数理解析研究所講究録
第 824 巻 1993 年 30-42

30

On the Discrete Boltzmann Equation
with Linear and Nonlinear Terms
MITSURU YAMAZAKI
満
山崎
東大・数理解析
Dept. Mathematical Science, Univ. Tokyo

Chapter 1

Formulation and results

The discrete. model of Boltzmann equations system in a thin infinite tube as follows :(B)$\{\begin{array}{l}\frac{\partial u_{i}}{\partial t}+c.\cdot\frac{\theta\tau\iota_{i}}{\theta W}=Q_{i}(\tau\iota)+L.\cdot(u)\tau\iota.\cdot(W,0)=\tau\iota_{i}^{0}(x)forx\in R,t\in B+\end{array}$where$Q_{i(u)=\sum_{j,h,1\in I}:}(A_{ij}^{h1}u_{h}u_{l}-A_{h1}^{ij}u\prime u_{j})$$L_{i}(u)= \sum_{h\in I}(\alpha_{i}^{h}u_{h}-\alpha_{h}:*)$.

The natural physical conditions are following:

Condition 1.–$A_{:j}^{h1}\geqq 0$,$A_{h1}^{:j}\neq 0$$A_{:j}^{h1}=A_{ji}^{hl}=A_{ij}^{1h}$$\Rightarrow$$\forall i\exists(j, k,t)$$i\neq j$such that

and$c;+c_{j}=c_{h}+c_{1}$$A_{h1}^{:j}\neq 0$$\alpha^{h}:\geqq 0$

Condition 2.–$\forall i\in I,$$\sum_{h\in I}\alpha_{h}^{:}(c_{h}-c_{i})=0$

31
We put$I_{h},$$k=0,1,$$\cdots$

as follows :for all $k\in I$}{{ ; particles with velocity ; don’t provoke any reflection}; there exists
such that{}$=\{i$ ; particles with velocity.
is transformed$i;\alpha_{h}^{i}=0$$I_{0}=$$i$$=$$I_{1}=$

(1.1)

$c$

$j\in I_{0}$$i\not\in I_{0}$$\alpha_{j}^{i}>0$$c_{\dot{\tau}}$

into a particle with velocity$I_{h+1}=${$i \not\in\bigcup_{l=0}^{h}I_{l}$Remark : As we see later,;$I_{0}$

there exists$c_{j},j\in I_{0}$$j\in I_{h}$by reflection

such that$\alpha_{j}^{i}>0$},}

is not empty, if we assume Condition 2.

Proposition 1.1.– Suppose Condition $s1$ an $d2$ . Le $tu:=u_{i}(x, t)\in C^{1}(R_{+}, S(R))$$(i\in I)$ a solu tion of $(B)$ . Then, for any $t\in n_{+}$ , we lnave(1.2)$\int_{R}\sum_{:}u:(x, t)dx=\int_{R}\sum_{:}u^{0}:(x)dx\equiv\mu$(mass cons ervation aw)
$l$

(1.3)

$\int_{R}\sum_{i}c_{i}u;(x, t)dx=\int_{R}c;u^{0}(x)dx$(momentum $con$ servation low)

Proposition 1.2.– Condition 2 implies tlat

$I_{0}$

is not empty.

The results of this note are following :under Condition 2

positive, summable
Theorem 1.– Suppose Conditi$ons1$ and 2. For a Cauchy data
and bounded, there exists a unique global bounded solution $u_{i}(x, t)\in L^{\infty}(R\cross n_{+})$ an$u_{i}^{0}$

$d$

(1.4)

where

$u_{i}(x,t) \leqq(1+\sup_{i,r}u^{0}:(x))\exp(a\mu^{2}+b\mu)$$a$an $db$ depend only on th $eeq$ uations system, and$\mu$

is th total mass.
$e$Corollary 2.– Suppose Conditions 1 and 2. For a Cauchy data
andthere exists a unique global solution$u_{i}^{0}$$u_{i}(x, t)\in L_{1oc}^{\infty}(R\cross n_{+})$

(1.5)

where

$u_{i}(x,t)\leqq\exp(A\mu^{2}t^{2}+B)$

$A$

and

$B$

don’t epend on time.
$d$positive and bounded,

32
Theorem 3.– Suppose hypo th eses like as in Theorem 1. $We$ have a asympto $tic$ al beAavior of a solu tion : there is a function $v^{T}:(z, t)$ which verifies that, for any $e>0$ , there
exists a large such th at for $t>T$
$T$

(1.6)

$\Vert u_{i}(\cdot,t)-v_{i}^{T}(t)\Vert_{L^{p}}<\epsilon(1\leqq p\leqq\infty)$

for all$i$and$\Vert v^{T}:(\cdot,t)\Vert_{L^{p}}$(1.7)$i\in Ii\not\in I_{0}^{0}$$\leqq\{\begin{array}{l}m(t)m(t)exp\{-\lambda t(1-\frac{1}{p})\}\end{array}$for ail , where $2\leqq p\leqq\infty$ and $m(t)$ is a strictly decreasing function.
$i$

without Condition 2 We put other hypotheses.
$C$

ondition 3.–
for$\alpha_{*}^{h}>0$$i\neq k$

Proposition 1.3.– Condition 2 is not ompatible with Condition 3.
$c$Moreover we assume the microreversibility of the reflections :Condition 4.–$\alpha^{h}:=\alpha_{h}^{:}$In this situation, we can apply the theory due to Shizuta and Kawashima [9] for a positive,
summable and bounded Cauchy data and obtain the following theorem :Theorem 4.– Assume Conditions 1,
and bound $edCau$ chy data : such thatfor the solu tion:$u^{0}$

(1.8)

$S$

and 4. Then, for a smooth, positive, summable
is defined and in , we Izave a decay estimate$L^{1}$$\hat{u}_{:}^{0}$$||u:(\cdot, t)||_{L^{\infty}}\leqq C_{*}(1+t)^{-1/2}(\Vert u^{0}\Vert_{L^{1}}+\Vert\hat{u}^{0}\Vert_{L^{1}})$

where the onstant$c$$C_{*}$

depends only on th equation system.
$e$

$s$

for small Cauchy data$i)Case$ with the binary coUision terms
In this case, we treat general form of the binary coUision terms :$(gQ)$$Q:(u)= \sum_{jh}B^{jh}u_{j}u_{h}:$ ’which is introduced by Bony [4]. In 1990, he showed that the global‘ existence of thesolution for small Cauchy data in the case of $L_{:}=0$ inand defined the corresponding
wave and scattering operators.
$R^{N}$The equations system is following :
(B)

with

$\{\begin{array}{l}\frac{\theta u}{\theta t}+c.\cdot\frac{\theta u}{\partial x}=Q_{i}(u)+L.\cdot(u)u_{i}|_{\ell=0}=u^{0}(\cdot)\end{array}$$L_{:}$

is of form as before. On this system, we impose some assumptions:33
Condition 5.–$B_{i}^{jh}\neq 0$$\Rightarrow$$j\neq k$$B_{i}^{jh}\neq 0$$\Rightarrow$

$j$

and

$k\not\in I_{0}$$\alpha_{i}^{h}\geqq 0$

Condition 6.–$\{$$i\in I\backslash ^{\emptyset}I_{0}I_{0\neq}\Rightarrow i\in I_{1}$Remark : Condition 6 means that the particles which don’t provoke any reflection don’t
make any binary collision.

Theorem 5.– Suppose Conditions 5 and 6. If the Cauchy data is suffi ciently smalI in$H^{\iota}(\ell=1,2, \cdots)$ , thsolu tion has a decay estimate as follows :
$e$

( so

$||u_{i}||_{H}.$$||u||_{L^{\infty}}$)(19)$\leqq\{C^{*}C\Vert_{u^{0}}^{u_{0}}\Vert_{H}^{H}.e^{-\iota_{\lambda t}}2$

where$C_{*}$$forfor$$i\in I_{1}^{0}i\in I$

depends only on the equa $tions$ system and onstant$c$$\lambda>0$.

with the multiple coUision terms
The case with the multiple collision terms is studied only in a few papers $[1][2][6]$ . We
consider the general multiple coUision terms as follows:$\ddot{u})Case$(R)$R_{i}(u)= \sum_{p=2}^{\sigma}\sum_{j_{1}}\cdots\sum_{j_{p}}\dot{F}_{i}^{1}j_{p}u_{j_{1}}\cdots u_{j_{p}}$where we permit the cases $j_{h}=j_{t},$ $k\neq t$ .
Then the equations system is following :

(M)

where

$\{\begin{array}{l}\frac{\partial u_{i}}{\theta t}+c.\cdot\frac{\partial u}{\partial W}=R.\cdot(u)+L_{i}(u)u.\cdot|_{t=0}=u_{i}^{0}(\cdot)\end{array}$$L$; is of form as before. On this system, we impose some assumptions :Condition 7.–$E^{-\ldots i_{r}}1\neq 0$$\Rightarrow$$\exists j_{a}\neq j\rho,j_{\alpha},j_{\beta}\in\{j_{1}, \cdots j_{p}\}$$E_{:}^{j_{1}\cdots j_{p}}\neq 0$$\Rightarrow$$\{\begin{array}{l}\exists j_{\alpha}\not\in I_{0}\exists j_{\alpha}\neq j\rho\not\in I_{0}\end{array}$$\alpha_{i}^{h}\geqq 0$

The similar result is then obtained :$ifi\not\in I_{0}^{0}ifi\in I$

34
Theorem 6.– Suppose Condition $s6$ and 7. If the Cauchy data is sufHciently smdl in$H^{\cdot}(\ell=1,2, \cdots)$ , thsolu tion has a decay estim ate as follows :$e$$||u_{i}||_{H},$( so$||\tau_{4}||_{L^{\infty}}$

)

(110)
$\leqq\{C_{*}C^{*}\Vert_{u^{0}}^{u_{0}}\Vert_{H}^{H}.e^{-\frac{1}{2}\lambda t}$

where$C_{*}$$forfor$$i\in I^{0}i\in I_{1}$depends only on the equa tions system and onstant$c$$\lambda>0$.

Chapter 2
On the proof\S 2.1 Estimations
In this section, assuming Conditions 1. and 2, we establish estimations of solutions,improving the method due to Bony [3]. We assume, for simplicity, thatfor $i\neq j$ .
This hypothesis is not essential at all and we can recover it by the usual argument.
Let’s define Bony’s function [3] and its variation:$c:\neq c_{j}$(2.1)$\varphi(t)=\sum_{i,j}(c;-c_{j})\int\int$(2.2)$\psi$( aeo,$t.;$$c_{0}$:) $= \sum_{:}(c:-c_{0})\int sgn\{x-(x_{0}+c_{0}t)\}u_{i}(x,t)dx$

Differentiating these functions, we have
LeInma 2.1.–$Su$ppose $T<T^{*}$ . Under Conditions 1 and 2, we Izave(23)$\Delta(0,T)\leqq C\mu^{2}$(2.4)$\delta(0, T)\leqq C\mu$

where

(2.5)

$T^{\cdot}$is the existence time of solutions and$\Delta(t_{1},t_{2})=\sup_{e:\neq c_{j}}\int_{1}^{\ell_{2}}\int_{R}u_{i}(x,t)u_{j}(x,t)dxdt$

35
(2.6)

$\delta(t_{1},t_{2})=\sup_{c:\neq c_{j}}\sup_{x\in R}\int_{1}^{\ell_{2}}u;(x+c_{j}t,t)dt$

Proposition 2.2.– Suppose Condition $s1$ and 2. Then there exists $p\geqq 0sucA$ tlat
(27)

$I=I_{0}\cup I_{1}\cup\cdots\cup I_{p}$For analyzing closely our partial differential equations system, now we consider simpler
equations system ; this is motivated by the dissipation of the effects due to the binary
coUision terms when time is going to infinity, which is suggested by definability of the
wave operator for the system without the linear term, due to Bony [4] :(0)$\{\begin{array}{l}\frac{df_{i}}{dt}=L.\cdot(f)f.\cdot|_{t=0}=f^{0}>0\end{array}$

Proposition 2.3.– Suppose Conditions 1 and 2. 1) for $dJi\in I,$
is positive.
$f:(t)$ is increasing and bounded, so tends to2) for
Iim $it>0$ as $tarrow+\infty$ .
3) fortends to exponentiaUy as $tarrow+\infty$ .
$f_{i}(t)$$i\in I_{0},$$i\not\in I_{0},$

$a$

$0$

$f_{i}(t)$Now we fix
and decompose : into the sum of $(quasi-)linear$ part““(essential-)nonlinear part” $w;$ . Leta solution for the equations system:$u$$t_{1}$$v_{i}$

and$v_{i}$.
(V)$v_{i}$$\{(\frac{\theta}{\theta t}+c_{i}\frac{\theta}{\theta x})::_{(\cdot,t)}v:|_{\ell=\ell=u}$where . is the solution of $(B)$ .
Then $w_{i}=u_{i}-v_{i}$ should satisfy$u_{*}$(W)$\{\begin{array}{l}w_{i}=L.\cdot(w)+Q.\cdot(w)+r_{i}\cdot\cdotswith^{i}r_{i}^{1}=\sum_{j_{\prime}k,l}A_{ij}^{hl}(v_{k}v_{1}+w_{k}v.\cdot v_{k}\tau v_{l})w|_{\ell=t}=0\epsilon_{i}=\sum_{j,k,1}A_{k^{j}1}w.\cdot v_{j}\end{array}$Definition.– The operator $P=(P_{i})$ : is said to be positively preserving if $ond$ only if
th solu tion ; is nonnega tive over $R\cross R+$ ’ where $u_{i}(x,t)$ is a solution for th equations
system :
$e$

(2.8)

$u$

$e$

$\{(\frac{\theta}{\theta t}+c_{i}\frac{\theta}{\theta x})u:u_{i}|_{t=0}=u_{i}^{0}(x)=\geqq 0\mathcal{P}(u)$Corollary 2.4.– The operat $ors(Q_{i})$ : and$(L_{i})_{i}$are positively preserving.

36
Proposition 2.5.–(2.9)$v:(x, t)\geqq 0$

and $w:(x, t)\geqq 0$ for any $x\in R$ and $t\in n_{+}$

Now we have some remarks:

Corollary 2.6.–(2.10)$\int_{1}^{t_{2}}\int_{R}r:dxdt\leqq\Delta(t_{1},t_{2}),$$\int_{1}^{\ell_{2}}\int_{R}\ell_{i}dxdt\leqq\Delta(t_{1},t_{2})$We are now going to estimate the “linear part” solution
as follows:

Proposition 2.7.– The function
The estimation for$v$$V(t) \equiv\sup_{l:}\frac{v_{i}(x,t)}{f:(t)}$$v:$$L^{\infty}$’

is stric $tlydecre$ asin .
$g$; and $w$; along the characteristic as following:

and $t_{1}<t_{2}$ ,Proposition 2.8.– For$c_{\dot{2}}\neq c_{j}$(2.11)$\sup_{a\in R}\int_{1}^{\ell_{2}}v;(x+c_{j}t,t)dt\leqq C_{*}\delta(t_{1},t_{2})$(2.12)$\sup_{x\in R}\int_{1}^{t_{2}}w;(x+c_{j}t,t)dt\leqq C_{*}\Delta(t_{1},t_{2})$where th $ese$ constants. Its estimation in$C_{*}$,

depend only on th equa $tfons$ system.
$e$

Now we would like to estimate more closely $M(t_{2})$ in terms of $M(t_{1})$ for $t_{1}<t_{2}<\tau*$ ,where(2.13).

$M(t)= \max\sup_{\leq tx}\sup_{\in R}u:(x, s)i\in I$

for$t<T^{*}$For the first, we integrate, along a characteristic curve, the equations system for$w_{i}$$w_{*} \cdot(t_{2}, x_{*})\leqq C_{*}\int_{1}^{t_{2}}(\sum_{j\neq\dot{*}}w_{j}+\sum_{h,j\neq i}w_{h}w_{j}$(2.14)$+ \sum_{h,j\neq i}v_{h}v_{j}+\sum_{h,j\neq:}w_{h}v_{j}+\sum_{h,j\neq:}v_{h}w_{j})(x+c_{i}t, t)dt$

where $x_{*}=x+c:(t_{2}-t_{1})$ . Then we have
(2.15)$\int_{1}^{t_{2}}\sum_{j\neq*}w_{j}(ae+c_{i}t,t)dt\leqq C_{*}\Delta(t_{1},t_{2})$: then37(2.16)$\int_{1}^{t_{2}}\sum_{h,j\neq i}w_{h}w_{j}(x+c_{i}t,t)dt\leqq C_{*}M(\ell_{2})\Delta(t_{1},t_{2})$(2.17)$\int_{1}^{t_{2}}\sum_{h,j\neq:}v_{h}v_{j}(x+c:t,t)dt\leqq C_{*}M(t_{2})\delta(t_{1},t_{2})$Hence we have

(2.18)$\sup_{x}w_{i}(x,t_{2})\leqq C$.

$(1+M(t_{2}))\cdot\Delta(t_{1},t_{2})+C.M(t_{2})\delta(t_{1},t_{2})$Consequently for $u_{i}(x, t)$ , we have

(2.19)$\sup_{l}u:(x, t_{2})\leqq\sup_{l}v_{i}(x, t_{2})+\sup_{l}w_{i}(x,t_{2})$$\leqq C.M(t_{1})+C_{*}(1+M(t_{2}))\cdot\Delta(t_{1},t_{2})+C_{*}M(t_{2})\delta(t_{1}, t_{2})$We take $T<T^{*}$ and thereafter a sequence $0=t_{0}<t_{1}<\cdots<t_{N}=T$ such that$\Delta(t_{j}, t_{i+1})<(4C_{*})^{-1}$ and $S(t_{j}, t_{j+1})<(4C_{*})^{-1}$ . Then, seeing that $\Delta(0, T)\leqq C.p^{2}$ and$\delta(0,T)\leqq C_{*}\mu$ by virtue of Lemma 2.1, we have $N=O(\mu^{2}+\mu)$ , and$M(t_{J+1})\leqq C_{*}M(t_{j})+C_{*}\Delta(t_{j},t_{j+1})$$(2.20)$$\leqq C.M(t_{j})+C_{*}$

therefore we obtain(2.21)$u_{i}(x,t) \leqq(1+\sup_{i,x}u_{i}^{0})\exp(a\mu^{2}+bp)$

38\S 2.2 Proof of the theorem 3
We examine in a more detailed way the argument developed in the last chapter, thatis, to decompose the solution
into the sum of $(quasi-)linear$ part“ and (essential-)
nonlinear part . Later on, we specify , which will be noted , and the dependence ofand ; on a cut time . Let’s write downof the form $u_{i}=v:^{T}+w_{:}^{T}$ :$u_{i}$$v_{i}$$T$$w$$v_{i}$$T$$t_{1}$$w_{i}$$(V^{T})$$u_{i}$$\{(\frac{\theta}{\theta t}+c_{i}\frac{\theta}{\theta ae})v^{T}:=L_{i}(v^{T})-\sum_{j,h,l}A_{hl}^{ij}u_{j}\cdot v_{j}^{T}$$\{\begin{array}{l}w^{T}=L.\cdot(w^{T})+Q.\cdot(w^{T})+r^{\tau_{-l}\tau}with^{T}=\sum_{jk,l}A_{ij}^{hl}(v_{k}^{T}v_{1}^{T}+w_{h}^{T}v_{1}^{T}+v_{h}^{T}w_{l}^{T})w_{i}^{T}|_{t=_{\prime}}r=0\ell_{i}^{T}=\sum_{j,h,l}A_{h^{j}1}w^{T}v_{j}^{T}\end{array}v^{T}:|=u(\cdot, T)$$(W^{T})$

Knowing that
(2.22)$u_{i}(x, t) \leqq M\equiv(1+\sup_{i,x}u_{i}^{0})\exp(ap^{2}+b\mu)$,

we have, for $t>T$(2.23)$\sup_{l}w_{i}^{T}(x,t)\leqq C_{*}(1+M(t))\cdot\Delta(T,t)+C.M(t)\delta(T,t)$$\leqq C_{*}(1+M)(\Delta(T,t)+\delta(T,t))$(2.24)$\Vert\sum_{:}w^{T}:(\cdot, t)\Vert_{L^{1}}\leqq C_{*}\Delta(T, t)$Using Lemma 2.1. which says
any $e>0$ , there exi ts such that$\Delta(0, \infty)\leqq C_{*}p^{2}$$s$(2.25)

and$\delta(0, \infty)\leqq C_{*}\mu$, we conclude that, for$T$$\Delta(T, \infty)+\delta(T, \infty)<C_{*}[(1+M)\cross(\#\{i\in I\})]^{-1}\cdot\epsilon$

Then we have, for $t>T$ ,(2.26)

hence we have
(2.27)$\Vert\sum_{:}w^{T}:(\cdot,t)\Vert_{L^{1}\cap L^{\infty}}<\epsilon$,$\sup_{:}\Vert w_{i}^{T}(\cdot,t)\Vert_{L^{p}}<\epsilon(1\leqq p\leqq\infty)$Le.

(2.28)$\sup_{i}\Vert u;(x,t)-v_{i}^{T}(x, t)\Vert_{L^{p}}<\epsilon(1\leqq p\leqq\infty)$,

save for trivial constants. Consequently we prove the first assertion of the theorem.
Now we would like to estimate

$v$

; :

39
Proposition 2.9.–$2)\sum_{:}||V_{i}(\cdot, t)||_{L^{2}}^{2}$$I$)

is strictly decreasing.

$V(t)= \sup_{i,ae}\frac{v_{i}^{T}(x,t)}{f_{i}(t)}$is also strictly decreasing where$V_{i}(x,t) \equiv\frac{v_{i}^{T}(x,t)}{\sqrt{f_{i}(t)}}$We pursue the proof of the theorem. By the above proposition, we have

(2.29)

and$\max_{i}\Vert\frac{v^{T}:(\cdot,t)}{f_{:}(t)}\Vert_{L^{\infty}}$$\max:\Vert\frac{v_{i}^{T}(\cdot,t)}{f_{:}(t)}\Vert_{L^{2}}$are strictly decreasing, where a positive bounded’ functions
verifies the following
condition :is increasing and tends to a limit $>0$ as $tarrow+\infty$ .
-for
-for
tends to exponentially as $tarrow+\infty$ . The interpolation between
andachieves then the proof.
$f_{i}(t)$$i\in I_{0},$ $f_{i}(t)$$i\not\in I_{0},$$f_{:}(t)$$L^{2}$$0$$L^{\infty}$\S 2.3 Proof of the theorem 5
Let’s consider the following equations system with$(B_{e})$$\epsilon$:$\{\begin{array}{l}\frac{\partial u}{\partial t}+c.\cdot\frac{\theta u_{i}}{\theta x}=\epsilon Q_{i}(u)+L_{i}(u)u.\cdot|_{t=0}=u^{0}(\cdot)\end{array}$where is a positive constant.
For this Cauchy problem, we would like to seek a solution $u_{i}(z, t)$ of type $u_{i}=$$e$$\sum_{m=0}^{\infty}\epsilon^{m}u!^{m)}$(so-called Hilbert-Chapman-Enskog expansion).

Remark : From a physical point of view, corresponds to the inverse of the “Knudsen
Number” andcorresponds to a fluid dynamical limit to a &ee molecular flow.
$e$$\epsilonarrow 0$

To prove Theorem 5, it is sufficient to show the following theorem:

Theorem 2.10.– Suppose Conditions 5 and 6. For $e\in[0,$$m(m)$

converge in $H$ “ $(s=1,2, \cdots)$ , so$u;= \sum_{m=0}^{\infty}eu_{i}$$L^{\infty}$$C_{*}.( \sum_{h=0}J_{h}^{2})^{-1}2[$, the series, uniformly with respec to $t\in R+$$t$

and then$||u;||_{H}.$( so$||u;||_{L^{\infty}}$)(2.30)$\leqq\{c^{*}(\sum^{h=0}J_{h}C_{*}(\sum_{h=0}J_{h_{2}\}_{e^{-}}^{2}}^{2}\frac{1}{2}2\iota_{\lambda t}\iota$$w.here$ $J$.

$=( \sum_{i}\Vert Du_{i}^{0}\Vert_{L^{2}}^{2})^{2}\iota$

and constant$\lambda>0$.

constants$C_{*}$

and$C_{**}$$i\in I_{1}i\in I_{0}$depend only on th equation system
$e$

$s$

40
Like as in Chapter 6, we use(0)$\{\begin{array}{l}\frac{df}{dt}=L.\cdot(f)f_{i}|_{\ell=0}=f^{O}>0\end{array}$Condition 6 implies a more preci se estimation for than Proposition 2.3. :Proposition 2.11.– Suppose Conditions 5 and 6. There exists $f_{i}^{0}>0$ such that, fortends to with th sam order i.e. th $ere$ is $\lambda>0sucA$ th atis increasing an bounded, so tends to a limit
polynomi oft and forwith$>0$ as $tarrow+\infty$ .
$f_{i}$$i\in I_{1},$$0$$f_{i}$$e$

Let put$i\in I_{0},$$w_{i}(x,t)= \frac{u:(x,t)}{\sqrt{f_{i}(t)}}$$w_{i}(x,t)= \sum^{\infty_{=}}\epsilon^{m}w^{(m)}(x, t)$(2.31)$f_{i}(t)=e^{-\lambda\ell}P_{i}(t)$

$e$

$aJ$

$P_{:}$$d$$f_{i}(t)$Now we write down the equation for $w_{i}(x,t)$ , and put. Then we have for $m=0,1,2,$$\cdots$$\{\begin{array}{l}w^{(m)}=\sum_{h}\{\alpha_{i^{k}}(\frac{f_{h}}{f})^{\frac{1}{2}}w_{h}^{(m)}-\alpha_{k}w_{i}^{(m)}\}-\frac{L.\cdot(f)}{2f_{i}}w_{i}^{(m)}+F^{(m)}(w)w_{i}^{(m)}|_{\ell=0}=\{u_{i}^{0}(W)0,m=0m=1,2,\cdots\end{array}$where$\cdots$(2.32)$F_{i}^{(m)}(w)= \{0\sum_{n=0}^{m-1}\sum_{jh}B_{i}^{jh}(\frac{f_{j}f_{h}}{f_{i}})^{2}\iota_{w_{j}^{(n)}w_{h}^{(m-n-1)}}$$form=0form=1,2,$

The energy estimation leads us:

Proposition 2.12

–

$Su$

ppose Condi $tions5$ and 6. For $s=0$ , 1 ’ 2,$\cdots$]$\frac{d}{dt}\sum_{i}\Vert Dw_{i}^{(m)}\Vert_{L^{2}}^{2}=-\sum_{ih}\alpha_{i}^{h}f^{h}\Vert D^{\iota}(\frac{w^{(m)}:}{f_{:^{2}}^{\iota}}-\frac{w_{h}^{(m)}}{f_{h^{2}}^{1}})$$||_{L^{2}}2$(2.33)$+2 \sum_{:}\{(D^{\iota}p_{:}^{(m)}(w),$ $Dw^{(m)}:)_{L^{2}}\}$

especialJy for $m=0$ ,(2.34)$\frac{d}{dt}\sum_{:}\Vert Dw^{(0)}:\Vert_{L^{2}}^{2}=-\sum_{ih}\alpha^{h}:f^{h}\Vert D(\frac{w^{(0)}:}{f_{i^{2}}^{1}}-\frac{w_{h}^{(0)}}{f_{h^{2}}^{\iota}})\Vert_{L^{2}}^{2}\leqq 0$

CoroMary 2.13.– Suppose Conditions 5 and 6. Then we have, for $s=0,1,2,$(2.35)

where$\Vert Dw^{(0)}:\Vert_{L^{2}}\leqq C_{*}J$.

$J$$=( \sum_{:}\Vert Du^{0}:\Vert_{L^{2}}^{2})^{\frac{1}{2}}$.

.

for all$i$$\cdots$

41
Proposition 2.14.– Suppose Conditions 5 and 6. For $s=1,2,$(2.36)$\cdots$$\int_{0}^{\infty}\sum_{:}\Vert p_{:}^{(1)}(w)\Vert_{H}.dt\leqq C.\sum_{h=0}^{l}J_{h}^{2}$

Proposition 2.15.– Suppose Conditions 5 and 6. Th en we have, for $s=1,2,$(2.37)$\Vert w_{i}^{(1)}\Vert_{H}$Now let’s put(2.38)$\Vert w_{i}^{(m)}\Vert_{H}$.

$\leqq a^{(m)}$.

$\leqq C.\sum_{h=0}^{l}J_{h}^{2}$

for $s=1,2,$$\cdots$

for $dI$$\cdots$$i$, then we have by induction$a_{l}^{(m+1)} \leqq C_{*}b^{(m)}\equiv C_{*}\sum_{n=0}^{m}a^{(n)}a^{(m-n)}$Let’s put $f(x)=f.(x)= \sum_{n=0}^{\infty}a^{(n)}x^{n}$ , the above inequality means that
(2.39)$\frac{f(ae)-f(0)}{l}=C_{*}\{f(x)\}^{2}$,then we have
(2.40)$f(x)= \frac{1-\sqrt{1-4C_{*}xf(0)}}{2C_{*}x}$.

$f(0)=C$$( \sum_{h=0}^{l}J_{h}^{2})^{\frac{1}{2}}$It is easy to see that the right-hand side can be written in infinite series with a positive
convergence radius, which achieve the proof.

\S 2.4 Proof of Theorem 6
Like as in the previous section, we consider the following equations system with$(M_{e})$$\{\begin{array}{l}\frac{\theta u_{i}}{\partial t}+c.\cdot\frac{\partial u_{i}}{\theta x}=\epsilon R.\cdot(u)+L_{i}(u)u_{i}|_{\ell=0}=u^{0}(\cdot)\end{array}$where is a positive constant.
The same argument shows$e$(2.41)$\Vert w_{i}^{(m)}\Vert_{H}$.

$\leqq b^{(m)}$for $s=1,2,$$\cdots$

where(2.42)$\{\begin{array}{l}b^{(m+1)}=C\sum_{p=2n_{1}+}^{\sigma}\sum_{n_{p}=m}b^{(n_{1})}\cdots b^{(n_{p})}b_{l}^{(0)}=C(\sum_{k=0}J_{h}^{2})^{\frac{1}{2}}\end{array}$

Similarly we achieve the proof.

$e$

:

42
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