Propagation Model
H.T. Banks
, H. Tran y
,S. Wynne z
December 14,2001
Abstract
We consider a nonlinearmodel for propagation of shear wavesin
viscoelastic tissue. Existence anduniqueness resultsfor solutionsare
established.
1 Introduction
Inthisnotewe examinethewell-posednessofaone-dimensionalshearwave
propagationmodel that arises in inverse problems related to the detection
and characterization ofcardiac artery stenoses. Ina previouspaper [6 ],we
deneda basicmodelto emulate shear waves propagating from acoronary
stenosis through a homogeneous, soft-tissue like medium. The medium is
considered viscoelastic, and the model uses internal strain variables (see,
e.g., [1 ], [3], [4 ], or [5 ]) to capture the nonlinear stress-strain relationship.
Anidealizedgeometry(basedonexperimentalprotocolstotestpiezoceramic
basedsurfacesensors) isdepictedin Figure1.
Asoutlinedin[1 ],theevolutionequationforone-dimensionalshearwave
propagationthrougha homogeneous, viscoelasticmediumis
@ 2
@t 2
u(t;x) @
@x
(t;x)=F(x;t); R
1
<x<R
2
; (1)
CenterforResearchinScienticComputation,Box8205,NorthCarolinaState
Uni-versity,Raleigh,NC27695-8205
y
CenterforResearchinScienticComputation,Box8205,NorthCarolinaState
Uni-versity,Raleigh,NC27695-8205
z
CenterforResearchinScienticComputation,Box8205,NorthCarolinaState
Sensor array
Acoustic disturbance
R
Gel mold
Tube
Figure1: The 1Dhomogeneous viscoelasticmodel.
whereurepresentsthesheardisplacement,representstheshearstress,and
F representsa bodyforcingterm. Forboundaryconditionsa pureshearing
force on the left boundary and a free surface on the right boundary were
assumed;hence,
(t;R
1
)=f(t); (t;R
2
)=0: (2)
Theinitial conditionswere u(0;x)=u
0
(x), and u
t
(0;x)=u
1 (x):
The focus of [1 ] concerns the choice of an eective constitutive
equa-tion forthis model. In that paper, the authors investigated internal strain
variablemodelsasalternativestothecomputationallyintensivequasi-linear
viscoelastic model proposed by Fung [9 ]. Specically, they assumed the
stress isgiven asa sumof internalstrain variables,
(t)= N
X
j=1
j
(t): (3)
Thedynamics ofeach internal strainvariable ismodeled dynamicallyas
d
j (t)
dt
=
j
j +C
j d
e
dt (u
x
(t));
j
(0;x)=0; j=1;:::;N; (4)
where
e
is the elastic response function dened in ([9 ],x7), and may be
givenas
e (u
x
(t;x))=+e ux
this formulation, with linear internal strain variable models, is equivalent
to Fung's formulation with a sum of exponential terms approximating the
relaxation function. More generally however, the internal strain variables
mightbe modeled by nonlineardynamics ofthe form
d
j (t)
dt =g
j (
j
(t))+C
j d
e
dt (u
x
(t));
j
(0;x)=0; j =1;:::;N: (6)
Allof these models correspond to a viscoelastic body under either loading
orunloading. Eachconstitutiveequation expressesthestressnonlinearlyin
termsof theinnitesimalstrainu
x .
The authors of [1] investigated three particular internal strain variable
modelsasconstitutiveequations: aone linearinternalstrainvariablemodel
( =
1
), a two linear internal strain variable model ( =
1 +
2 ), and
onepiece-wise linearinternalstrainvariablemodel. Numerical experiments
veried theeectiveness of the internal strain variable models and
demon-strated good agreement with simulated data in the case of the two linear
internalstrain variable model.
In thisnote, we focus, forsimplicity, on theoretical foundationsfor the
one linear internal strain variable formulation. The case of multiple linear
internalstrainvariablesisreadilytreatedinthesameway. Theshearstress,
,isgiven by theequation
=
1 +C
D u
tx ;
where we have added a Kelvin-Voigt damping term withC
D
>0 asa rst
approximation to damping present inviscoelastic materials. The terms
1
and
e
areassumed given by
d
dt
1
+
1 =c
d
dt
e (u
x
(t;x));
1
(0;x)=0 (7)
e (u
x
(t;x))=+e ux
: (8)
We thenanalyze thesystem
u
tt C
D u
txx
1x
=F inV
(9)
u(0;x)=u
0
2V (10)
u
t
(0;x)=u
1
2H; (11)
where H = L 2
(), and V =H 1
(), and = [R
1 ;R
2
]. The inner product
embeddedin H,and H is continuously embedded inV
,the dualspace of
V.
The organization of thispaper is as follows. We rst deneweak
solu-tionsto system(7)-(11), and listsomeassumptionsinSection2. InSection
3 we develop the Galerkinapproximation, utilizingseveral lemmas, and in
Section4weestablishtheexistence anduniquenessofbothlocalandglobal
weaksolutions. Thisworkadaptsthetechniquesof[1 ]and[2 ]to oursystem
withlinearinternalstrain variablesbutwithnonlinearstress-strain
interac-tion.
2 Preliminaries
We will interpret system(7)-(11) in theV
sense. In developing a general
theory, we will at various times invoke several from among the following
assumptions:
(AF) Theforcing termsatisesF 2L 2
(0;T;V
)
(Af)The innerboundaryconditionsatises f 2L 2
(0;T)
(AL)Theelasticresponsefunction
e
satisesalocalLipschitz
condi-tion,
k
e
(u)
e
(v)k L
B
r
ku vk
forsome positiveconstant L
B
r
andforallu,v inB
H
(0;r),theballin
H centered at 0 ofradius r.
(AG) ThereexistsconstantsC
1 andC
2
such that
k
e
(u)kC
1
kuk+C
2
forevery u2H.
Notethat assumption(AL) canbeveriedfor
e
in[1](i.e.,
e
given in
(5)above)byrstcomputing d
dt
e
(tu+(1 t)v),integratingwithrespecttot
overtheinterval[0;1],thentakingthenormofbothsides. Assumption(AG)
isaphysicalboundonthegrowth of
e
priorto rupture. Itis satisedbya
modied versionof
e
in(5), callit~
e
,inwhich~
e
accounts forsaturation
before rupture and agrees with
e
up to this saturation. However, (AG) is
notsatisedbythe
e
of (5)asit isdenedthere.
We have the following denition of weak solutions for the one linear
Denition 2.1 LetL
T
=fw:[0;T]!H :w2C
W
([0;T];V)\L ([0;T];V)
and w
t 2C
W
([0;T];H)\L 2
(0;T;V)g. We dene u2L
T
to be a weak
so-lution of system(7)-(11) if it satises
Z
t
0 [ (u
s (s);
s
(s))+C
D (u
sx (s);
x
(s))+(
1 (u
x (s));
x
(s))]ds
+(u
t
(t);(t)) (u
1 ;(0))
= Z
t
0
[<F(s);(s)>
V
;V
f(s)(s;R
1
)]ds (12)
for any t 2[0;T] and 2L
T
, with the initial conditions u
0
2 V, u
1 2 H,
and
1 (u
x
(t;x))=c
e (u
x
(t;x)) e t
e (u
0x )
Z
t
0 e
(t s)
e (u
x
(s;x))ds
:
(13)
Notethisnotion ofweaksolutionfor system(7)-(11) agrees withtheusual
oneinthatityieldsu
tt 2L
2
([0;T];V
)withequation(9)holdinginthesense
ofL 2
([0;T];V
). Here, C
W
([0;T];V) refersto the setof weakly continuous
functionsinV on [0;T].
We rst establish existence of local weak solutions under only the
as-sumptions(AF),(Af),and(AL).Todealwiththenonlinearelasticresponse
term, we rst denetheoperatorP astheradial retractionfrom thespace
H onto the ballB
H (u
0
x
;1) of radius 1 centered at u
0x
. Then we dene a
newelasticresponse function^
e by
^
e
(u)=
e
(Pu); 8u2H:
Thus, fromassumption (AL),one can easilyargue theglobalconditions:
(AL2)k^
e
(u) ^
e
(v)k L
B
1+ku
0x k
ku vkforall u;v2H,
(AG2) k^
e
(u)kC
1
kuk+C
2
forallu2H.
We also denea modiedinternalstrain^
1
asfollows
^
1 (u
x
(t;x))=c
^
e (u
x
(t;x)) e t
^
e (u
0x )
Z
t
0 e
(t s)
^
e (u
x
(s;x))ds
:
We begin our arguments by developing the standard Galerkin
approxima-tion and establishing a priori bounds for them. Let f
i
g be any linearly
independenttotalsubsetofV. Foreachm,letV m =spanf 1 ; 2 ;:::; m g. Choose fu m 0
g and fu m
1 g2V
m
such thatu m
0 !u
0
inV andu m
1 !u
1 inH,
andlet M
0
andM
1
beconstants suchthat
ku m 0 k V M 0 and ku m 1
kM
1
: (15)
We dene the Galerkin approximation u m (t) = P m k=1 a m k (t) k as the
uniquesolutionofthe followingm-dimensionalintegro-dierentialsystem
<u m tt ; j > V ;V +C D (u m tx ; jx )+c( ^
1 (u m x ); jx
) = <F(t);
j > V ;V f(t) j (R 1 ) (16)
forj=1;:::;montheinterval[0;T]forsomeT >0. Wehavethefollowing
a priori estimateforthe Galerkinapproximation.
Lemma3.1 Let u m
(t) bethe Galerkin approximation on [0;T]. There
ex-istsa constant K >0 such that
ku m t (t)k 2 +ku m x (t)k 2 +C D Z t 0 ku m sx (s)k 2
dsK (17)
where K depends on the problem data (i.e., u
0 , u
1
, f, F, c, , , , C
D ,
andT), but isindependent of m.
Proof. Toobtainthea prioriestimatewemultiply(16)by d
dt a
j
(t)andsum
forj=1;:::;m,
1 2 d dt ku m t k 2 +C D ku m tx k 2
=<F(t);u m t > V ;V f(t)u m t (R 1 ) c(^ e (u m x );u m tx ) +ce t (^ e (u m 0x );u m tx )+c(
Z t 0 e (t s) ^ e (u m x (s))ds;u m tx ): Adding(u m x ;u m tx
) to each sideofthe above equation,wehave
1 2 d dt ku m t k 2 + 1 2 d dt ku m x k 2 +C D ku m tx k 2
j<F(t);u m t > V ;V
j+jf(t)u m t (R 1 )j +j(c^ e (u m x )+u
m
x ;u
m
tx
)j+cj(^
(AG2), estimates (15), the embedding H 1
() ,! C(), and standard
in-equalitiesto obtain
d dt (ku m t k 2 +ku m x k 2
)+C
D ku m tx k 2 C D ku m t k 2 +10(cC 1 +1) 2 C 1 D ku m x k 2 +5C 1 D kF(t)k 2 V +5c 2 3 C 1 D jf(t)j 2 +10c 2 C 2 2 C 1 D +5c 2 C 1 D (C 1 M 0 +C 2 ) 2 +5(c) 2 C 1 D (C 1 (M 0
+1)+C
2 ) 2 T 2 : (18)
Integrating from 0 to t,we obtain
ku m t k 2 +ku m x k 2 +C D Z t 0 ku m sx (s)k 2
dsku m 1 k 2 +ku m 0x k 2 + ~ K Z t 0 (ku m s (s)k 2 +ku m x (s)k 2
) ds+5C 1 D Z T 0 kF(s)k 2 V ds +5c 2 3 C 1 D Z T 0 jf(s)j 2
ds+10c 2 C 2 2 C 1 D T +5c 2 C 1 D (C 1 M 0 +C 2 ) 2
T+5(c) 2 C 1 D (C 1 (M 0
+1)+C
2 ) 2 T 3 :
ApplyingGronwall'sinequalitywecanconcludethatthesequencesfku m t )k 2 g and fku m x k 2
garebounded. Hencethere existsa positive constant
K = K(M
0 ;M
1
;c; ;T;kFk
L 2
(0;T;V
) ;kfk
L 2
(0;T)
) independent of m such
that ku m t (t)k 2 +ku m x (t)k 2 +C D Z t 0 ku m sx (s)k 2
dsK (19)
foreach t2[0;T]. Thisproves thelemma.
Lemma3.2 Letu m
(t)betheGalerkinapproximation on[0;T]. Thenfu m
g
isbounded uniformlyin C([0;T];H)L 2
([0;T];H).
Proof. Forfu m
t gL
2
([0;T];H), we have
(u m
(t
2
);) (u m (t 1 );)= Z t2 t 1 (u m s (s);)ds
for all 2 H and forany t
1 ;t
2
2[0;T]. We take a sequence ft k
1
g 2 [0;T]
such thatt k
1
!0 ask!1, then
(u m
(t
2
j(u m
(t);)j j(u m
0
;)j+ Z
t
0 j(u
m
s
(s);)jds
ku
m
0
kkk+ Z
T
0 ku
m
s
(s)kkkds
M
0 +TK
1=2
kk:
Thus ku m
(t)k M
0 +TK
1=2
, and fu m
(t)g is boundeduniformlyin H for
t2[0;T]and hence inC([0;T];H).
In thefollowing we repeatedly take subsequencesof sequencesof fu m
g.
Ineach case we againdenote thesubsequence asfu m
g.
Lemma3.3 There exist functions u 2L 2
([0;T];V) and u^ 2L 2
([0;T];V),
and a subsequencefu m
g such that
u m
!u weakly in L 2
([0;T];V)
u m
t
!u^ weakly in L 2
([0;T];V) :
Proof. From Lemma 3.1 and Lemma 3.2 we have that fu m
g is bounded
uniformlyin C([0;T];V) L 2
([0;T];V). We also have, from Lemma 3.1,
thatfu m
t
gisboundeduniformlyinL 2
([0;T];V). Wethenapplythe
Banach-AlaogluTheoremto obtainthedesired resultsinthelemma.
Lemma3.4 Theset fu m
g is an equicontinuous andbounded subsetof
C([0;T];V), moreover,
u m
(t)!u(t) weakly in V
uniformlyin t2[0;T], i.e.,u m
!u weakly in C
W
([0;T];V).
Proof. The boundedness result follows from Lemma 3.1 and Lemma 3.3.
To prove theequicontinuity,we have
u m
(t+t) u m
(t)= Z
t+t
t u
m
s (s)ds
fort;t+t2[0;T]. UsingLemma3.1, we obtain
ku m
(t+t) u m
(t)k
V
Z
t+t
t
ku m
s (s)k
V ds
t
1=2
(TK+C 1
D K)
1=2
Thus, forany >0 and t2[0;T], 9 Æ(;t) =(=K(T +C
D ))
2
such that
jt 0
tj<Æ impliesku m
(t 0
) u m
(t)k<.
For theconvergence, we usea version ofthe Arzela-Ascolitheorem(see
[2 ] or [10 ], Thm. 3.17.24). Let Y = B
V (0;K)
V
, the closure in V of the
ball centered at zero with radius K taken with the weak topology. Y is
a complete metric space. Let F = fu m
g C([0;T];Y). By the above
estimate,equicontinuityintheV senseimpliesequicontinuityintheY sense.
Also,foreach t2[0;T], theset fu m
(t);u m
2Fgis relativelycompactin Y
(Banach-Alaoglu Theorem). Then F is relatively compact in C([0;T];Y),
i.e.,thereexistsasubsequence,denotedfu m
gagain,suchthatu m
(t)!u(t)
inY uniformlyint2[0;T],i.e.,
u m
(t)!u(t) weakly inV uniformlyint2[0;T]:
Lemma3.5 The derivative u
t
exists in the V sense and u
t
= u^ a.e. in
[0;T].
Proof. Now u
t 2L
2
([0;T];V)if thereexists av2L 2
([0;T];V) such that
u(t
2 ) u(t
1 )=
Z
t2
t
1
v(s)ds; t
1 ;t
2
2[0;T]
where R
t2
t
1
v(s)dsor R
t2
t
1 u
s
(s)dsisdened usingduality,i.e.,
<u(t
2 ) u(t
1 );>
V
;V =
Z
t2
t
1 <u
s
(s);>
V
;V ds
forall 2V and fort
1 ;t
2
2[0;T]. Now
<u m
(t
2 ) u
m
(t
1 );>
V
;V =
Z
t
2
t1 <u
m
s
(s);>
V
;V ds
for all 2 V and for t
1 ;t
2
2 [0;T]. By Lemma 3.3 u m
t
! u^ weakly in
L 2
([0;T];V), hence
Z
t2
t
1 <u
m
s
(s);>
V ds!
Z
t2
t
1
<u(s);^ >
V
ds 82V:
ByLemma 3.4u m
(t)!u(t)weakly inV uniformlyin t2[0;T], hence
<u(t
2 ) u(t
1 );>
V =
Z
t
2
t1
<u(s);^ >
V ds
forall 2V and fort
1 ;t
2
2[0;T]. Thereforeu
t
(t)=u(t),^ a.e. t2[0;T]in
Lemma3.6 Thesequence fu
t
g converges to u
t
strongly in L ([0;T];H).
Proof. Use Aubin's Theorem ([7 ], Lemma 8.4). We have fu m
t
g bounded
uniformlyinL 2
([0;T];V). If we can show thatfu m
tt
g is bounded uniformly
inL 2
([0;T];V
), then, sinceV ,!H compactly and H,!V
continuously,
there existsasubsequence,denoted fu m
t
gthat converges inL 2
([0;T];H).
NowL 2
([0;T];V
)=L 2
([0;T];V)
. FixM andlet
M = P M k=1 c k (t) k withc k
(t)2C 1
([0;T]). Then formM,
j<u m tt ; M > L 2 ([0;T];V) ;L 2
([0;T];V) j= Z T 0 <u m ss (s); M (s)> V ;V ds :
Usingequation (16)to substitute infor<u m ss (s); M (s)> V ;V
,wehave
j<u m tt ; M >j R T 0
j<F(s);
M (s)>
V
;V jds+
R T 0 j f(s) M (s;R 1 )jds + R T 0 j(C D u m sx (s); M x
(s))jds+ R
T
0 j ( ^
1 (u m x (s)); M x (s))jds C D ku m tx k L 2 ([0;T];H) +k ^
1 k
L 2
([0;T];H) +kFk
L 2 ([0;T];V ) k M k L 2 ([0;T];V) + R T 0 jf(s)jk M (s;R 1 )k L 1 ds;
where we used Cauchy-Schwartz and Young's inequality in the last step.
Now k ^
1 k 2 L 2 ([0;T];H) = R T 0 k ^ 1 (u m x (s))k 2
ds and, using the denition of ^ in
equation (14), we have
k
1 (u
m
x
(t))kc k^ e (u m x
(t))k+k^
e (u m 0 x )k+ Z t 0 k^ e (u m x (s))kds c C 1 (ku m x
(t)k+ku m
0
x
k)+2C
2 + Z T 0 C 1 ku m x
(s)k+C
2 ds c 2C 2 +C 1 (M 0
+K+T(C
1 K 1=2 +C 2 )
wherewehaveusedtheboundforu m
0
andLemma3.1inthelaststep. Thus,
we have R T 0 k ^ 1 (u m x (s))k 2
ds C(c;C
1 ;C
2 ;M
0
;K ;T) bounded independent
ofm. Also,usingthefactthat,inonedimension,H 1
(),!C 0
(),wehave
Z T 0 jf(s)jk M (s;R 1 )k L 1ds kfk L 2 ([0;T]) k M k L 2
([0;T];V)
: (20)
Hence,
j<u m tt ; M > L 2 ([0;T];V) ;L 2
([0;T];V) j
K+C+kFk
L 2
([0;T];V
) +kfk
Sincef
M g
M=1
forma dense subsetof L ([0;T];V),wehave
ku m
tt k
L 2
([0;T];V
)
K+C+kFk
L 2
([0;T];V
) +kfk
L 2
([0;T])
and thus fu m
tt
g isuniformlyboundedinL 2
([0;T];V
).
Lemma3.7 Thefunctionsfu m
t
gareboundedinC([0;T];H)andaree
quicon-tinuous in C
W
([0;T];H); moreover, for each t,
u m
t
(t)!u
t
(t) weakly in H:
Proof. The boundedness statement follows from Lemma3.1. The
conver-gencestatement willfollowfrom an applicationof Arzela-Ascoliin
C
W
([0;T];H) oncewe establishtheequicontinuity. To dothiswe rst note
that, forv2V,
j<u m
t
(t+t) u m
t
(t);v>
V
;V
j
Z
t+t
t
j<u m
ss
(s);v>
V
;V jds
ku
m
tt k
L 2
([0;T];V
) kvk
V t
1=2
~
Ckvk
V t
1=2
wherewehaveusedtheresultfromtheLemma3.6toboundtheku m
tt kterm
intermsof ~
C=K+C+kFk
L 2
([0;T]
V )
+kfk
L 2
([0;T])
,independentof m.
Assume nowthat2H and x>0. Forv2V and t;t+t2[0;T],
we have
j(u m
t
(t+t) u m
t
(t);)j j<u m
t
(t+t) u m
t
(t);v>
V
;V j
+j(u m
t
(t+t) u m
t
(t); v>j
~
Ct 1=2
kvk
V
+2Kk vk:
Choose v suchthat Kk vk=4and choose Æ=(=2 ~
Ckvk
V )
2
. Then
j(u m
t
(t+t) u m
t
(t);)j:
Thisprovesequicontinuity inC
W
([0;T];H). LetY =B
H (0;K)
H
with the
weak topology, and let F = fu m
t
g C([0;T];H). It follows that F is
equicontinuousinC
W
([0;T];Y)andfu m
t (t):u
m
t
2Fgisrelativelycompact
in Y for each t 2 [0;T]. We then use an application of Arzela-Ascoli in
C
W
([0;T];H) to obtainF relativelycompactinC
W
([0;T];Y). Thus,there
existsasubsequence, denotedfu m
t
g,such that
u m
t
(t)!u
t
(t) weakly inH
Lemma3.8 There existsan h2L ([0;T];H) such that
^
1 (u
m
x
)!h weakly in L 2
([0;T];H):
Proof. We needtoshowthat^
1 (u
m
x
)isboundeduniformlyinL 2
([0;T];H).
In Lemma 3.6 we computed R
T
0 k ^
1 (u
m
x (s))k
2
ds C(c;C
1 ;C
2 ;M
0
;K ;T):
Thus f ^
1 (u
m
x
)g is bounded in L 2
([0;T];H) independent of m, and there
existsa subsequenceand an h2L 2
([0;T];H) such that
^
1 (u
m
x
)!h weaklyin L 2
([0;T];H):
4 Existence and Uniqueness Theorems
We now state andprovethetheorem regardingexistence anduniqueness of
weak solutions,rstestablishing alocalresult.
Theorem 4.1 Under the assumptions (AF), (Af), and (AL), system
(7)-(11) has a unique local weak solution on [0;t
] for some t
>0.
Proof. Denote by P
M
(M =1;2;:::) the class of functions 2L
T which
can be representedintheform
(t)= M
X
k=1 c
k (t)
k
wherec
k 2C
1
([0;T]). LetP =[ 1
M=1 P
M
. Note P isdense inL
T .
We start with equation (16), multiply by c
k
(t), sum from 1 to M, and
integrateover (0;t) to obtain
Z
t
0 (<u
m
ss
(s);(s)>
V
;V +C
D (u
m
sx (s);
x
(s))+( ^
1 (u
m
x (s));
x
(s)))ds
= Z
t
0
(<F(s);(s)>
V
;V
f(s)(s;R
1 ))ds
or,integratingthe rst termbyparts, we nd
Z
t
0 [ (u
m
s (s);
s
(s))+C
D (u
m
sx (s);
x
(s))+( ^
1 (u
m
x (s));
x
(s))]ds
+(u m
t
(t);(t)) (u m
1
;(0))= Z
t
0
[<F(s);(s)>
V
;V
f(s)(s;R
M
Now x 2P
M
with M m and pass to the limit as m ! 1, using
Lemma3.3,Lemma3.6,Lemma3.7,Lemma3.8,andtheconvergenceu m
1 !
u
1
inH. Hence,onanyinterval[0;t], withtT,we obtain
Z t 0 [ (u s (s); s
(s))+C
D (u
sx (s);
x
(s))+(h(s));
x
(s))]ds+(u
t
(t);(t))(21)
(u
1
;(0))= Z
t
0
[ <F(s);(s) >
V
;V
f(s)(s;R
1 )]ds:
We nowneed to showthat
Z t 0 (h(s); x (s))ds= Z t 0 ( ^ 1 (u x (s)); x
(s))ds; 82L
T :
This is accomplished by establishing the strong convergence of u m
x (t) !
u
x
(t) in H as m ! 1. To do thiswe take u m
t
and u
t
as test functions in
equations(16)and d
dt
(21)respectively,andadda(u m
x ;u
m
tx )or(u
x ;u
tx )term
to bothsides oftheirrespectiveequations. We have
<u m tt (t);u m t
(t)>+C
D (u m tx (t);u m tx
(t))+( ^
1 (u m x (t));u m tx
(t))+(u m x (t);u m tx (t)) =(u m x (t);u m tx
(t))+<F(t);u m t (t)> V ;V f(t)u m t (t;R 1 ) and <u tt (t);u t
(t)>+C
D (u
tx (t);u
tx
(t))+(h(t);u
tx
(t))+(u
x (t);u tx (t)) =(u x (t);u tx
(t)+<F(t);u
t (t)> V ;V f(t)u t (t;R 1 ): Letz m
(t)=u m
(t) u(t)and subtract thetwo equationsto obtain
1 2 d dt kz m t k 2 +kz m x k 2 +C D kz m tx k 2 = d dt <z m t ;u t > 2C D <z m tx ;u tx >
+<F(t);z m t > V ;V <^ 1 (u x );z m tx
> <^
1 (u
m
x
) h;u
tx > f(t)z m t (R 1 )+<z
m
x ;z
m
tx
> <^
1 (u m x ) ^ 1 (u x );z m tx >:
We then use the Cauchy-Schwartz and Young's inequalityon the last two
termson theright and integrateon (0;t) to obtain
kz m t k 2 +kz m x k 2 ku m 1 u 1 k 2 +ku m 0 x u 0x k 2 +2C 1 D Z t 0 kz m x (s)k 2 ds +2C 1 D Z t 0 k ^ 1 (u m x
(s)) ^
1 (u
x (s))k
2
ds+X
m (t)+Y
X
m
(t)=2<u m 1 u 1 ;u t
(t)> 2<z m t (t);u t (t)> 4C D Z t 0 <z m sx (s);u sx
(s)>ds+2 Z
t
0
<F(s);z m s (s)> V ;V ds 2 Z t 0 <^ 1 (u x (s));z m sx
(s)>ds 2 Z t 0 <^ 1 (u m x
(s)) h(s);u
sx >ds
and
Y
m (t)=2
Z t 0 jf(s)z m s (s;R 1 )jds:
Note that X
m
(t)! 0 as m ! 1 byLemma 3.3, Lemma 3.7, Lemma 3.8,
and theconvergenceu m
1 !u
1
inH. To see thatY
m
(t)!0 asm!1, we
usetheembeddingV ,!C 0
and Agmon'sinequality(see [11 ])to obtain
Y
m
(t) 2 Z t 0 jf(s)jkz m s (s)k L 1 ds 2ckfk L 2 (0;T) Z t 0 kz m s (s)kkz m s (s)k H 1 ds 1=2 2ckfk L 2 (0;T) kz m t k 1=2 L 2 ([0;T];H) kz m t k 1=2 L 2 ([0;T];V) : Since z m t
! 0 strongly in L 2
([0;T];H) by Lemma 3.6, and since fz m
t g is
boundeduniformlyinL 2
([0;T];V),wehave Y
m
(t)!0 asm!1. Forthe
integraltermontherightside,weuseassumptions(AL)and(AL2)to show
that Z t 0 k ^ 1 (u m x (s)) ^ 1 (u x (s))k 2
ds 4L 2 Tku m 0 x u 0x k 2 +2L 2
(2+T) Z t 0 ku m x (s) u x (s)k 2 ds:
SeetheAppendixfortheproof.
We then have
kz m t k 2 +kz m x k 2 2C 1 D
(1+4L 2 +2L 2 T) Z t 0 kz m x (s)k 2
ds+ku m 1 u 1 k 2
+(1+8C 1 D L 2 T)ku m 0 x u 0 x k 2 +X m (t)+Y
m (t)
We may apply the generalized Gronwall inequality to the above equation
to obtain kz m
x
(t)k ! 0 a.e. t 2 [0;T]. Thus, ^
1 (u m x ) ! ^ 1 (u x
) strongly in
L 2
([0;T];H),and bytheuniquenessof thelimitwehave h=^
1 (u
x
Z
t
0 [ (u
s (s);
s
(s))+C
D (u
sx (s);
x
(s))+( ^
1 (u
x (s));
x
(s))]ds+(u
t
(t);(t))
(u
1
;(0))= Z
t
0
[<F(s);(s)>
V
;V
+f(s)(s;R
1 )]ds
whichholdsforanyinterval[0;t],fortT,andforall2L
T
. Thussystem
(7)-(11)with
1
replacedby^
1
hasasolutiononanyarbitraryinterval[0;T].
Uniqueness of this weak solution can be shown in the standard way (see,
e.g.,[1 ], [4],[8 ]).
To show thatsystem (7)-(11) with
1
has a local uniqueweak solution,
we notethat theweaksolutionu has thepropertythat u
x
is continuousin
t. Thus,there existsa t
with0t
T such that
ku
x (t) u
0x
k1 forall t2[0;t
];
and therefore,fromthe denitionof ^
1
,we have
^
1 (u
x
(t;x))=
1 (u
x
(t;x)) forallt2[0;t
]:
Henceuisa weaksolutionofsystem(7)-(11) on[0;t
]. Uniquenessis again
shown in the standard way. This completes the proof of local existence
underassumptions(AF), (Af),and (AL).
If we add the growth condition (AG) on
e
(recall that (8) does not
satisfy this condition unless it is modied for large u
x
), then we can use
argumentssimilarto those aboveto establish globalexistence.
Theorem 4.2 Undertheassumptions(AF),(Af),(AL),and(AG),system
(7),(9)-(11) has a unique global weaksolution on any nite interval [0;T].
Proof. Undertheadditionalassumption(AG),wecanargueasinTheorem
2 of [1] to establish that this local unique solution actually exists on any
arbitraryinterval[0,T].Essentially,oneusesthecondition(AG)toestablish
a prioriboundssimilarto(17)forapproximationsinvolving
e ,not^
e ,and
thenarguesapointwiseboundon u
x
(t). ThelocalLipschitzcondition(AL)
can then be used on
e
and arguments similar to those above carried out.
We argue theinequality Z t 0 k ^ 1 (u m x (s)) ^ 1 (u x (s))k 2
ds 4L 2 Tku m 0 x u 0x k 2 +2L 2
(2+T) Z t 0 ku m x (s) u x (s)k 2 ds:
Usingequation(14) to dene,^ we have
Z t 0 k ^ 1 (u m x (r)) ^ 1 (u x (r))k 2 dr Z t 0 fk^ e (u m x (r)) ^ e (u x (r))k +ke r (^ e (u m 0 x ) ^ e (u 0x ))k +k Z r 0 e (r s) (^ e (u m x
(s)) ^
e (u x (s))dskg 2 dr 4 Z t 0 k^ e (u m x
(r)) ^
e (u
x (r))k
2
dr+4 Z
t
0 k(^
e (u m 0 x ) ^ e (u 0x ))k 2 dr +4 Z t 0 Z r 0 2 e 2(r s) k(^ e (u m x (s)) ^ e (u x (s))kds 2 dr 4L 2 Z t 0 ku m x (r) u x (r)k 2
dr+4L 2 Z t 0 ku m 0 x u 0x k 2 dr +4 Z t 0 Z r 0 e (r s) Lku m x (s) u x (s)kds 2 dr;
where L=L
B
1+ku
0x k
and we have used assumption (AL2) in thelast step.
Now Z t 0 ku m 0 x u 0x k 2
drku m 0 x u 0x k 2 T
fort2[0;T]. Also,usingCauchy-Schwartz, we have
Z
t
0 k ^
1 (u
m
x
(s)) ^
1 (u
x (s))k
2
ds 4L 2
Tku m
0
x u
0x k
2
+2L 2
(2+T) Z
t
0 ku
m
x
(s) u
x (s)k
2
ds:
5 Acknowledgments
This research was supported in part (H.T.B. and S.W.) by the U.S. Air
Force OÆceof Scientic Research under grant AFOSR F-49620-00-1-00 26.
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