On evolutionary Navier-Stokes-Fourier
type systems in three spatial dimensions
Miroslav Bul
ek,Roger Lewandowski, Josef M
alek
Abstrat.Inthispaper,weestablishthe large-dataandlong-timeexisteneof
a suitableweaksolutiontoaninitialandboundaryvalue problemdrivenbya
systemofpartialdierentialequationsonsistingoftheNavier-Stokesequations
withthevisositypolynomiallyinreasingwithasalarquantitykthatevolves
aordingtoanevolutionaryonvetiondiusionequationwiththerighthand
side (k)jDDD(v)j 2
that ismerely L 1
-integrable over spae and time. We also
formulateaonjetureonerningregularityofsuhasolution.
Keywords:large data existene, suitable weak solution, Navier-Stokes-Fourier
equations, inompressibleuid, thevisosityinreasingwitha salarquantity,
regularity,turbulentkinetienergymodel
Classiation: 35Q30,35Q35,76F60
1. Introdution
LetR 3
beanopenbounded setandT 2(0;1). Ourgoalistoprovethe
existeneofatriple(v;k;p):(0;T)!R 3
R
+
Rwhihsolves,in(0;T),
thefollowingnonlinearsystemofvepartialdierentialequations
divv=0;
(1.1)
v
;t
+div (vv) div((k)D D
D (v))= rp;
(1.2)
k
;t
+div (kv) div((k)rk)+"(k)=(k)jD D
D (v)j 2
: (1.3)
Weompletethesystem(1.1){(1.3)bythefollowinginitialandboundaryondi-
tions:
v(0;x)=v
0 (x)
k(0;x)=k
0
(x) and k
0 (x)0
a.e.in ; (1.4)
MiroslavBulekthankstheJindrihNeasCenterforMathematialModeling,theprojet
LC06052nanedbyM
SMTforitssupport. RogerLewandowski'sworkispartiallysupported
bytheANRprojet08FA300-01. RogerLewandowskiisalsogratefultoJindrihNeasCenter
forMathematialModelingforthesupportandthehospitalityduringhisstaysattheCharles
UniversityinPragueinNovember2008andMay2009. JosefMalek'sontributionisapartofthe
vn=0
v
+(1 )((k)DDD(v)n )
=0
on(0;T); (1.5)
k=0 on(0;T)
D
; (1.6)
rkn=0 on(0;T)
N : (1.7)
Here,D D
D(v)denotes thesymmetripartof thegradientof thevetoreldv, i.e.,
2D D
D(v) = rv+(rv) T
, n =n(x) is the outer normal to the boundary loated
at x 2 , w
:= w (wn)n denotes the projetion of a vetor w = w(x)
to thetangentplaneofthe boundaryat x,
D
and
N
aresmoothsubsetof
satisfying
D
[
N
=and
D
\
N
=;. Theparameter2[0;1℄
homotopially onnets a homogeneous Neumann type boundary ondition for
=0withthehomogeneousDirihletboundaryonditionfor=1. If0<<1,
then(1.5)
2
isalled Navier'sslip boundaryonditions. Inthis paperweassume
thatisanynumberfrom[0;1).
Conerningthefuntions ; ;": R
+
!R
+
, werequirethat theyare ontin-
uousand that for ertain;; 2[0;1)and two positive onstantsC
1
;C
2 the
followinginequalitieshold forallk2R
+ :
C
1 (1+k)
(k)C
2 (1+k)
;
C
1 (1+k)
(k)C
2 (1+k)
;
C
1 k
1+
"(k)C
2 k
1+
: (1.8)
Withintheframeworkofweaksolutionsthetermontherighthandsideof(1.3)
isnoteasytohandle. Thus,itismoreappropriateto\equivalently"reformulate
thesystem(1.1){(1.3)inthefollowingway. DeningthesalarquantityE as
(1.9) E:=
1
2 jvj
2
+k;
wededuetheequationforE bytakingthesalarprodutof(1.2)andv andby
addingtheresultto(1.3). Doingso,wearriveattheequation
(1.10) E
;t
+div(v(E+p)) div((k)rk) div((k)DDD(v)v)+"(k)=0:
Of ourse, assuming that the multipliation of (1.2) by v is meaningful (or in
otherwords,assumingthat v isapossibletest funtionin theweakformulation
of (1.2))theidentities (1.3)and(1.10) areequivalent. However,in three spatial
dimensions we usually do not know that v is an admissible test funtion and
weannotonludetheequivaleneof (1.3)and(1.10). Themainmathematial
reasonwhyweprefer(1.10)to (1.3)isthefat thatin (1.10)allnonlinearterms
areindivergeneformandbelongtoabetterspaethanL 1
whilein(1.3)theterm
onthe righthand side belongs usually to L 1
only. Consequently, itis easier to
identifyweaklimitsofallnonlinearquantitiesin(1.10)thanin(1.3). Thesefats
seemto berst speied and exploited in [13℄. On the otherhand, onsidering
system (1.1){(1.3) by using divergene-free test funtions in (1.2). Moreover,
assumingthatwehaveaweaksolutionto (1.1){(1.2)and(1.10) thatinaddition
satises
(1.11) k
;t
+div (kv) div((k)rk)+"(k)(k)jDDD (v)j 2
;
inaweaksense,thenitisnaturaltoallsuhasolutionasuitableweaksolution
in senseofCaarelli,Kohn, Nirenberg, see[10℄. Indeed,subtrating(1.11) from
(1.10),one dedues
(1.12) jvj
2
;t
+div v(jvj 2
+2p)
div(2(k)DDD(v)v)0;
thatistheformofloalenergyinequalityasappearedinthedenitionofsuitable
weaksolutiontoNavier-Stokessystem,see[10℄.
Inthisstudyweestablishthefollowingresult.
Theorem1.1. Assumethat , and"satisfy (1.8)with
(1.13) 0<
2
5 +
2
3
; 0<+ 2
3 :
Thenforany2C 1;1
,T >0,v
0 2L
2
n;div andk
0 2L
1
(),k
0
0a.e.in,there
existsasuitableweaksolution(v;p;k)to Problem(1.1){(1.7),that inpartiular
fulls (1.1){(1.2) and(1.9){(1.11)inthesenseofdistributions.
Thepreisedenition ofthesolutionandformulationoftheresultisgivenin
Theorem2.1below,seeSetion2.
Thesystem(1.4){(1.7)with,and"oftheform(1.8)isinterestingfromthe
pointofviewofmathematialanalysis ofPDEs,in partiular,from thepointof
viewofregularitytheory. Weshalladdressthispointnext.
Tosimplifydisussionbelow,weassumethat ,and"areoftheform
(1.14) (k):=
0 k
; (k):=
0 k
and "(k)="
0 k
2
;
where
0 and
0
arepositiveonstantsand"
0 0.
Weformulatethefollowingonjeture.
Conjeture 1.1. Let2R, , and"beof theform (1.14). Thenthere exist
Æ>0andC
>0suhthatforanytriple(v;p;k)solving(1.1){(1.2)and(1.10){
(1.11)inthesense ofdistributionthefollowingimpliationholds:
If
(1.15)
Z
0
1 Z
B1(0)
(k)jDDD (v)j 2
dxdtÆ
then
(1.16) jv(t;x)jC
in ( 1
2
;0)B 1
2 (0):
This onjeture ertainly holds for 0 sine then the system (1.1){(1.2)
redues toNavier-Stokesequation forwhih Conjeture 1.15wasprovedin [10℄,
seealso[31℄. Toourbestknowledge,Conjeture1.15isopenforgeneralvaluesof
positive's. Inwhat follows, wewill showhowConjeture 1.1 impliesthat, for
ertain's,anysuitableweaksolutionhasboundedveloity.
Indeed, assume that a triple (v;k;p) solve (1.1){(1.2) and (1.10){(1.11) on
someneighborhoodof(0;0)thatontainsforsome`
0
>0aset( ` A
0
;0)B
`
0 (0)
withsomeA>0speiedbelow. Thenweresalethetripleinthefollowingway.
Forany``
0
wedeneforsomeB >0
v
`
(t;x):=` B
v(`
A
t;`x);
p
`
(t;x):=` 2B
p(`
A
t;`x);
k
`
(t;x):=` 2B
k(`
A
t;`x):
Itiseasytoshowthat ifwehooseA,B suhthat
A:=
2 2
1 2
; B :=
1
1 2
and assumethat 6=
1
2
then thetriple(v
`
;p
`
;k
`
)solves(1.1){(1.2) and(1.10){
(1.11)inthesenseofdistributionin( 1;0) B
1
(0). Next,weapplyConjeture1.1
onthe resaledveloity v
`
. Hene,using thestandardsubstitution theorem we
seethatweneedto showthat
Æ Z
0
1 Z
B1(0) (k
` )jDDD(v
` )j
2
dxdt
= Z
1
1 Z
B
1 (0)
`
2B+2B+2
(k(`
A
t;`x))
jD D
D(v(`
A
t;`x))j 2
dxdt
= Z
0
` A
Z
B
` (0)
`
2B+2B+2 A 3
(k(t;x))
jDDD(v(t;x))j 2
dxdt
=` 6 1
1 2 Z
0
` A
Z
B`(0) k
jDDD(v)j 2
dxdt:
(1.17)
Interestingly,weseethatfor 1
6 <
1
2
weanhoose`sosmallthatthepremise
ofConjeture1.1isfullled. Asitsonsequene,weonludethat v
`
isbounded
in( 1=2;0)B
1=2
(0)andv isboundedin( (`=2)
;0)B
`=2
(0). Evenmore,it
followsfrom(1.17),Conjeture1.1andthestandardoveringargumentproedure
that,for<
1
6
,theHausdordimensionofthesetSofpossiblesingularitiesofv
(here,thepointof singularityisdenedsuh(t;x) thatv is notboundedin any
neighborhoodof(t;x))isbounded by
(1.18) d(S)<
1 6
;
whih is onsistent with the standard estimate of possible singular set for the
Navier-Stokesequations.
Tosummarize,thesystem(1.4){(1.7)with, and"oftheform(1.14)isan
interestingsystemfromthepointofviewofregularitytheory. Beforehoweverone
startstostudyregularitypropertyofanysolutiononeneedstoestablishitsexis-
tene,andthis isthesubjetofthis paper. WhilethestatementofTheorem1.1
for=0wasinvestigatedin [8℄,thease>0isanalyzed in thisstudy. Note
that for "0and =, Theorem 1.1guaranteesthe existeneof solutionfor
0<
10
9 .
Therearetwomain reasonsmotivatingusto analyzetheproblem(1.1){(1.8).
The rstoneomes from thelarge-data analysis of turbulent models. The se-
ondreasonisonnetedwiththequestionoflarge-dataqualitativemathematial
propertiesofowsofinompressibleheat-ondutingNewtonianuids. Weshall
disussthebothissuesinwhat follows.
(1)Kolmogorovmodel. Theproblemin onsideration(1.1){(1.3)islosely
related to the so-alled turbulent kineti energy model; then v represents the
statistialmean (averaged)veloityoftheuid, pis assoiatedto thestatistial
mean normalstress - the averagedpressure, standsfor the visosity, is the
eddydiusionandkdenotestheturbulentkinetienergydenedas 1
2 P
3
j=1 jv
0
j j
2
,
whereas v 0
is the veloityof utuationsand z stands for the averagingof the
quantityz. Thetermonthe righthand side of (1.3)representstheenergy that
thelargesalestransmitonto thesmallsales,andthelast termofthelefthand
side of (1.3)measures the energyrate returnedby thesmall sales to the large
sales. Usually, the quantities ; and " are depending on the mixing length
sale ` that is a positivegiven funtion or it is driven by another evolutionary
equation.
Infat,oneoftherstmodelsofthistypewasproposedbyKolmogorovin[15℄,
seealsothepaperNo.48in[16℄orAppendixin[30℄. Basedonloalpropertiesof
turbulene andinorporating,asKolmogorovlearlystates, (unspeied) rude
approximations,heformulates alosedsystemofequationsoftheform
divv=0 (1.19)
v
;t
+div (vv)= r
p
% +b
+Adiv
2 b
! D D
D(v)
; (1.20)
!
;t
+div(!v)= 7
11
! 2
+A 0
div
b
! r!
; (1.21)
b
;t
+div(bv)= b!+ 4
3 A
b
! jDDD(v)j
2
+A 00
div
b
! rb
; (1.22)
wherethe veloityof theuidis thesumoftheaveragedveloityv andtheve-
loityofutuationsv 0
,pistheaveragedpressure,b:=
1
3 P
3
j=1 jv
0
j j
2
isone third
ofthe sumofaveragedsquareof theomponents oftheveloityofutuations,
p
A 0
and A 00
are onstants. Equations (1.20){(1.22) oinide exatly 1
with equa-
tions(1){(3)in[15℄,[16℄,[30℄,wemerelyompletedthesystembytheonstraint
ofinompressibility(1.19). Thus, weobtainalosedsystemof sixequationsfor
(v
1
;v
2
;v
3
;p;`;b).
Next,assumingthat`isagivenknownfuntion,equation(1.21)isredundant.
Thus,settingv:=v,p:=
p
%
+bandnotiingthatb!= C
` b
p
band b
!
=
`
C p
b,the
system(1.19){(1.22)simpliesto
divv=0 (1.23)
v
;t
+div (vv)= rp+div
2`A
C p
bD D
D(v)
; (1.24)
b
;t
+div (bv)= C
` b
p
b+ 4`A
3C p
bjDDD(v)j 2
+div
`A 00
C p
brb
: (1.25)
Setting k :=
3
2
b, (k) :=
2`A
C p
b = 2
p
2`A
p
3 C p
k, (k) =
` p
2A 00
p
3C p
k and "(k) =
p
2C
p
3`
p
kkwearriveatthesystemoftheform(1.1){(1.3)that issubjetofinves-
tigationin thispaper. Notethat thequantity E introdued in (1.9)(that plays
animportantrolein ouranalysis)is thesumofthekinetienergyassoiatedto
theaveragedveloityandtheturbulentkinetienergyk= 1
2 P
3
j=1 jv
0
j j
2
.
Although the model (1.1){(1.3) desribes ompliated turbulent behavior in
asimplied manner (seefor exampledisussion in [30℄), it is quitepopularand
eÆientinvariousappliations. Itisusedforinstaneinoeanography([5℄,[32℄,
[20℄),inmarineengineering([22℄,[28℄),et., andsurprisinglygivesveryaurate
numerialresultsin omparisonwithexperimentaldata. Inertainappliations,
this model thus \prevents" the omputational analysts from dealing with the
(k ")model(seethe originalwork dueto LaunderandSpalding[17℄, andalso
[26℄formoredetails) thatisfromtheomputationalpointofviewveryostly.
Thederivation of models suh as(1.1){(1.3) is mainly based on dimensional
analysisandphysialassumptionsontheturbulene(see[26℄and[20℄)that lead
tothefollowingformsfor and
(1.26) (k)=
0 +
1 p
k and (k)=
0 +
1 p
k;
where
0 0,
1 0,
0
>0and
1
>0areonstants. Notethatthease(1.26)
with
0 ,
1 ,
0 ,
1
positive is overedby Theorem 1.1. There are also works
towards themathematial justiationof the k-equation (1.3) from theNavier-
Stokesequations([11℄, [25℄,[12℄), but atransparentand onsistentderivationof
thesemodelsis,toourbestknowledge,missing.Thelimitationsandappliability
of the model in onsideration are one of the topis studied in our forthoming
paper.
1
Infat,wefollowthe translationgivenbySpaldingin[30℄. There seemstobeamisprint
Fromthepointofviewofanalysisofturbulentkinetienergymodelstheresult
presentedinthispaperanbeonsideredasanaturalontinuationofTheorem4
in [21℄ sine it solves the problem formulated in [21℄ that has been left open.
Also,Theorem 4in[21℄that onernstheasewhenboth and arebounded
funtion of k proves that in three spatial dimensions the limit equation for k
deduedfromapproximatedsolutionssatisesavariationalinequality. Thispaper
givestwo essentialnovel ontributions to the analysis of (1.1){(1.3). First, the
unknownk is shownto fulll theequation forE (see (1.10) above) ratherthan
equation (1.3),and seond,it investigates three-dimensionalowswith and k
thatareunboundedfuntionsofk. Thepriewepayfordealingwith(1.10)rather
thanwith(1.3)isthat weneedtointroduegloballyintegrablepressureandthis
isthe reasonwhywearenotableto extend, at theurrent state,thetheory to
Dirihletboundaryonditionfortheveloity(the ase=1in (1.5)
2 ).
We nish this partby realling several related results and approahes. The
system (1.1){(1.3) was rst studied in [19℄ and [21℄. Assuming that the eddy
visosityisaboundedfuntion ofk,theauthorestablishestheexisteneofweak
(distributional)solutionsinthesteady-stateaseandintheevolutionary2Dase
ifbothkandvsatisfyhomogeneousDirihletboundaryonditions. Theseresults
have been generalized in many ways and for other boundary onditions, asfor
instanetoowsoftwointeratinguidssuhastheOeanandtheAtmosphere
([3℄, [4℄, [1℄). There are very few uniqueness results that are mainly obtained
under smallness assumptions onthe total variationof the eddy visosityorthe
soure term, and they onern steady-stateows ([2℄, [6℄). In order to analyze
models withunbounded eddyvisosities(that are important, see(1.26)) several
dierenttoolsweredeveloped,mostlyforsomesimpliedmodels(suhassteady-
statemodels, models withoutonvetiveterms, andevenwithout thepressure).
We refer the interested reader to Lewandowski and Murat [20, Chapter 5℄ for
details onerningrenormalized solutions, orto [14℄ (energysolutionsin speial
funtionspaes)orto[18℄(energysolutionswithperiodiboundaryonditions).
(2)Navier-Stokes-Fourier system.Assoiatingkwiththeinternalenergy
(or temperature) and setting " 0, the system (1.1){(1.3) desribes unsteady
owsof inompressibleheat-ondutinguidsin whih theCauhystressTTT and
theheatuxq aregivenbytheonstitutiveequationsoftheform
(1.27) TTT:= pIII+(k)DDD (v) and q:=(k)rk:
Thesystemofequations(1.1){(1.3)togetherwith(1.27)isalledtheinompress-
ible Navier-Stokes-Fouriersystem, where denotes the kinematial visosity of
theuid and is the heat ondutivity. In most liquids, that are well approx-
imated as inompressible materials, the internal energy is proportional to the
temperatureandthevisositydereases withinreasingtemperature. Thisisjust
opposite senario than that desribed by the assumptions (1.8). Although the
Navier-Stokes-Fouriersystemwith the visosity satisfying (1.8)is notreeting
areunsteadyowsofalassofNewtonianuidsthatexistforlargedataandthe
veloityisbounded.
Thelargedata existeneresultpresentedhereanbeviewedastheextension
oftheapproah(thatisbasedontheappropriateform ofthebalaneofenergy)
originallydevelopedin [13℄and[8℄where theNavier-Stokes-Fouriersystemwith
thebounded visosityandtheheatondutivityistreated;thespatially-periodi
problemisanalyzedin[13℄whileowsinboundeddomainssatisfyingtheNavier's
slipboundaryonditionsarestudiedin[8℄. Naumann[27℄studiedthemodelwith
thetemperaturedependentvisosityand theheatondutivity,he howeveruses
equation(1.3)insteadof(1.10);duetodiÆultiestoidentifythelimitthedissipa-
tivetermattheright-handsideof(1.3)hisoneptofsolutionisweakerthanthat
introduedin [13℄, [8℄ andused in this paperaswell. Forthesakeof omplete-
ness,weremarkthat Lions[23,Setion 3.4℄studies theasewhere thevisosity
andtheheatondutivityarepositiveonstants(temperatureindependent)and
providestwoapproahes(dierentfromthatpresentedhere)howtheprobleman
beinvestigatedinorderto establishlong-timeandlarge-dataexisteneresults.
Thepaperisorganizedasfollows. Afterintroduing relevantfuntion spaes,
weestablish, inSetion 2,themain resultthat inludes thepreisedenition of
suitableweaksolutionsto(1.1){(1.3). Then,inSetion3,weintroduetwo-level
approximations depending on parameters n and m and prove the main result.
Sine the existene of solutions to the (m;n)-approximation, for a xed n and
m,isgivenin [8,Appendix℄ wefousontheanalysisofthelimitbehaviorofthe
solutions(v m;n
;p m;n
;k m;n
)rstasn!1andthenasm!1.
2. Mainresult
Inordertostatethemainresultwithalldetailsweneedtolarifythenotation
ofrelevantfuntion spaes. Fortheveloityeld,wedene
W 1;p
n :=
v2W 1;p
() 3
: vn=0on ;
W 1;p
n ;div :=
v2W 1;p
n
: divv=0in ;
W 1;p
0
n
:= W 1;p
n
; W 1;p
0
n;div :=
W 1;p
n;div
;
L 2
n;div :=W
1;2
n;div kk
2
:
Wealsointroduethenaturalspaefork;forsomexed2R
+ weset
E
:=
n
k2L 1
(0;T;L 1
()): k0a.e.;
((1+k) s
1)2L 2
(0;T;W 1;2
D
()) foralls<
+1
2 o
;
whereW 1;2
():=fk2W 1;2
(); k=0on
D g.
Notethat byusing standardinterpolationtehniquethefollowingontinuous
embeddingholds(weshowitin theproofof themain theorem)for2[0;1℄
E
,!L
r
(0;T;L r
() 3
)\L q
(0;T;W 1;q
D ()
3
) forallr<
3+5
3
andq<
3+5
4 :
If>1then q=2intheaboveembedding.
Moreover, in what follows weuse the abbreviation(a;b)
A :=
R
A
ab whenever
ab2L 1
(A). InasethatA=wealsoomitwritingthesubsript. Thesame
notationisusedforvetor-and tensor-valuedfuntions aswell.
Weformulatethemainresultof thispaper.
Theorem 2.1. Let 2 C 1;1
, T >0, v
0 2 L
2
n;div and k
0 2 L
1
(), k
0
0 a.e.
in , be givenarbitrarily. Assume that ; and" satisfy (1.8)with ; and
fullling
(2.1) 0<
2
5 +
2
3
; 0<+ 2
3 :
Thenthereexistatriple(v;p;k)and Egivenas
E= 1
2 jvj
2
+k;
satisfying
v2C
weak (0;T;L
2
n;div )\L
2
(0;T;W 1;2
n;div );
(2.2)
v
;t 2L
q 0
(0;T;W 1;q
0
n
)forallq<min
5
3
;2
2
++ 5
3
; (2.3)
k2E
; (2.4)
k
;t
2M(0;T;W 1;1+Æ
)forertainÆ>0small;
(2.5)
p2L q
(0;T;L q
()) forallq<min
5
3
;2
2
++ 5
3
; (2.6)
p
(k)DDD(v)2L 2
(0;T;L 2
() 33
);
(2.7)
E
;t 2L
1+Æ
(0;T;W 1;1+Æ
D
())forertainÆ>0small;
(2.8)
andfullling
Z
T
0 hv
;t
;wi (vv;rw)+
1
(v;w)
+((k)DDD (v);DDD(w))dt
= Z
T
(p;divw)dt forallw2L 1
(0;T;W 1;1
n );
(2.9)
Z
T
0 hE
;t
;wi (v(E+p);rw)+((k)rk;rw)+("(k);w)dt
= Z
T
0
((k)DDD (v)v;rw)dt forallw2L 1
(0;T;W 1;1
D ());
(2.10)
and
Z
T
0 hk
;t
;wi (kv;rw)+((k)rk;rw)+("(k);w)dt
Z
T
0
((k)jDDD (v)j 2
;w)dt forallw2C(0;T;W 1;1
D ()):
(2.11)
Moreover,theinitial onditionsareattainedinthefollowingsense
(2.12) lim
t!0+
kv(t) v
0 k
2
2
+kk(t) k
0 k
1
=0:
Itisworthofnotiing thatTheorem2.1overstheinterestingase= =
for0<10=9. Inpartiular,thease(1.26) isinluded.
Wealsoremarkthat alltermsin(2.9){(2.11)aremeaningful;themostritial
term is the last term in (2.10) and the L 1
-integrability of this term leads to
the restrition (2.1)
1
. Indeed, notiing that (k)D D
D(v)v = p
(k)D D
D(v)v p
(k)
and p
(k)D D
D(v) 2 L 2
(0;T;L 2
() 33
), v 2 L 10=3
(0;T;L 10=3
() 3
) and p
(k) 2
L 3+5
3 s
(0;T;L 3+5
3 s
())weobserve,byapplyingtheHolderinequalitythat
(k)DDD(v)v2L 1
(0;T;L 1
()) () 0<
2
5 +
2
3
;
whih is therst ondition in (2.1). The seond ondition(2.1)
2
is required in
ordertoknowthat"(k)belongstoabetterspaethan L 1
(0;T;L 1
()),whihis
neededtoestablishtheompatnessofthetermsinvolving"(k).
3. Proof ofTheorem 2.1
Firstweintrodueanotationofvarioustrunatedfuntions. Foranym2R
+ ,
wedenethefuntionT
m
through
(3.1) T
m (y):=
(
y ifjyjm;
msgn(y) ifjyj>m;
andweusethesymbol
m
todenotetheprimitivefuntionto T
m ,i.e.,
(3.2)
m (y):=
Z
y
T
m ()d:
For introduedin(1.8)
2
andforarbitrarys0,wealsointroduethefuntion
s
bytheformula
(3.3)
s (y):=
Z
y
0
(1+) s 1
2
d = 2
s+1 h
(1+y) s+1
2
1 i
:
Finally,weonsiderasmoothnon-inreasingfuntionGsuhthatG(y)=1when
y2[0;1℄andG(y)=0fory2,anddeneG
m as
(3.4) G
m
(y):=G
y
m
:
Theprimitivefuntion toG
m
isthendened through
(3.5)
m (y):=
Z
y
0 G
m ()d:
The rst part of the proof takesinspiration in the method developed in [8℄.
We start with a \semi"-Galerkin approximation. Let fw
k g
1
k =1
be a basis of
W 1;2
n;div
\W 2;4
() d
,whihexistsduetotheseparabilityofthisspae. Welookfor
(v n;m
;k n;m
),where
v n;m
:=
n
X
i=1
n;m
i (t)w
i
(x); and k
n;m
0 a:e:
fulll theequations
(v n;m
;t
;w
i
) G
m (jv
n;m
j 2
)v n;m
v n;m
;rw
i
+
1
(v n;m
;w
i )
+((T
m (k
n;m
))DDD(v n;m
);DDD(w
i
))=0 foralli=1;:::;n;
(3.6)
Z
T
0 hk
n;m
;t
;wi (v n;m
k n;m
;rw)+((k n;m
)rk n;m
;rw)+("(k n;m
);w)dt
= Z
T
0 ((T
m (k
n;m
))jD D
D(v n;m
)j 2
;w)dt forallw2L 2
(0;T;W 1;2
D ());
(3.7)
aswellastheinitialonditionsoftheform
v n;m
(0;x):=v n
0 (x):=
n
X
i=1
0
i w
i with
0
i :=(v
0
;w
i );
lim
t!0 kk
n;m
(t) k n
0 k
2
2
=0withk n
0 :=k
0 1
n
; (3.8)
where 1
n
is thestandardregularizing kernelof radii 1
n and k
0
is extended by 0
outsideof. Notethatv n
0
!v
0
stronglyinL 2
()andthatk n
0
!k
0
stronglyin
1
Theexisteneofthesolutionto(3.6){(3.8)isestablishedin[8,Appendix℄and
here we merely state the result onerning large-data and long-time existene
provedtherein.
Theorem3.1. Letarbitraryn;m2N bexed. Assumethatallassumptionsof
Theorem2.1hold. Thenthereexist ( n;m
;k n;m
)solving (3.6){(3.8)suhthat
n;m
2W 1;2
(0;T) n
; (3.9)
k n;m
2L 1
(0;T;L 1
())\L 2
(0;T;W 1;2
D ());
(3.10)
k n;m
;t 2L
2
(0;T;W 1;2
0
()):
(3.11)
3.1 Limit n ! 1. Sine m 2 N is xed in this subsetion, we write (v n
;k n
)
insteadof(v n;m
;k n;m
),where(v n;m
;k n;m
)denotesasolutionto(3.6){(3.8). Our
goal is to study the onvergene in equations (3.6){(3.7) if n ! 1. We will
follow the proedure developed in [8℄ that we have to modify in order to treat
unboundedoeÆients and. Thisiswhyweinvestigatethislimitingproess
hererigorouslyandindetail.
3.1.1 Uniformestimateson v n
. Multiplying thei-th equationin(3.6)by n
i
andthensummingoveri=1;:::nweget
1
2 d
dt kv
n
k 2
2 1
2 (G
m (jv
n
j 2
)v n
;rjv n
j 2
)+
1
kv n
k 2
;2
+ Z
(T
m (k
n
))jD D
D(v n
)j 2
dx=0:
(3.12)
Next,usingthefat thatv n
n=0onanddivv n
=0in wededuethat
1
2 (G
m (jv
n
j 2
)v n
;rjv n
j 2
)= 1
2 (v
n
;r
m (jv
n
j 2
))= 1
2 (divv
n
;
m (jv
n
j 2
))=0:
Thus,weonludefrom (3.12)that
(3.13)
sup
t2(0;T) kv
n
(t)k 2
2 +2
Z
T
0 Z
(T
m (k
n
))jD D
D (v n
)j 2
dxdtkv n
0 k
2
2 C(v
0 )<1:
Itthenfollowsfrom(1.8)
1
andtheKorninequalitythat
(3.14)
Z
T
0 kv
n
(t)k 2
1;2
dtC(C 1
1
;v
0 )<1:
Moreover,usingthestandardinterpolationinequality,(3.13){(3.14)impliesthat
(3.15)
Z
T
kv n
k 10
3
10
3
dtC :
Notenallythat itfollowsfrom(3.6)and(3.13){(3.14)that
(3.16)
Z
T
0 kv
n
;t k
2
W 1;2
n;div
C(m):
3.1.2 Estimateson k n
uniform w.r.t.both m and n. Setting w:=T
1 (k
n
)
in(3.7)(notethatT
1 (k
n
)isapossibletest funtion)weobtaintheidentity
d
dt Z
1 (k
n
)dx (v n
;r
1 (k
n
))+((k n
)rk n
;T 0
1 (k
n
)rk n
)
+("(k n
);T
1 (k
n
))=((T
m (k
n
))jDDD(v n
)j 2
;T
1 (k
n
)):
(3.17)
Sinedivv n
=0inandv n
n=0on,theseondtermonthelefthandside
vanishes. Moreover,using (1.8),weseethat thethird termontheleft handside
is nonnegative. Thus, integrating (3.17) overtime, using (1.8)
3
to estimatethe
last term onthe left hand side from belowand using (3.13) to bound the right
handsideof (3.17), weonludethat
(3.18) sup
t2(0;T) k
1 (k
n
(t))k
1 +C
Z
T
0 kk
n
k +1
+1
dtC+k
1 (k
n
0 )k
1 :
Finally, usingthesimpleestimateforthegrowthof
1
wegetthat
(3.19) sup
t2(0;T) kk
n
(t)k
1 +C
Z
T
0 kk
n
k +1
+1
dtC+kk
0 k
1
<1:
Next,reallingthatk n
0a.e. inweonsiderw=(1+k n
) s
1withs>0
smallandobservethatsuhwisanadmissibletestfuntionin(3.7),inpartiular
kwk
1
2andw2L 2
(0;T;W 1;2
D
())foreahn2N. Insertingsuhwinto(3.7),
usingthefat thatdivv n
=0andtheestimatesestablishedin (3.13)and(3.19),
weget
Z
T
0 Z
(k
n
)(1+k n
) s 1
jrk n
j 2
dxdtC(s 1
):
(3.20)
Consequently, usingtheassumption(1.8)
2
andreallingthedenitionof
s ,see
(3.3),weonludethat (usingthefat that
s
haszerotraeon
D )
Z
T
0 k
s (k
n
)k 2
1;2 dtC
Z
T
0 kr
s (k
n
)k 2
2 dt
C Z
T
0 Z
(k
n
)(1+k n
) s 1
jrk n
j 2
dxdtC(s 1
):
(3.21)
Usingtherstinequalityin
(3.22)
1
((1+x) s+1
2
1) (x)(1+x) s+1
2
; (x0)
theembeddingW 1;2
D
(),!L 6
()and(3.21)
1
weobservethat
(3.23)
Z
T
0 kk
n
k s+1
3( s+1)
dtC(1+ Z
T
0 k
s (k
n
)k 2
1;2
dt)C(s 1
)foralls>0small:
Then,referringto thestandardinterpolationinequality
(3.24) kuk
s+
5
3 kuk
1 a
1 kuk
a
3( s+1)
witha:=
s+1
s+
5
3
;
appliedontok n
weonludefrom(3.19)and(3.23)that
Z
T
0 kk
n
k s+
5
3
s+
5
3 dt
Z
T
0 kk
n
k 2
3
1 kk
n
k s+1
3( s+1) dt
(3.20)
(3.23) C(s
1
)foralls>0small:
(3.25)
Notiethattheestimate(3.25)isbetterthantheseondestimatein(3.19) sine
weassumethat<+ 2
3
,see(2.1)
2
. Moreover,usingtheHolderinequalityand
theestimates (3.15)and (3.25), it iseasy to deduethat (note that thespei
valueofasmallparametersdiersfromsin (3.25))
(3.26) Z
T
0 kv
n
k n
k 10
9 3+5
+5 s
10
9 3+5
+5 s
dtC(s 1
) foralls>0small:
Conerningtheestimateonthegradientofk n
,weonsiderrstthease2[0;1℄
and we set q:=
3 3s+5
4
. Combining theestimates stated in (3.20) and (3.25),
weonludethat
Z
T
0 krk
n
k q
q C
Z
T
0 Z
(k
n
)(1+k n
) s 1
jrk n
j 2
q
2
(1+k n
) q (s+1 )
2
dxdt
C Z
T
0 Z
(k
n
)(1+k n
) s 1
jrk n
j 2
dxdt
! q
2 Z
T
0 k1+k
n
k +
5
3 s
+ 5
3 s
dt
! 2 q
2
C(s 1
):
If >1weanalwaysnd s>0small enoughso that s 1>0. Conse-
quently 2
,
Z
T
0 krk
n
k 3+5 s
4
3+5 s
4
C(s 1
) foralls>0small for2[0;1℄;
Z
T
0 krk
n
k 2
2
C for>1:
(3.27)
2
Similarly,theestimates(3.21){(3.25)togetherwith(1.8)
2
implythat
(3.28) Z
T
0 k(k
n
)rk n
k 3+5
3+4 s
3+5
3+4 s
C(s 1
) foralls>0small:
Finally,usingtheaboveestablishedestimatesitisnotdiÆulttoobserve(see[7℄
fordetails)that
(3.29)
Z
T
0 kk
n
;t k
1;r s
dtC(s 1
) foralls>0small
withrgivenby
(3.30) r:=min
3+5
3+4
; 10
9 3+5
+5
:
3.1.3 Limitn!1. Lettingn!1andusing(3.13),(3.15),(3.16),(3.25)and
(3.27),andusingtheonventionthataseletedsequeneisdenotedagainasthe
originalone,weanndasubsequenesuhthat 3
v n
*
v weakly
inL 1
(0;T;L 2
n;div );
(3.31)
v n
*v weaklyinL 2
(0;T;W 1;2
n;div )\L
10
3
(0;T;L 10
3
() 3
);
(3.32)
v n
;t
*v
;t
weaklyinL 2
(0;T;W 1;2
n;div );
(3.33)
k n
*k weaklyinL q
(0;T;W 1;q
D
()) forallq<min
3+5
4
;2
; (3.34)
k n
*k weaklyinL
!
(0;T;L
!
())forall1!<
3+5
3
; (3.35)
v n
*v weaklyinL 8
3
(0;T;L 8
3
() 3
):
(3.36)
Inaddition,usingthegeneralizedversionoftheAubin-Lionsompatnesslemma
(see[29℄)togetherwith(3.33)and(3.29)leadstotheonlusionsthat
v n
!v stronglyinL q
(0;T;L q
() 3
)forallq<
10
3
; (3.37)
v n
!v stronglyinL q
(0;T;L q
() 3
)forallq<
8
3
; (3.38)
k n
!k stronglyinL q
(0;T;L q
()) forallq<
3+5
3
; (3.39)
andonsequentlyweshowthat (atleastforasuitablesubsequene)
v n
!v a.e. in(0;T); (3.40)
k n
!k a.e. in(0;T); (3.41)
3
s (k
n
)*
s
(k) weaklyinL 2
(0;T;W 1;2
D
())foralls>0small:
(3.42)
Moreover,usingtheFatoulemma,(3.19) and(3.41)weanonludethat
(3.43) sup
t2(0;T) kk(t)k
1 C :
Conerninglimitsinthenonlineartermsin(3.6)and(3.7)wersteasilyobserve
(reallthat(T
m (k
n
))isaboundeda.e.onvergentsequeneasn!1)that
p
(T
m (k
n
))DDD(v n
)* p
(T
m
(k))DDD(v) weaklyin L 2
(0;T;L 2
() 33
);
(3.44)
(T
m (k
n
))DDD (v n
)*(T
m
(k))DDD (v) weaklyin L 2
(0;T;L 2
() 33
):
(3.45)
Next,havingtheassumptionon,see(1.8)
3
,oneanalsoobtainbyusing(3.34),
(3.39)andtheVitalitheoremthat
"(k n
)!"(k) stronglyinL q
(0;T;L q
()) forallq<
3+5
3(+1) : (3.46)
Also,itisaonsequeneof (3.28)thatthereissomeq suhthat
(k n
)rk n
*q weaklyin L q
(0;T;L q
() 3
)forallq<
3+5
3+4 : (3.47)
Inordertoidentifyq,werstremarkthatitisenoughtoshowthat
lim
n!1 Z
T
0 ((k
n
)rk n
;')dt= Z
T
0
((k)rk;')dt forall'2D((0;T)):
However,using the assumption(1.8)
2
onerning and theonvergene results
(3.39)and(3.42)weobservethat
Z
T
0 ((k
n
)rk n
;')dt= Z
T
0 ((k
n
)(1+k n
) s 1
2
| {z }
stronglyinL 2
r
s (k
n
)
| {z }
weaklyinL 2
;')dt
n!1
! Z
T
0
((k)(1+k) s 1
2
r
s
(k);')dt= Z
T
0
((k)rk;' )dt:
Consequently,q=(k)rk.
AllaboveestablishedonvergeneresultsarenotsuÆienttotakethelimitin
thenonlineartermattherighthand sideof (3.7). However,sinemisxedand
m
here. First,wenotiethatitfollowsfrom(3.31){(3.33),(3.37) and(3.45)that
Z
T
0 hv
;t
;wi G
m (jvj
2
)vv;rw
dt+ Z
T
0 ((T
m (k))D
D
D(v);D D
D(w))dt
+
1
Z
T
0 (v;w)
dt=0 forallw2L 2
(0;T;W 1;2
n;div ):
(3.48)
Moreover,using(3.31){(3.33)and(3.44)itisstandardtodedue(seeforexample
[24℄)that
v2C([0;T℄;L 2
n;div
) and v(0)=v
0 :
Next, weshall showthat wean replae the weak onvergenein (3.45) bythe
strongone. Forthispurpose, werstintegrate (3.12)w.r.t. timet 2(0;T)and
obtain
Z
T
0 k
p
(T
m (k
n
))D D
D(v n
)k 2
2 dt=
1
2 kv
n
(T)k 2
2 +
1
2 kv
n
0 k
2
2 Z
T
0
1
kv n
k 2
2;
dt
= 1
2 kv
n
(T) v(T)k 2
2 +
1
2 kv
n
0 v
0 k
2
2 Z
T
0 hv
;t
;v n
vi+hv n
;t
;vidt
Z
T
0
1
kv n
k 2
2;
dt:
Therefore,lettingn!1wededuefrom(3.32), (3.33),(3.38)and(3.8)that
limsup
n!1 Z
T
0 k
p
(T
m (k
n
))DDD(v n
)k 2
2 dt
Z
T
0 hv
;t
;vidt
Z
T
0
1
kvk 2
2;
dt:
(3.49)
Next,settingw:=v in(3.48)andusing(3.49) weobtain
limsup
n!1 Z
T
0 k
p
(T
m (k
n
))DDD(v n
)k 2
2 dt
Z
T
0 k
p
(T
m
(k))DDD(v)k 2
2 dt:
(3.50)
Consequently,as(3.44)impliesthat
Z
T
0 k
p
(T
m (k))D
D
D(v)k 2
2
dtliminf
n!1 Z
T
0 k
p
(T
m (k
n
))D D
D(v n
)k 2
2 dt (3.51)
wenallyonludethat
p
(T
m (k
n
))DDD(v n
)! p
(T
m
(k))DDD(v) stronglyinL 2
(0;T;L 2
() 33
);
(3.52)
orsayingdierently
(T (k n
))jDDD(v n
)j 2
!(T (k))jDDD(v)j 2
stronglyinL 1
(0;T;L 1
()):
(3.53)
Finally, using(3.7),(3.29)and(3.53)weobservethat
k n
;t
*k
;t
weaklyin L 1
(0;T;W 1;r s
D
()) foralls>0small;
(3.54)
with r given by (3.30). At this point, it is easy to take the limit in (3.7) and
arriveat
Z
T
0 hk
;t
;wi (vk;rw)+((k)rk;rw)+("(k);w)dt
= Z
T
0 ((T
m
(k))jDDD (v)j 2
;w)dt forallw2L 1
(0;T;W 1;1
D ()):
(3.55)
3.1.4 Attainmentofinitialdatak
0
. Werstintegrate(3.17)w.r.t.timeover
(0;t) and obtain (note that the seond termvanishes and the third and fourth
termsarenonnegative)
k
1 (k
n
(t))k
1
Z
t
0 (T
m (k
n
))jDDD (v n
)j 2
dxd+k
1 (k
n
0 )k
1 :
Next,weletn!1. Usingthenonnegativityof
1
, thepoint-wiseonvergene
ofk n
, see (3.41), andthe Fatou lemma weare ableto takelimitin the termat
theleft hand sidewith orrespondinginequalitysign. Ontheother hand,using
(3.53) we are ableto identify limit of therst term onthe right hand side and
thereforeweobtainforalmostalltimet2(0;T)
(3.56) k
1 (k(t))k
1
Z
t
0 (T
m (k))jD
D
D(v)j 2
dxd +k
1 (k
0 )k
1
;
whih impliesthat
(3.57) limsup
t!0+
k
1 (k(t))k
1
k
1 (k
0 )k
1 :
Next,settingin(3.55)w:=T
1 (k
n
)(
1 (k
n
)) 1
2
'
[0;t℄
where'2D(),'0,we
obtain(notethat wisanadmissibletestfuntion)
2(
p
1 (k
n
(t)) ;') 2 Z
t
0 (v
n p
1 (k
n
) ;r')d
+ Z
t
0 Z
(k
n
)
T 0
1 (k
n
)(
1 (k
n
)) 1
2 1
2 (T
1 (k
n
)) 2
(
1 (k
n
)) 3
2
jrk n
j 2
'dxd
+ Z
t
0 ((k
n
)T
1 (k
n
)(
1 (k
n
)) 1
2
rk n
;r')d
+ Z
t
0 ("(k
n
);T
1 (k
n
)(
1 (k
n
)) 1
2
')d
= Z
t
((T
m (k
n
))jD D
D (v n
)j 2
;T
1 (k
n
)(
1 (k
n
)) 1
2
')d+2(
q
1 (k
n
0 );'):
Observing that the integrand in the third integral is non-positive and the rst
integralon the right hand side is nonnegative, we anneglet both of them by
replaingtheequalitysignbytheinequality 4
. Then weletn!1. Applyingall
onvergeneresultsestablishedabove,it isstandardtoonludethat foralmost
alltimest2(0;T)
( p
1
(k(t)) ;') Z
t
0 (v
p
1
(k);r')d + 1
2 Z
t
0 ((k)T
1 (k)(
1 (k))
1
2
rk;r')d
+ 1
2 Z
t
0
("(k);T
1 (k)(
1 (k))
1
2
')d 2(
p
1 (k
0 ) ;'):
Finally, lettingt!0
+
weobservethat
liminf
t!0+
( p
1
(k(t)) ;')( p
1 (k
0
);') forall'2D(); '0:
Thus, using the density argument, (3.43) and the fat that
1
(k) has at most
lineargrowthink, wenally deduethat
(3.58)
liminf
t!0+
( p
1
(k(t)) ;')( p
1 (k
0
) ;') forall'2L 2
(); '0a.e.in :
Consequently,itistheneasyto observethat
lim
t!0+
k p
1 (k(t))
p
1 (k
0 )k
2
2
= lim
t!0+
k
1 (k(t))k
1 +k
1 (k
0 )k
1 2(
p
1 (k(t));
p
1 (k
0 ))
(3.57);(3.58)
k
1 (k
0 )k
1 +k
1 (k
0 )k
1 2(
p
1 (k
0 );
p
1 (k
0 ) )=0;
whih nallyleadsto
(3.59) lim
t!0+
kk(t) k
0 k
1
=0:
3.2 Limit m!1. Intheprevioussubsetion,weestablishedtheexisteneof
(v m
;k m
)fullling,foreverym2Nxed,theweakformulations(3.48)and(3.55).
Beforesummarizingtheestimatesfor(v m
;k m
)thatare uniformwithrespetto
m, we takethe advantageof onsideredslip boundaryonditions (0<1in
(1.5))andintroduetheintegrablepressure.
Foranyw2W 1;2
n
weobservethattheHelmholtzdeompositionw=w
div + r'
with'havingzeromeanoverandsolving '=divwinandhomogeneous
Neumannproblemonisompatiblewith(1.5)for0<1. Indeed,notiing
4
Atthislevelofapproximation,weevendonotneedthissimpliationbeauseweareable
toidentifythe limitoforrespondingquantities. However,it willnot betheaseinthenal
that
(3.60)
Z
T
0 hv
m
;t
;widt= Z
T
0 hv
m
;t
;w
div idt;
weanextendthedenitiondomainforv m
;t
andobservethatv m
;t 2L
2
(0;T;W 1;2
n ).
Letusintroduep m
asthesolutionofthefollowingproblem
(p m
;4')=((T
m (k
m
))D D
D(v m
);r (2)
')+
1
(v m
;r')
(G
m (jvj
2
)v m
v m
;r 2
') forall'2W 2;2
(); r'2W 1;2
n : (3.61)
Taking w2L 2
(0;T;W 1;2
n
)arbitrarily,applying theHelmholtzdeompositionon
suhw,takingthesumof(3.48)withthetestfuntionw
div
and(3.61)andusing
(3.60)weobtainthefollowingidentity
Z
T
0 hv
m
;t
;wi G
m (jv
m
j 2
)v m
v m
;rw
+((T
m (k
m
))DDD(v m
);DDD(w))dt
+
1
Z
T
0 (v
m
;w)
dt= Z
T
0 (p
m
;divw)dtforallw2L 2
(0;T;W 1;2
n ):
(3.62)
It is easyto hek from (3.62) that suh normalizedp m
is uniquelydetermined
byagivensolution(v n
;k n
).
We also reall that the m-approximation satises (3.55) that we repeat for
brevity. It readsas
Z
T
0 hk
m
;t
;wi (v m
k m
;rw)+((k m
)rk m
;rw)+("(k m
);w)dt
= Z
T
0 ((T
m (k
m
))jDDD (v m
)j 2
;w)dt forallw2L 1
(0;T;W 1;1
D ()):
(3.63)
Next,werealltheuniformboundon(v m
;p m
)andderivetheuniformbound
onthepressurep m
thatwill beneededinwhat follows. First,referringto lower
semiontinuityofthenormsandtheFatoulemmawegetfrom(3.13)and(3.19)
sup
t2(0;T) kv
m
(t)k 2
2 +kk
m
(t)k
1
+ Z
T
0 Z
(T
m (k
m
))jD D
D (v m
)j 2
dxdt
+ Z
T
kk m
k +1
+1
dtC : (3.64)
Moreover,using (3.64) and thestandardembedding ofSobolev funtions tothe
spaeoftraestogetherwith thestandardinterpolationinequalitiesoneande-
due,see[9,Lemma1.12℄fordetails,that
(3.65)
Z
T
0 Z
jv
m
j 8
3
dSdt+ Z
T
0 kv
m
k 10
3
10
3
dtC :
Inaddition, referringagain to thelower semiontinuityof the norms we obtain
from(3.21)and(3.25){(3.28)
Z
T
0 k
s (k
m
)k 2
1;2 +kv
m
k m
k 10
9 3+5
+5 s
10
9 3+5
+5 s
+kk m
k +
5
3 s
+ 5
3 s
+krk m
k min(2;
3+5
4 ) s
min(2;
3+5
4 ) s
dt
+ Z
T
0 k(k
m
)rk m
k 3+5
3+4 s
3+5
3+4 s
dtC(s 1
) foralls>0small:
(3.66)
Next,observingthat
(T
m (k
m
))D D
D(v m
)= p
(T
m (k
m
))D D
D(v m
) p
(T
m (k
m
));
andreallingthataordingto(3.64) p
(T
m (k
m
))DDD(v m
)isuniformlyboundedin
L 2
(0;T;L 2
() 33
) and aording to (3.66) p
(T
m (k
m
)) , whih grows as
(1+k m
)
=2
, is bounded uniformly in L 2
(+
5
3 s)
(0;T;L 2
(+
5
3 s)
()), we on-
ludethat
Z
T
0 k(T
m (k
m
))DDD (v m
)k q0 s
q0 s
dtC(s 1
) foralls>0small;
withq
0 :=
2(3+5)
3+3+5 : (3.67)
Similarly,inorporatingalsotheseondestimatein(3.65), weobservethat
Z
T
0 k(T
m (k
m
))DDD (v m
)v m
k w0 s
w0 s
dtC(s 1
) foralls>0small;
withw
0 :=
10(3+5)
15+24+40 : (3.68)
Notethattheassumption(2.1)
1
guaranteesthatw
0
>1.
Atthis point, weandedue from (3.61)the estimatesfor fp m
gthat will be
uniform with respet to m. We onsider ' with zero mean over solvingthe
homogeneous Neumann problem 4' = jp m
j q 2
p m
1
jj R
jp
m
j q 2
p m
dx and
m
weobtain
Z
T
0 kp
m
k z
0 s
z0 s
dtC(s 1
) foralls>0small;
withz
0 :=min
5
3
;
2(3+5)
3+3+5
: (3.69)
Finally,using equation(3.62)andtheaboveestimatesweonludethat
(3.70)
Z
T
0 kv
m
;t k
z0 s
W 1;z
0 s
n
dtC(s 1
) foralls>0small:
Similarlyasintheprevioussubsetion,using(3.55),(3.64)and(3.66)wededue
that
(3.71) Z
T
0 kk
m
;t k
W 1;r s
D
dtC(s 1
) foralls>0smallandrdened in(3.30):
Havingalluniformestimates(3.64),(3.65), (3.66),(3.69),(3.70) and(3.71),and
using the generalized version of the Aubin-Lions ompatness lemma we nd
subsequenesthat weagainlabelinthesamewayastheoriginal sequenessuh
that(weusetheonventionthat s>0issmallbut arbitrary)
v m
*
v weakly
in L 1
(0;T;L 2
n;div );
(3.72)
v m
*v weaklyinL 2
(0;T;W 1;2
n;div )\L
10
3
(0;T;L 10
3
() 3
);
(3.73)
v m
;t
*v
;t
weaklyinL z0 s
(0;T;W 1;z0 s
n
)forz
0
from(3.69);
(3.74)
p m
*p weaklyinL z
0 s
(0;T;L z
0 s
())forz
0
from(3.69);
(3.75)
k m
*k weaklyinL q
(0;T;W 1;q
D
()) forallq<min(2;
3+5
4 );
(3.76)
k m
;t
*
k
;t
weakly
in M(0;T;W 1;r s
D
())forrfrom (3.30);
(3.77)
v m
*v weaklyinL 8
3
(0;T;L 8
3
() 3
);
(3.78)
v m
!v stronglyin L q
(0;T;L q
() 3
)forallq<
10
3
; (3.79)
v m
!v stronglyin L q
(0;T;L q
() 3
)forallq<
8
3
; (3.80)
k m
!k stronglyin L q
(0;T;L q
()) forallq<
3+5
3
; (3.81)
v m
!v a.e. in(0;T);
(3.82)
k m
!k a.e. in(0;T);
(3.83)
s (k
m
)*
s
(k) weaklyinL 2
(0;T;W 1;2
()):
(3.84)
Moreover,usingthesameproedureasintheprevioussubsetionweanonlude
that
(3.85) sup
t2(0;T) kk(t)k
1 C :
Similarly,asintheprevioussubsetion,see(3.47),weanverifythat
(k m
)rk m
*(k)rk weakly inL q
(0;T;L q
() 3
)forallq<
3+5
3+4 : (3.86)
Moreover,itfollowsfrom(3.64)thatthereisanSSS2L 2
(0;T;L 2
() 33
)suhthat
p
(T
m (k
m
))DDD(v m
)*SSS weaklyinL 2
(0;T;L 2
() 33
):
(3.87)
To identify SSS werst observethat (3.83), the growthassumption (1.8)
1 , (3.66)
andVitali'stheoremimplythat
(T
m (k
m
))!(k) stronglyinL q
(0;T;L q
())forallq<
3+5
3 : (3.88)
Sinetheassumption (2.1)guaranteesthat 3+5
3
>2,itfollowsfrom (3.73)and
(3.88)that
(3.89) S
S
S= p
(k)D D
D(v)a.e.in (0;T):
Similarly,using (3.67),weandeduethat
(3.90) (T
m (k
m
))D D
D (v m
)*S S
S
2
weaklyin L q
(0;T;L q
() 33
) forall q<q
0 :
To identify S S
S
2
it is then enough to ombine (3.87), (3.89) and (3.88) to obtain
that
S S
S
2
=(k)D D
D (v)a.e.in (0;T):
At this point, we an omplete the proof of Theorem 2.1. First note that
(3.72){(3.88)impliesthatthetriple(v;k;p)satises(2.2){(2.7). Next,theabove
establishedonvergenes(3.72){(3.90)suÆetoprove(2.9)bylettingm!1in
(3.62). Similarly,lettingm!1in(3.63)wededue(2.11),usingtheweaklower
semiontinuityofthelast termin (3.63).
Then,settingin(3.62) w:=v m
wwitharbitraryw2L 1
(0;T;W 1;1
D
())and
addingtheresultto(3.63) wearriveat
Z
T
0 hE
m
;t
;wi (v m
(p m
+k m
);rw) (G
m (jv
m
j 2
)v m
v m
;r(v m
w))dt
+ Z
T
((T
m (k
m
))D D
D(v m
)v m
;rw)+((k m
)rk m
;rw)+("(k m
);w)dt=0;
(3.91)
whereweset
E m
:=
1
2 jv
m
j 2
+k m
:
Notiing that thethird term in (3.91)an besimplied by using integrationby
partsandalsothefatthat divv m
=0in,weget
(G
m (jv
m
j 2
)v m
v m
;r(v m
w))
= 1
2 (wv
m
;r
m (jv
m
j 2
)+(G
m (jv
m
j 2
)jv m
j 2
v m
;rw))
=((G
m (jv
m
j 2
)jv m
j 2
1
2 m
(jv m
j 2
))v m
;rw):
From (3.91) we an obtain the estimate on the time derivative of E m
and by
seletingasubsequeneobservethat
(3.92) E m
;t
*E
;t
weaklyin L q
(0;T;W 1;q
D
()); where E:=
1
2 jvj
2
+k;
forall1<q<min n
10
9
;w
0
; 3+5
3+4 o
;w
0
isintroduedin(3.68).
Finally,setting m!1in(3.91) itisstandardtoobtain(2.10).
3.2.1 Attainment of initial ondition. We aim to prove (2.12). The rst
part, i.e., the attainment of theinitial veloity v
0
is standard and we referthe
reader to [24℄. Toestablish theseond partweuse the similar proedure as in
the previous subsetion with only one essential hange. First part follows the
proedurefrom thepreeding subsetionandwededuethat
(3.93)
liminf
t!0+
( p
1
(k(t));')( p
1 (k
0
);')forall'2L 2
(); '0a.e.in :
Tonishtheproofof (2.12)itisthenenoughto obtain
(3.94) limsup
t!0+
k
1 (k(t))k
1
k
1 (k
0 )k
1
andthesameargumentsasinpreedingsubsetionthenleadsto(2.12). Toprove
(3.94)wehavetoproeeddierently. Rewriting(3.56)againas
(3.95) k
1 (k
m
(t))k
1
Z
t
0 Z
(T
m (k
m
))jD D
D (v m
)j 2
dxdt+k
1 (k
0 )k
1
;
weanreplae thersttermon therighthand side byusing w:=v n
[0;t℄
asa
testfuntionin (3.62). Hene,after negletingtheboundaryintegral,beauseof
orretsign,weget
(3.96) k
1 (k
m
(t))k
1
kv
m
(t)k 2
+kv
0 k
2
+k
1 (k
0 )k
1 :
Therefore,passingtothelimitw.r.t.m wegetafterusing theFatou lemmaand
weaklowersemiontinuityofnormthat
(3.97) k
1 (k(t))k
1
kv(t)k 2
2 +kv
0 k
2
2 +k
1 (k
0 )k
1 :
Consequently, using also therst partof (2.12) thenleadsto (3.94). Thus, the
proofisomplete.
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MiroslavBulek, JosefMalek:
CharlesUniversity,FaultyofMathematisandPhysis,Mathematial
Institute,
Sokolovsk
a 83,18675Prague 8,CzehRepubli
E-mail: mbul8060karlin.m.uni.z
malekkarlin.m.uni.z
RogerLewandowski:
IRMAR,UMR CNRS6625,
Universit
edeRennes1,CampusBeaulieu,
35042Rennesedex,Frane
E-mail: Roger.Lewandowskiuniv-rennes1.fr