数理解析研究所講究録
1009 巻 1997 年 1-21

1

The Linearized Boltzmann Equation with an External Force
Minoru Tabata
Department of Applied Mathematics, Faculty of Engineering,
Kobe University, Kobe 657 Japan田畑稔

神戸大学工学部情報知能工学科数学系\S 1 INTRODUCTION
The nonlinear Bolfzm
with an extern force describes theevolution of the density of rarefied gas acted upon by the external force. We
admit that the force depends on the space variable and the velocity variable.
by
We assume that the gas particles are confined in a bounded domain
being reflected perfectly from the boundary. The equation has the form,$al$$\partial Imequ\mathrm{a}ti_{\mathrm{o}n}$$\Omega\subset \mathbb{R}^{3}$a$\Omega$(NBE),$\partial \mathrm{f}/\partial \mathrm{t}+\Lambda \mathrm{f}=\mathrm{Q}(\mathrm{f},\mathrm{f}\rangle$

where we denote by
the unknown function which represents thedensity of gas particles that have a velocity
at timeand at a point$\mathrm{f}=\mathrm{f}(\mathrm{t},\mathrm{x}, \xi)$$\xi\in \mathbb{R}^{3}$$\mathrm{x}\in\Omega$.

$\mathrm{Q}=\mathrm{Q}(\cdot\cdot, \cdot)$$\mathrm{t}\geqq 0$

denotes the nonlinear collision operator1

2

and A is adifferential operator defined as follows:(1. 1)$\Lambda\equiv\xi\cdot\nabla_{\mathrm{x}}+\mathrm{F}\cdot\nabla_{\xi^{\gamma}}$where
has the form,$\mathrm{F}=\mathrm{F}(\mathrm{x}, \xi)$denotes the external force. We assume that$\mathrm{F}(\mathrm{x}, \xi)\equiv-\nabla\oint(\mathrm{x})+\mathrm{b}(\mathrm{x}, \xi)$

where
of

(1.2)

,

, and
is a sufficiently smooth, real-valued function ofis a sufficiently smooth, 3-dimensional-real-vector-valued function$\mathrm{x}\in\Omega$$\oint=\oint(\mathrm{x})$$=\mathrm{b}(\mathrm{x}, \xi)$$\mathrm{F}=\mathrm{F}(\mathrm{x}, \xi)$$(\mathrm{x}, \xi)\in\Omega\cross \mathbb{R}^{3}$such that$\mathrm{b}$2for each }$\xi\cdot \mathrm{b}(\mathrm{x}, \xi)=0$

Under the assumption of cut-off

.

(1.3)

$\in\Omega \mathrm{X}\mathbb{R}^{3}$$\xi\rangle$$\mathrm{x},$,rd potentials in the sense of$\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{l}2$$\mathrm{h}\mathrm{a}$, welinearize (NBE) around the equilibrium state. Substitutingin (NBE), applying (1.2-3), and dropping the nonlinear term, we
obtain the linearized Boltzmann equation with an external forc ,$\mathrm{M}\equiv \mathrm{e}^{-\phi-}(\mathrm{x})|\epsilon|^{2}/2$$\mathrm{f}=$$\mathrm{M}+_{\mathrm{M}^{1/2}\mathrm{u}}$$\mathrm{e}$a$\mathrm{u}/\mathrm{a}_{\mathrm{t}}=\mathrm{B}\mathrm{u}$

(LBE)

,

$\mathrm{B}\equiv-\Lambda+_{\mathrm{e}^{-\phi}(-}\nu+\mathrm{K})$

(1.4)

,

where $\nu=\nu(\xi\rangle$ is a multiplication operator, and is an integral operator.
andact only on the velocity variable, and have the following$\mathrm{K}$$\mathrm{K}$

$\xi$

$\nu$

properties1 2:
Lemma$1.1\alpha(\mathrm{i})$$\nu(\xi)\leqq\nu_{1}(1+|\xi|)$(ii)$\mathrm{K}$There exist positive constantsfor each$\xi\in \mathbb{R}^{3}$$\mathrm{L}^{23}(\mathbb{R}_{\xi})$(iii) $(-\nu+\mathrm{K})$ is a nonpositive operator in$\xi_{\mathrm{j}}\mathrm{e}^{-\mathrm{I}\epsilon}|^{2}/4,$

the j-th component of$\xi$$\mathrm{j}=1,2,3,$by$\xi_{\mathrm{j}},$, such that$\nu_{0}\leqq$.

is a self-adjoint compact operator inspanned by$\mathrm{j}^{=}\mathrm{O},1$$\nu_{\mathrm{j}},$$\mathrm{e}^{-1\xi}\mathrm{I}^{2}/4$$\mathrm{j}=1,2,3$, and, i.e.,.

$\mathrm{L}^{2}(\mathbb{R}\xi \mathrm{s}_{)}$

whose null space is$|\xi|^{22}\mathrm{e}^{-\mathrm{I}\xi|/4}$, where$\xi=(\xi_{1}, \xi_{2}, \xi_{3})$we denote.

The purpose of the present paper is to study decay of solutions to themixed problem for (LBE) with the perfectly reflective boundary condition. In
subject, we need to investigate the structure of theorder to study
spectrum of the operator B. We can prolong solutions of the following systemof ordinary differential equations in time by the law of perfect reflection:’$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$$\mathrm{d}\mathrm{x}/\mathrm{d}\mathrm{t}=\xi$,$\mathrm{d}\xi/\mathrm{d}\mathrm{t}=\mathrm{F}(\mathrm{x}, \xi)$,(SODE)and the continued characteristic curves of A is described by these prolonged

3
solutions (cf. p. 3913). Noting that spectral properties of A are deeply
connected to the continued characteristic curves of A and that the operator
is strongly \’influenced by A , we can reasonably conclude that spectral
properties ofmust be closely related to the behavior of the prolonged
solutions of (SODE).. Therefore we need to investigate the behavior of the
prolonged solutions of (SODE) (cf. \S 1 pp. $742- 746^{5}$ , pp. 754-756, and .
$\mathrm{B}$$\mathrm{B}$$\mathrm{p}$12736).
However it must be noted that the behavior of the prolonged solutions of(SODE) is very complicated; it is difficult to observe the solutions globally intime. Hence it is advisable to simplify the behavior of the prolonged solutions
by imposing some assumptions. We note that the complexity of the behavior of
the prolonged solutions depends largely on geometry of the boundary surface. For example, as the geometry of
becomes more complex, thesolutions of (SODE) are prolonged by being reflected fromin a more
complicated manner. Hence the behavior of the prolonged solutions is also
more complex. Conversely, as the geometry of
becomes simpler, thebehavior of the prolonged solutions is also simplified. For these reasons wewill impose the following assumption, in addition to the assumption thatisbounded:aa$\Omega$$\Omega$aa$\Omega$$\Omega$$\Omega$

Assumption$\Omega$.

$\partial\Omega$

is a 2-dimensional piecewise.ear manifold.

$\mathrm{l}\mathrm{i}\mathrm{n}$We note that the solutions of (SODE) considerably changes at the time(see \S 7 for the details). In orderwhen the particle
collides withto prevent the
curves of A from being very largely influenced
by the external force when the characteristic curves collide with, wewill impose the following assumption:$\mathrm{x}=\mathrm{x}(\mathrm{t})$a$\Omega$$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{C}l\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}_{\mathrm{S}}\mathrm{t}\mathrm{i}\mathrm{c}$aAssumption F. For almost all (X,$—$)$\in \mathrm{F}_{+}$,$\{(\gamma_{+}\mathrm{F}(\cdot,\cdot))(\mathrm{X},\overline{=}\rangle+(\gamma-\mathrm{F}(\cdot,\cdot\rangle)(\mathrm{x}_{-2\mathrm{X}},---(\mathrm{n}()\cdot\overline{=}\rangle \mathrm{n}(\mathrm{x}))\mathrm{I}\perp \mathrm{n}(\mathrm{x})$,$\Omega$4.

$\{(\gamma_{+}\mathrm{F}\mathrm{t}\cdot,\cdot))(\mathrm{X},--)--(\gamma_{-}\mathrm{F}(\cdot,\cdot)\rangle(\mathrm{X},----2(\mathrm{n}(\mathrm{X})\cdot\overline{=})\mathrm{n}1^{\mathrm{x}}\rangle\rangle\}//\mathrm{n}(\mathrm{X}\rangle$

. .

..

Here we denote byof A onto$i$$\gamma\pm^{\mathrm{t}\mathrm{h}\mathrm{e}}$

.

$..=$

.

$t.$

.

trace operators along the characteristic curves
5.

$\mathrm{F}_{\pm}\equiv|(\mathrm{x},\overline{=})\in \mathrm{F}(\partial\Omega)\cross \mathbb{R}^{3};\pm \mathrm{n}(\mathrm{x}_{)}\cdot--->0|$

a

.

(1.5)

We denote by
the outer unit normal ofat(), where (\dag er . By
is contained in an edge ofvirtue of Assumption and Assumption , we can simplify the behavior of
the prolonged solutions (in Remark 7.5 we will fully discuss the roles of theseassumptions). The main result of this paper is Main Theorem in \S 3, which is
as follows: the semigroup generated by decays exponentially in time.
Under the spatial periodicity condition, we have already investigateddecay of solutions of (LBE) in the case where. Hence, it seems tobe promising to attempt to apply the method in [4] also to the problem of thispaper. However, if we try to do so, then we immediately encounter thedifficulty which is caused not only by the fact that the external force depends
on the velocity variable but also by the fact that there is a possibility thatsome continued characteristic curves of tend to follow(see p. 3913).
The possibility presents a great difficulty in attacking our problem (we will
discuss this difficulty in \S 7 fully).
$\mathrm{n}=\mathrm{n}(\mathrm{x})$$\mathrm{E}$a$\mathrm{E}$$\Omega\rangle$a$\Omega$$\equiv\{\mathrm{x}\in$a$\mathrm{x}\in \mathrm{F}(\partial\Omega)\equiv\partial\Omega\backslash$$\Omega$a$\Omega;\mathrm{x}$$\Omega$$\Omega$$\mathrm{F}$$B$$\mathrm{b}(\mathrm{x}, \xi)\equiv 0^{4}$a$\Lambda$$\Omega$\S 2 PRELIMINARIES(1) Assumptions. We impose the following two assumptions on theexternal force in addition to Assumption

Assumption

denote by$\mathrm{x}_{\mathrm{i}}$$\oint$.

$\sup_{\mathrm{x}\mathrm{e}\Omega}|$a2$\oint(\mathrm{x})/\partial \mathrm{x}_{\mathrm{i}}$the i-th component ofAssumption$\mathrm{b}$.

$(\mathrm{i})$$\mathrm{F}$$\mathrm{x},$

a

(see

$\mathrm{r}\geqq 1$:$\mathrm{x}_{\mathrm{j}}|<+\infty,$$\mathrm{i}=1,2,3$$\sup|\mathrm{a}\mathrm{b}(\mathrm{x}, \xi)/\mathrm{a}_{\mathrm{x}_{\mathrm{i}}|}$$\mathrm{x}\in\Omega,\S^{\mathrm{g}_{\Gamma}}$$=1,2,3$ , for each$(1.2)\rangle$, where we denote by, i.e.,,$\mathrm{i},$$\mathrm{j}=1,2,3$$\mathrm{x}=(\mathrm{x}_{1},\mathrm{x}_{2^{\mathrm{X}_{3})}},$, where we.

$\sup|\partial \mathrm{b}(\mathrm{x}, \xi)/\partial\xi_{\mathrm{i}}|<+\infty,$$\mathrm{i}$$\mathrm{x}\in\Omega.\mathrm{e}\mathrm{g}\mathrm{r}$$\xi_{\mathrm{j}}$the i-th component of$\xi,$$\mathrm{i}=$51,2,3.
(ii)$\nabla_{\xi}\cdot \mathrm{b}(\mathrm{x}, \xi)=0$

for each$(\mathrm{x},$.

$\xi\rangle\in\Omega \mathrm{X}\mathbb{R}^{3}$

we denote theset of all bounded (compact, respectively) linear operators from a Banach
space X to a Banach space Y. For simplicity, we writeandas
B(X,X) and C(X,X) respectively. By,, we denote a Hilbert space of(2) Function spaces. By B(X,Y)$(\mathbb{C}(\mathrm{X},\mathrm{Y})$, respectively)$\mathrm{B}(\mathrm{X})$$\alpha\geqq 0$$\mathrm{E}_{\mathrm{a}}$$\mathrm{c}\mathrm{o}.\mathrm{m}$plex-valued functions of(recall Remark 2.1,$(\mathrm{i}\mathrm{i})$$(\mathrm{x}, \xi)\in\Omega\cross \mathbb{R}^{3}$

with the following inner product):$( \mathrm{u},\mathrm{v})_{\mathrm{a}}\equiv\int\Omega\cross \mathrm{R}^{3}\mathrm{u}(\mathrm{x}, \xi)\overline{\mathrm{v}(\mathrm{x},\xi\rangle}(1+\mathrm{E}(\mathrm{X}, \xi))^{\mathrm{a}}$where$\mathrm{E}(\mathrm{x}, \xi)\equiv\oint(\mathrm{x})+|\xi|^{2}/2$

norm of operators of$\mathbb{C}(\mathrm{X}\rangle$$\mathrm{B}(\mathrm{E}_{\mathrm{a}})$. Define. Writedxd$||\mathrm{u}||_{\mathrm{a}}\equiv((\mathrm{u},\mathrm{u})_{\mathrm{a}})^{1/2}$$||\cdot||,$as$|||\cdot|||$$||\cdot||_{0},$

$\xi$

(2.1)

,

. Write$|||\cdot|||_{0}$,$|||\cdot|||_{\mathrm{a}}$as therespectively forsimplicity.

. We denote the domain of an operator. Let us define the domain of A (seeas follows:(3) The domains ofby$D(\mathrm{L}\rangle$$D(\Lambda)\equiv|\mathrm{u}=$$(1.1)\rangle$$\mathrm{u}(\mathrm{x}, \xi)\in \mathrm{E}_{\mathrm{a}}$; A$\mathrm{u}\in$$\mathrm{E}_{\alpha}$$\mathrm{L}$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t},\mathrm{o}\mathrm{r}\mathrm{S}$, and$\mathrm{u}=\mathrm{u}(\mathrm{x}, \xi)$satisfies the perfectly reflective

boundary condition,,$(\gamma_{+}\mathrm{u}(\cdot,\cdot\rangle)(\mathrm{x},--)-=(\gamma-^{\mathrm{u}}(\cdot,\cdot))(\mathrm{X},----2(\mathrm{n}(\mathrm{X})\cdot\overline{=}\backslash )\mathrm{n}(\dot{\mathrm{x}}_{)\rangle}$for.
operator,$\mathrm{a}.\mathrm{e}$. See (1.5) for$(\mathrm{X}, ---)\in \mathrm{F}_{+}|$

A

as follows:(PRBC) for$\mathrm{n}(\mathrm{X})$.

$(\mathrm{X},---)\in \mathrm{F}_{+}|$$\mathrm{F}_{\pm}$$\equiv-\Lambda+_{\mathrm{e}^{-t}(}-\nu$$D(\mathrm{A}\rangle\equiv|\mathrm{u}=\mathrm{u}(\mathrm{x}, \xi)\in \mathrm{E}_{\mathrm{a}}$$\mathrm{a}.\mathrm{e}$

and

; Au

(PRBC). We define the domain of the),$\in \mathrm{E}_{\alpha}$

(2.2)
, and

$\mathrm{u}=\mathrm{u}(\mathrm{x}, \xi)$satisfies. Applying Remark 2.1, (ii), we deduce that$\mathrm{K}\equiv \mathrm{e}^{-_{\phi}}\mathrm{K}\in \mathrm{B}(\mathrm{E}_{\mathrm{a}},\mathrm{E}_{\mathrm{a}}+_{1})$, for each$\mathrm{a}\geqq 0$

,

(2.3)

6
in the same way as Lemma 2.1,$(\mathrm{i}\mathrm{v})^{4}$. Hence, we can define$D(\mathrm{B}\rangle$$\equiv D(\mathrm{A})$(see(1.4) .
$)$

(4) The purely imaginary point spectrum of B. Noting thatisbounded, and applying Assumption, we see thathas no axis of symmetry. Making use of this fact, and performing calculations similar to those in, we can obtain the following lemma (see (2.1) for
Theorem):$\Omega$$\Omega$$4.1^{6}$$\mathrm{E}(\mathrm{x},$Lemma 2.2. The intersection of \dager
spectrum of is equal to }. The null space of$\mu\in \mathbb{C};{\rm Re}\mu\geqq$$|0$$\mathrm{B}$$\Omega$.

$0$} and the pointis spanned by$\mathrm{B}$$\xi)$$\mathrm{e}^{-\mathrm{E}(\mathrm{x},\xi)/}2$

and$\mathrm{E}(\mathrm{x}, \xi)\mathrm{e}^{-\mathrm{E}(\xi}\mathrm{x},)/2$

We denote by the projection operator (in ) upon the null space of B. It
follows from the lemma above and Remark 2.1, (ii) that$\mathrm{P}$$\mathrm{E}_{0}$$\mathrm{P}\in \mathbb{C}(\mathrm{E}\alpha’ \mathrm{E}_{\beta}\rangle$

for each$\alpha$, $\beta\geqq 0$ .

(2.4)

\S 3 MAIN THEOREM
We denote: by(in$\mathrm{E}_{0}$$\mathrm{E}_{\mathrm{a},\perp}$

the set of all functions of$\mathrm{E}_{\mathrm{a}}$which are perpendicular) to the null space of B. In what follows throughout the paper, we denote

some positive constants by

and we use the letter as a generic constantreplacing any other constants by . The following theorem is the main resultof this paper:$\mathrm{c}$,$\mathrm{c}$$\mathrm{c}$, the operator
generates a
Theorem. For each
, which satisfies that, for eachcontinuous semigroup in,$\alpha\geqq 0$$\mathrm{M}\mathrm{a}\mathrm{i}\acute{\mathrm{n}}$$\mathrm{B}$$\mathrm{E}_{\mathrm{a}}$$\mathrm{u}_{0}\in \mathrm{E}_{\mathrm{a}.\perp}$, for each$\mathrm{e}^{\mathrm{t}\mathrm{B}}\mathrm{u}_{0a\perp}\in \mathrm{E},’||\mathrm{e}^{\mathrm{t}\mathrm{B}}\mathrm{u}_{0}||_{\mathrm{a}}\leqq \mathrm{c}||\mathrm{u}|0|_{\mathrm{a}}\exp(-\mathrm{C}_{3.0}\mathrm{t})$$\mathrm{W}^{\vee}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{C}3.0$

is a positive constant’$\mathrm{i}\dot{\mathrm{n}}\mathrm{d}\mathrm{e}_{\mathrm{P}}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}$of and$\mathrm{t}$$\mathrm{u}_{0}$.

$\mathrm{t}\geqq 0$,$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}.\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}$7
In order to prove the Main Theorem, we will apply the following lemma,
which will be proved in \S 5 (see (2.2) for A):

Lemma 3. 1. (i) For any, the operator A generates a strongly
continuous semigroup in , which satisfies (see Remark 2.1, (ii), for ),$\alpha\geqq 0$$\mathrm{E}_{\mathrm{a}}$$\mathrm{c}_{2.1}$, for each$|||\mathrm{e}^{\mathrm{t}\mathrm{A}}|||_{\alpha}\leqq\exp(^{-\mathrm{c}_{2.1}\mathrm{t}})$(ii) Let$0$. If$\alpha,$$\mathrm{c}_{3.1}$

and$\mathrm{C}$be constants such thatis a continuous function from$\mathrm{f}=\mathrm{f}(\mathrm{t})$$\mathrm{t}\geqq 0$$\mathrm{a}\geqq 0,$$[0,+\infty)_{\mathrm{t}}$to.

$\mathrm{c}_{2.1}>_{\mathrm{C}_{3.1}}>0$, and$\mathrm{C}>$such that (see (2.3)$\mathrm{E}_{\mathrm{a}}$for K)$||\mathrm{f}(\mathrm{t})\mathrm{N}_{\mathrm{a}}\leqq \mathrm{C}\exp(^{-}\mathrm{c}_{3}.\mathrm{t})1$’

for each$\mathrm{t}\geqq 0$

,

then

Il

$\int_{0}^{\mathrm{t}}\mathrm{e}^{(\mathrm{t}-\mathrm{s}\rangle}\mathrm{K}\mathrm{A}\mathrm{f}(\mathrm{s})\mathrm{d}_{\mathrm{S}}||_{\mathrm{a}}+1\leqq \mathrm{c}\mathrm{C}\exp(-\mathrm{c}_{3}.\mathrm{t})1$’(iii) Write$\mu^{=_{\mathit{7}^{+\mathrm{i}\delta}}}$,$\gamma$$\int_{arrow}^{+\infty}||(\mu-\mathrm{A})^{-}\mathrm{f}1||^{2}\mathrm{d}$

,

a

a,

$\in \mathbb{R}$. Iffor each$\beta\equiv\gamma^{+}\mathrm{c}_{2.1}>0$$\int_{\infty}^{+\infty}||(\mu-\mathrm{A}^{*})^{-1}\mathrm{f}||^{2}\mathrm{d}$a$\mathrm{t}\geqq 0$

and.

$\mathrm{f}\in \mathrm{E}_{0}$$\leqq \mathrm{c}||\mathrm{f}||^{2}/\beta$, then,.

In addition to the lemma above, we need the following key lemma:

Lemma 3.2.
function offor each$\mathrm{P}$$\mu\in$$\mu\in$]$\mathrm{D}$is an nalytic operator-valued}, and satisfies that$\mathrm{L}=\mathrm{L}(\mu)\equiv(\mathrm{K}-\mathrm{P})(\mu-\mathrm{A})^{-1}J$]$\mathrm{D}\equiv|\mu\in \mathbb{C};{\rm Re}\mu\geqq-\mathrm{C}_{2.1^{/}}2$

and that$|||\mathrm{L}^{4}(\mu)|||arrow 0$

as$\dot{\mathrm{a}}$$\mathrm{L}^{4}(\mu)\in \mathbb{C}(\mathrm{E})0$$|\mu|arrow+\infty,$$\mu\in$]$\mathrm{D}$.

roof of the Main Theorem. It follows from Lemma 3.1, (i), thatis an analytic operator-valued function ofresolvent equation,$-\mathrm{A})^{-1}$$\mu\in$$(\mu$

D. Hence we can set the( $\mu-\mathrm{B}\rangle^{-1}=(\mu-\mathrm{A})+(\mu--1\mathrm{A})^{-1}(1-\mathrm{L}(\mu))^{-}\mathrm{L}(\mu\rangle 1$,

(3.1)

8
. We consider the operator in place of , inandorder to remove the null space of (cf. p. $1833^{4}$). By (2.3-4) and Lemma 3.1,(i), we easily see thatgenerates a strongly continuous semigroup in,which is represent-ed in terms of the inverse Laplace transformation of (3.1).
Applying Lemmas 3.1-2 and Lemma 2.2 in the same way as pp. $1833- 1834^{4}$ ,where$\mu\in \mathrm{D}$$\mathrm{B}\underline{=}\mathrm{B}-\mathrm{P}$$\mathrm{B}$$\mathrm{B}$$\mathrm{B}$$\mathrm{B}$$\mathrm{E}_{\alpha}$

we obtain the theorem.
Remark 3.3. (i) In$\mathrm{p}\mathrm{p}.\cdot$

1833-18344$\mathrm{w}\mathrm{e}..,\mathrm{d}\mathrm{o}\wedge$

not.eed to directly apply the$\mathrm{n}$spatial periodicity condition. Henc. , without the aid of the condition, we can$\mathrm{e}$apply Lemmas 3.1-2 in the proof above. For the same reason as

consider the 4-th power(ii) We can$\mathrm{L}_{\mathrm{K}}(\mu\rangle$$\mathrm{L}^{4}(\mu)$and$6^{4}$, wein Lemma 3.2. Cf. pp. 433-4343.

$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}-\mathrm{l}\mathrm{e}\mathrm{L}\langle\mu$$\equiv \mathrm{K}(\mu-\mathrm{A})$\S)

as follows:$\mathrm{L}(\mu)=\mathrm{L}_{\mathrm{K}}(\mu-1)+\mathrm{L}_{\mathrm{P}}(\mu)$, where. We can easily derive$\mathrm{L}_{\mathrm{P}}(\mu)\equiv-\mathrm{P}(\mu-\mathrm{A})$

Lemma 3.2 from the.following (3.2-5):$\mathrm{L}_{\kappa^{4}}(\mu)\in \mathbb{C}(\mathrm{E}_{0})$$|||\mathrm{L}\kappa^{4}(\mu)|||arrow 0$

as$\mathrm{L}_{\mathrm{P}}(\mu)\in \mathbb{C}(\mathrm{E}_{0})$$|||\mathrm{L}_{\mathrm{P}}(\mu)|||arrow 0$

asfor each$\mu\in 1\mathrm{D}$$|\mu|arrow+\infty,$

for each$\mu\in \mathbb{D}$$\mu\in \mathrm{D}$$|\mu|arrow+\infty,$

(3.2)

,

,

(3.3)

(3.4)

,

$\mu\in 1\mathrm{D}$

.

(3.5)

Making use of (2.4) and Lemma 2.2, and performing calculations similar to,

but much easier than, those in proving (3.2-3), we can obtain (3.4-5). Hence
we will prove (3.2-3) only. (3.2-3) will be proved in \S 6-9.

\S 4. THE SOLUTIONS TO (SODE)
By$(\mathrm{C}\mathrm{P})$

we denote the Cauchy problem for (SODE) with the initial data,$(\mathrm{x}, \xi)(0_{)}=(\mathrm{X},---)\in(\Omega\cross \mathbb{R}^{3})\mathrm{U}\mathrm{F}_{-}$.

$(4.1\rangle$9
See (1.5) for$\mathrm{F}_{-}$. Recall Remark 2.1, (iv). In the same way as pp. $1284- 1285^{6}$ ,we can prolong the solution of$(\mathrm{C}\mathrm{P})$by the law of perfect reflection (cf.

(PRBC) , i.e., by$)$$(\mathrm{x}(_{\mathrm{S}^{+_{0}}}), \xi(_{\mathrm{S}^{+}}0))$$=(\mathrm{x}(\mathrm{s}^{-\mathrm{o}_{)}}, \xi(\mathrm{s}-0)-2(\mathrm{n}(\mathrm{x}(\mathrm{s}-0))\cdot\xi(\mathrm{s}-\mathrm{O}))\mathrm{n}\langle_{\mathrm{X}}(\mathrm{S}-\mathrm{o})))$

where$\mathrm{x}(\mathrm{s}\pm \mathrm{O})\in \mathrm{F}$solution of$(\mathrm{C}\mathrm{P})$(a). See$\Omega$\S 1 for$\mathrm{F}$(a$\Omega$,(4.2)). We denote the prolongedby$(\mathrm{x}, \xi)=(\mathrm{x}(\mathrm{t},\mathrm{X},---), \xi(\mathrm{t},\mathrm{X},\Xi))$

a

.

(4.3)

a

into four disjoint subsets as follows:;, whereand, see \S 1. If (. For
does not go) an infinite, and if
into
does not collide with (number of times in a finite time interval, then we can prolong the solution of(
globally in timein the same way as pp. 1284-12856.
We can decompose$\Omega\cross \mathbb{R}^{3}$$\mathbb{R}^{3}=\mathrm{E}\mathrm{U}\mathrm{F}_{0-}\mathrm{U}\mathrm{F}_{+}\mathrm{U}\mathrm{F}$$\mathrm{n}(\mathrm{x})\cdot\xi=0\}$$\mathrm{E}\mathrm{U}\mathrm{F}_{0}$$\mathrm{E}\equiv \mathrm{E}(\partial\Omega\rangle$$\mathrm{E}(\partial\Omega)$$\mathrm{x},$$\mathrm{x}=\mathrm{x}(\mathrm{t},\mathrm{X},$$\mathrm{X}\mathbb{R}^{3}$$\Omega\cross$$\mathrm{F}_{0}\equiv|(\mathrm{x}, \xi)\in \mathrm{F}(\mathrm{a}\Omega)\mathrm{X}\mathbb{R}^{3}$$\xi\rangle=(\mathrm{x}(\mathrm{t},\mathrm{X}, ---), \xi(\mathrm{t},\mathrm{X}, ---))$a$\mathrm{F}$$—\rangle$$\Omega$$\mathrm{t}\in \mathbb{R}$$\mathrm{C}\mathrm{P}\rangle$\S 5. THE OPERATOR A
Proof of Lemma 3.1. Applying the conservation law of energy (see(2.1) ,
$)$

(5.1)

,

$\mathrm{E}(\mathrm{X},---)=\mathrm{E}(\mathrm{x}(^{-\mathrm{t},\mathrm{X},),\xi}\overline{=}(-\mathrm{t},\mathrm{X},\Xi))$

and Lemma 4.2, (ii), we easily see that A (see (2.2)) generates a strongly. The semigroup
continuous semigroup in
for each
has the form,$\mathrm{E}_{\mathrm{a}}$$(\mathrm{e}^{\mathrm{t}\mathrm{A}}\mathrm{f}(\cdot,\cdot))(\mathrm{X},---\rangle$

where$=\mathrm{f}(\mathrm{x}(^{-\mathrm{t}},\mathrm{X},---\rangle, \xi(-\mathrm{t},\mathrm{X},--)-)\mathrm{e}(\mathrm{t},\mathrm{X},\overline{=}\rangle$$\mathrm{e}(\mathrm{t},\mathrm{X}, ---)\equiv\exp(-\int_{0}^{\iota}\mathrm{e}^{-}\phi \mathrm{t}\mathrm{x}(-\mathrm{s}\lambda\Xi))\nu(\xi(-\mathrm{s},\mathrm{X},$implies that$\mathrm{e}^{\mathrm{t}\mathrm{A}}$$\mathrm{a}\geqq 0$$\overline{=}\rangle\rangle$$\mathrm{d}\mathrm{s}\rangle$

,

(5.2)

. Remark 2.1, (ii),10,$\mathrm{e}(\mathrm{t},\mathrm{X},---)\leqq\exp(^{-\mathrm{c}_{2.1}\mathrm{t}})$for each$\mathrm{t}\geqq 0$

.

(5.3)

Applying (5.3), (2.3), Lemma 4.2, (ii), and (5.1), we can obtain Lemma 3.1 inthe same way as Lemma 3.14 and Lemma$3.3^{4}$.

Restricting the domain of integration of the Laplace transformation of (5.2)

within a Lebesgue measurable set$\mathrm{M}\subseteq[0,+\infty)$,

we define the following

operator:$( \mathbb{R}(\mu,\mathrm{M})\mathrm{f}(\cdot,\cdot))(\mathrm{x},---)\equiv\int_{\mathrm{t}\in \mathrm{M}}\mathrm{R}(\mu,\mathrm{t},\mathrm{X},\underline{=})\mathrm{f}(\mathrm{X}(-\mathrm{t},\mathrm{X},---),$$\xi(-\mathrm{t},\mathrm{X},---\rangle\rangle \mathrm{d}\mathrm{t},$$(5.4)$

where
D. See Lemma 3.2 for D. We canobtain the following lemma in the same way as Lemma 3.2,:$\mathrm{R}(\mu,\mathrm{t},\mathrm{X},---)\equiv \mathrm{e}(\mathrm{t},\mathrm{x},$$\overline{=}\rangle$$\exp(-\mu \mathrm{t}),$$\mu\in$$(\mathrm{i})^{4}$

Lemma 5.1. If$\beta\equiv{\rm Re}\mu^{+_{\mathrm{C}_{2.1}}}>0$

and$\mathrm{M}\subseteq[\mathrm{o},+\infty)$$||| \mathbb{R}(\mu,\mathrm{M}\rangle|||\leqq\int \mathrm{e}^{-\rho \mathrm{t}}\mathrm{t}\in \mathrm{M}\mathrm{d}\mathrm{t}$, then.

\S 6. DISCUSSION ON (3.2-3)
We will seek estimates which imply (3.2-3). Consider operators of thefollowing form:
(6.1)

,

$\mathbb{G}(\mu,\mathrm{T})\equiv\prod_{\mathrm{j}^{=}1}^{4}|\mathrm{e}^{-}\phi(\mathrm{x}\mathrm{k}_{\mathrm{j}}\mathbb{R}(\mu,[0,\mathrm{T}])\}$

where$\mu\in 1\mathrm{D},$$1\leqq \mathrm{T}<+\infty$

Here we denote by($\cdot,$$\mathrm{Q}(\mathbb{R}_{\epsilon^{3}})$,

and$\xi\in \mathbb{R}^{3}$$\mathrm{L}^{2}(\mathbb{R}_{\epsilon^{3}})$.

$\mathrm{f}^{=}\mathrm{f}(\xi)$anddenote the inner
are infinitely differentiable functions of, where the brackets$\mathrm{g}^{=}\mathrm{g}(\xi\rangle$

which have compact supports. By$\mathrm{A}_{2}\mathrm{A}_{1}$for operators$\mathrm{A}_{\mathrm{j}},$. See Lemma 3.2 for D.

the set of all the one-rank operators of the form, (ku$\rangle\rangle(\mathrm{x}, \xi)^{=}(\mathrm{u}(\mathrm{x}, \cdot),\mathrm{f}(\cdot))\mathrm{g}(\xi\rangle$

product in$\mathrm{j}=13,\ldots,4$$\mathrm{k}_{\mathrm{j}}\in \mathrm{Q}(\mathbb{R}_{\xi}),$$\mathrm{j}_{-}=1,\ldots,\mathrm{m}$$\prod_{\mathrm{j}=1}\mathrm{A}\mathrm{m}\mathrm{j}$$(\cdot, \cdot)$

we denote the product$\mathrm{A}_{\mathrm{m}}\mathrm{A}_{\mathrm{m}-1}$. Making use of Lemma 1.1, (ii), and11
Theorem$\mathrm{V}\mathrm{I}.13^{16}$, we deduce that the operator

with a finite sum of operators of$\mathrm{Q}(\mathbb{R}_{\xi}\rangle 3$$\mathrm{K}$can be approximated in$\mathrm{B}(\mathrm{E}_{0})$. Applying this result and Lemma 5.1,

we can derive (3.2-3) from the following (G.1-2):
for each$\mu\in \mathrm{D}$as $|\mu|arrow+\infty,$$\mu\in \mathrm{D}$$\mathrm{G}(\mu,\mathrm{T}\rangle\in \mathbb{C}(\mathrm{E}_{0})$$|||\mathbb{G}(\mu,\mathrm{T})|||arrow \mathrm{o}$and$\mathrm{T}\in[1,+\infty)$

(G.1)

,

, for each $\mathrm{T}\in[1,+\infty)$ .

(G.2)

Hence, we have only to show (G.1-2), which will be proved in \S 9.
, i.e., we write
as the variable of. In the same way as p. $1837^{4}$ , we canextract the integration kernel ofas follows:

We write$(\mathrm{X}_{4}, \xi_{4})\in\Omega\cross \mathbb{R}^{3}$$\mathbb{G}$$\mathbb{G}(\mu,\mathrm{T})\mathrm{u}$$(\mu,\mathrm{T})\mathrm{u}=(\mathbb{G}(\mu,\mathrm{T})\mathrm{u}(\cdot , ))(\mathrm{x}_{4}, \xi 4)$$\mathbb{G}(\mu,\mathrm{T})$$(\mathbb{G}(\mu,\mathrm{T})\mathrm{u}(\cdot$.

$\cdot\rangle\rangle$($\mathrm{x}_{4},$, (6.2)$\xi_{4}\rangle=\int_{0\leqq \mathrm{g}|\leqq}\leqq \mathrm{T},|\eta,\Gamma,$ $\mathrm{j}^{=1},\ldots,4\mathrm{t}\mathrm{G}\mathrm{u}(\mathrm{X}\xi_{0}0’)\mathrm{d}\mathrm{d}\eta$$\mathrm{c}=\mathrm{G}(\mu,\mathrm{x}_{4}, \xi 4;\eta_{4},\mathrm{t}\eta_{3}4" \mathrm{t}\eta_{2}3" \mathrm{t}_{2}, \eta 1’)\mathrm{t}_{1}$$\equiv\prod_{\mathrm{j}=1}^{4}|\mathrm{e}-\emptyset(\mathrm{X}_{\mathrm{j}})_{\mathrm{k}(\mathrm{j}\xi_{\mathrm{j}}},$$\mathrm{x}_{\mathrm{j}}\equiv \mathrm{x}(^{-\mathrm{t}_{\mathrm{j}}}+_{1},\mathrm{X}_{\mathrm{j}}+_{1}, \mathrm{v}_{\mathrm{j}}+1),$

where dt$—\mathrm{d}\mathrm{t}_{1}\cdots \mathrm{d}\mathrm{t}_{4}$, anddenote the integration kernel of$\mathrm{k}_{\mathrm{j}}(\xi_{\mathrm{j}}, \eta_{\mathrm{j}})\mathrm{f}(\mathrm{x}\eta_{\mathrm{j}}\mathrm{j}’\rangle$$=1,\ldots,4$$\mathrm{d}\eta \mathrm{j}’ \mathrm{j}=1,\ldots,4$.

$\mathrm{j}=0,\ldots,3$$\xi_{\mathrm{j}}\equiv\xi(-\mathrm{t}_{\mathrm{j}}+1^{\mathrm{X}}’ \mathrm{j}+_{1}, \eta \mathrm{j}+_{1}),$$\equiv \mathrm{d}\eta 1\ldots \mathrm{d}\eta_{4}$$\mathrm{d}\eta$$\eta \mathrm{j})\mathrm{R}(\mu,\mathrm{t}_{\mathrm{j}^{\mathrm{X}}\mathrm{j}},,$$\mathrm{k}_{\mathrm{j}}\in \mathrm{Q}(\mathbb{R}^{3}),$. Recall (4.3). By$\mathrm{j}^{=1}\xi’\ldots,4$$\mathrm{r}>0$, i.e.,\S 7. ESTIMATES FOR$|\eta|\leqq \mathrm{r}$$(6.3)$(6.4),$\mathrm{k}_{\mathrm{j}}(\xi_{\mathrm{j}}, \eta_{\mathrm{j}})$$(\mathrm{k}_{\mathrm{j}}\mathrm{f}(\cdot, ))(\mathrm{x}_{\mathrm{j}},$is a constant so large that, can be contained in $|\xi;|\xi|\leqq\Gamma|\cross|\eta$ ;$\eta_{\mathrm{j}}\rangle\},$we$\xi_{\mathrm{j}}\rangle=\int$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{k}_{\mathrm{j}}(\cdot, )$,$\mathrm{j}$}.

$\mathrm{J}=\mathrm{J}(\mathrm{t},\mathrm{X},$$\overline{=}\rangle$. See (4.4).
The purpose of this section is to obtain estimates for, X, $—,$ respectively,
By
we denote the j-th components of$\mathrm{J}=\mathrm{J}(\mathrm{t},\mathrm{x},---)$$\xi_{\mathrm{j}},$$\mathrm{X}_{\mathrm{j}},\underline{\overline{-}}_{\mathrm{j}}$$\mathrm{x},$$\xi$$\mathrm{x}_{\mathrm{j}},$$\mathrm{j}=1,2,3$. We denote the$=1,\ldots,6$ ,andi.e., if

$4\leqq \mathrm{j}\leqq 6,$$1\leqq \mathrm{i},$the.

$(\mathrm{i},\mathrm{j})$$\mathrm{j}\leqq 3$

component of, then$\mathrm{n}\mathrm{m}_{\mathrm{i}\mathrm{j}}(\mathrm{t},\mathrm{X},$$—\rangle$$\equiv$$\mathrm{r}=\mathrm{J}(\mathrm{t},\mathrm{x}, ---)$

by$\mathrm{m}_{\mathrm{i}\mathrm{j}}(\mathrm{t},\mathrm{X}, ---)\equiv\partial \mathrm{X}_{\mathrm{i}}(^{-\mathrm{t}},\mathrm{X},$a$\mathrm{x}_{\mathrm{i}}(^{-\mathrm{t}},\mathrm{X}, \overline{=})/\partial_{-}^{-}-_{\mathrm{j}-3}$$\mathrm{m}_{\mathrm{i}\mathrm{j}}=\mathrm{m}_{\mathrm{i}\mathrm{j}}(\mathrm{t},\mathrm{X}, ---),$$—\rangle$ $/\mathrm{a}\mathrm{X}_{\mathrm{j}}$

. If

. If

$\mathrm{i},$$\mathrm{j}$$1\leqq \mathrm{i}\leqq 3$$4\leqq \mathrm{i}\leqq 6$

and 1

12

$\leqq \mathrm{j}\leqq 3$$\underline{=})\equiv$

bya, then$\xi_{\mathrm{i}3}-(-\mathrm{t},\mathrm{X},\underline{\prime-\neg}\rangle$$\mathrm{J}_{\mathrm{i}}=_{\mathrm{J}_{\mathrm{i}(\mathrm{X}}}\mathrm{t},,---$

Let$\mathrm{m}_{\mathrm{i}\mathrm{j}}(\mathrm{t},\mathrm{X}, ---)\equiv$$\mathrm{b}_{\mathrm{j}},$a)$/\mathrm{a}\mathrm{x}_{\mathrm{j}}$. We denote the$/\mathrm{a}---\mathrm{j}-3$), i.e., we define$\mathrm{j}=1,\ldots,\mathrm{N}$$\xi_{\mathrm{i}^{-3(-\mathrm{t},\mathrm{x}}},\overline{=}$$\mathrm{i}\prime \mathrm{t}\mathrm{h}$. If$4\leqq \mathrm{i},$$\mathrm{j}\leqq 6$

row vectors of)$\mathrm{i}6’ \mathrm{i}=1,\ldots,6$$\mathrm{J}_{\mathrm{i}}=\mathrm{J}_{\mathrm{i}(\mathrm{t},\mathrm{x},)(}---\equiv \mathrm{m}_{\mathrm{i}}1’\ldots,\mathrm{m}$linearly independent vectors in, be, thenorthogonalize these vectors, i.e., we define$\mathrm{b}_{\grave{\mathrm{J}}}.\perp’ \mathrm{j}=1,\ldots,\mathrm{N}$,$\mathrm{m}_{\mathrm{i}\mathrm{j}}(\mathrm{t},\mathrm{X}$$\mathrm{J}=\mathrm{J}(\mathrm{t},\mathrm{X}, ---)$.

$\mathrm{n},\mathrm{N}\in \mathrm{N}$$\mathbb{R}^{\mathrm{n}},$. We

as follows (we do,

not normalize them):$\mathrm{b}_{1,\perp}\equiv \mathrm{b}_{1}$,$\mathrm{b}_{\mathrm{j}1.\perp}+\equiv \mathrm{b}_{\mathrm{j}}+1-\sum_{\mathrm{k}=1}^{\mathrm{j}}(\mathrm{b}\mathrm{j}+1\mathrm{k},\perp \mathrm{b})\mathrm{b}_{\mathrm{k}},\perp/|\mathrm{b}_{\mathrm{k}},\perp|^{2},$Lemma 7.1. Let $1\leqq$ r,T $<+\infty$ . If
(4.6-7), then, for,$\mathrm{t}\in[0,\mathrm{T}]$$\mathrm{j}=1,\ldots,\mathrm{N}$and (X,

$—$

)

–1.

$\in \mathrm{D}(\mathrm{r})$satisfy$\mathrm{i}=1,\ldots,6$$6-5/2\exp(-5\mathrm{C}\mathrm{t})4.2\leqq|\mathrm{J}_{\mathrm{i},\perp}(\mathrm{t},\mathrm{x},---)|\leqq|\mathrm{J}_{\mathrm{i}}(\mathrm{t},\mathrm{x}, \Xi)|\leqq 6^{1/2}\exp(\mathrm{c}_{4}.2\mathrm{t})$Remark 7.2. (i) It does not(II) is essentially bounded in.

offrom Lemma 4.1 that the integer. In fact, there is a possibility that this$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}_{-}\mathrm{w}$$\Omega \mathrm{X}\mathbb{R}^{3}$$\mathrm{m}$,number goes to infinity as (X, ) moves, i.e., that the trajectory of. See p. 391. Hence we need totends to follow the boundary surface$\mathrm{x}=\mathrm{x}(-\mathrm{t},\mathrm{X}$$\overline{=}$a$—\rangle$$\Omega$prove that the inequalities of Lemma 7.1 hold uniformly for . Therefore webefore and after
have to carefully inspect the change of$\mathrm{m}$$\mathrm{x}=\mathrm{x}($$\mathrm{J}=\mathrm{J}(\mathrm{t},\mathrm{X}, ---)$$-\mathrm{t},\mathrm{X},$and$\overline{=}\rangle$

collides with-),$\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{x},\overline{=})^{-0},\mathrm{x},=$$\mathrm{F}(\partial\Omega)$, i.e., the difference between$\mathrm{n}=1,\ldots,\mathrm{m}$$\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{X},---)+0,\mathrm{X},$$—\rangle$.

(ii) Considering the methods employed in prolonging solutions of\S 4 and pp. $1284- 1285^{6}$), we easily see that if$\mathrm{t}=\mathrm{t}$and$\mathrm{t}=\mathrm{s}$$(\mathrm{C}\mathrm{P})$(seesatisfy (4.7),

then$\mathrm{x}(-\mathrm{t},\mathrm{X},\overline{=})=\mathrm{x}(-\tau,\mathrm{x}(^{-}\mathrm{s},\mathrm{X},---), \xi(^{-\mathrm{S}},\mathrm{X},---))$$\xi(^{-_{\mathrm{t},\mathrm{X},\overline{=}}})=\xi(-\tau,\mathrm{x}(^{-\mathrm{S}},\mathrm{x},---), \xi(-\mathrm{s},\mathrm{x},---))$

where
of
and

$\tau\equiv \mathrm{t}-\mathrm{s}$$\tau$. Therefore we can regard$(\mathrm{x}(-_{\mathrm{S},\mathrm{x}}, ---),$$\xi(-\mathrm{s},\mathrm{X}, --- ))$(7.1.1),,$(\mathrm{x}(-\mathrm{t},\mathrm{x}, ---), \xi(^{-\mathrm{t},\mathrm{X},-}-)-)$(7.1.2)

as function. Hence we can define the following

13
Jacobian matrix:$\mathrm{J}(\mathrm{t},\mathrm{s},\mathrm{X}^{-)\equiv},--\mathrm{a}(\mathrm{x}(-_{\mathrm{t},\mathrm{x}_{-}},--),\xi(^{-}\mathrm{t},\mathrm{x},---\rangle)/\partial(\mathrm{x}(-\mathrm{s},\mathrm{X},---),\xi(-\mathrm{s},\mathrm{x}_{\overline{=}},)\rangle$

From (4.4) and (7.2), we easily obtainandin this equality. Then we have$\mathrm{J}(\mathrm{t},\mathrm{X}, ---)=\mathrm{J}(\mathrm{t},\mathrm{s},\mathrm{X},$ $—\rangle \mathrm{J}(\mathrm{S},\mathrm{x}, ---)$$\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{X},\overline{=})+_{0,\mathrm{X}},\underline{=})=\mathrm{U}^{\mathrm{n}}(\mathrm{X},---)\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{x},---)-\mathrm{o},\mathrm{x},\Xi),$$—),$$(7.2)$. Let$\mathrm{n}=1,\ldots,\mathrm{m}$$\mathrm{U}^{\mathrm{n}}(\mathrm{X}, ---)\equiv\iota\downarrow \mathrm{t}^{\mathrm{n}\frac{1}{--}}(\mathrm{X}.)\mathrm{i}\mathrm{m}-\mathrm{J}S\uparrow \mathrm{c}^{\mathrm{n}}(\mathrm{x},--)(\mathrm{t}_{\mathrm{S}},,\mathrm{x}, \overline{=}),$$\mathrm{n}=1,\ldots,\mathrm{m}$$\mathrm{n}=1,\ldots,\mathrm{m}$

(7.3)

,

. We will prove that

(7.2). Hence we assume that$\mathrm{t}$is an orthogonal matrix. In order to, we let
and
in

and

$\mathrm{s}$$\mathrm{t}\downarrow \mathrm{t}^{1}(\mathrm{X}, ---)$$\mathrm{s}\uparrow \mathrm{t}^{1}(\mathrm{X},\overline{\underline{-}})$

satisfy.

(7.4)$0\leqq \mathrm{s}<\mathrm{t}^{1}(\mathrm{x},---)<\mathrm{t}<\mathrm{t}(2\mathrm{x},\overline{=})$For simplicity, we write (as$\mathrm{j}=1,2$$i(\mathrm{X}, ---)$. Hence,and$\mathrm{X}_{-}\overline{-}arrow-A$$i(_{-}\mathrm{X}_{d^{-}}.\overline{-}\alpha$as$(\mathrm{x}(-\mathrm{S},\mathrm{X}, ---), \xi(-\mathrm{s},\mathrm{X}, ---))$

respectively,$\mathrm{j}=1,2$$\tau\equiv \mathrm{t}-\mathrm{s}$$\tau\downarrow 0$

where

we write. We easily see that$i$$i-\mathrm{s}=\underline{\dot{i}}$

,

, and

$(\mathrm{X}arrow\overline{-}A-arrow(\mathrm{X}^{1}, ---^{1-\mathrm{c}})$

,

(7.5)

,

. Moreover we deduce that if$\Gamma^{1}(\mathrm{X},---)\equiv|(\mathrm{x}, \xi)=(\mathrm{x}(^{-\mathrm{S}},\mathrm{X},---\rangle$$\Gamma^{1}(\mathrm{X}, ---)$, andit- follows from (7.4) that-.
$0<\underline{\mathrm{t}}^{1}<\tau<\underline{\mathrm{t}}^{2}$

where

,

$\mathrm{U}^{1}(\mathrm{x}, \underline{=} )$$\mathrm{U}^{1}(\mathrm{X},---)$$\underline{i}$$\mathrm{U}^{\mathrm{n}}(\mathrm{X}$, are orthogonal matrices.

First we will prove thatobtain the concrete form ofand$\mathrm{t}\downarrow$$\mathrm{s}\uparrow \mathrm{t}^{\mathrm{n}}(\mathrm{X},\overline{=})$$\mathrm{t}^{\mathrm{n}}(\mathrm{X}, ---)$

where

.

$\mathrm{t}\downarrow \mathrm{t}^{1}$with (

and$\mathrm{s}\uparrow \mathrm{t}^{1}$, then$\underline{\mathrm{X}},\overline{--}A\in\Gamma^{1}(\mathrm{x},\underline{=}\rangle$$\xi(^{-\mathrm{s},\mathrm{X}},\overline{=})\rangle;\mathrm{s}$satisfies (denotes a characteristic curve of A which connects (X,$(\mathrm{X}^{1}, ---^{1^{-\mathrm{c}}})\equiv(\mathrm{x}(^{-\mathrm{t}^{1}},\mathrm{x},---), \xi(-(\mathrm{t}^{1}-\mathrm{o}),\mathrm{x},--)-)$

.

(7.6)

,

$7.4\rangle$$—$(, that is,

) and
(7.7)

14
Substituting$(\underline{\mathrm{X}},--\simeq)$and (7.1.1-2) in (7.2), and considering (4.4), we obtain

From (7.5-6) we see that$\mathrm{J}(\mathrm{t},\mathrm{s},\mathrm{X}, ---)=\mathrm{J}(\tau,\underline{\mathrm{X}},---)-\cdot$

is equivalent to$\lim_{\mathrm{c}\Downarrow^{1_{\S}}.\uparrow_{\mathrm{t}}}1$. Hence,$\lim$$\tau\downarrow_{0(,-)arrow}\mathrm{J}^{1}<T<\mathrm{t}^{2},\mathrm{x}^{-}-(\mathrm{X}1,---1-0),(\mathrm{x},---)\in\Gamma 1(\mathrm{x}---)$$\mathrm{U}^{\mathrm{i}}(\mathrm{X},---)=\tau\downarrow_{0,\mathrm{t}\mathrm{x},-}\mathrm{t}^{1}<T<\iota,--)arrow(\mathrm{X}^{1},--^{1\triangleleft}-),(\mathrm{X}^{-)\Gamma},--\in(\mathrm{X}^{-}--)^{\mathrm{J}(}2\lim 1T,\underline{\mathrm{X}},--\simeq)$.

$(7.8\rangle$

Let us calculate the right hand side of (7.8). In order to simplify thecalculations, we will introduce a 3-dimensional rectangular coordinate system
(
plane
,
in such a way that the origin coincides with , that the$\mathrm{X}^{1}$$\mathrm{x}_{1},\mathrm{x}_{2^{\mathrm{X}_{3})}}$$(\mathrm{x}_{1}=0)$$\mathrm{x}_{2}\mathrm{x}_{3}$See \S 1 forof$\mathrm{x},$

,

$\xi$

$\mathrm{n}=\mathrm{n}(\mathrm{x})$$—1-0,$

component of$\rho_{2}\equiv 1$

, and

. By

$\mathrm{F}(\mathrm{x}, \xi),$$\rho_{3}\equiv 1$$\mathrm{J}(\tau,\underline{\mathrm{X}},--\simeq)$awhich contains$–\mathrm{j}-1-0,$$\underline{\mathrm{X}}_{\mathrm{j}},$respectively,$\mathrm{a}\mathrm{n}\mathrm{d}----\cdot$$\mathrm{j}=1,2,3$$1=1,2,3$ . We define. Definewith (7.5), and if$\tau\downarrow 0$$\mathrm{x}_{\mathrm{i}}(-\tau,\mathrm{x}---)/\mathrm{a}\underline{\mathrm{X}}arrow-\cdot \mathrm{j}$$\rho_{\mathrm{i}},$’$\mathrm{X}^{1}$, and that$\mathrm{n}(\mathrm{X}^{1}\rangle^{=}(1,0,0)$

we denote the j-th components. We denote by, as follows:.

$\mathrm{F}_{\mathrm{i}}(\mathrm{x}, \xi)$$\mathrm{i}=1,2,3$

the i-th$\rho_{1}\equiv-1$

a$6\cross 6$

matrix$(\underline{\mathrm{X}}_{fA-}\overline{-}arrow(\mathrm{X}^{1}, ---^{1-0})$along$\Gamma^{1}(\mathrm{X},$a.s. follows:$\mathrm{i},\mathrm{j}=1,2,3$$\xi_{\mathrm{i}}(^{-\tau},\underline{\mathrm{X}}_{f}\overline{--}A/\partial^{-}---\cdotarrow\rho_{\mathrm{i}}\delta_{\mathrm{i}\mathrm{j}},$$\mathrm{x}_{\mathrm{i}}(^{-}\tau_{- \mathrm{a}},\mathrm{x}\overline{-}\alpha d^{-}--/\partial--\cdotarrow 0,$$\mathrm{i},$$\mathrm{j}=1,2,3$,,$arrow(\mathrm{F}_{\mathrm{i}}(\mathrm{X}1,1+0\underline{=})^{-\mathrm{F}}\rho \mathrm{i}\mathrm{i}(\mathrm{x}^{11^{-0}},---))/---1^{-0}1,$$\xi_{\mathrm{i}}(-\tau,arrow\overline{-}\Delta \mathrm{X}-/\partial\underline{\mathrm{X}}\mathrm{j}arrow 0,$

Applying Assumptiona,$—\rangle$$\xi_{\mathrm{j}}(-\tau,\mathrm{x}-\alphaarrow\overline{-}/\partial\underline{\mathrm{x}_{1}}$a.

$–1+_{0}-\equiv\xi(-(\mathrm{t}^{1}+\mathrm{o}),\mathrm{X},\overline{=})$

converges to aaa$\xi_{\mathrm{j}},$$\Omega$$\mathrm{x}_{\mathrm{j}},$$arrow \mathrm{X}\mathrm{a}\mathrm{n}\mathrm{d}----$

Lemma 7.3. Ifthenaincludes the face of$\mathrm{F}$$\mathrm{i}=1,2,3,$$\mathrm{i}=1,2,3$$\mathrm{j}=2,3$

,

(7.9)

.

to (7.9) of this lemma, we have$\xi_{\mathrm{i}}(-\tau,\mathrm{X}arrow\overline{--}\alpha/\partial\underline{\mathrm{x}_{1}}arrow 0,$$\mathrm{i}=1,2,3$.

(7.10),

15
It follows from Lemma 7.3 and (7.10) thatis an orthogonal matrix.
Performing the same calculations as above, we see that, also are orthogonal matrices. Applying these results to (7.3), we have$\mathrm{U}^{1}(\mathrm{X},---)$$\mathrm{U}^{\mathrm{n}}(\mathrm{X},$$—\rangle,$$\mathrm{n}=$$2,\ldots,\mathrm{m}$$|\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{X}, ---)+0,\mathrm{X},$$—)|\cdot=1\mathrm{J}(\mathrm{t}^{\mathrm{n}}(\mathrm{x}, ---)-0,\mathrm{X},---\rangle|,$$\mathrm{n}$$=1,\ldots,\mathrm{m}$. Combining theseequalities and Lemma 4.2, (i), we see that$|\mathrm{J}(\mathrm{t},\mathrm{x},---)|\leqq|\mathrm{r}(0,\mathrm{X},---\rangle$$|\mathrm{e}^{\mathrm{c}_{4}.\mathrm{t}}2$,

for each$\mathrm{t}\geqq 0$.

Making use of this inequality, Lemma 4.2, (ii), and the following inequalities

and equalities, we can obtain Lemma 7.1:

II$\mathrm{J}_{\mathrm{i}.\perp}(\mathrm{t},\mathrm{X},---)|\leqq$$\mathrm{J}_{1.\perp(\mathrm{t}},\mathrm{x},---$)$|\cdots$

I

I$]_{\mathrm{i}(\mathrm{t},\mathrm{X},\overline{\underline{-}}}\rangle|\leqq 1\mathrm{J}(\mathrm{t},\mathrm{X}^{-},--\rangle 1,$$\mathrm{i}=_{1},\ldots,6$,$\mathrm{J}_{6,\perp}(\mathrm{t},\mathrm{x}_{\overline{=}},)|=|\det(\mathrm{J}(\mathrm{t},\mathrm{x},---\rangle\rangle|, |_{\mathrm{J}(0},\mathrm{x}_{-},--)1=61/2$\S 8. ESTIMATES FOR$\mathrm{J}=\mathrm{J}(\omega_{1})$The purpose of this section is to calculate the rank of$\mathrm{J}=\mathrm{J}(\omega_{1})$(see$(6.6)\rangle$.

In this section we will prove Lemmas 8.1-4. By making use of Lemma 8. , weprove Lemma, respectively. The main result of this section is$\mathrm{j}$$8.\mathrm{j}+1,$$\mathrm{j}=1,2,3$

Lemma 8.4, which will be employed in \S 9 to prove (G.1-2).
, be vectors of
, m.N
Let
. Let
. By
$\mathbb{R}^{\mathrm{m}}$$\mathrm{j}=1,\ldots,\mathrm{N}$$\epsilon>0$$\in \mathrm{N}$$\mathrm{u}_{\mathrm{j}},$denote the set of all vectors ofby$\mathrm{j}=1,\ldots,\mathrm{N}$$\mathrm{u}_{\mathrm{j}},$Write$\Omega_{4}(\mathrm{r},\mathrm{T})$$\Omega\cross \mathbb{R}^{3}\cross[0,\mathrm{T}]$$\mathbb{R}^{\mathrm{m}}$$\epsilon$. Let$\mathrm{T}$be a positive constant.

as the set of all vectors of the following form:$|\eta_{4}|\leqq \mathrm{r};\mathrm{r}$$\omega_{4}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4})\in$

is the positive constant defined in (6.2). If$\omega_{4}\in\zeta 2_{4(\epsilon},\mathrm{r},\mathrm{T})\equiv|\omega_{4}=(\mathrm{x}_{4},$$\eta_{4},\mathrm{t}4\rangle\in\Omega_{4}(\Gamma,\mathrm{T});\eta_{4}\in \mathrm{U}(0;\epsilon)\}$then we can obtain estimates for the rank of the Jacobian matrix
as follows (we can regard in (6.4) as function of ):$\mathrm{t}_{4})$$\mathrm{x}_{3}$wewhose distance from the subspace spanned, is greater than or equal to, where$\mathrm{U}(\mathrm{u}_{1},\ldots,\mathrm{u}_{\mathrm{N}};\epsilon)$$\omega_{4}$a(,$\mathrm{x}_{3}\rangle/\mathrm{a}(\eta 4$’

16
Lemma 8.1. Let
vector from the Jacobian matrix

be constants. We can choose one column
almostfor$\epsilon,\mathrm{T}>0$a$\omega_{4}\in \mathrm{f}2_{4}(\epsilon,\mathrm{r},\mathrm{T})$$(\mathrm{x}_{3})/\partial(\eta 4’ \mathrm{t}_{4})$) of the norm of theeverywhere so that the essential infimum (in$\ovalbox{\tt\small REJECT} 4\langle\epsilon,\mathrm{r},\mathrm{T}$$)$chosen column vector is positive, i.e., there exists a Lebesgue measurable,
integer-valued function
which satisfies the following (i-ii):$\mathrm{c}=\mathrm{G}(\omega_{4})$(i)$\mathrm{G}(\omega_{4})\in|1,\ldots,4|$(ii)$\mathrm{e}\mathrm{s}\mathrm{s}$inf

We denote byi.e.,

for almost all$||\mathrm{a}(\mathrm{c}(\omega_{4}), \omega)4|$$\mathrm{a}(\mathrm{k}, \omega_{4})$;f24$\omega_{4}\in$$\omega_{4}\in S2_{4(\epsilon,\mathrm{r}},\mathrm{T}$$(\epsilon,\mathrm{r},\mathrm{T}\rangle$,) $|>0$ .

athe k-th column vector of$(\mathrm{x}_{3})/\partial(\eta 4’ 4\mathrm{t}),$,$\mathrm{k}=1,\ldots,4$$\mathrm{k}=1,2,33,2’ 3’ \mathrm{a}(4,\omega_{4})\equiv(\partial/\mathrm{a}_{\mathrm{t}_{4})(}\mathrm{t}\mathrm{x}_{3},\mathrm{x}1’ 3,2’ 3\mathrm{x}.)\mathrm{s}$$\mathrm{a}(\mathrm{k},\omega_{4})\equiv(\partial/\partial\eta_{4}.\mathrm{k})^{\mathrm{t}}(\mathrm{x}_{3,1},\mathrm{x}\mathrm{x}_{3},),$’

where by$\mathrm{x}_{\mathrm{i},\mathrm{j}},$respectively, i.e.,

Let
form:

$\epsilon,$

we denote the j-th components of.

$\eta_{\mathrm{i},\mathrm{j}}$$\eta_{\mathrm{i}}\in \mathbb{R}^{3},$$\mathrm{x}_{\mathrm{i}},$$\mathrm{j}=1,2,3$,$\mathrm{x}_{\mathrm{i}}=(\mathrm{X}_{\mathrm{i},1},\mathrm{X}_{\mathrm{i}.2},\mathrm{X})\mathrm{i},3’\eta_{\mathrm{i}}=(\eta_{\mathrm{i},1}, \eta_{\mathrm{i}}.2’\eta \mathrm{i},3)$$\mathrm{T}>0$. Write$\omega_{3}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta_{3},\mathrm{t}_{3})$$\Omega_{3}(\mathrm{r},\mathrm{T})$, whereas the set of all vectors of the following$\mathrm{x}_{4}\in\Omega,$$|\eta_{\mathrm{j}}|\leqq \mathrm{r},$$\eta_{\mathrm{j}}\in \mathbb{R}^{3},0\leqq \mathrm{t}_{\mathrm{j}}\leqq \mathrm{T},$$\mathrm{j}=3,4$.

We define$\zeta 2_{3(}\epsilon,\mathrm{r},\mathrm{T})\equiv|\omega_{3}=(\mathrm{x}_{4}, \eta_{4’ 4}\mathrm{t}, \eta_{3},\mathrm{t}_{3})\in\Omega(3\mathrm{r},\mathrm{T})$;$\omega_{4}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4})\in\zeta \mathrm{p}_{4}(\epsilon ,\mathrm{r},\mathrm{T}),$

where$\mathrm{a}(\cdot, \omega_{4})$and$\mathrm{c}=\mathrm{G}(\omega_{4})$,are those in Lemma 8.1. Ifthen we can obtain estimates for the rank
(we can regard

in 6.4) as function of$0\dot{\mathrm{f}}$$\langle$$\mathrm{x}_{2}$Lemma 8.2. Let
Jacobian matrixa$\epsilon,\mathrm{T}>0$$\eta_{3}\in \mathrm{U}(\mathrm{a}(\mathrm{G}(\omega)4’\omega_{4});\epsilon )\}$$\omega_{3}$a$\omega_{3}\in$,$\mathrm{f}\S_{3(}\epsilon,\mathrm{r},\mathrm{T}$.

$(\mathrm{x}_{2})/\partial(\eta_{4},\dot{\mathrm{t}}_{4}, \eta 3’ \mathrm{t})3\mathrm{a}\mathrm{S}\backslash$),follows):. We can choose two column vectors from the$(\mathrm{x}_{2})/\mathrm{a}(\eta_{4},\mathrm{t}_{4}, \eta_{3’ 3}\mathrm{t})$for$\omega_{3}\in\zeta 2_{3}(\mathrm{g},\mathrm{r},\mathrm{T})$almost everywhere

so that the chosen column vectors a..re uniformly linearly in.dependent for, i.e., there exist Lebesgue measurable, integeralmost all,, such that, for almost allvalued functions$\omega_{3}\in\Omega_{3}(\epsilon,\Gamma,\mathrm{T})$$\mathrm{H}_{\mathrm{j}}=\mathrm{H}_{\mathrm{j}}(\omega_{\vee}\mathrm{q}),$$\mathrm{j}=1,2$$\omega_{3}\in\Omega_{3}(\epsilon$

17$\mathrm{r},\mathrm{T}\rangle$,(i)$\mathrm{H}_{\mathrm{j}}(\omega_{3}\rangle$(ii)$\mathrm{b}(\mathrm{H}_{\mathrm{j}}(\omega_{3}), \omega_{3}),$

We denote by$1,\ldots,8$},$\in|1,\ldots,8$$\mathrm{j}=1,2$$\mathrm{j}=1,2$$\mathrm{b}(\mathrm{k}, \omega 3)$

,

and

,

$\mathrm{H}_{1}(\omega_{3})\neq \mathrm{H}_{2}(\omega_{3})$, are uniformly linearly independent.

the k-th column vector ofa$(\mathrm{x}_{2})/\partial(\eta 4’ 4\mathrm{t}, \eta 3’ 3\mathrm{t}),$$\mathrm{k}=$, i.e.,$\mathrm{k}2,2’ 2,3--1,2,3$ ,$\mathrm{b}(\mathrm{k}, \omega_{3})\equiv(\partial/\partial\eta_{4,\mathrm{k}})^{\mathrm{t}}(\mathrm{X}_{2,1},\mathrm{X}\mathrm{X}\rangle,$,$\mathrm{b}(4, \omega_{3})\equiv(\mathrm{a}/\partial \mathrm{t}_{4})\mathrm{t}(_{\mathrm{X}_{2}\mathrm{x}\mathrm{X}_{2}}.1’ 2,2" 3)$$\mathrm{b}(\mathrm{k}, \omega_{3})\equiv(\partial/\partial\eta_{3,\mathrm{k}-4})^{\mathrm{t}}(\mathrm{X}\mathrm{x}\mathrm{X}_{2.\mathrm{s}}2,1’ 2.2’),$$\mathrm{k}=5,6,7$$\mathrm{b}(8,\omega_{3})\equiv(\partial/\mathrm{a}\mathrm{t}_{3}\rangle^{\mathrm{t}}(\mathrm{x}\mathrm{x}\mathrm{x})2,1’ 2,2’ 2,3$

Let e,T $>0$ . Write
form:$\Omega_{2}(\mathrm{r},\mathrm{T})$$\omega_{2}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta_{3’ 3}\mathrm{t}, \eta_{2},\mathrm{t}_{2})$$=2,\ldots,4$

as the set of all vectors of the following, where$\mathrm{x}_{4}\in\Omega,$$|\eta_{\mathrm{j}}|\leqq \mathrm{r},$$\eta_{\mathrm{j}}\in \mathbb{R}^{3},0\leqq \mathrm{t}_{\mathrm{j}}\leqq \mathrm{T},$$\mathrm{j}$;$\equiv|\omega_{2}=(\mathrm{x}\eta_{4}4" \mathrm{t}\eta_{3}4" \mathrm{t}\eta_{2}3")\mathrm{t}_{2}\in\Omega_{2}(\mathrm{r},\mathrm{T})$,$\eta_{2}\in\dot{\mathrm{U}}(\mathrm{b}(\mathrm{H}_{1()}\omega 3’\omega\rangle 3’ \mathrm{b}(\mathrm{H}_{2}(\omega)3’\omega_{3});\epsilon)|$$\omega_{3}(=\mathrm{x}_{4}, \eta 4’ \mathrm{t}4’\eta 3’ \mathrm{t}_{3})\in\ovalbox{\tt\small REJECT} 3(\epsilon,\mathrm{r},\mathrm{T}),$$\mathrm{r},\mathrm{T})$.

. Define$E2_{2}(\epsilon,\mathrm{r},\mathrm{T}\rangle$

where

,

$\mathrm{b}(\cdot, \omega_{3})$and$\mathrm{H}_{\mathrm{j}}=\mathrm{H}_{\mathrm{j}}(\omega_{\mathrm{s}}),$$\mathrm{j}=1,2$,

are those in Lemma 8.2. If, then we can obtain estimates for the rank ofas follows (we can regard
Lemma 8.3. Let
Jacobian matrixa$\mathrm{x}_{1}$in (6.4) as function of$\epsilon,\mathrm{T}>0$

everywhere so that the chosen$\omega_{2}\in\ovalbox{\tt\small REJECT}_{2}(\epsilon,\mathrm{r},\mathrm{T})$,$(\mathrm{x}_{1})/\partial(\eta 4’ 4\mathrm{t}, \eta_{3},\mathrm{t}_{3}, \eta 2’ 2\mathrm{t})$$\omega_{2}$

):

for

$\omega 2\in\zeta 22(\epsilon,\mathrm{r},\mathrm{T})$almostcol..umn vectors are uniformly linearly$\omega 2^{-}\in\Omega 2(\epsilon,\mathrm{r},\mathrm{T})$measurable, integer-valued functions

almost all,. We can choose three column vectors from the$(\mathrm{x}_{1})/\partial(\eta 4’ \mathrm{t}_{4}, \eta 3’ 3\mathrm{t}, \eta 2’ 2\mathrm{t})$independent for almost alla$\omega_{2}\in\zeta 2_{2}(\epsilon$,$\mathrm{I}_{\mathrm{j}}=\mathrm{I}_{\mathrm{j}}(\omega 2),$

i.e., there exist$\mathrm{j}=1,2,3$,$\dot{\mathrm{L}}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{g}\mathrm{u}\mathrm{e}$such that, for

18

(i)

$\in|1,\ldots,12$$\mathrm{I}_{\mathrm{j}}(\omega_{2}\rangle$} for$\mathrm{j}=1,2,3$

, and

if

$\mathrm{I}_{\mathrm{i}}(\omega_{2})\neq \mathrm{I}_{\mathrm{j}}(\omega_{2}.)$$\mathrm{i}\neq \mathrm{j}$,$\mathrm{i},$$\mathrm{j}=$$1,2,3$ ,(ii)$\mathrm{j}=1,2,3$$\mathrm{c}(\mathrm{I}_{\mathrm{j}}(\omega_{2}), \omega_{2}),$where we denote by$\eta_{2},\mathrm{t}_{2}),$$\mathrm{k}=1,\ldots,12$

Let
where

$\epsilon,\mathrm{T}>0$$\mathrm{x}_{4}\in\Omega,$$(\epsilon,\Gamma,\mathrm{T})$, then$\mathrm{c}(\mathrm{k},$, are uniformly linearly independent,athe k-th column vector of$\omega_{2}\rangle$$(\mathrm{x}_{1})/\partial(\eta_{4},\mathrm{t}_{4},$$\eta_{3},\mathrm{t}_{3}$,.

. Write,$\Omega_{1}=\Omega_{1(\mathrm{T})}\Gamma$

as the set of all vectors of the form (6.5),. Noting that if, we can decompose
into four$\mathrm{j}=1,\ldots,4$$\eta_{\mathrm{j}}\in \mathbb{R}^{3},0\leqq \mathrm{t}_{\mathrm{j}}\leqq \mathrm{T},$$|\eta_{\mathrm{j}}|\leqq \mathrm{r},$$\omega_{\mathrm{j}}+_{1}\in\Omega_{\mathrm{j}}+_{1(}\mathrm{S},\mathrm{r},\mathrm{T}\rangle$

disjoint subsets as follows:,$\omega_{\mathrm{j}}\in 62_{\mathrm{j}}$$\mathrm{j}=2,3$$\Omega_{1}(\mathrm{r},\mathrm{T})$4$\Omega_{1}(\mathrm{r},\mathrm{T})\equiv \mathrm{N}(\epsilon,\mathrm{r},\mathrm{T})\cup(\cup \mathrm{S}_{\mathrm{j}}(\mathrm{j}=2\epsilon,\mathrm{r},\mathrm{T}))$, where;$\mathrm{S}_{4}(\epsilon,\mathrm{r},\mathrm{T})\equiv|\omega_{1}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta 3’ \mathrm{t}_{3}, \eta_{2’ 2}\mathrm{t}, \eta 1’ 1)\mathrm{t}\in\Omega_{1}(\mathrm{r},\mathrm{T})$,$\omega_{4}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4})\not\in\zeta 2_{4}(\epsilon,\mathrm{r},\mathrm{T})\}$;$\mathrm{s}_{3(\epsilon,\mathrm{r}},\mathrm{T})---\{\omega_{1}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta_{3},\mathrm{t}\eta 2’ \mathrm{t}\eta_{1’ 1}2’)3’\in \mathrm{t}\Omega_{1}(\mathrm{r},\mathrm{T}\rangle$$\omega_{4}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t})4\mathrm{f}2_{4(\epsilon,\mathrm{r}}\in,\mathrm{T}),$$\mathrm{S}_{2}(\epsilon,\mathrm{r},\mathrm{T})\equiv\rfloor\omega_{1}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta \mathrm{s}’ \mathrm{t}_{3}, \eta 2’ \mathrm{t}2’\eta 1’ 1)\mathrm{t}\in\Omega_{1}(\mathrm{r},\mathrm{T})$$\omega_{3}=(\mathrm{x}_{4},$$\mathrm{N}(\epsilon$,$\mathrm{r},\mathrm{T}\rangle$$\eta_{4},\mathrm{t}_{4},$$\eta 3’ \mathrm{t}\rangle 3\in\Omega_{3(}\epsilon,\Gamma,\mathrm{T}\rangle,$$\equiv|\omega_{1}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta.

$\eta_{\mathrm{j},\mathrm{k}}$,;3’\eta 2’\eta 1’ 1)\mathrm{t}_{3},\mathrm{t}_{2},\mathrm{t}\in\Omega_{1}(\mathrm{r},\mathrm{T}\rangle$

the k-th component of$\mathrm{j}=1,\ldots,4,$$\eta_{\mathrm{j},\mathrm{k}},$;$\omega_{2}(=\mathrm{x}_{4}, \eta_{4’ 4}\mathrm{t}, \eta_{3’ 3}\mathrm{t}, \eta_{2},\mathrm{t})2\not\in\zeta 2_{2}(\epsilon,\mathrm{r},\mathrm{T})\}$$\omega_{2}=(\mathrm{x}_{4’\eta_{4’ 4}}\mathrm{t},$We denote by$=1,\ldots,15$ , as,$\omega_{3}=(\mathrm{x}_{4}, \eta_{4},\mathrm{t}_{4}, \eta 3’)\mathrm{t}_{3}\not\in\Omega_{3}(\epsilon ,\mathrm{r},\mathrm{T})\}$$\mathrm{k}=1,2,3,$$\mathrm{t}_{\ell},$$\eta_{3},\mathrm{t}_{3},$$\mathrm{k}=1,2,3,$$\eta 2’ \mathrm{t}2\rangle^{\xi}=\Omega_{2}(\epsilon,\mathrm{r},\mathrm{T})\}$$\mathrm{j}=1,\ldots,4$$\eta_{\mathrm{j}},$. Write$\ell=2,3,4$ , respectively.

$\mathrm{W}\mathrm{e}\sim$$\zeta(\mathrm{i}),$.

$\mathrm{i}$do notneed .
$\mathrm{t}_{1}$Lemma 8.4. (i) Let
that, for each ,$\epsilon$

meas \dager$\mathrm{T}>0,$. There exists a positive constant}, and} },}$\mathrm{j}=2,3,4$$\mathrm{x}_{4},$$\mathrm{k}\in$$\mathrm{t}_{\mathrm{k}},$$1,\ldots,4$$\eta_{\mathrm{i}}$.

$\mathrm{i}\in \mathfrak{i}1,\ldots,4$$\mathrm{c}_{\mathrm{a}4}$such$\backslash |\mathrm{j}$$\eta_{\mathrm{j}}\in \mathbb{R}^{\mathrm{s}};\omega_{1}(\mathrm{x}_{4}, \eta 4’ 4’\eta_{3’ 3}\mathrm{t}\mathrm{t}, \eta=\mathrm{t}2’ 2’\eta_{1},\mathrm{t}1)\in \mathrm{S}_{\mathrm{j}}(\epsilon,\mathrm{r},\mathrm{T})\}\leqq \mathrm{c}8.4\mathrm{r}^{4}-j\mathrm{j}^{-1}\epsilon$

19
where we denote the Lebesgue measure of a set(ii) Letfunctions($\epsilon,$$\mathrm{T}>0$$\mathrm{i}=1,\ldots 6-,$

$1\rangle \mathrm{K}_{\mathrm{i}}(\omega_{1})\in|1,\ldots,15\}$$1,\ldots,6$by meas Y.

. Then, there exist Lebesgue measurable, integer-valued,$\mathrm{K}_{\mathrm{i}}=\mathrm{K}_{\mathrm{i}}(\omega_{1}\rangle$$\mathrm{Y}\subseteq \mathbb{R}^{3}$for, such that, for almost all$\mathrm{i}=1,\ldots,6$,

and$\mathrm{K}_{\mathrm{i}(\omega_{1}}$)$\omega_{1}\in \mathrm{N}(\epsilon,\mathrm{r},\mathrm{T})$$\neq \mathrm{K}_{\mathrm{j}}(\omega_{1})$if,$\mathrm{i}\neq \mathrm{j},$$\mathrm{i},$$\mathrm{j}=$,(2)$\mathrm{e}\mathrm{s}\mathrm{s}$

inf

a

(

$|\det$$(\mathrm{x}_{0}, \xi 0)/\mathrm{a}(\zeta(\mathrm{K}_{1}(\omega_{1})),\ldots, \zeta(\mathrm{K}_{6}(\omega_{1})))\rangle|>0$.

$0)1^{\in \mathrm{N}(}\mathrm{e}x,\mathrm{T})$\S 9. PROOF OF (G. 1-2)
Decomposeas follows (see$\mathbb{G}(\mu,\mathrm{T}\rangle$$(6.2- 3 \rangle\rangle:\mathbb{G}(\mu,\mathrm{T}\rangle=\sum_{\mathrm{k}^{=}1}^{4}\mathbb{G}_{\mathrm{k}}(\epsilon , \mu,\mathrm{r},\mathrm{T})$

,

where
(

$\mathrm{k}=1,\ldots,4$of$\xi_{0}\rangle$$\mathrm{d}\mathrm{t}\mathrm{d}\eta\mathfrak{h}|\leqq\Gamma,$ ’$\mathrm{j}=_{1,\ldots,4}$$\mathbb{G}_{\mathrm{k}}(\epsilon,\mu,\mathrm{r},\mathrm{T}\rangle \mathrm{u}(\cdot,\cdot))(_{\mathrm{X}\xi_{4})}4’\overline{=}\int_{0\leqq\leqq \mathrm{T},\eta \mathrm{i}|}\emptyset \mathrm{k}\mathrm{u}\mathrm{c}(\mathrm{x}_{0},$. By$\mathrm{N}(\epsilon,\mathrm{r},\mathrm{T}),$$\mathbb{G}(\mu,\mathrm{T})$parts$\psi_{\mathrm{k}}=_{\psi_{\mathrm{k}}}..(\omega)1’ \mathrm{k}=1,\ldots,4,$$\mathrm{S}_{\mathrm{k}}(\epsilon,\mathrm{r},\mathrm{T}\rangle$, $\mathrm{k}=2,3,4,$$\mathrm{w}_{\wedge}\mathrm{e}$denote the characte

respective.ly. The lemma belo.w implies thatis decomposed into the principal part$\mathbb{G}_{\mathrm{k}}(\epsilon , \mu,\mathrm{r},\mathrm{T}),$$\mathrm{k}=2,3,4$$\mathrm{r}\mathrm{i}\mathrm{s}\dot{\mathrm{t}}\mathrm{i}\mathrm{c}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}$$\mathbb{G}_{1}(\epsilon,$$\mu,\mathrm{r},\mathrm{T}\rangle$

and the negligible. Applying this lemma and Theorem VI.12, (a),p. $200^{16}$ , we can easily obtain (G.1-2).

Lemma 9.1. (i), andbut is independent of$\mathrm{T}\in[1,+\infty)$$\mu\in \mathrm{D}$(ii) If(1)(2)$\mathrm{k}1=2,3,4$$|||\mathbb{G}_{\mathrm{k}}(\epsilon,\mu,\mathrm{r},\mathrm{T})|||\leqq \mathrm{c}_{9.1}\epsilon^{\mathrm{k}-},$$1\leqq \mathrm{T}<+\infty$$\mathbb{G}_{1}(\epsilon,\mu,\mathrm{r},\mathrm{T}\rangle$, where$\epsilon,$

and

$\mu$

$\mathrm{c}_{9.1}$, for each$\epsilon>0$

is a positive constant which depends on,$\mathrm{r}$,, and T.
$\epsilon>0$, then$\in \mathbb{C}(\mathrm{L}^{2}(\Omega\cross \mathbb{R}^{3})\rangle$$|||\mathrm{G}_{1}(\in,\mu,\mathrm{r},\mathrm{T})|||arrow 0$

asfor each$|\mu|arrow+\infty,$$\mu\in 1\mathrm{D}$$\mu\in \mathrm{D}$,.

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