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PureMathematicsnØêÆ,2021,11(7),1400-1415
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.117157
šàgBurgers•§iùЊ6įK)
ìC-½5
ÜÜÜîîŒ§§§ooo
þ°“‰ŒÆ§þ°
Email:1914939880@qq.com
ÂvFϵ2021c612F¶¹^Fϵ2021c714F¶uÙFϵ2021c721F
Á‡
©Ì‡ïÄšàgBurgers•§…ܯK§ÐŠ•iùЊ±Ï6Äž§ÄÅ(ìC
-½5"·‚uy-Å)6Ä§3k•žmTE•-Å)§3?¿ž•t>T§§†mGE
•±Ï¼ê§… 3L
∞
‰ê¿ÂeP~–0"AO/§6Ä-Å3-Åüý{ħ6Ä
DÕÅ)3L
∞
‰ê¿ÂeP~–0"
'…c
šàgBurgers•§§-ŧDÕŧ±Ï6ħ2ÂA‚
AsymptoticStabilityofShockWavesand
RarefactionWavesunderPeriodic
PerturbationsforInhomogeneous
BurgersEquation
ZhaoxiangZhang,YueLi
ShanghaiNormalUniversity,Shanghai
Email:1914939880@qq.com
©ÙÚ^:ÜîŒ,o.šàgBurgers•§iùЊ6įK)ìC-½5[J].nØêÆ,2021,11(7):
1400-1415.
DOI:10.12677/pm.2021.117157
Üo
Received:Jun.12
th
,2021;accepted:Jul.14
th
,2021;published:Jul.21
st
,2021
Abstract
In this paper we study large time behaviors toward shock waves and rarefaction waves
underperiodicperturbationsforinhomogeneousBurgersequation. Weshowthatfor
shockwaves, afterafinitetime,theperturbedshockactuallyconsistsoftwoperiodic
functionscontactingeachotheratashock,andthisshockcurveoscillatesonboth
sidesofthebackgroundshockcurve. Bothofperturbedshockwavesandperturbed
rarefactionwavestendtozerointheL
∞
norm.
Keywords
InhomogeneousBurgersEquation,ShockWaves,RarefactionWaves,Periodic
Perturbations,GeneralizedCharacteristics
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©ïÄXeЊ¯KŒžm1•
u
t
+(
u
2
2
)
x
= −u,x∈(−∞,+∞),t>0,(1.1)
u(x,0) =



¯u
l
+ω
0
(x),x<0,
¯u
r
+ω
0
(x),x>0,
(1.2)
Ù¥¯u
l
,¯u
r
•~ê,ω
0
(x)´±Ï•p(>0)¼ê.••BOŽ,·‚-
1
p
Z
p
0
ω
0
(x)dx≡0.
DOI:10.12677/pm.2021.1171571401nØêÆ
Üo
•§(1.1)´XešàgÅðÆ•§AÏœ¹
u
t
+f(u)
x
= g(u),x∈(−∞,+∞),t>0,(1.3)
Ù¥f(u) ∈C
2
(R) …f
00
(u) >0,g: R→R•1w¼ê.
éu˜„Њ
u(x,0) = u
0
(x),(1.4)
·‚•¯K(1.3)(1.4) =BЊ1w, Ù)•ŒUduA3k•žmSƒ».Ïd3Ï
~œ¹e(1.3)(1.4)1w)´Ø•3.·‚•Ä(1.3)(1.4)f).XJ¼êu(x,t) ´¯K
(1.3)(1.4)f),¿…éA¤kt>0,u(x,t) ÷ve¡^‡
u(x−,t) ≥u(x+,t),(1.5)
@o·‚¡u(x,t) ´¯K(1.3)(1.4) ).
,éu¯K(1.3)(1.4), Kruzkov 3[1]¥|^Ê5ž”{y²éu?¿ÛÜk.CÐ
Š, ¯K(1.3)(1.4) •3•˜ŒNNN). AO/,‘؇L‚5O•ž, é?¿ÛÜk.
CЊu
0
,¯K(1.3)(1.4) 3BV˜m¥k•˜ŒNNN).
g(u) = 0 ž,¯K(1.3)(1.4) )ìC/®²2•/ïÄ. ~X,u
0
(x) ∈L
∞
∩L
1
ž,
)3L
∞
‰ê¿Âe±t
−1
„ÇP~–0. u
0
(x)k.¿äk;|8ž, )3L
1
‰ê
¿Âe±t
−1/2
„ǪuN−Å, „[2][3]. Њäk±Ï5ž, Glimm ÚLax[4]y²)
±t
−1
„ÇP~–Њ3˜‡±Ïþ²þ. éuiù¯K, "[5]y²‰iùЊ±Ï6Ä,
-Å)6Ä±t
−1
„Ǫu-Å), DÕÅ)6Ä3L
∞
‰ê¿Âe±t
−1/2
„Ǫu
DÕÅ). 3"Ä:þ, [6]‰iùЊØÓ±Ï6Ä, ¿y²-Å6Äƒéu-Å
) £.DÕÅ)6Ä3L
∞
‰ê¿Âe±t
−1/2
„ǪuDÕÅ).
g(u) 6= 0ž,šàg¯K(1.3)(1.4)•kXŒþïÄ.Lyberopoulos [7]y²3g(u) = u
œ¹e, eЊ´ëY±Ï¼ê, 3˜‡±ÏþþŠ•", ¿…f(0) =f
0
(0) =0,uf
0
(u) >0,u6=
0, @o¯K(1.3)(1.4) )ªu1Å)…Å„•". Fan[8]ïÄ•\˜„‘, 3g(u) äkk
•‡":…•3~êM
0
>0, ¦é?¿|u|>M
0
Ñkug(u)<0b^‡e, eЊ´Û
ÜCk.±Ï¼ê, @o¯K(1.3)(1.4) )‡oÂñug(u),‡":, ‡oÂñuÅ„
•f
0
(a
2n+1
) 1Å), Ù¥a
2n+1
L«g(u) 12n+1‡":. ²[9]3dÄ:þé‘‡¦?
1~f,•Äg(u) ∈C(R)äkk•‡":¿…3a
2n+1
NC؇L‚5O•, y²eЊ´Û
ÜCk.±Ï¼ê¿…þŠua
2n+1
, @o¯K(1.3)(1.4) )ªuÅ„•f
0
(a
2n+1
) 1
Å).
6Äω
0
(x) = 0ž,¯K(1.1)(1.2) •iù¯K.Mascia[10]:
¯u
l
>¯u
r
ž,(1.1)(1.2))•-Å){¡µ-Å), LãXe.
u
S
(x,t) =



¯u
l
e
−t
,x<S(t),
¯u
r
e
−t
,x>S(t),
(1.6)
Ù¥-Å-‚S(t) = (1−e
−t
)(¯u
l
+ ¯u
r
)/2.
DOI:10.12677/pm.2021.1171571402nØêÆ
Üo
¯u
l
<¯u
r
ž,(1.1)(1.2))•DÕÅ){¡µDÕÅ), LãXe,
u
R
(x,t) =







¯u
l
e
−t
,x<(1−e
−t
)¯u
l
,
˜u(x,t),(1−e
−t
)¯u
l
<x<(1−e
−t
)¯u
r
,
¯u
r
e
−t
,x>(1−e
−t
)¯u
r
,
(1.7)
Ù¥˜u(x,t) d[10]¥½Â2.4 ‰Ñ.
·‚uyt→∞ž,µ-Å)†müà±e
−t
„ÇP~–0,µ-Å-‚ªu†
‚x= (¯u
l
+ ¯u
r
)/2.µDÕÅ)üà±e
−t
„ÇP~–0,´¥mÜ©˜u(x,t) vkw«Lˆª,
Ã{ä)ìC5.
®k(JL²,ü‡ýš‚5ÅðÆ•§|±ÏЊ¯K)¬ªu~Š,iùЊ±
Ï6Ä)ªu µ); éu²ïÆ•§, ±ÏЊ¯K)ªu~Š½ö1Å), ùL²‘é
¯KkŸK•. ©•Ä²ïÆ•§iùЊ±Ï6įK)•žm5, ïÄÙ)†µ
)ƒm'X,&¢‘é)(K•.éuCauchy ¯K(1.1)(1.2), :
½n1.1éu¯u
l
>¯u
r
,XJ±Ï¼êω
0
(x)3˜‡±Ï(0,p]þCk.,¿…•3•˜˜
:a∈(0,p],¦ω
0
(a−)≤0≤ω
0
(a+), @o•3k•žmT
s
Ú•˜-‚X(t)∈Lip(0,+∞), ¦
é?Ût>T
s
, Cauchy ¯K(1.1)(1.2) )÷v
u(x,t) =



u
l
(x,t),x<X(t),
u
r
(x,t),x>X(t).
(1.8)
t→+∞ž,sup
x<X(t)
|u(x,t)|+sup
x>X(t)
|u(x,t)|→0.
AO/,(¯u
l
−¯u
r
)(1−e
−t
)/p´êž,6Ä-ņµ-Ń,=X(t)=S(t).
:‡ê† Њƒ',(¯u
l
−¯u
r
)=Np,N•êž, 6Ä-Ūuµ-Å.Np<
(¯u
l
−¯u
r
) <(N+1)pž, §‚•3N‡:. (¯u
l
−¯u
r
) <pž, §‚Ø•3:.
½n1.2éu¯u
l
<¯u
r
,XJ±Ï¼êω
0
(x)3˜‡±Ï(0,p]þCk.,¿…•3•˜˜
:a∈(0,p],¦ω
0
(a−)≤0≤ω
0
(a+).@ot→+∞ž,Cauchy¯K(1.1)(1.2)
)u(x,t) 3L
∞
‰ê¿ÂeP~–0.
2.2ÂAnØ
Äk·‚0˜e2ÂA‚nØ,ùÑŒ±3[8][9][11]¥é. 3«m[a,b]þ˜^
Lipschitz-‚¡•'u(1.1) )u(x,t) A‚,XJéA¤kt∈[a,b],k
˙
ξ(t) ∈[f
0
(u(ξ(t)+,t)),f
0
(u(ξ(t)−,t))].
é?¿(¯x,
¯
t)∈(R(0,+∞)),–•3˜‡½Â30≤s≤
¯
tþ•2ÂAξ(t;¯x,
¯
t),¦
DOI:10.12677/pm.2021.1171571403nØêÆ
Üo
ξ(
¯
t;¯x,
¯
t) =¯x. ²¡þ¤k²L:(¯x,
¯
t)•AÑ•›3•Ú•Œ•2ÂA¤Ü¤Ï
¥. ·‚^ξ
−
(t;¯x,
¯
t),ξ
+
(t;¯x,
¯
t) ©OL«• Ú•Œ•2ÂA. t>
¯
tž•3•˜c•
A,Xã1¤«.
½Â3[a,b] þAξ(·), eéA¤kt∈[a,b] Ñ÷vu(ξ
−
(t),t) =u(ξ
+
(t),t), K¡•ý
A.
'u•§(1.3)2ÂA, ·‚k±en‡5Ÿ,•õ[!Œ±ë•[11].
Figure1.Forward/backwardcharacteristic
ã1.c•/•2ÂA
Ún2.1XJξ: [a,b] →R´˜^A, @oéuA¤kt∈[a,b], Ñk
˙
ξ(t) = f
0
(u(ξ(t)±,t)),u(ξ(t)+,t) = u(ξ(t)−,t)ž,
˙
ξ(t) =
f(u(ξ(t)+,t))−f(u(ξ(t)−,t))
u(ξ(t)+,t)−u(ξ(t)−,t)
,u(ξ(t)+,t) <u(ξ(t)−,t)ž.
(2.1)
Ún2.2eξ: [a,b] →R´˜^ýA, @o•3˜‡¼êv: [a,b] →R÷v



˙
ξ(t) = f
0
(v(t)),
˙v(t) = g(v(t)),
(2.2)
¿…éuA¤kt∈(a,b), Ñk
v(t) = u(ξ(t)+,t) = u(ξ(t)−,t).
Ún2.34Ú4Œ•Aξ
−
(t) Úξ
+
(t) Ñ´´ýA.§‚©OÏL¦)(2.2)3t=
¯
tž•±(¯x,u(¯x−,
¯
t)) Ú(¯x,u(¯x+,
¯
t)) •ÐŠ•¯K, …u3à:?÷v,
u(ξ
−
(0)−,0) ≤u(ξ
−
(0)+,0),u(ξ
−
(
¯
t)−,
¯
t) ≥u(ξ
−
(
¯
t)+,
¯
t),
u(ξ
+
(0)−,0) ≤u(ξ
+
(0)+,0),u(ξ
+
(
¯
t)−,
¯
t) ≥u(ξ
+
(
¯
t)+,
¯
t).
íØ2.1(1) eü^ØÓýAƒ,Kdž)-Å,@oT:7´ùü^ýAª
:.
(2) 
¯
t>0 ž, 4•Aξ
−
(t) Ú4Œ•Aξ
+
(t)-Ü…=u(¯x−,
¯
t) = u(¯x+,
¯
t).
(3)é?¿ü^lx¶Ñu4c•Aξ
−
(t) Ú4Œc•Aξ
+
(t),XJ§‚3,˜ž
•t
0
>0ƒ, @o§‚3t>t
0
-Ü.
DOI:10.12677/pm.2021.1171571404nØêÆ
Üo
Ún2.4-ξ(t) Ú
˜
ξ(t) •©O†(1.1)‘kÛÜCk.Њu(x,0)Ú˜u(x,0))u(x,t)
Ú˜u(x,t)ƒ'ü^4Š•A,Xã2¤«,§‚l½:(¯x,
¯
t) ∈R(0,+∞)Ñu.e
˜
ξ(0) <ξ(0),
@o¤áXeª
Figure2.Minimalbackwardcharacteristic
ã2.••A
Z
ξ(0)
˜
ξ(0)
(u(x,0)−˜u(x,0))dx
=
Z
¯
t
0
(−e
t
[u(ξ(t),t)−˜u(ξ(t),t)]
˙
ξ(t)+e
t
[f(u(ξ(t),t))−f(˜u(ξ(t),t)))])dt
+
Z
¯
t
0
(e
t
[u(
˜
ξ(t),t)−˜u(
˜
ξ(t),t)]
˙
˜
ξ(t)−e
t
[f(u(
˜
ξ(t),t))−f(˜u(
˜
ξ(t),t)))])dt.
(2.3)
y ½Â«•Ω := (x,t)|0 ≤t≤
¯
t,
˜
ξ(t) ≤x≤ξ(t). Äkò(1.1) =†•
(e
t
u)
t
+(e
t
f(u))
x
= 0.(2.4)
-u= u(x,t)−˜u(x,t), ¿òÙ3Ω þÈ©,$^Green úª´(2.3).
Ún2.5eЊu
0
3[0,p] þCk.¿…±p•±Ï, -u´¯K(1.1)(1.4)). @o
(1) éu?¿t>0, u(·,t) ´±p•±Ï¼ê.
(2) éuz‡½T>0,
1
p
Z
p
0
u(x,T)dx=
e
−T
p
Z
p
0
u
0
(x)dx.(2.5)
y(1)½Âv(x,t) = u(x+p,t).éuz‡φ∈C
1
0
(R×(0,+∞)),XJ
˜
φ(x,t) = φ(x−p,t),@
o
ZZ
t>0
vφ
t
+f(v)φ
x
+vφdxdt+
Z
t=0
u
0
φdx
=
ZZ
t>0
u(x+p,t)φ
t
+f(u(x+p,t))φ
x
+u(x+p,t)φdxdt
=
ZZ
t>0
u(x,t)
˜
φ
t
+f(u(x,t))
˜
φ
x
+u(x,t)
˜
φdxdt
=0.
DOI:10.12677/pm.2021.1171571405nØêÆ
Üo
Ïd,v•´T¯K). u´d)•3•˜5•u(x,t) ≡v(x,t).
(2) ½ÂÝ/«•Ω={(x,t)|0≤x≤p,0≤t≤
¯
t}. ò(2.4) 3Ω þÈ©. $^Green úªÚ
(1)
0 =
ZZ
Ω
(e
t
u(x,t))
t
+(e
t
f(u(x,t)))
x
dxdt
=
I
−e
t
u(x,t)dx+e
t
f(u(x,t))dt
=
Z
p
0
−u(x,0)dx+
Z
T
0
e
t
f(u(p,t))dt+
Z
p
0
e
T
u(x,T)dx−
Z
T
0
e
t
f(u(0,t))dt
=
Z
p
0
−u(x,0)dx+
Z
p
0
e
T
u(x,T)dx.
y..
-R(u(·,t)) = {u(x±,t) : x∈R,t>0}, Ù¥u÷vÚn2.5 ^‡.'uR(u(·,t)) ·‚kX
eü‡5Ÿ,Ùy²Œ±3[8]¥é.
Ún2.6R(u(·,t)) ´˜‡4«m.
Ún2.7eR(u(·,t))=[α(t),β(t)],@oé?¿γ∈(α(t),β(t)),•3˜‡ëY:x
0
∈R¦
u(x
0
,t) = γ.
·‚Pu
l
(x,t),u
r
(x,t),u(x,t)•(1.1)©O‘kXeЊ):
u
l
(x,0) =¯u
l
+ω
0
(x),x∈R,
u
r
(x,0) =¯u
r
+ω
0
(x),x∈R,
u(x,0) =



¯u
l
+ω
0
(x),x<0,
¯u
r
+ω
0
(x),x>0.
dFan [8]¥½n1.1 Œ•,t→+∞ž,
|u
l
(x,t)|→0,|u
r
(x,t)|→0.(2.6)
eω
0
(x)÷v"þŠ^‡,@odÚn2.5(2) •,3½T>0ž•k
1
p
Z
p
0
u
l
(x,T)dx=
e
−T
p
Z
p
0
u
l
(x,0)dx=¯u
l
e
−T
.
qdÚn2.6•,R(u(·,T)) ´˜‡4«m. Ïd¯u
l
e
−T
3T«mS. |^u(·,T) ±Ï5ÚÚn
2.7,·‚íä•3ëY:x
0
∈(0,p], ¦u(x
0
,T) =¯u
l
e
−T
.
Ún2.8b:(¯x,
¯
t)∈R×(0,+∞) ÷vu
l
(¯x,
¯
t)=¯u
l
e
−
¯
t
, lT:ÑuŠ4Š•Aξ(t). u´
t>
¯
tž, ξ(t) •´k½Â, ¿…÷vu
l
(ξ(t),t) =¯u
l
e
−t
.
DOI:10.12677/pm.2021.1171571406nØêÆ
Üo
yæ^‡y{.¿©>0,b3t=
¯
t+ž•,u(ξ(t),t)6=¯u
l
e
−t
.´•
3x
0
∈(ξ(t)−p/2,ξ(t)+p/2), ¦u(x
0
,t) =¯u
l
e
−t
.lT:ÑuŠ4Š•Aη(t), Xã3¤«.
duξ(t),η(t)´÷vÚn2.2,Ïd
ξ(t) = ξ(0)+¯u
l
(1−e
−t
),t∈(0,
¯
t],
η(t) = η(0)+ ¯u
l
(1−e
−t
),t∈(0,
¯
t+],
…÷Xξ(t) ku(ξ(t),t) =¯u
l
e
−t
,÷Xη(t) ku(η(t),t) =¯u
l
e
−t
.
Figure3.Specialcharacteristic
ã3.AÏA
y3·‚•I‡y²ξ(0) = η(0) =Œ.du3(0,p]S•k•˜˜:a÷v
ω
0
(a−) ≤0 ≤ω
0
(a+),
…dÚn2.3•
¯u
l
+ω
0
(ξ(0)−) = u(ξ(0)−,0) ≤¯u
l
≤u(ξ(0)+,0) =¯u
l
+ω
0
(ξ(0)+),
¯u
l
+ω
0
(η(0)−) = u(η(0)−,0) ≤¯u
l
≤u(η(0)+,0) =¯u
l
+ω
0
(η(0)+),
η(0)=ξ(0) +Np,Ù¥N•ê. w,t∈(0,
¯
t) ž,ξ(t)²1uη(t),=η(
¯
t)=ξ(
¯
t) + Np.
-→0,dALipschitzëY5η(
¯
t+)→η(
¯
t).duη(
¯
t)∈(ξ(
¯
t) −p/2,ξ(
¯
t) + p/2),Ï
dN= 0, =η(0) = ξ(0).y..
Ún2.9¯u
l
>¯u
r
ž,
u(x,t) =



u
l
(x,t),x<X
+
(t),
u
r
(x,t),x>X
−
(t),
(2.7)
Ù¥X
−
(t) •l:Ñu'uu4c•A, X
+
(t) •l:Ñu'uu4Œc•A.
y Ø”˜„5,·‚y²x<X
+
(t)œ¹.
éu?Û½(¯x,
¯
t)…¯x<X
+
(
¯
t),
¯
t>0.·‚Äky²u(¯x+,
¯
t) = u
l
(¯x+,
¯
t).L:(¯x+,
¯
t)©
OŠ'u)u,u
l
4Œ•Aξ
+
(t) Úη
+
(t), Xã4¤«. Ï•4Œ•A´ýA,
DOI:10.12677/pm.2021.1171571407nØêÆ
Üo
Figure4.Perturbedshockwave
ã4.6Ä-Å)
ξ
+
(t)Úη
+
(t)3[0,
¯
t]þ÷vXe'Xª



˙
ξ
+
(t) = f
0
(v(t)),
˙v(t) = −v(t),
Ú



˙η
+
(t) = f
0
(˜v(t)),
˙
˜v(t) = −˜v(t),
¿…éuA¤kt∈(0,
¯
t),Ñk
v(t) = u(ξ
+
(t)+,t) = u(ξ
+
(t)−,t),˜v(t) = u
l
(η
+
(t)+,t) = u
l
(η
+
(t)−,t).
dut>0 ž,l?¿:Ñuc•A´•˜, ξ
+
(t)†X(t)؃,=ξ
+
(0) <0.
e¡æ^‡y{:eu(¯x+,
¯
t) >u
l
(¯x+,
¯
t),=v(
¯
t) >˜v(
¯
t).du



v(t) = v(
¯
t)e
−(
¯
t−t)
,
˜v(t) =˜v(
¯
t)e
−(
¯
t−t)
,
v(t) >˜v(t). dfà5´
˙
ξ
+
(t) = f
0
(v(t)) >f
0
(˜v(t)) =˙η
+
(t).(2.8)
•Äùü^4Œ•AÑ´l(¯x,
¯
t)Ñu,Ïdξ
+
(0) <η
+
(0).|^Ún2.4 ´
Z
¯
t
0
e
t
[u
l
(ξ
+
(t),t)−u(ξ
+
(t),t)]
˙
ξ
+
(t)−e
t
[f(u
l
(ξ
+
(t),t)))−f(u(ξ
+
(t),t))]dt
−
Z
¯
t
0
e
t
[u
l
(η
+
(t),t)−u(η
+
(t),t)]˙η
+
(t)−e
t
[f(u
l
(η
+
(t),t)))−f(u(η
+
(t),t))]dt
=
Z
η
+
(0)
ξ
+
(0)
u
l
(x,0)−u(x,0)dx.
(2.9)
η
+
(0)≤0ž, (2.9) mýu0;η
+
(0)>0ž, (2.9) mýu
R
η
+
(0)
0
(¯u
l
−¯u
r
)dx>0.|
^fà5ØJuy, (2.8)†ý´š, Ïdη
+
(0) ≤0, •Ò´`éu?¿t∈(0,
¯
t),Ñk
u
l
(ξ
+
(t),t) = u(ξ
+
(t),t) = v(t),
u
l
(η
+
(t),t) = u(η
+
(t),t) =˜v(t).
DOI:10.12677/pm.2021.1171571408nØêÆ
Üo
džη
+
(t)•´†uk'•A.´ξ
+
(t)´l(¯x,
¯
t)Ñu'uu4Œ•A,ù
†ξ
+
(0) <η
+
(0)w,gñ.Ïdu(¯x+,
¯
t) ≤u
l
(¯x+,
¯
t).
Ón, bu(¯x+,
¯
t) <u
l
(¯x+,
¯
t)•´Ø¤á. Ïdu(¯x+,
¯
t) = u
l
(¯x+,
¯
t).d(¯x,
¯
t)?¿5, =
yx<X
+
(t)ž,u(¯x+,
¯
t) = u
l
(¯x+,
¯
t).
aq,l(¯x,
¯
t)Ñu©O‰Ú)u,u
l
k'4•Aξ
−
(t)Úη
−
(t),u´Œyu(¯x−,
¯
t) =
u
l
(¯x−,
¯
t).
(Üþã?Ø,·‚x<X
+
(t)ž,é?¿t>0, Ñku(x,t) = u
l
(x,t).
y..
íØ2.2e¯u
l
<¯u
r
, @o
u(x,t) =



u
l
(x,t),x<X
l+
(t),
u
r
(x,t),x>X
r−
(t),
(2.10)
Ù¥X
r−
(t)•l:Ñu'uu
r
4c•A, X
l+
(t)•l:Ñu'uu
l
4Œc•A
.
y -
u
l
(x,t= 0) =



u(x,t= 0),x<0,
¯u
l
−¯u
r
+u(x,t= 0),x>0,
u
r
(x,t= 0) =



¯u
r
−¯u
l
+u(x,t= 0),x<0,
u(x,t= 0),x>0.
òÙ“\Ún2.9=y.
3.y²½n1.1
Äky²-Å-‚•˜5:
eé?¿t∈[0,+∞) ÑkX
−
(t) ≡X
+
(t),dÚn2.9, Œá=¦(1.8).
eé?¿t∈[0,+∞)ÑkX
−
(t)<X
+
(t), @o3X
−
(t)<x<X
+
(t) þðku
l
(x,t)=
u
r
(x,t). ùp‡¦X
+
(t)−X
−
(t)<p,ÄKe•3t
0
>0 ¦ª¤á, u´dÚn2.5(2) á•
Ñgñ. ½
¯
t>0,?¯x∈(X
−
(
¯
t),X
+
(
¯
t)). l(¯x,
¯
t) Ñu‰'uu
l
(x,t)4•Aξ(t). d
Ún2.3•, ξ(t) 3t∈[0,
¯
t]þ´ýA,



˙
ξ(t) = f
0
(v(t)),
˙v(t) = g(v(t)),
…
v(
¯
t) = u
l
(¯x−,
¯
t).
DOI:10.12677/pm.2021.1171571409nØêÆ
Üo
?˜Ú,éuA¤kt∈(0,
¯
t),Ñk
v(t) = u
l
(ξ(t)+,t) = u
l
(ξ(t)−,t),
…u
l
(ξ(0)−,0)≤u
l
(ξ(0)+,0).bξ(t)†X
−
(t)u:(X
−
(τ),τ),τ>0(†X
+
(t)ƒœ
¹aq).u´ξ(t)3(τ,
¯
t)þ•´'uu
r
ýA.-t
0
=(τ+
¯
t)/2,x
0
=ξ(t
0
),ØJu
y, u
l
(x
0
−,t
0
)=u
l
(x
0
+,t
0
). u´l:(x
0
,t
0
)ÑuŠ'uu
r
4Š•Aη(t). e¡ò©œ¹
?Ø:
(1)eξ(t)Úη(t) 3[0,t
0
]þØ-Ü,@oùw,†íØ2.1 gñ.
(2)eξ(t)Úη(t)3[0,τ)þØ-Ü,3[τ,t
0
]þ-Ü.du3[0,t
0
]þ,ξ(t)•:(x
0
,t
0
)Ñu
'uu
l
4Š•A,η(t)´l:(x
0
,t
0
)Ñu'uu
r
4Š•A.u
l
(x
0
,t
0
) = u
r
(x
0
,t
0
),
ù†Ún2.2gñ.
(3)eξ(t) Úη(t) 3[0,t
0
] þ-Ü, ù`²3[0,t
0
] þξ(t)Q´'uu
l
(x,t)4•A,
•´'uu
r
(x,t)ý•A.Ïd, ω
0
(ξ(0)) 7•mä:.ÄK,
¯u
l
+ω
0
(ξ(0)) = u
l
(ξ(0),0) = u
r
(ξ(0),0) =¯u
r
+ω
0
(ξ(0)),
w,gñ. q3(X
−
(t),X
+
(t)) þkØŒ‡:, lù:Ñu©OŠ4•A, §‚†x¶
uØŒ‡:…þ3˜‡±ÏS.duù:þ•mä:,w,†ω
0
(x)3[0,p] þCk.gñ.
nÜþã©Û, 7½•3,‡¿©T
s
>0, ¦t>T
s
žðkX
−
(t) =X
+
(t). u´·‚y
²(1.8).
e5´'u6Ä-Å-‚X(t)Úµ-Å-‚S(t) ƒm'X(Ø:
5¿Њu
l
(x,0) =¯u
l
+ω
0
(x)Ú¯ω= 0, |^Ún2.5 Œ,é?¿t>0,
1
p
Z
p
0
u
l
(x,t)dx=
e
−t
p
Z
p
0
u
l
(x,0)dx=¯u
l
e
−t
.
dÚn2.6ÚÚn2.7´,•3x
0
∈(−∞,X
−
(T))Úx
00
∈(X
+
(T),+∞),¦u(x
0
,T)=
¯u
l
e
−T
,u(x
00
,T) =¯u
r
e
−T
.l(x
0
,T)Ú(x
00
,T)Ñu©O‰'uu4Š•Aξ(t)Úη(t),Xã
5¤«,§‚÷v
Figure5.Perturbedshockcurve
ã5.6Ä-Å-‚
DOI:10.12677/pm.2021.1171571410nØêÆ
Üo
ξ(t) <X
−
(t),η(t) >X
+
(t).
|^(2.2)Œ



ξ(t) = ξ(0)+¯u
l
(1−e
−t
),t∈[0,T],
η(t) = η(0)+ ¯u
r
(1−e
−t
),t∈[0,T],
(3.1)
Ù¥u(ξ(0)−,0) ≤¯u
l
≤u(ξ(0)+,0) Úu(η(0)−,0) ≤¯u
r
≤u(η(0)+,0). dω
0
(x)b^‡,-
ξ(0) = a−N
1
p,N
1
•ê,
η(0) = a+N
2
p,N
2
•šKê.
.(3.2)
½ÂF/«•:
Ω(T) ,{(x,t) : 0 ≤t≤T,ξ(t) ≤x≤η(t)}.(3.3)
3Ω(T) ¥,d‚úªŒ
0 =
ZZ
Ω
(e
t
u(x,t))
t
+(e
t
f(u(x,t)))
x
dxdt
=
Z
η(0)
ξ(0)
−u(x,0)dx+
Z
η(T)
ξ(T)
e
T
u(x,T)dx+
Z
T
0
−e
t
u(η(t),t)˙η(t)+e
t
f(u(η(t),t))dt
+
Z
T
0
e
t
u(ξ(t),t)
˙
ξ(t)−e
t
f(u(ξ(t),t))dt,I
1
+I
2
+I
3
+I
4
.
|^|η(0)−ξ(0)|•±Ïê,´
I
1
=
Z
0
η(0)
(¯u
r
+ω
0
(x))dx+
Z
ξ(0)
0
(¯u
l
+ω
0
(x))dx= ξ(0)¯u
l
−η(0)¯u
r
.(3.4)
Š³|ÑC†
t
0
= t,x
0
= x−¯u
l
(1−e
−t
),
u
0
(x
0
,t
0
) = u(x−¯u
l
(1−e
−t
),t)−¯u
l
e
−t
.
N´yu
0
(x
0
,t
0
)E÷v•§(1.1), Ù•´T•§),…éu?¿½t
0
>0
1
p
Z
p
0
u
0
(x
0
,t
0
)dx
0
=
1
p
Z
p
0
u(x
0
,t
0
)−¯u
l
e
−t
0
dx
0
= 0.
Ón,-x
0
= x−¯u
r
(1−e
−t
)ž•kaq(Ø.Ïd, 1‘U•
I
2
=
Z
X(T)
ξ(T)
e
T
u
l
(x,T)dx+
Z
η(T)
X(T)
e
T
u
r
(x,T)dx
=
Z
X(T)
ξ(T)
e
T
[u
0
(x−¯u
l
(1−e
−T
),T)+¯u
l
e
−T
]dx+
Z
η(T)
X(T)
e
T
[u
0
(x−¯u
r
(1−e
−T
),T)+¯u
r
e
−T
]dx
= X(T)(¯u
l
−¯u
r
)−¯u
l
ξ(T)+ ¯u
r
η(T)+e
T
Z
X(T)−¯u
l
(1−e
−T
)
X(T)−¯u
r
(1−e
−T
)
u
0
(y,T)dy.
.
(3.5)
DOI:10.12677/pm.2021.1171571411nØêÆ
Üo
X
I
3
+I
4
=
Z
T
0
−e
t
u(η(t),t)˙η(t)+e
t
f(u(η(t),t))dt+
Z
T
0
e
t
u(ξ(t),t)
˙
ξ(t)−e
t
f(u(ξ(t),t))dt
=
Z
T
0
−e
t
¯u
r
e
−t
f
0
(¯u
r
e
−t
)dt+
Z
T
0
e
t
¯u
l
e
−t
f
0
(¯u
l
e
−t
)dt−
Z
T
0
e
t
[f(¯u
l
e
−t
)−f(¯u
r
e
−t
)]dt
=
Z
T
0
¯u
r
f
0
(¯u
r
e
−t
)dt+
Z
T
0
¯u
l
f
0
(¯u
l
e
−t
)dt−
Z
T
0
e
t
[f(¯u
l
e
−t
)−f(¯u
r
e
−t
)]dt.
.
(3.6)
5¿ξ(t) = ξ(0)+¯u
l
(1−e
−t
),η(t) = η(0)+ ¯u
r
(1−e
−t
),(Ü(3.4)(3.5)(3.6) Œ
X(T)−S(T) =
e
T
¯u
l
−¯u
r
Z
X(T)−¯u
r
(1−e
−T
)
X(T)−¯u
l
(1−e
−T
)
u
0
(y,T)dy.
éu?ÛêN, e(¯u
l
−¯u
r
)(1−e
−T
) = Np, @oX(T) = S(T). Xe(Ø
(1)(¯u
l
−¯u
r
) = Np,N•êž, 6Ä-Ūuµ-Å.
(2)Np<(¯u
l
−¯u
r
) <(N+1)pž,6Ä-ņµ-Å•3N‡:.
(3)(¯u
l
−¯u
r
) <pž, 6Ä-ņµ-ÅØ•3:.
4.y²½n1.2
-X
r−
(t)•l:Ñu'uu
r
4c•A, dÚn2.8 •,•3ü^†u
r
ƒ'A, ¦

a−p+ ¯u
r
(1−e
−t
) ≤X
r−
(t) ≤a+¯u
r
(1−e
−t
),∀t>0.
Ón•3ü^†u
l
ƒ'A,¦
a−p+ ¯u
l
(1−e
−t
) ≤X
l+
(t) ≤a+¯u
l
(1−e
−t
),∀t>0.
-ξ(t) = a−p+¯u
l
(1−e
−t
),η(t) = a+¯u
r
(1−e
−t
).díØ2.2•,x<ξ(t)ž,u(x,t) = u
l
(x,t),
x>η(t)ž, u(x,t)=u
r
(x,t). ¤±é?¿t>0,x∈(ξ(t),η(t)), l(x,t)Ñu4Š•A
†ξ(t),η(t)Ã:.¯¢þ,u(ξ(t),t)=u
l
(ξ(t),t)≡¯u
l
e
−t
.XJ•3(ξ(t),η(t))ƒm˜
:(¯x,
¯
t), ¦lT:Ñu'uu4•Aγ
−
(t) †ξ(t)ƒu:(ξ(τ),τ),Ù¥τ>0,
u´f
0
(u(ξ(τ)+,τ))>f
0
(¯u
l
e
−τ
), dfà5,u(ξ(τ)+,τ)>¯u
l
e
−t
. w,gñ.ÓnŒy,é?
¿t>0, l(x,t) Ñu4Œ•Aγ
+
(t)†η(t) Ã:.Xã6¤«.
•Äγ
±
(t) ´ýA, |^(2.2) Œγ
±
(t)=γ
±
(0)+ C(1−e
−t
),t∈[0,
¯
t], Ù¥C´
†u(¯x±,
¯
t)k'~ê.¿…3t∈(0,
¯
t)þ,ku(γ
±
(t),t) = v(t) = Ce
−t
.
A^þã(Ø,é?¿½t>0 ·‚k
DOI:10.12677/pm.2021.1171571412nØêÆ
Üo
Figure6.Perturbedrarefactionwave
ã6.6ÄDÕÅ)
(1)ex<ξ(t),@od(1.7)(2.6) Œ•,t¿©Œž
|u(x,t)−u
R
(x,t)|= |u
l
(x,t)−¯u
l
e
−t
|→0.
(2)eξ(t) <x<¯u
l
(1−e
−t
),duγ
±
(t)†ξ(t)Ã:,·‚
a−p+ ¯u
l
(1−e
−t
) ≤γ
±
(0)+C(1−e
−t
) ≤¯u
l
(1−e
−t
).
duγ
±
(0) ∈(a−p,p), þªŒU¤
−p+ ¯u
l
(1−e
−t
) ≤C(1−e
−t
) ≤p−a+¯u
l
(1−e
−t
).
ØªüýÓر(1−e
−t
),2~¯u
l
,
−
p
1−e
−t
≤C−¯u
l
≤
p−a
1−e
−t
.
ØªüýÓ¦e
−t
,du÷XA‚ðkv(t) = Ce
−t
,Œ
−
p
e
t
−1
≤v(t)−¯u
l
e
−t
≤
p−a
e
t
−1
.
d(¯x,
¯
t)?¿5,·‚
|u(x,t)−u
R
(x,t)|≤
p
e
t
−1
.
(3)e¯u
l
(1−e
−t
) <x<¯u
r
(1−e
−t
),@ou
R
(x,t) =˜u(x,t).duéJ‰Ñ˜u(x,t)w«Lˆ
ª,·‚d?•éu(x,t) ?1O
¯u
l
(1−e
−t
) ≤γ
±
(0)+C(1−e
−t
) ≤¯u
r
(1−e
−t
).
duγ
±
(0) ∈(a−p,p), þªŒU¤
−a+ ¯u
l
(1−e
−t
) ≤C(1−e
−t
) ≤p−a+¯u
r
(1−e
−t
).
DOI:10.12677/pm.2021.1171571413nØêÆ
Üo
ØªüýÓ¦e
−t
/(1−e
−t
),÷XA‚ðk
−
a
e
t
−1
+ ¯u
l
e
−t
≤v(t) ≤
p−a
e
t
−1
+ ¯u
r
e
−t
.
d(¯x,
¯
t)?¿5,·‚
|u(x,t)|≤Me
−t
,M>0•~ê.
(4)e¯u
r
(1−e
−t
) <x<η(t), †1«œ¹aq,Œ
|u(x,t)−u
R
(x,t)|≤
p
e
t
−1
.
(5)ex>η(t), †1˜«œ¹aq,Œ
|u(x,t)−u
R
(x,t)|= |u
r
(x,t)−¯u
r
e
−t
|→0.
y..
ë•©z
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