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(1)

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

(2)

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

(3)

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):

(4)

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

;

(5)

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

(6)

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

(7)

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

(8)

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.

(9)

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)

(10)

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:

(11)

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

(12)

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 :

(13)

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)

(14)

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

(15)

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

(16)

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

(17)

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)

(18)

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 );'):

(19)

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

(20)

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)

(21)

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

(22)

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)

(23)

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)

(24)

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 :

(25)

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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formationandvariability,J.Phys.Oeanogr.28(1998),1089{1106.

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

参照

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