数理解析研究所講究録
第 1905 巻 2014 年 132-143

132

A computer assisted proof of the Kolmogorov problem ofincompressible viscous fluid
Yoshitaka Watanabe
Kyushu University
Abstract

A computer-assisted proof which proves the existence of non-trivial steady-state solutions for the Kolmogorov flows is presented. The method is based on the infinite-dimensional
fixed-point theorem using Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact
non-trivial solution with mathematical rigorous error bounds. All discussed numerical results are taken into account of the eﬀects of rounding errors in the floating point computations.

1

Introduction

Consider the Navier-Stokes equations:$u_{t}+uu_{x}+vu_{y}= \nu\Delta u-\frac{1}{\rho}p_{x}+\gamma\sin(\frac{\pi y}{b})$$v_{t}+uv_{x}+vv_{y}= \nu\triangle v-\frac{1}{\rho}p_{y}$$u_{x}+v_{y}=0$

,

,

,

(1)
(2)
(3)

and are velocity vector, mass density, pressure and kinematic viscosity,
is a constant representing the strength of the sinusoidal outer force. Also
. The flow region is a rectangle $[-a, a]\cross[-b, b]$andperiodicareboundary conditions
and theimposed in both directions. The aspect ratio is
where $(u, v)$ ,respectively and$\rho,$$*\xi$$p$$\nu$$\gamma$$:=\partial/\partial\xi(\xi=t, x, y)$$\triangle$$:=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$denoted by $:=b/a.$
The above equations (1-3) describe the Navier-Stokes flows in a two-dimensional flat torus
under a special driving force proposed by Kolmogorov [1, 5], [6, Chapter 5] and have a basic
solution which is written as$(u, v,p)=(k\sin(\pi y/b), 0, d)$ ,$\alpha$where
and is any constant. It is known that non-trivial solutions bifurcatefrom the basic solution at a certain Reynolds number, which is defined below, if and only if$0<\alpha<1[1]$ . Okamoto-Shoji [5] computed numerically bifurcation diagrams with the Reynolds
number as a bifurcation parameter varying the aspect ratio as a splitting parameter. They alsostrongly suggested stability of the bifurcating solutions for all $0<\alpha<1$ . Nagatou [2] took
an another approach to this stability problem by employing the theory of verified computationand showed that the stability of the bifurcating solutions is mathematical rigorously assured
for the cases of $\alpha=0.4$ , 0.7 and 0.8.
In the previous paper [10], we proposed a method to prove the existence and the local uniqueness of the steady-state solutions of the Navier-Stokes equations (1-3) for a given Reynolds$k$$:=b^{2}\gamma/(\pi^{2}v)$

$d$

133
number and aspect ratio by a computer-assisted proof with some verified results. It was also
the first theoretical results to the non-trivial solutions of the equations (1–3).
The aim of this paper is to apply our other verification method: FN-Int[3, 4, 9] to provethe existence of the steady-state solutions of problem (1-3) and to ascertain its eﬀectiveness inthe actual numerical computation.
In FN-Int, the equation is decomposed into the finite-dimensional part and the infinitedimensional error part, and if the both part lead to the retraction maps under suitable assumptions, an infinite-dimensional fixed-point theorem implies the existence of the solution
in a certain function set. In the self-validating process in computer, Newton-like iteration is
executed for the finite-dimensional part, and the computation comes down to solving interval
linear systems. Note that we have also proposed some verification algorithms which assure thelocal uniqueness of the solution in the enclosed set [11, 12] We will discuss about them infuture works. We also note that our verification methods described above can be formulated asa more general form and one may apply it to many kind of diﬀerential equations and integral
equations which can be transformed into fixed-point equations.
We admit that our study in this paper has some restrictions (a driving force, two-dimensional
rectangle region, boundary condition, etc however, we belieVe that our idea, not our results
themselves, will pave the way to a tool to study the global bifurcation structure for partial
diﬀerential equations arising in more practical, or even industrial problems.

2

Nondimensionalization and function spaces

The letter$0<\alpha<1$$T_{\alpha}$denotes the rectangular region $(-\pi/\alpha, \pi/\alpha)\cross(-\pi, \pi)$ for a given aspect ratio
satisfying $u=\phi_{y}$ and $v=-\phi_{x}$ so(see Fig. 1). Introducing the stream function

Figure 1: Shape of$\phi$$T_{\alpha}$that $u_{x}+v_{y}=0$ , the equations (1-3) can be rewritten as(4)$( \triangle\phi)_{t}-\nu\triangle^{2}\phi-J(\phi, \Delta\phi)=\frac{\gamma\pi}{b}\cos(\frac{\pi y}{b})$by cross-diﬀerentiating equations (1) and (2) and eliminating the pressure
form defined by$J(u, v) :=u_{x}v_{y}-u_{y}v_{x}$.

$p$. Here is a bilinear
$J$

(5)

134
The equation (4) is nondimensionalized by using change of variables$(x’, y’)=( \frac{\pi x}{b}, \frac{\pi y}{b}) , t’=\frac{\gamma b}{\nu\pi}t, \phi’(t’, x’, y’)=\frac{\nu\pi^{3}}{\gamma b^{3}}\phi(t, x, y)$and the Reynolds number$R:= \frac{\gamma b^{3}}{\nu^{2}\pi^{3}}$. After dropping the primes, an equation(6)$( \Delta\phi)_{t}-\frac{1}{R}\Delta^{2}\phi-J(\phi, \Delta\phi)=\frac{1}{R}\cos(y)$is obtained.
is equated to
We shall find steady-state solutions, whereproblem:following
nonlinear, namely consider theregion$(\Delta\phi)_{t}$$0$in equation (6) in the$T_{\alpha}$$\Delta^{2}\phi=-RJ(\phi, \Delta\phi)-\cos(y)$Assume that$\phi$

is subject to periodicity conditions in

$x$

in

$T_{\alpha}$.

(7)and , and the symmetry condition
$y$

(8)

$\phi(x, y)=\phi(-x, -y)$as well as the normalization $\int_{\Omega}\phi dxdy=0[2]$ , then the equation (7) has a trivial solution$\phi=-\cos(y)$ for any $R>0$ (Fig. 2).

The aim of this paper is to enclose a non-trivial solution

Figure 2: Shape of the trivial solution $\phi=-\cos(y)$ and stream line of$[\phi_{y}, -\phi_{x}]^{T}.$of (7) by computer-a.ssisted proof.

3

Function spaces

From the periodicity, the stream function. can be expanded to double Fourier series by$\phi$$\phi(x, y)=\sum_{(0,0)\neq(mn)\in \mathbb{Z}},a_{m,n}e^{im\alpha x+iny}, a_{m,n}\in \mathbb{C}.$

Note that if$\phi$

is a solution of (7),$\phi+c(\forall c\in \mathbb{C})$is also the solution. Then we exclude thecase $(m, n)=(O, 0)$ . By using Euler’s formula and symmetry condition $\phi(x, y)=\phi(-x, -y)$ , itholds that$\phi(x, y)=\sum_{(0,0)\neq(mn)\in \mathbb{Z}},a_{m,n}(\cos(m\alpha x+ny)+i\sin(m\alpha x+ny))$,

135$\phi(-x, -y)=\sum_{\backslash }a_{m,n}(\cos(m\alpha x+ny)-i\sin(m\alpha x+ny))(0,0)\neq(m,n)\in \mathbb{Z}.$Then adding equations and translating coeﬃcient$a_{m,n}$, we have$\phi(x, y)=\sum_{(0,0)\neq(m,n)\in \mathbb{Z}}a_{m,n}\cos(m\alpha x+ny)$.

Now decomposing indices and using the property of cosine together with replacing
obtain$\phi(x, y)=\sum_{1\leq n\leq\infty}a_{n}\cos(ny)+ \sum \sum a_{m,n}\cos(m\alpha x+ny)$$a_{m,n}$, we.

$1\leq m\leq\infty-\infty\leq n\leq\infty$

Consequently, we can define function space $X^{k}(k\geq 0)$ by the closure in
linear hull of all functions$\cos(m\alpha x+ny) , m\in \mathbb{N}_{0}, n\in \mathbb{Z}, (m, n)\neq(O, O)$$H^{k}(T_{\alpha})$of the.

Especially we define$X:=X^{3}.$For each$\psi\in X^{k}$can be represented by$\psi=\sum_{(m,n)\in Q}A_{mn}\cos(m\alpha x+ny) , A_{mn}\in \mathbb{R},$

where

,

(9)

$(\cos(m\alpha x+ny), \cos(k\alpha x+ly))_{L^{2}}=\{\begin{array}{ll}\frac{2\pi^{2}}{\alpha} if k=m and l=n0 else\end{array}$(10)$Q:=\{(m, n)\in \mathbb{Z}\cross \mathbb{Z}| \langle m=0andand-\infty\leq n\leq\infty"1\leq n\leq\infty^{J/}or\}$and it is noted that the base function of

for any $(m, n)$ , $(k, l)\in Q$ , where$(\cdot, \cdot)_{L^{2}}$$X^{k}$

satisfiesmeans the usual$L^{2}$-inner product in$T_{\alpha}.$Projection and an a priori error estimate

4

Letbe the finite-dimensional subspace of $X$ , which depends on a non-negative integer
parameter $N$ , defined by$X_{N}$$X_{N} := \{\sum_{(m,n)\in Q_{N}}A_{mn}\cos(m\alpha x+ny) A_{mn}\in \mathbb{R}\}$

,

(11)

where
$QN$

$:=\{(m, n)\in \mathbb{Z}\cross \mathbb{Z}|$$\langle m=0and1\leq m\leq Nand-N\leq n\leq N"1\leq n\leq N"or\}$Then$K :=\dim X_{N}=2N(N+1)$ .

.

(12)

136
Let

$X_{*}$

denote the orthogonal complement of
can be represented by$X_{N}$

in $X$ such that $X=X_{N}\oplus X_{*}$ , then for any$\psi_{*}\in X_{*}$$\psi_{*}=\sum_{(m,n)\in Q_{*}}A_{mn}\cos(m\alpha x+ny) , A_{mn}\in \mathbb{R}$

,

(13)

where
$Q_{*}:= \{(m, n)\in \mathbb{Z}\cross \mathbb{Z} (1\leq m\leq Nand0\leq m\leq NandN+1\leq m\leq\infty and-\infty\leq n\leq\infty\frac{N}{}\infty\leq n\leq-N-1"+1\leq n\leq\infty"or, or \}.$

Figure 3 indicates the area of(14)$Q_{*}.$

Areal $=\{(m, n)\in \mathbb{Z}\cross \mathbb{Z}|0\leq m\leq N$ and $N+1\leq n\leq\infty\},$$A_{I}e_{c}’\iota 2=\{(m_{\}}n)\in \mathbb{Z}\cross \mathbb{Z}|1\leq m\leq NaIld-\infty\leq n\leq-N-1\},$

Area3 $=\{(m, n)\in \mathbb{Z}\cross \mathbb{Z}|N+1\leq m\leq\infty$ andFigure 3: Area of$-\infty\leq n\leq\infty\}.$$Q_{*}$

Now we define the projection $Xarrow X_{N}$ by the N-th truncation of Fourier expansion. Note-projection:coinsides with usualthat by the orthogonality of of the basis$P_{N}$$H_{0}^{2}$$(\Delta(\psi-P_{N}\psi), \Delta\psi_{N})_{L^{2}}=0, \forall\psi_{N}\in X_{N}$,and we obtain the following a priori error estimate for the liner problem ofLemma 4.1 For each $g\in X^{0}$ let$\xi\in X^{4}$

the solution of$\Delta^{2}\xi=g$(15)$\Delta^{2}\xi=g.$, then$\Vert\xi-P_{N}\xi\Vert_{X}\leq C_{5}\Vert g\Vert_{L^{2}}$holds, where$C_{5}= \frac{1}{\alpha(N+1)}$

.

(16)

137
Some estimations for $X$

5
Since

$|\psi|_{H^{3}(\Omega)}^{2}=\Vert u_{xxx}\Vert_{L^{2}}^{2}+3\Vert u_{xxy}\Vert_{L^{2}}^{2}+3\Vert u_{xyy}\Vert_{L^{2}}^{2}+\Vert u_{yyy}\Vert_{L^{2}}^{2}$$= \frac{2\pi^{2}}{\alpha} \sum (\alpha^{6}m^{6}+3\alpha^{4}m^{4}n^{2}+3\alpha^{2}m^{2}n^{4}+n^{6})A_{mn}^{2}$$(m,n)\in Q$$= \frac{2\pi^{2}}{\alpha} \sum (\alpha^{2}m^{2}+n^{2})^{3}A_{mn}^{2},$$(m,n)\in Q$

we have(17)

Therefore since semi-norm
product of $X$ by$|\psi|_{H^{3}(\Omega)}$becomes norm of $X$ , we define the norm and the inner-$\Vert\phi\Vert_{X}:=|\psi|_{H^{3}(\Omega)},$$(u, v)_{X} :=(u_{xxx}, v_{xxx})_{L^{2}}+3(u_{xxy}, v_{xxy})_{L^{2}}+3(u_{xyy}, v_{xyy})_{L^{2}}+(u_{yyy}, v_{yyy})_{L^{2}},$

respectively.
For the norm$\Vert\cdot\Vert_{X}$, the following estimations hold.

Lemma 5.1 For any $\psi\in X$ , and$\forall\psi_{*}\in X_{*}$, it can be checked that$\Vert\psi\Vert_{L^{2}}\leq\alpha^{-3}\Vert\psi\Vert x, \Vert\psi_{*}\Vert_{L^{2}}\leq C_{1}\Vert\psi_{*}\Vert_{X},$$\Vert\psi_{x}$I$L^{2}\leq\alpha^{-2}\Vert\psi\Vert_{X},$$\Vert(\psi_{*})_{x}\Vert_{L^{2}}\leq C_{2}\Vert\psi_{*}\Vert_{X},$$\Vert\psi_{y}\Vert_{L^{2}}\leq C_{3}\Vert\psi\Vert_{X}, \Vert(\psi_{*})_{y}\Vert_{L^{2}}\leq C_{4}\Vert\psi_{*}\Vert_{X},$$\Vert\nabla\psi\Vert_{L^{2}}\leq\alpha^{-2}\Vert\psi\Vert_{X}, \Vert\nabla\psi_{*}\Vert_{L^{2}}\leq C_{2}\Vert\psi_{*}\Vert_{X},$$\Vert\nabla\psi_{x}\Vert_{L^{2}}\leq a^{-1}\Vert\psi\Vert_{X}, \Vert\nabla(\psi_{*})_{x}\Vert_{L^{2}}\leq C_{5}\Vert\psi_{*}||x,$$\Vert\nabla\psi_{y}\Vert_{L^{2}}\leq C_{6}\Vert\psi\Vert_{X}, \Vert\nabla(\psi_{*})_{y}\Vert_{L^{2}}\leq C_{7}\Vert\psi_{*}\Vert_{X},$$\Vert\triangle\psi\Vert_{L^{2}}\leq\alpha^{-1}\Vert\psi\Vert_{X}, \Vert\Delta\psi_{*}\Vert_{L^{2}}\leq C_{5}\Vert\psi_{*}\Vert_{X},$$\Vert\triangle\psi_{x}\Vert_{L^{2}}\leq\Vert\psi\Vert x, \Vert\triangle(\psi_{*})_{x}\Vert_{L^{2}}\leq \Vert\psi_{*}\Vert_{X},$$\Vert\Delta\psi_{y}\Vert_{L^{2}}\leq\Vert\psi\Vert_{X}, \Vert\triangle(\psi_{*})_{y}\Vert_{L^{2}}\leq \Vert\psi_{*}\Vert_{X},$where$C_{1}$$=$$\frac{1}{\alpha^{3}(N+1)^{3}},$$C_{2}$$=$$\frac{1}{\alpha^{2}(N+1)^{2}},$$C_{3}$$=$$\max\{1,$$C_{4}$$=$$\max\{\frac{1}{(N+1)^{2}},9\alpha^{2}(N+1)^{2}2\sqrt{3}\},$$C_{5}$$=$$\frac{1}{\alpha(N+1)},$$C_{6}$$=$$\max\{1,$$C_{7}$$=$$\max\{\frac{1}{N+1},$$\frac{2\sqrt{3}}{9\alpha^{2}}\},$In actual calculations,$\frac{1}{2\alpha}\},$$\frac{1}{2\alpha(N+1)}\}.$$L^{\infty}$-estimates proposed by Plum [7] are also needed.

138
Lemma 5.2 (Plum,1992 [7]) For any $\psi\in X$ , the following assertion holds true:$\Vert\psi\Vert_{L\infty}\leq C_{8}\Vert\psi\Vert_{L^{2}}+C_{9}\Vert\nabla\psi\Vert_{L^{2}}+C_{10}\Vert\Delta\psi\Vert_{L^{2}}$where$\Vert\cdot\Vert_{L}\infty$,

is the $\sup$-norm and$C_{8}= \frac{\sqrt{\alpha}}{2\pi}, C_{9}=\frac{1.1548}{\sqrt{3}}\sqrt{\frac{\alpha^{2}+1}{\alpha}}, C_{10}=\pi\frac{0.44722}{3}\sqrt{\frac{9\alpha^{4}+10\alpha^{2}+9}{5\alpha^{3}}}.$

Lemma 5.1 and Lemma 5.2 imply
Lemma 5.3 For$\forall\psi\in X$

and$L^{\infty}$-estimates immediately.

$\forall\psi_{*}\in X_{*}$, it is ture that$\Vert\psi\Vert_{L\infty}\leq C_{11}\Vert\psi\Vert_{X}, \Vert\psi_{*}\Vert_{L}\infty\leq C_{12}\Vert\psi\Vert_{X},$$\Vert\psi_{x}\Vert_{L}\infty\leq C_{13}||\psi\Vert_{X}, \Vert(\psi_{*})_{x}\Vert_{L}\infty\leq C_{14}\Vert\psi_{*}\Vert_{X},$$\Vert\psi_{y}\Vert_{L}\infty\leq C_{15}\Vert\psi\Vert_{X}, \Vert(\psi_{*})_{y}\Vert_{L^{\infty}}\leq C_{16}\Vert\psi_{*}\Vert_{X},$where$C_{11} = a^{-3}C_{8}+a^{-2}C_{9}+\alpha^{-1}C_{10}, C_{12} = C_{1}C_{8}+C_{2}C_{9}+C_{5}C_{10},$$C_{13} = \alpha^{-2}C_{8}+\alpha^{-1}C_{9}+C_{10}, C_{14} = C_{2}C_{8}+C_{5}C_{9}+C_{10},$$C_{15} = C_{3}C_{8}+C_{6}C_{9}+C_{10}, C_{16} = C_{4}C_{8}+C_{7}C_{9}+C_{10}.$

We mention about partial integrations at finite-dimensional part. Let$Y^{1} := \{v=\sum_{(m,n)\in Q}A_{mn}\sin(\alpha mx+ny)|A_{mn}\in \mathbb{R}, \Vert\nabla v\Vert_{L^{2}}<\infty\},$then it holds true.

Lemma 5.4$(\psi_{x}, \phi)_{L^{2}}=-(\psi, \phi_{x})_{L^{2}}, \psi\in X^{1}, \phi\in Y^{1}.$$(\psi_{y}, \phi)_{L^{2}}=-(\psi, \phi_{y})_{L^{2}}, \psi\in X^{1}, \phi\in Y^{1}.$$(\Delta\psi, \phi)_{L^{2}}=(\psi, \Delta\phi)_{L^{2}}, \psi, \phi\in X^{2}.$$(\triangle\psi, \Delta\phi)_{L^{2}}=(\Delta^{2}\psi, \phi)_{L^{2}}, \psi\in X^{4}, \phi\in X^{2}.$$(J(u, v), w)_{L^{2}}=(J(w, u), v)_{L^{2}}=-(J(u, w), v)_{L^{2}}, u, v, w\in X^{2}.$The following is an important property of Jacobian for (7).

Lemma 5.5$\forall\psi_{1},$$\psi_{2}\in X^{1},$$J(\psi_{1}, \psi_{2})\in X^{0},$namely$J(\psi_{1}, \psi_{2})$can be re-expanded by $\cos(m\alpha x+ny)((m, n)\in Q)$ .

(18)

139
6

Verification procedure

This section is devoted to detailed verification procedure of the steady-state Kolmogorov problem (7).

6.1

Matrices

For fixed$\phi_{N}\in X_{N}$, define $H,$ $D,$

$L,$ $G\in \mathbb{R}^{K\cross K}(1\leq i,j\leq K)$by$H_{ij}:=((\phi_{j})_{xxx}, (\phi_{i})_{xxx})_{L^{2}}+3((\phi_{j})_{xxy}, (\phi_{i})_{xxy})_{L^{2}}+$$3((\phi_{j})_{xyy}, (\phi_{i})_{xyy})_{L^{2}}+((\phi_{j})_{yyy}, (\phi_{i})_{yyy})_{L^{2}}$$D_{ij}:=(\triangle\phi_{j}, \triangle\phi_{i})_{L^{2}}$$L_{ij}:=(\phi_{j}, \phi_{i})_{L^{2}}$

,

(19)

(20)

,

(21)

,

$G_{ij}:=(\triangle\phi_{j}, \Delta\phi_{i})_{L^{2}}+R(J(\phi_{N}, \Delta\phi_{j})+J(\phi_{j}, \Delta\phi_{N}), \phi_{i})_{L^{2}}$

.

(22)

Residual and fixed-point formulation

6.2

By setting$r_{2N}:=-\triangle^{2}\phi_{N}-RJ(\phi_{N}, \triangle\phi_{N})-\cos y$$r_{2N}$,(23)

is able to be re-expanded as an element in $X_{2N}$ and we can compute its inner-product withand -norm by interval arithmetic.
of (7), setting
For fixed approximate solution$\{\phi_{i}\}_{i=1}^{K}$$L^{2}$$\phi_{N}\in X_{N}$$\phi=\phi_{N}+\psi$

we try to find residual term$\psi$,(24). Substituting (24) to (7), we obtain a residual equationin$\triangle^{2}\psi=-RJ(\phi_{N}+\psi, \Delta\phi_{N}+\triangle\psi)-\cos(y)-\triangle^{2}\phi_{N}$$T_{\alpha}$

.

(25)

By denoting the right hand side of (25) by$f(\psi):=-RJ(\phi_{N}+\psi, \triangle\phi_{N}+\triangle\psi)-\cos(y)-\Delta^{2}\phi_{N}$

,

(26)

from Lemma 5.5,
: $Xarrow X^{0}$ is continuous and maps any bounded set of $X$ to a bounded setofDenote $F:=\triangle^{-2}f$ : $Xarrow X$ , then $F$ becomes compact operator and problem (25) isequivalent to a fixed-point equation $\psi=F(\psi)$ in $X.$$f$

$X^{0}.$

6.2.1

Newton-like operator

By using the projection
finite-dimensional part

, the fixed-point residual equation (25) can be decomposed intoasand infinite-dimensional part

$P_{N}$$X_{N}$$X_{*}$$\{\begin{array}{l}P_{N}\psi = P_{N}F(\psi) ,(I-P_{N})\psi = (I-P_{N})F(\psi) .\end{array}$

Now we define Newton-like operator$\mathcal{N}_{N}$: $Xarrow X_{N}$ by$\mathcal{N}_{N}(\psi) :=P_{N}\psi-[I-F’[0]]_{N}^{-1}P_{N}(\psi-F(\psi))$,

140
and re-formulate the finite-dimensional part equivalently to$P_{N}\psi=\mathcal{N}_{N}(\psi)$.

Here $[I-F’[0]]_{N}^{-1}$ : $X_{N}arrow X_{N}$ is the inverse operator of $P_{N}(I-F’[O])$ : $Xarrow X_{N}$ whose
definition is restricted to. Next we definie Newton-like operator on $X$ by$T$$X_{N}$$T(\psi):=\mathcal{N}_{N}(\psi)+(I-P_{N})F(\psi)$

which is the compact map.
Candidate set

6.2.2

Let
be the set of $K$ -dimensional interval vector. A finite-dimensional set
to be a set of linear combinations of base functions inwith interval coeﬃcientsuch as$U_{N}\subset X_{N}$$\mathbb{I}\mathbb{R}$$X_{N}$$U_{N}:= \sum_{i=1}^{K}B_{i}\phi_{i}$,is taken$\{B_{i}\}_{1\leq i\leq K}$

(27)

where
. Herehas upper and lower bounds such thatis interpreted as
the set of functions in which each element is linear combination of
whose coeﬃcientof belongs to the corresponding interval
for each $1\leq i\leq K$ , namely,$\sum_{i=1}^{K}B_{i}\phi_{i}$$B_{i}=[\underline{B}_{i}, \overline{B}_{i}]$$B_{i}$$\{\phi_{i}\}_{1\leq i\leq K}$$[\underline{B}_{i}, \overline{B}_{i}]$$\phi_{i}$$U_{N}= \{\sum_{i=1}^{K}v_{i}\phi_{i}\in X_{N} v_{i}\in \mathbb{R}, v_{i}\in B_{i}, 1\leq i\leq K\}$

For $\alpha>0$ , a infinite-dimensional set$U_{*}\subset X_{*}$

6.2.3

(28)

and a candidate set $U\subset X$ is taken to be$U_{*}:=\{\psi_{*}\in X_{*}|\Vert\psi_{*}\Vert_{X}\leq\beta\}$$U:=U_{N}+U_{*}$

.

,

(29)
(30)

.

Verification condition

Theorem 6.1 Assume that the candidate set $U\subset X$ is defined by (29) and (27) with(30), and that any element $\psi\in U$ is represented by$\psi=\psi_{N}+\psi_{*}, \psi_{N}\in U_{N}, \psi_{*}\in U_{*}.$

Letof$d=[d_{i}]\in \mathbb{I}\mathbb{R}^{K}$

denote an interval enclosure of the set whose i-th component consists$\{(f(\psi)-f’[0]\psi_{N}, \phi_{i})_{L^{2}}\in \mathbb{R}|\psi\in U\}, 1\leq i\leq K$

If, for an interval vector $v=[v_{i}]\in I\mathbb{R}^{K}$ enclosing the solution
equation$Gx=d$ ,.

$x\subset \mathbb{I}\mathbb{R}^{K}$(31)

for the linear(32)

the conditions$v_{i}\subset B_{i}, 1\leq i\leq K$

,

(33)

and

Usup

$\Vert(I-P_{N})F(\psi)\Vert_{X}\leq\beta$$\psi\in$hold, then there exists a fixed-point of $F$ in $U.$

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7

Some verification results

We use Sun Fortran 95 Ver.8.6 Linux-i386 (supporting interval arithmetic) and the interval arithmetic toolbox INTLAB [8] Version 6 with MATLAB 7.14.0.739 $(R2012a)$ on Fujitsu
PRIMERGY TX300 S5 (CPU: Intel Xeon E55202. $27GHz$ , OS: Red Hat Enterprise Linux
Server release 5.6).
In the case of $\alpha=0.7$ , the basic flow (trivial solution) $-\cos(y)$ loses stability at a critical
Reynolds number
and another steady state bifurcates. Okamoto-Shoji strongly suggested
that there is no secondary bifurcation from this branch. Nagatou [2] also enclosed theinthe interval [3.011528364444, 3.011528364446].
Figure 4 shows the bifurcation diagram, wheremeans the absolute value of the coefficient to the base $\cos(ny)$ for obtained approximate solution.
$R_{c}$$R_{c}$$|A_{0,1}|$$|A_{0,1 ,\downarrow}|$$arrow R$

Figure 4: Bifurcation diagram for $\alpha=0.7$
Figure 5 shows verification results for $R=3.015$ , 3.02, 3.05, 3.1, 3.2, and 3.5 by IN-Linz.
We will report on various verification results for various a in another article.

Acknowledgments
This work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports,
Science and Technology of Japan (No. 24340018).

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142

R $=3.015$$\Vert\phi-\phi_{N}\Vert_{L}\infty\leq 2.0677\cross 10^{-11}$

R $=3.02$$\Vert\phi-\phi_{N}\Vert_{L}\infty\leq 1.2599\cross 10^{-11}$

R $=3.05$$\Vert\phi-\phi_{N}\Vert_{L}\infty\leq 9.8987\cross 10^{-12}$

R $=3.1$$\Vert\phi-\phi_{N}\Vert_{L}\infty\leq 1.085610^{-11}$

R $=3.2$$\Vert\phi-\phi_{N}\Vert_{L\infty}\leq 1.1318\cross 10^{-11}$

R $=3.5$$\Vert\phi-\phi_{N}\Vert_{L^{\infty}}\leq 1.4785\cross 10^{-11}$

Figure 5: Shape of the stream line of$[(\phi_{N})_{y}, -(\phi_{N})_{x}]^{T}$

when $\alpha=0.7,$ N $=20.$

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