Asymptoic Behavior of Solutionsforthe …EquationsofIsothermal Gas Dynamics FUMIOKI ASAKURA 0. Introduction 71 We study the Cauchy problemfora hyperbolicsystem of conservation laws

Otemon Economic Studies，24（1991）

Asymptoic Behavior of Solutions for the

　Equationsof Isothermal Gas Dynamics

FUMIOKI　ASAKURA

0. Introduction

71

　　We study the Cauchy problem for a hyperbolicsystem of conservation

laws of the form:

い（ｻyTO’仇‾硲゜O

in R×尺＋

u(x, O)＝Moズ，v(x, 0)=Vo(x)>v＞0　on　尺

(1)

(2)

which　is　the　isothermal (model)equations　of gas dynamics　in　Lagrangian

coordinates.　Here V is the specific volume, u　the speed of gas and んa

positive　constant. The pressure is eχpressed as ）=k佃几　We set U＝(m, v).

F(の= (l/v, －u); we may assume that た= 1. The system is strictlyhyperbol-

ic for ﾉ＞0 (see section　1)and genuinly nonlinear in the sense of Lax [71.

Such genuinly nonlinear, strictly hyperbolic systems do not have in general

smooth global solutions. We say that U{x，t)(E:£゛is a weak global solution

to (1)with initial data びq(%)＝(mo(x), V(,(x))，ifthe following integral identity

holds:

于L

2O(び゛φ汁F(U)･φ)dxdt十Jし防{x)・φ(x, O)dx = O (3)

　　　　　　　　　　　　　　for every　φeLip。（i?xi?十）.

The existence of　weak　global solutions to the system （1）is established　by

Nishida［14］（see also （15]）using the Glimm difference scheme （［5］）:he

showed that if initial data are bounded and the total variation of which is

locally bounded, then there exists a weak global solution, which is the first

existence theorem for large initialdata （［14],［15]). In［O］the author showed

　　　　　　　　　　　　　　　　　　　　　（∩

72　　　　　　　　　　　　　　　　　FUMIOKI　ASAKURA

that the total amount of interaction of simple waves in the Nishida solution,

measured by the difference of Riemann invariants, is bounded and hence the

total variation of the solution decays to zero provided :

　　　　　　　　　　　　　　防圃）＝び。:constantfor　X j >趾　　　　　　　　　　（4）

　The aim of this report is to prove that the solution with initial data ひo（x）

satisfying

　　　　　　　　　　　　　　　　　　　　　T.V. Uo<L,　　　　　　　　　　　　　　（5）

which do not satisfy（4）, converges to the solution for the Riemann problem

with initialdata :

U{x,,)≪二了ズく0

X＞0㈲

In　order　to　study　the　aysmptotic　behavior　of　the　solution　following　the

argument of Liu［9], we have to and only to prove that the total amount of

interaction of simple waves measured by the defference of characteristic roots

is　bounded.　First　we　study　the　Riemann　problem　for　the　system （1）in

section 1. We shall show that the Riemann problem has a unique solution

for arbitrary （large）initial data. In section 2, we review Glimm's construction

of approximate　solutions （［5], （岡）and　Nishida's　result （［14], （15]）. We

shall prove, in section 3, that the total　amount of interaction measured by

defference of characteristic roots is uniformly bounded. Section　4 summarizes

basic properties　of approximate characteristics　which are first introduced　in

Glimm-Lax［6]. In section 5, we make a brief review of Liu's theory［9］on

which our theorem relies｡

　　　In［9］Liu　announced　that　he　had　proved　the　finiteness　of　the　total

amount of interaction in Nishida's solution even with large initial data, hence

convergence　of　Nishida's　solution　to　the　solution　for the　Riemann　problem,

indicating a recipe for proof. However l have not yet succeeded in proving

the finiteness following the recipe. Thus one can say that this report gives

another proof to Liu's result.

1

The Riemann problem

Let us denote び＝(祐v), F(U)= (ソ柘一m). We first study the associated

　　　　　　　　　　　　　　　　　(2)

　　ASYMPTOIC BEHAVIOR OF SOLUTIONS FORTHE EQUATIONS OF ISOTHERMAL GAS DYNAMICS

quasi-linear system :

妬十呪ダ(の＝0 (x,t)^Rx孔．

If v＞0, F (U)has two real distincteigenvalues:

　　　　　　　　　　　　λ.(の＝一言，λ2(のV

Th111

1

ご

a

ごｽﾂﾞでlj1ご1こｲiで

r

lV

t

Jご1111(7)is hyperbolicfor v＞0

　　　　　　　　　　　　切＝u ＋logv.　z=u －logV.

(7)

(8)

These functions can be used as a new coordinate system. By (8）we have

u ＝士叫十z)，v＝e十(゛z)

73

Hence the correspondence （m, v）→（w，z）is a （global）diffeomorphism from

尺×R＋to R^ and V＞Vo corresponds to w －z＞Wo－2o.

　　　Alsowe find that

匹

∂耀=召

÷(゛^)＞o，等＝召÷(w -z』＞0，

which shows that the associated quasi-linear system （7）is genuinly nonlinear

in i?xi?.ト.

　　Let U, =（u,, V,）, 隔＝（㈲，晦）（伽，v.:positive）be constant vectors. The

Riemann problem for

　　　　　　　　　　　　　　　　　　　　萌十F（U）。＝O　　　　　　　　　　　　　　　　　（9）

is the Cauchy problem with initial data :

び収0)=j^でｽﾞ＜0

ズ＞0

(10)

To our particular system, we shall see that a solution eχists even if　l び尺－び£1

may large, which is proved by Nishida ［141.

　　Let び(,＝（痢，砺）be a constant state in R×i? ＋. We introduce a parameter :

e = log三

び0

The 戸rarefaction curve （y= 1, 2）through びîs defined by

　　　　　　　　　　　　　　　　　　　　∩）

74 FUMIOKI　ASAKURA

i?,(防)= {(M, v) ; u一痢＝E，

瓦(防)＝{u，v) ; u一痢＝ε，

首＝が,eGi?}

首＝召二と(E?｝

In a similar manner, we define the /-shock curve （ﾌﾞ＝1, 2 ）through びn by

s,(びo)＝{(M, v); u一Mo＝2 sinh･1ﾚ，寺＝が,ε∈i?}

S2(びo)＝{(w, v); u一両＝2 sinh2 '首=eミe(ER}

By direct calculation, we obtain the following proposition:

Proposition 1.3.　Let U^Sj（びo）（i=l, 2）. Then U, Ua satisfy

　　　　　　　　　　　　　s(U, びo)(びーびo)＝F(び)－れびo)

with s(U, Un)＝(一丿(回n)尚'^ which is called the j-shock speed

　　Now　assume　that　び, = (M,, vC)<^罵（防）with ﾉ＞むq（e＞0）.　Then we find

that 防and t/、are connected by 戸rarefaction wave. Next assume that t/庶

吊（びo）（ﾌﾞ=l, 2）, む＜ﾉo （£＜0）.　In this case　Un and　び,　are connected by i-

shock wave with x = ts its shock front. Moreover we can verify that the Lax

entropy condition holds in the following way:

　　　　　　　　　　　　　　　　λ,(び,)＜s(びo, び)＜λ;(碩).　　　　　　　　　　　　(11)

　Now we study the geometry of curves 馬，S, (ブ= 1, 2)in tcz-plane. The

following proposition is easily verified.

　　Proposition 1.4.　Let W(,＝（友a，2o）bethe point corresponding to Uo＝（痢，

ひn）and Rj（Wo）the image of the curve Rj（びo）（ﾌﾞ=l,2）. Then Rχ（Wn）is the

horizontal line through Wq and■^2（W'ô）is

the verticalline through Wn.

　Let S,（Wô）theimage of SXUr,）in wz-plane. SAWo）is expressed as the

following form:

　　　　　　　　　　　　　　（4）


75

　　　　　　　　　　　　　　　　　　扨－Wa＝2 sinh号十ε　　　　　　　　　　　(12)

　　　　　　　　　　　　　　　　　　z－Z(s＝2sinh号－£.　　　　　　　　　　　　(13)

Since　φ(ε)＝2sinh(£/2)十E　is　a　monotonically　increasing　C "-function　in 尺

satisfyingφべε)＞0，then φ‾‰)is also monotonically increasing C 町function.

Thus there exist a C "-function / such that S(Wo)is expressed as

　　　　　　　　　　　　　　　　　　Z－ZQ＝/叫－WQ).

Similarly, using same /, Si(Wa)is expressed as

　　　　　　　　　　　　　　　　　　w一Wq＝バZ－Zq).

Summarizing above discussion, we obtain the following proposition :

　　　Proposition　1.5 （Nishida［141. （15]).

pressed as

Curves　S.（M/o）（i＝l，2）are ex-

S,(Wo)＝{叫，z}(E?'； z－Zo＝f佃－Wo)}

S2(Wo)＝{(w，z)e^?"； g一心＝バZ－Zn)}

Here f(ぎ)is a C゛づ皿ction with properties:

　　(i)/(ーぎ)＝一八ぎ)

　　(ii)　バO)=が(O)＝丿(O)＝0，バ(O)＝1/4

　　(m)O＜が(ぎ＜l，sgn(ぎ)/"(ど)＞0 for ぎ≠0

The following remark is important in later argument.

　　Remark 1.6（see Nishida［141).

the paralleltranslationw.→w‰

(14)

(15)

We can identify 吊（Wo）and 妬（庄o）by

Let us define wave curves Tj（W）（y＝l, 2）as the following :

ぺ寸言言Wo）

大川二≒。O）（5）

for　拓三倒(i

for　勿<耀n

for　2≧Zq

for　Z<Zn

(㈲

(17)


Then, by above argument, we find that W＝W(のbelongs to 石(P7o)，if and

only if U is connected to びn by a f-rarefaction wave or a i-shock wave.

　　Now, we can solve the Riemann problem in the following way. Let 呪＝

防and Wô＝M^(びo).Curves T,(Wa)，乃(W^)divide the luz-vlane into four

parts. Then for any U＝Ur，there eχistsa unique intermediate state びm such

that U, and Uu are connected by 瓦or s. and Um and 隔by 鳥or s. accord-

ing to the region to which 隔belongs (c.f. Remark 1.6). Thus we have the

following theorem :

　　Theorem 1.7（Nishida［14],［151).　　For any　Ur， Ul^以　there exists a

weak solution to the Riemann )roblem. The solution consists of three constant

states connected by rarefaction waves or shock waves. Moreover the solution of

this form is 皿ligue and the solution satisfies

w(U{x, i))>mm{w(U,)，w隔)}

z(U{x, t))<max位(t7r), z(Ur))

　The magnitude of waves in a solution to the Riemann problem are defined

as the following:

6,=w(Um)－W(U,):

ε2＝zみ)－z(U,;)　I

the magnitude ofレwave

the magnitude of 2-wave.

We abuse e.口= 1,2）to indicate the wave itself and denote e = (eい£2）. Also

we define e/ = max {土ら，0} which is called the stre昭th of /--rarefaction（or

shock）wave.

2. The Glimm difference scheme

Let 昌)収)＝(痢口)，ﾉnW)be the initial data. We set

　　　　　　　　　Wo ＝inf tﾉ(Uo(ズ))，2o＝sup zび.(x))

　　　　　　　　　塙＝{U^RX?パw(の＞Wo, z{のくZo}

Let h，んbe mesh length satisfyingthe stabilitycondition:

　　　　　　　　　　　　　　　　　　kλ＞召÷（to.･ヤ.

We hold suchλfiχed. Let d＝{On) be a sequence of random numbers uniform-

　　　　　　　　　　　　　　　　　　　　　（6）


77

ly　distributed　in (－1, 1) ；　we　may　assume　that　θ　is　an　equidistributed

sequence in (－1. 1).　Let m, n　be integers so chosen as m十n = even, n≧O。

We set α,,n＝((四十d,,)h, nk)for 万>0, which will be our mesh points. The

half　plane　尺×尺や　is　divided　by　a　countable　number　of　diamond　shaped

domains∠i抱｡ .defined by vertices :

　　　　　　　　　　　　Nこa ,刀十,,/^＝a, ＿,，, S＝a ,,刀｡_i,h＝a^＋i.

which is called the interaction diamond centered at (mh, nk).

　　　Now we define Glimm approximations U＝防いas the following. We take

Ui,,9(a,,,,o)＝防((四十θo)ん).　Assume that U,.e is defined at　(x. )=a,,,－,, and

flm+l. ■　To define 叫｡θ(α,n.n-i)，we solve the Riemann problem :

　　　　　　　　　‰十F(V入＝O (m－l)h<x<(m ＋l)ん, nh＜t＜(n＋l)た,　　　　(18)

　　　　　　　び(ら－i.。)(m－l)h<x＜mhVix. nk)ニ↓び(ら＋い,)mhくx≦(jn十l)ね。

By Theorem 1.7, a solution V eχists and then we define

　　　　　　　　　　　　　　　　　f/,い(ら,,,+|)＝^(α。,.,,+i)｡

(19)

It　is　convenient　to　set　び,,e(x，t)＝V{x,　t)for(m－l)h<x＜(m＋l)h, nh<t＜

(n ＋＼)k. Since rarefaction and shock waves issued from (mh, {n－＼)か)and

((m ＋2)h，(n－l)ん)do not intersect for (n一＼)た<t<肋, we find that U,,e亀. t)

thus defined is an eχact solution in the strip. In this way, we construct an

approximate solution 萌.o巳　O for x(ER, 0<t<T.　Also we can see easily by

Theorem 1.7 that U,,.o亀，t)isin 塙｡

　　In order to prove the convergence of the approximate solutions，it is suf-

ficient to establish a uniform bound :

　　　　　　　　　　　　　　　Totalvariation of U,,.e(x，t)<C,　　　　　　　　　　(20)

where C is chosen independent of h, e. One can show that the approximate

solutions satisfy

　　　　　　　　　　　　　　　　　　　lim{Uい ,:c十F(U,,,g),}＝Q　　　　　　　　　　　　　　　　　　　*→0

in　the　weak　sense　for　almost　all　choice　of d　and　then　employing　Helly's

theorem, one can prove that U,,.a thus selected really converge to a weak solu-

tion.

　　　　　　　　　　　　　　　　　　　　　　　　　(7)


　　To get such a bound（20），we have to estimate the magnitude of element-

ary waves in the solution to the Riemann problem （18）,（19）. The elementary

waves issuing from （（m－l)h，(n－l）ん）and （（四＋l)h,(n－l）た）and entering

∠^刀しny denoted byβand ｱ，are called the incoming waves with respect to∠^詞,n｡’

Also the elementary waves issuing from （批h, nk）, denoted by a, are called

the outgoing waves. We say in short that βand アinteract in ∠]。,。and gener-

ate a｡

　　Glimm［5］proved（20）for initial data with small oscillation and Nishida

［14］proved for arbitrary initial data in the isothermal equations. Nishida's

estimates are based on the following lemma :

　　Lemma 2.1（Nishida［14])　　Letβ, r be incoming waves and a outgoing

waves with respect to an interaction diamond∠1.Then it follows that

jFj√< 耳 (β√十rp (21)

　　This lemma says that the negative variationof the approximate solution

does not increase. The estimate（20）is an easy consequence of the lemma.

3. Interaction of waves.

　　　Leta be outgoing waves and βand 7 incoming waves with respect to an

interaction diamond　∠J＝∠臨,n･　Equivalently, we say that a　are produced by

interaction ofβand r in　∠J,which is denoted　by β十ｱ→a.　Suppose thatβ

connect Uo and びm and that アconnect Um and Ud.Since the correspondences

びルto Um and 防m to f/^ are c with piecewise continuous third derivatives,両

is also a c-一function of β＝(S,,β,)and r＝(ri,γ).The aim of this section to

study how a depend onβand ｱ. Throughout in this section 7 denotes l for i

= 2 and denotes 2 forﾌﾟ＝l. We define the quantities e-OS,-, r,)as the follow-

ing (c. f. Asakura[O]):

Q’β:,･r,-)＝thestrength of the 升rarefaction wave generated by the interac-

　　　tionof the shock wavesβ), 乃ﾌﾟ＝l，2).

Q'拓r;)＝the strength of the トshock wave generated by the interaction

　　　ofa rarefaction waveβ',■and a shock wave 乃(ﾌﾟ＝l, 2).

We set

（8）


　　　　　　Q(∠1)≒ (22)

79

　　Proposition 3.1（Asakura［O］）.　　Let a，β，r,Q be as above. Then it fol-

lows that

　　　　　　　　　　　　　町=β汁乃十〇(i)e(∠1)　(ﾌﾟ＝l，2)

where 0(1)depends only on the magnitude of incoming waves.

(23)

　　The main result in Asakura[O]is the following theorem which shows

the boundedness of the total amount of interaction of simple waves.

　　Theorem 3.2　　Let Q(∠)be the total amount of interaction 仇∠j面fined by

(22). Then it follows that

　　　　　　　　　　　　　　　　　　　ΣQ(∠)<c，

where C is independent of h, k, 6.

　　The magnitude of 友一waves are also measured by

　　　　　　　　　　　　　　　　　どi＝λ,(びJ－λI(呪)

　　　　　　　　　　　　　　　　　ら＝λ2(隔)－λi防)

which are equivalent to those defined in section 1.　Clearly ら＞O(＜O)，if and

only ifら＞O(＜O)．Let

　　　　　　　　　　　　Q(β，ア)＝Q(^)＝β2llr,十e(β，r).

We shall show that the amount of interaction in ∠jmeasured byらis equival-

ent to that measured by ら(c. f. Liu (13], Theorem 3.3).

　　Proposition 3.3.　Let a,β，ｱ, Q be as above. Then it follows that

　　　　　　　　　　　　ら＝烏十乙＋o（）Q（且r）（i=i,2）

where 0（1）dependsonly on the magnitude of incoming waves

　　We firstsettlethe case of two wave interaction.

　　　　　　　　　　　　　　　　　　　　（9）

(24)


　　Lemma 3.4　(i）//β斤ｱi＝0, then α-i＝β1，α2＝γ2.

（ii）//β＼＝γt＝0, then

　　　　　　　　　　　　　　　　　at＝ﾃ＼十0（1）β2II ri I

　　　　　　　　　　　　　　　　　&i＝鳥十0（1）β2卜ll.

　　　　　　　　　　　　　　　　　　　　　　　　　　　－　　Proof.∩）is obvious（no interaction）. Let U denote the intermediate

state connecting a. and a,. We find that

　　　　　　　　　　　　　　　　　　　　－　　　　　　　　　　　　　　　　≪,=λ,(のーλI(呪)

　　　　　　　　　　　　　　　　　＝fl十の(β2，7l),

where のis a C ^-function defined by

　　　　　　　　　の(β2,r,)＝{λ,(のーU呪}－{λl(ら)－λ,(びm)}.

SinceくZ)(β2，O)=の(O, ri)=O, we have IKβ2，ri)=O(i)β2lけiト　Hence

　　　　　　　　　　　　　　　　&^＝ﾃﾞ汁0(1)I β2II7＼

1n a similar manner we get

　　　　　　　　　　　　　　　　尚＝爪十0(1) Iβ2IIr①

　　Lemma 3.5.　　S以節>osethatβ'―乃―n･

∩)が0j＞0, rj＞0, then ≪i＝爪十万&,＝0･

(ii)が高＜0，乃･く0, then

　　　　　　　　　　　　　　ら＝爪十乙十0(1)0 'βﾌﾟ，石)

　　　　　　　　　　　　　　ら＝O(l)Q＋貼ry).

（ffi）//鳥＜0, r,＞0, then

弟=爪十fj＋O(l)Q一隅,ri)

a,＝○(＼)Q-爪, ry).

Proof.　(i)is obvious. Ifβj＜0, 乃＜0, then

　　　　　　　　　　　　　　　　　　　　　　　　一　　　　　　　　　　　　　　　島＝λJ(隔)－λ;(の

　　　　　　　　　　　　　　　　=○(l)両

　　　　　　　　　　　　　　　　=o(i)Q ＋(s,-,r,),

　　　　　　　　　　　　　　　　　　　　いい


Ifβj＜0，乃＞0，then

の={λ,■(の一λi(U,)}十隔(隔)－λj(の)

　　　　　－　　十{λ;■(のーλ/(隔)}

　=爪十方十0(＼)a,

　=/3i十f,-＋0(1)ぐ隅,r,-).

　　　　　　　　－a戸λ-,(曝)－λ,(の

　＝O(i)Q－鳥，乃)，

a戸{λjび)－λj(呪)}十亀(隔)－λj(の)

　　　　　－　　十{λj(∽－λi(隔)}

　゜爪十乙十〇(l)Q＋03,.,7/).

Thus we have proved the lemma.

81

　　Proof of Proposition 3.3.　　Let Bo, and ri interact first separately. Then

we have outgoing waves ∂i，∂2such that

　　　　　　　　　　　　　　5,＝f汁0(1)β2 11r, I

　　　　　　　　　　　　　　尽＝八十0(1)I β2IIr工

Next let βl and ∂,interact and let∂2 and ｱ2 interact. These interactions gene-

rate /3' and 7' satisfying

　　　　　　　　　/3,'＝β,十ﾁﾞﾄO(l)Qニ(β|，5,)

　　　　　　　　　　　＝βi十ﾃﾞﾄ0(1)I β2IIr,十〇(1)0ﾆ^:(S,, r,)

　　　　　　　　　応＝o(i)qにβ|，s)

　　　　　　　　　　　＝0(1)|β, IIr,十〇(1)Qﾆt:(β1，r,)

　　　　　　　　　f[＝O(!)Qﾆにγ2，∂,)

　　　　　　　　　お＝鳥十ﾁﾞ汁O(1)Qここ(β2, δ9).

By uniqueness of the Riemann problem, we can see that a are produced by

the interaction of ダ　with Λ　Thus we obtain

　　　　　　　　　d,=/3,'十〇(i)β汗o(i)r,'

　　　　　　　　　　　＝β|十ﾁﾞ汁0(1)β2II r＼十〇(l)0に(β，r)

　　　　　　　　　励＝八十〇(1)βj十〇(i)r,'

　　　　　　　　　　　＝i3|十ﾁﾞ汁0(1)β2 11川十〇(1)Qてβ. r),

　　　　　　　　　　　　　　　　　　∩∩

82

which proves the proposition.

FUMIOKI　ASAKURA

　　　The principal result in this section is the following theorem　which says

that the total amount of interaction measured by the defference of characteri-

stic roots is also uniformly bounded.

　　Theorem 3.6　Let Q（∠）bethe totalamou戒of interactionin A defined in

Proposition3.3、Then it follows that

ΣQ(A)<C，

where C is independent of h，k，6.

　　Proof.　By Theorem 3.2, we have only to show that

Σ|β2 11アx I <C.

To the sum for crossing two shocks, argument in the proof of Lemma 5.3 in

［O］is also　valid.　To　the　sum　for other crossing waves, we　proceed　in　a

similar way as in the proof of Lemma 5.4 in ［O］.Let / be an /-curve. Denote

that

0(/) ＝β:-. 7･

Σ　　(1β2！榴一十β∩r, I）.

crossingJ.

Let /'　be another /-curve such that戸J is composed of WN and NE sides of

　　　　　　　　　　　　-an interaction diamond ∠jand /一丿　iscomposed of ws and SE sides of ∠I

Obviously /'＞/.　Now suppose that 戸waves /3;,jj enter∠j and ay leaves∠I

With £,being an arbitraryshock wave crossing/－∠1，wehave by Theorem 4.5

　　　　　　　け2 Iα＼＋<I£9β1＋十け■21ぐ十ぽ2l Q＋β2，ｱ^)

　　　　　　　e, I爾七旧|βよ十ぽ1 Iが十ぽ1 I Q＋(βi，ri).

Then we have

　　Σ　　ε2 Iα,゛<　Σ　(｜ε2 Iβi＋十ε･21 か)十現゛(β2,址)と:crossing/　　　　　　し:crossingj

　　Σ　　ぽ!レド≦　Σ　(ぽ1！βﾀﾞ十けi絃)十徊＋βi，rよ　　　　　　　と:crossi昭/

Hence, denoting

ε:crossing /

∩2）


Q(∠)＝Iβ2 I it十β2∩ｱi

83

we get

　　　　　　　　　　　　　　　　Q(ア)<Q(/)－Q(∠J)十飼＋(β;r).　　　　　　　　　　(25)

Let 0 be a unique /-curve lying between t=0 and t＝んand ブalso an /-curve

lying t＝nk and (n＋1 )た. Summing the inequalities (34)over all diamonds in

the region 0<nk, we obtain

　　　　　　　　　　　　　　　　ΣQ(A)<Q(J)－Q(6)＋IΣぐ(β，r)

　　　　　　　　　　　　　　　　j　　　　　　　　　　　　　　』　　　　　　　　　　　　　　　　　　　　　くゼ-'十CP.

Thus we have the lemma.

4. Generalized Characteristics

　　The　notion　of generalized　characteristics　was　introduced　by　Glimm-Lax

［6]. They showed that a generalized characteristic 石け）口＝1,2）exists as a

limit　of　a　sequence　of　approximate 升characteristics　Xj.h（t）in　the　Glimm

approximations Ui,,g.These approximate characteristics converge uniformly to

a　Lipschitz　function　Xj（t）on　bounded　intervals　of　time.　The　derivatives

converge pointwisely with exception of a certain countable set of values of Z :

　　　　　　　　　　　　　　　え()＝lim χii()　a. e. t.　　　　　　　　　　　　　　　　　　　か→0

They　also　showed　that　for　all　but　a　countable　set　of　f，the　approximate

solutions are equicontinuous on either side of the approximate characteristics

([6]Lemma 3.4), which guarantees that the following limit exists for a.e. t：

　　　　　　　　　　　　　　　　　　　　　じﾆ＝び吊(t)士O).

0n points of continuity of U at x=x(t)(i.e. U白＝Uつ, the characteristic 石

propagates at characteristic speed :

　　　　　　　　　　　　　　　　Xi(t)＝λ,-(t/凧)),t)ﾌﾞ＝＼，2)

and on points of discontinuity, 為･propagates at shock speed :

　　　　　　　　　　　　　　　　　　　右t)＝sび＋，Uつ.

Glimm-Lax's　proof of　the　results　above　is shifted　to　our case　with　minor

　　　　　　　　　　　　　　　　　　　　　　　　(13 )


modification. Modification is needed in constructing approximate characteri-

sties. In our case, when the amplitude of incoming waves is large, the interac-

tion terms Q犬may dominate the additive terms. If a /-approximate charac-

teristicenters a diamond∠j with two 斤rarefaction waves and a y-shock wave

leaves, then the approximate /-characteristicis continued in a unique manner

along the outgoing shock　wave. Similarly, if a /-approximate characteristic

enters∠1 with two 戸shock waves and a ﾌﾞｰrarefactionleaves, then the /-appro-

ximate characteristicis continued, for example but in a definite manner, along

the　left　edge　of　the　outgoing　rarefaction　wave.　In　the　proof of one-sided

equicontinuity at approximate characteristics, we have only to note that the

following quadratic quantity is uniformly bounded :

βバ,

?ﾐj

hocks

l β>.y.｜＋:shocks,

r,: rarefactions

|β痢｜

Thus all properties of generalized characteristics proved in ［6］are also verified

in our case.

5. Proof of decay via Liu ［9］

　　In this section, we shall make a brief sketch of Liu's work［9］on which

our proof of decay completely relies. Let xl and xl be generalized k一charac-

teristicsissued from 乙＝らwhere xt lies to the left of χぶ　We set

that

Z)t()＝distance between χl and xi- at time t, t>tn

ば侑)＝び(x抑)士0, t), λン＝几(俳壇))．

Proposition 5.1 There exist e'＝d'(t)(z＝l, 2)satisfying 0くe'(f)＜＼such

仄(t)＝λよ^(0－λよ'≪

十古(1十θ'){λよ'(わーλご(わ十万(i－θ')}λ,r'()－λ√'()} (26)

　Moreover since we may assume that there exists θ0 such that　Oくθ'くθ0，we

find that

爪(0 <-と{λごt)－λ√'()}十万{λご(£)十λこ1(Z)}

　　　　　　　　　　　∩4 )


十手(KKt)－λ:Kわ)十孚(λに(t)－λグ(だ)

＝万(1十e,){λ,^Ktトλご≪}十万(1一e,){λごt)－λk'(t)}

Hence setting θ＝十(1十θo), we obtain a propopsition :

Proposition 5.2.

　仄(f)≦θ{λkKわーλご≪ト(i→)榴犯}－λ.-'(f)}(0くe＜i). (27)

85

Denote that

χに（0＝total　amount　of　友一rarefaction　and　た-shock　waves, respectively,

　　　　　between（but not on）返and 坦at time Z

丸＝total amount ofトwaves （ﾌﾞ≠k）between χl and xi at time Z

K（t,バ）＝total amount of シWaves （i≠k）crossing xl or vl

店（ら，）＝totalamount of interaction in the region between 肩and xl for

　　　　　to＜s＜t.

We have by approximate conservation laws :

Since

we have by （24）

瓦()＝0(1)J[dQ(to，s)十dh(tn, s)]

脳聯)－λよ|(£)＝x,:(t)十X√(t)十O(D瓦i)，

玖()＝x/()十xよ(わ十万'十的strxK)十万l→')str x,l(t)

　　　　十〇(i)J言Q(ら,s)十dh(f),s)].

Integrating　above　expression　from　ら　to　ら　and　applying　the　argument　in

Glimm-Lax［6], section 3，we obtain the following lemma :

　　Lemma 5.3.　　Leによi，be any time afterwhich 示and 回do not intersect.

Then it follows thatfor t>t.

x川)<ﾃﾞﾄﾄ0(l){Qム, }十駈(U,t)}

　　　　　∩5 )

(28)

86 FUMIOKI ASAKURA

Obviously the characteristic speeds are strictly separated, i.e. there exist fi,

(0</<3) and a constant 5>0 such that

1u0<minU1(&r); U(EQ}-8

maxU,(£/); UGQ} +8<nl<min{12QJ) ＼U(EQ}~6

max{A2(f7); U^Q) +6<fx3.

Hence the half plane RxR+ is divided by the regions £?,･defined by

£o={Ot, t); 0<j<^0}

£>･={(*,t); n^<j<vi], (y=i,2)

fiS,={(x, t); A3<f}

We have proved, in section 3, that the total amount of interaction is finite.

Then it follows that for any e> 0, there exist t=ta(e) and M=M(e) such that

cOo. °°)=c{(*, t) ; t>to}<e

T. V.{U(x, Q ; |x | >M} <£>

Let x*1 and x* denote the &-th characteristicsissued from (―M, ft0)and (M, f0)

respectively. Also let

Fk : the region between Xiland xl (k = l, 2)

Ao : the region left of xi

/I,: the region between X＼and xi

Ao '■the region right of y|

By approximate conservation laws, we can see easily the following lemma :

Lemma 5.4 ([9], Lemma 5.1).

( 1 ) The amount of k-waves outside Pk at time t is 0(1) s.

(2) The totalvariation of U in regions Aj at time t is O(l)e.

(3) For any (xu U) and (x2, h) in Aj: | U(xu tx)-U(x2, t2) ＼=O(l)e.

(4) Z/(O<^f +O(l)e.

Following two lemmas are decisivein estimating totalamount of fe-waves

in the region Fk.

( 16 )

ASYMPTOIC BEHAVIOR OF SOLUTIONS FORTHE EQUATIONS OF ISOTHERMAL GAS DYNAMICS

87

Lemma 5.5 ([ 9 ], lemma 5.4) Suppose that

lkW(xk, t,^<Xk(JJ{Xk-i, 4-i))-?7

for some (xk, tk)Âk, (xk-i, tk-i)Âkî and tj>0. Then there exist a constant eo

>0 independent of t such that for sufficientlylarge t and 0<£<e0,

(1) X,;{i)=O(S)e.

(2) xl o,nd xt come together and unite into a k-shock with magnitude

h(X(xk, tk))-Xk(U(jKk-lt **_,))-O(l)e

Lemma 5.6 ([ 9 ], Lemma 5.5) Suppose that

h(U(xk, tk))>lk(JJQxk-x, tk-,))-de

for some (xk, tk)Âk, (xk-＼,tk-＼)Âk-＼and d>Q, Then for sufficientlylarge t,

it follows that

( 1 ) I Xk'{i) I + I strx^ | + | strxl | =O(l)e.

(2) U(xk, tk)^Rit(.UOck-i, ^-.))+O(l)e.

Using these lemmas, we have the following theorem which is the main

resultin thisreport.

Theorem 5.7 ([ 9 ], Theorem 5.7) Suppose that the Riemann problem with

initialdata (6) is solved by centered waves ex,£2connecting UL, UM, UR. Then

it follows that

( 1 ) U(x, t)-~>UL,UM, Ur as I -* 00 for j =p.i(f=l, 2, 3).

(2) // £,-is a rarefaction wave, the amount of i-shock wave in Q{ approaches

zero as t -* °°and U(x, i) approaches the rarefaction waves £,･pointwise in

Qt as t -> °°.

(3) // £,-is a shock wave, then there exists an i-shock wave approaching the

shock wave e,-both in strength and speed. Moreover, the total variation of

the solution in <2,-outside of this shock wave approaches zero as I -> °°.

References

[ 0 ] F. Asakura, Decay of solutions for the equations of isothermal gas dynamics, to

appear J.J.I.A. M. 1992

[ 1 ] R. Courant and K. 0. Friedrichs, supersonic Waves and Shock Waves. Wiley-Inter-

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　　　science New York，1948.

[2]R. J.DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic

　　　systems of conservation laws. Indiana. Univ. Math T.,24 (1975), 1047－1071.

[3]R. J.DiPerna, Decay of solutions of hyperbolic systems of conservation laws with a

　　　convex extensions. Arch. Rat. Mech. Anal, 64 (1977), 1－46.

[4]K. 0. Friedrichs and p. D. Lax, Systems of conservation laws with a convex exten-

　　　sion. Proc. Nat. Acad, Sci.,U.S.A., 68 (1971), 1686－1688.

[5]J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations.

　　　Comm. Pure Appl. Math., 18 (1965), 697－715.

[6]J. Glimm and　p. D. Laχ, Decay　of solutions of systems of nonlinear hyperbolic

　　　conservation laws. Amer. Math. Soc. Memoir, No. 101. A. M.S. Providence, 1970.

[7]p. D. Lax, Hyperbolic systems of conservation laws n. Comm. Pure Appl. Math｡10

　　　(1957), 537－566.

[8]J. D. Laχ, Shock waves and entropy. Contributions to nonlinear Functional Analy-

　　　sis, edited by E. Zarantonello. Academic Press New York, 1971, 603－634.

[9]T. P. Liu, Large time behavior of initial and initial-boundary-value problems of

　　　general systems of hyperbolic conservation laws. Comm. Math, Phys., 55 (1977),

　　　163 －177.

[10]T, P. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic

　　　conservation laws. Comm. Pure Appl. Math., 30 (1977), 585－610.

[11]T. P. Liu, Linear and nonlinear large time behavior of general systems of hyperbo-

　　　lie conservation laws. Comm. Pure Appl. Math., 30 (1977), 767 －796.

[12]T. P. Liu, The deterministic version of the Glimm scheme. Comm. Math. Phys., 57

　　　(1977)，135－148.

[13]T. P. Liu, Asymptotic behavior of solutions of general system of nonlinear hyperbo-

　　　lie conservation laws. Indiana Univ. Math. J.,27 (1978), 211－253.

[14]T. Nishida, Global solution for an initial boundary value problem of a quasilinear

　　　hyperbolic system. Proc. Japan Acad｡44 (1968), 642－646.

[15]T, Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics.

　　　Publ. Math, No. 79－02.Univ. Paris-Sud，1978.

[16]J. Smoller, Shock Waves and R〉eaction一DiffusionEquations. Springer Verlag New

　　　York, 1983.

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Asymptoic Behavior of Solutionsforthe …EquationsofIsothermal Gas Dynamics FUMIOKI ASAKURA 0. Introduction 71 We study the Cauchy problemfora hyperbolicsystem of conservation laws