Waves damped wave and observation

Description

Niveau: Supérieur, Licence, Bac+2
1Waves, damped wave and observation? Kim Dang PHUNG Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China. E-mail: kim dang Abstract This talk describes some applications of two kinds of obser- vation estimate for the wave equation and for the damped wave equation in a bounded domain where the geometric control con- dition of C. Bardos, G. Lebeau and J. Rauch may failed. 1 The wave equation and observation We consider the wave equation in the solution u = u(x, t) ? ? ? ∂2t u?∆u = 0 in ?? R , u = 0 on ∂?? R , (u, ∂tu) (·, 0) = (u0, u1) , (1.1) living in a bounded open set ? in Rn, n ≥ 1, either convex or C2 and connected, with boundary ∂?. It is well-known that for any initial data (u0, u1) ? H2(?) ? H10 (?) ? H10 (?), the above problem is well-posed and have a unique strong solution. Linked to exact controllability and strong stabilization for the wave equation (see [Li]), it appears the following observability problem which consists in proving the following estimate ?(u0, u1)?2H10 (?)?L2(?) ≤ C ∫ T 0 ∫ ? |∂tu (x, t)|2 dxdt ?This work is supported by the NSF of China under grants

• weight function


• wave equation


• damped wave


• french-chinese summer


• null initial


• equation implies


• unique strong


• ct ?


• also give theirs

Sujets

• Fudan University
• Weight function
• Wave equation
• Damped wave
• Syllable onset

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

∗Waves,dampedwaveandobservation
KimDangPHUNG
YangtzeCenterofMathematics,SichuanUniversity,
Chengdu610064,China.
E-mail:[e-mail]
Abstract
Thistalkdescribessomeapplicationsoftwokindsofobser-
vationestimateforthewaveequationandforthedampedwave
equationinaboundeddomainwherethegeometriccontrolcon-
ditionofC.Bardos,G.LebeauandJ.Rauchmayfailed.

1Thewaveequationandobservation

1

Weconsiderthewaveequationinthesolution
u
=
u
(
x,t
)

∂
t
2
u
−
Δ
u
=0inΩ
×
R
,

u
=0on
∂
Ω
×
R
,(1.1)
(
u,∂
t
u
)(
∙
,
0)=(
u
0
,u
1
),
livinginaboundedopensetΩin
R
n
,
n
≥
1,eitherconvexor
C
2
and
connected,withboundary
∂
Ω.Itiswell-knownthatforanyinitialdata
(
u
0
,u
1
)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω),theaboveproblemiswell-posed
andhaveauniquestrongsolution.
Linkedtoexactcontrollabilityandstrongstabilizationforthewave
equation(see[Li]),itappearsthefollowingobservabilityproblemwhich
consistsinprovingthefollowingestimate
ZZTk
(
u
0
,u
1
)
k
2
H
1
(Ω)
×
L
2
(Ω)
≤
C
|
∂
t
u
(
x,t
)
|
2
dxdt
0ω0∗
ThisworkissupportedbytheNSFofChinaundergrants10525105and10771149.
PartofthistalkwasdonewhentheauthorvisitedFudanUniversitywithafinan-
cialsupportfromthe”French-ChineseSummerInstituteonAppliedMathematics”
(September1-21,2008).

2KimDangPHUNG
forsomeconstant
C>
0independentontheinitialdata.Here,
T>
0
and
ω
isanon-emptyopensubsetinΩ.Duetofinitespeedofpropa-
gation,thetime
T
havetobechosenlargeenough.Dealingwithhigh
frequencywavesi.e.,waveswhichpropagatesaccordingthelawofge-
ometricaloptics,thechoiceof
ω
cannotbearbitrary.Inotherwords,
theexistenceoftrappedrays(e.g,constructedwithgaussianbeams(see
[Ra])impliestherequirementofsomekindofgeometricconditionon
(
ω,T
)(see[BLR])inorderthattheaboveobservabilityestimatemay
.dlohNow,wecanaskwhatkindofestimatewemayhopeinageometry
withtrappedrays.Letusintroducethequantity
k
(
u
0
,u
1
)
k
H
2
∩
H
01
(Ω)
×
H
01
(Ω)
,=Λk
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)
whichcanbeseenasameasureofthefrequencyofthewave.Inthis
paper,wepresentthetwofollowinginequalities
ZZ2
C
Λ
1
/β
T
2
k
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)
≤
e
|
∂
t
u
(
x,t
)
|
dxdt
(1.2)
ω0dnaZ
C
Λ
1
/γ
Z
22k
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)
≤
C
|
∂
t
u
(
x,t
)
|
dxdt
(1.3)
ω0where
β
∈
(0
,
1),
γ>
0.Wewillalsogivetheirsapplicationstocontrol
theory.
Thestrategytogetestimate(1.2)isnowwell-known(see[Ro2],[LR])
andasketchoftheproofwillbegiveninAppendixforcompleteness.
Moreprecisely,wehavethefollowingresult.
Theorem1.1.-
Forany
ω
non-emptyopensubsetin
Ω
,forany
β
∈
(0
,
1)
,thereexist
C>
0
and
T>
0
suchthatforanysolution
u
of
(1.1)withnon-identicallyzeroinitialdata
(
u
0
,u
1
)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω)
,theinequality(1.2)holds.
Now,wecanaskwhetherisitpossibletogetanotherweightfunction
ofΛthantheexponentialone,andinparticularapolynomialweight
functionwithageometry(Ω
,ω
)withtrappedrays.Herewepresentthe
followingresult.
Theorem1.2.-
Thereexistsageometry
(Ω
,ω
)
withtrappedrays
suchthatforanysolution
u
of(1.1)withnon-identicallyzeroinitialdata

Waves,dampedwaveandobservation3
(
u
0
,u
1
)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω)
,theinequality(1.3)holdsforsome
C>
0
and
γ>
0
.
TheproofofTheorem1.2isgivenin[Ph1].WiththehelpofTheorem
2.1below,itcanalsobededucedfrom[LiR],[BuH].

2Thedampedwaveequationandourmo-
tivation

Weconsiderthefollowingdampedwaveequationinthesolution
w
=
w
(
x,t
)
½
2∂
t
w
−
Δ
w
+1
ω
∂
t
w
=0inΩ
×
(0
,
+
∞
),(2.1)
w
=0on
∂
Ω
×
(0
,
+
∞
),
livinginaboundedopensetΩin
R
n
,
n
≥
1,eitherconvexor
C
2
and
connected,withboundary
∂
Ω.Here
ω
isanon-emptyopensubsetin
Ωwithtrappedraysand1
ω
denotesthecharacteristicfunctionon
ω
.
Further,forany(
w,∂
t
w
)(
∙
,
0)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω),theabove
problemiswell-posedforany
t
≥
0andhaveauniquestrongsolution.
¢¡¢¡Denoteforany
g
∈
C
[0
,
+
∞
);
H
01
(Ω)
∩
C
1
[0
,
+
∞
);
L
2
(Ω),
Z´³1E
(
g,t
)=
|r
g
(
x,t
)
|
2
+
|
∂
t
g
(
x,t
)
|
2
dx
.
2ΩThenforany0
≤
t
0
<t
1
,thestrongsolution
w
satisfiesthefollowing
formula
ZZt12E
(
w,t
1
)
−
E
(
w,t
0
)+
|
∂
t
w
(
x,t
)
|
dxdt
=0.(2.2)
ωt0

2.1Thepolynomialdecayrate

Ourmotivationforestablishingestimate(1.3)comesfromthefollowing
result.
Theorem2.1.-
Thefollowingtwoassertionsareequivalent.Let
.0>δ

4KimDangPHUNG
(i)
Thereexists
C>
0
suchthatforanysolution
w
of(2.1)withthe
non-nullinitialdata
(
w,∂
t
w
)(
∙
,
0)=(
w
0
,w
1
)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω)
,wehave
´³Z
C
EE
((
∂tw,w
0
,
)0)1
/δ
Z
k
(
w
0
,w
1
)
k
2
H
01
(Ω)
×
L
2
(Ω)
≤
C
|
∂
t
w
(
x,t
)
|
2
dxdt
.
ω0(ii)
Thereexists
C>
0
suchthatthesolution
w
of(2.1)withtheinitial
data
(
w,∂
t
w
)(
∙
,
0)=(
w
0
,w
1
)
∈
H
2
(Ω)
∩
H
01
(Ω)
×
H
01
(Ω)
satisfies
C2E
(
w,t
)
≤
t
δ
k
(
w
0
,w
1
)
k
H
2
∩
H
01
(Ω)
×
H
01
(Ω)
∀
t>
0.
Remark.-
Itisnotdifficulttosee(e.g.,[Ph2])byaclassicalde-
compositionmethod,atranslationintimeand(2.2),thattheinequality
(1.3)withtheexponent
γ
forthewaveequationimpliestheinequality
of(
i
)inTheorem2.1withtheexponent
δ
=2
γ/
3forthedampedwave
equation.Andconversely,theinequalityof(
i
)inTheorem2.1withthe
exponent
δ
forthedampedwaveequationimpliestheinequality(1.3)
withtheexponent
γ
=
δ/
2forthewaveequation.
ProofofTheorem2.1.-
(
ii
)
⇒
(
i
).Supposethat
C2E
(
w,T
)
≤
T
δ
k
(
w
0
,w
1
)
k
H
2
∩
H
01
(Ω)
×
H
01
(Ω)
∀
T>
0.
Thereforefrom(2.2)
ZZTCE
(
w,
0)
≤
δ
k
(
w
0
,w
1
)
k
2
H
2
∩
H
01
(Ω)
×
H
01
(Ω)
+
|
∂
t
w
(
x,t
)