MI2008-13 Jacques FARAUT and Masato WAKAYAMAHermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek poly-nomials

MI2008-14 Takashi NAKAMURARiemann zeta-values, Euler polynomials and the best constant of Sobolev inequality

MI2008-15 Takashi NAKAMURASome topics related to Hurwitz-Lerch zeta functions

MI2009-1 Yasuhide FUKUMOTOGlobal time evolution of viscous vortex rings

MI2009-2 Hidetoshi MATSUI & Sadanori KONISHIRegularized functional regression modeling for functional response and predictors

MI2009-3 Hidetoshi MATSUI & Sadanori KONISHIVariable selection for functional regression model via the L1 regularization

MI2009-4 Shuichi KAWANO & Sadanori KONISHINonlinear logistic discrimination via regularized Gaussian basis expansions

MI2009-5 Toshiro HIRANOUCHI & Yuichiro TAGUCHIIFlat modules and Groebner bases over truncated discrete valuation rings

MI2009-6 Kenji KAJIWARA & Yasuhiro OHTABilinearization and Casorati determinant solutions to non-autonomous 1+1 dimen-sional discrete soliton equations

MI2009-7 Yoshiyuki KAGEIAsymptotic behavior of solutions of the compressible Navier-Stokes equation aroundthe plane Couette flow

MI2009-8 Shohei TATEISHI, Hidetoshi MATSUI & Sadanori KONISHINonlinear regression modeling via the lasso-type regularization

MI2009-9 Takeshi TAKAISHI & Masato KIMURAPhase field model for mode III crack growth in two dimensional elasticity

MI2009-10 Shingo SAITOGeneralisation of Mack’s formula for claims reserving with arbitrary exponents forthe variance assumption

MI2009-11 Kenji KAJIWARA, Masanobu KANEKO, Atsushi NOBE & Teruhisa TSUDAUltradiscretization of a solvable two-dimensional chaotic map associated with theHesse cubic curve

MI2009-12 Tetsu MASUDAHypergeometric τ-functions of the q-Painleve system of type E

(1)8

MI2009-13 Hidenao IWANE, Hitoshi YANAMI, Hirokazu ANAI & Kazuhiro YOKOYAMAA Practical Implementation of a Symbolic-Numeric Cylindrical Algebraic Decompo-sition for Quantifier Elimination

MI2009-14 Yasunori MAEKAWAOn Gaussian decay estimates of solutions to some linear elliptic equations and itsapplications

MI2009-15 Yuya ISHIHARA & Yoshiyuki KAGEILarge time behavior of the semigroup on Lp spaces associated with the linearizedcompressible Navier-Stokes equation in a cylindrical domain

MI2009-16 Chikashi ARITA, Atsuo KUNIBA, Kazumitsu SAKAI & Tsuyoshi SAWABESpectrum in multi-species asymmetric simple exclusion process on a ring

MI2009-17 Masato WAKAYAMA & Keitaro YAMAMOTONon-linear algebraic differential equations satisfied by certain family of elliptic func-tions

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

MI2014-12 Chien-Chung HUANG, Naonori KAKIMURA & Naoyuki KAMIYAMAExact and approximation algorithms for weighted matroid intersection

MI2014-13 Yusuke SHIMIZUMoment convergence of regularized least-squares estimator for linear regression model

MI2015-1 Hidetoshi MATSUI & Yuta UMEZUSparse regularization for multivariate linear models for functional data

MI2015-2 Reika AOYAMA & Yoshiyuki KAGEISpectral properties of the semigroup for the linearized compressible Navier-Stokesequation around a parallel flow in a cylindrical domain

MI2015-3 Naoyuki KAMIYAMAStable Matchings with Ties, Master Preference Lists, and Matroid Constraints

MI2015-4 Reika AOYAMA & Yoshiyuki KAGEILarge time behavior of solutions to the compressible Navier-Stokes equations arounda parallel flow in a cylindrical domain

MI2015-5 Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokes-Korteweg system on R3

MI2015-6 Naoyuki KAMIYAMAPopular Matchings with Ties and Matroid Constraints

MI2015-7 Shoichi EGUCHI & Hiroki MASUDAQuasi-Bayesian model comparison for LAQ models

MI2015-8 Yoshiyuki KAGEI & Ryouta OOMACHIStability of time periodic solution of the Navier-Stokes equation on the half-spaceunder oscillatory moving boundary condition

MI2016-1 Momonari KUDOAnalysis of an algorithm to compute the cohomology groups of coherent sheaves andits applications

MI2016-2 Yoshiyuki KAGEI & Masatoshi OKITAAsymptotic profiles for the compressible Navier-Stokes equations on the whole space

MI2016-3 Shota ENOMOTO & Yoshiyuki KAGEIAsymptotic behavior of the linearized semigroup at space-periodic stationary solu-tion of the compressible Navier-Stokes equation

MI2016-4 Hiroki MASUDANon-Gaussian quasi-likelihood estimation of locally stable SDE

MI2016-5 Yoshiyuki KAGEI & Takaaki NISHIDAOn Chorin’s method for stationary solutions of the Oberbeck-Boussinesq equation