; TeX output 2000.12.28:0949yi"u src:27Ana.TEXXQ cmr12Svreuscilia?steuZagrebuPriroSdoslorvno-matematisckifakultetT7 src:28Ana.TEXMatematirsckioSdjelzJ src:34Ana.TEXN cmbx12AnaVukelic,dipl.@inz.mat. & src:38Ana.TEXNff cmbx12Rubneffimje%HsovitezadaYcezakvazinewtonovske uideJʙ src:44Ana.TEXMagistarskiradd" src:50Ana.TEXVVoSditelj:8doc.Wdr.sc.EduardMarua?sirsc-PalokXag4 src:56Ana.TEXZagreb,2000.*yi* src:61Ana.TEXNH cmbx12Sadr8RzajGj0 src:2Ana.toc` 1Fizik@alnainterpretacijakvazinewtonovskog uidaK1]F> src:3Ana.tocY1.1` Osnorvnede nicije8č. C........................֚1]F> src:4Ana.tocY1.2` KrivuljatokXa. C..........................֚1]F> src:5Ana.tocY1.3` Prolimernetalineiotopine-. C...................֚3]F> src:6Ana.tocY1.4` KvXanrtitativniopispSonaa?sanjatokaC. C...............֚4j0 src:7Ana.toc` 2Stacionarnikv@azinewtonovskiStokesovsustav_5]F> src:8Ana.tocY2.1` FVunkrcionalniprostoric. C......................֚5]F> src:9Ana.tocY2.2` De nicijainekXasvrojstvag cmmi12rS-StokresovogoperatoraY. C.......֚9]F> src:10Ana.tocY2.3` EgzistencijaGijedinstvrenostrjea?senjakvXazinewtonovskogStoke-KJsorvogsustavXaX. C..........................150 src:11Ana.toc` 3Stacionarnikv@azinewtonovskiNavier-Stokesovsustav,;17]F> src:12Ana.tocY3.1` De niranjeproblemaipretpSostarvkeOq. C..............17]F> src:13Ana.tocY3.2` Egzistencijaaproksimativnogrjea?senja. C.............21]F> src:14Ana.tocY3.3` RezultatiegzistencijevrloslabSogrjea?senja. C...........29]F> src:15Ana.tocY3.4` DokXazjedinstvrenostiǍ. C......................340 src:16Ana.toc` 4Navier-Stokesov(sustavsviskoznoIfscukojaovisiodeformacijil36]F> src:17Ana.tocY4.1` Linearnislurscaj׍. C.........................36]F> src:18Ana.tocY4.2` Jedanteoremokrompaktnosti;. C.................45]F> src:19Ana.tocY4.3` Klasirscnirezultatizainicijalniproblem. C............46^:iޠyxiii0 src:20Ana.toc` AMeto`damonotonihoperatora53]F> src:21Ana.tocYA.1` MonotoniopSeratorik. C.......................53]F> src:22Ana.tocYA.2` Stacionarniproblem 8. C......................53]F> src:23Ana.tocYA.3` PseudomonotoniopSeratori. C...................56j0 src:24Ana.toc` BTop`oloIfskistupanjinjegov@aprimjena58]F> src:25Ana.tocYB.1` OsnorvnasvojstvXar. C........................58]F> src:26Ana.tocYB.2` Konstrukrcijastupnja5. C......................60ܠyi* src:63Ana.TEXPredgo8vor; src:64Ana.TEXUorvomraduprourscavXamoXJcistiviskoznitokkvXazinewtonovskog uidasvi-skroznoa?sscukojaovisiodeformaciji. src:68Ana.TEXUprvromUpSoglavljuopisujemomoSdelezakvXazinewtonovske uide.oUinszenjer-skrojliteraturipSostojinizmoSdeladokrojihsedoa?slonaosnorvuekspSerimen-talnihpSodatakXa.TKakromatematisckXakompleksnostjednadszbirastesbrojemparametara,Kuinrszenjerskimprorascunimanajscea?scesekoristejednostavnimo-delizakronapSotencijeiCarreauovXazakona. src:75Ana.TEXUn{drugomnpSoglarvljuprouscavXamotokstacionarnogkvazinewtonorvskogn u-idaopisanogStokresovimjednadszbamasnelinearnomviskoznoa?sscukojajedanaTdzakronompSotencijeiCarreauovimzakonom.vUTIprvojtosckide niramofunkrcionalne\prostore,kdokudrugojtosckide niramorS-StokesovopSeratorzakvXazinewtonorvske& uideidokazujemonekasvrojstvaizkrojihzsceslijeditimo-notonostr>2n=(n)+2)adobivXamoegzistencijubaremjednogvrloslabSogrjea?senja.3xLStorvise,za?r`3n=(n%+2)dokXazujemodasvaslabarjea?senjalerszeukugliizWOƿ1;r0HsapSolumjerommanjimodkritirscnevrijednosti. src:99Ana.TEXUPscetvrtom84pSoglarvljuprouscavXamoevolucijskiNavier-Stokesovsustavsvi-skroznoa?sscukojajedanazakonompSotencijeiCarreauovimzakonom.Upr-vrojtosckidokXazujemoegzistencijuiregularnostrjea?senjalinearnogsluscajakroristesciYGaljorkinovuproSceduru,autrescojtosckidokXazujemoegzistencijurjea?senjazanelinearnislurscajpSodpretpostavkomdajer>UR1+2n=(n+2).iiiڠyivisrc:107Ana.TEXZadnjaYdvXapSoglarvljasudodatciukrojimaiznosimonekevXasznerezultatekojisukroria?steniuprethoSdnimpoglarvljima.src:110Ana.TEXNaPkrajukroristimprilikudasenajtoplijezahrvXalimdoSc.dr.sc. jEduarduMarua?sirscu-PalokikojimijesvesrdnopSomogaodanascinimovuradnju.)]yiosrc:115Ana.TEXP8oglavlje T{16oFizikqalna T{in8terpretacijakvqazinewtono8vskog T{ uidaBo*NG cmbx121.1KGOsnouvnezde nicije0src:118Ana.TEXWx+@ cmti12Cisti2viskoznitok)se~de nirakXaoproScesukromesesvaupSotrebljenamehanirsckaenergijaku uidunepSorvratnodifuziralakXaotoplinaprekomolekularnogtrenja.OvXaj(?proScesjepoznatkXaodifuzijakviskozneenerffgije.?ܟxCisti(?viskroznitokjeumnogim:slurscajevimadobraaproksimacijazarazrjeUЉFfdenepSolimerneotopine,̷irscestozakroncentriraneotopineitalinegdjeseotpSor uidanemijenjavreomabrzo;tojekXadaimamopriblirsznuravnotezutokXa.Nosrc:127Ana.TEXViskroznost uidamjeriopiranje uidatoku.$KvXantitativnoonasede nirauterminimadvXajuosnorvnihparametara:$ smi3Lcnoffg naprezanjaideformacije,tj.8simetriziranogdijelagradijenrtabrzinee(u)UR=Fu1z@2 (ru+ru2tʹ).,1.2KGKrivuljaztokasrc:132Ana.TEXKadavrescinu uidapSodvrgnemokronstantnojdeformaciji(ilikonstantnomsmirscnom>naprezanju)prikonstantnojtempSeraturi,T odgovXarajusca>vrijednostsmirscnognaprezanja(ilideformacije)semoszeizmjeriti.TVajednoznascnarela-cija0WizmeUЉFfdusmirscnognaprezanjaideformacijeprikonstatnojtempSeraturijepSoznatakXaokrivulja35toka.Nosrc:139Ana.TEXNewtonorv%RzakonoviskoznostikXaszedajesmiscnonaprezanjepropSorcionalnodeformaciji,akronstantapropSorcionalnostipostajeviskroznostn9:*~yo=URn9e(u):(1.1)1+ay 2isrc:145Ana.TEXFluidikrojizadovoljavXajuovupretpSostavkusunewtonovski uidi.4OvXapret-pSostarvkXadostadobroopisujenekenepSolimerske uide,LLkXaofstosuplinovi,vroSdaitoluol.OvXakavoblikpSonaa?sanjatokXamoszeseoscekivXatizamale,!rela-tivno9simetrirscnemolekule,gdjesustrukturai/iliorijentacijanepromjenljivesainrtezitetomdeformacije.src:147Ana.TEXAritmetirscki[JkrivuljatokXazanewtonovski uidjepravXackrozishoSdia?stesanagibSomAz(SlikXa1.1).ĪKakrop^ie(u)ųscestovarirajupSojakrozsirokojAskXali,uobirscajeno٠y 4istanjima, zamra?senosti,|diimajunajvresci, otpSorutoku. KadasedeformacijapSorvescavXaY,umolekulepostajunezamra?seneirasporeUЉFfdeneupoljudeformacije,reducirajursci0*svoj0*otpSorumeUЉFfdusobnomodmicanju. fKodstrogedeformacije,oni<sceSbitivreomalijepSokompletnorazmra?seneiraspSoreUЉFfdene,idosegnuti<sceminimalniotpSorutoku.8TVojeilustriranou(Slici1.2)., 1.4KGKvanutitativnizopisp=onaYNVsanjatokaύsrc:200Ana.TEXDa9biopisalianalitirsckikvXazinewtonovski uid,moramomatematisckiopisativrezuizmeUЉFfduW,n9,ie(u),kXao+stojenaprarvljenouNewtonovomzakonuzanewtonorvske uide.src:202Ana.TEXTVradicionalniinrszenjerskimoSdelzacistiviskroznikvXazinewtonovskitokjeta-krozvXanizakon35pffotencije尹:c'o=URje(u)j rUR2 uidjedilatoran.src:212Ana.TEXAkrodizrascunamoovisnostviskoznostiodeformacijiza uidkojiispunjavXazakronpSotencijedobivXamo:n9(e(u))",=URje(u)j rUR1 UZ0;>0iiQ1035takodajeN덒AhjujLr,p( )URcjrujLr,p( )4T;8u2Wr:#bsrc:316Ana.TEXDokazj: #PretpSostarvimosuprotno:f(8ko2URN)(9u kx2Wƹ)t.d.|ju k#jLr,p( )kgjru kjLr,p( )4T:src:320Ana.TEXMorszemojoa?suzetidajeju k#jLr,p( )=UR1;(2.2)}@src:324Ana.TEXpaVslijedidajejru2k#jLr,p( )Fuz1zk !g0kXadakΝ!1,pafu2k#gimakronvergentan*ӍpSodnizTOuL2rb( )injegaoznarscimosafu2k60UYgiu2k60>!u20 SuL2r( ). uSa(2.2)dobivXamodajeju20jLr,p( )=UR1.}src:331Ana.TEXMorszemozapisati΍v訟甆Z}T _ 6u k60ō @ [z, ΍@xi=!UR甆Z _ u 0ō.@7[z, ΍@xi;jza,nUR2DUV( ):"xsrc:333Ana.TEXKakroFuO1۟zk jw@x33z 萟@xx8:iNjLr,p( )!UR0slijedidajeFu@xu-:0۟z ꍑ@xx8:i=0,paondau20(x)=CF(s.s.).׍src:338Ana.TEXZbSogu20V2URWnvrijedidaje ̿0(u20)=0napaslijedidajeu20V=0.src:341Ana.TEXTVonemorszebitijerjeju20jLr,p( )=UR1,pasmodoa?slidokontradikcije.,;;Nap`omena2.1ysrc:345Ana.TEXKonstanta35csemoffzeizabratitakodabude΍ccUR=ō d[z ΍rS1=rm;;_}src:346Ana.TEXgdje35jedUR=diam . Prop`ozicija2.3(Necasov@anejednakost) src:349Ana.TEXNekaG"je URR2nrotvorffen,vZome-#!ЉFdenskupslokalnoLipschitzovimrubffom.OA2kodistribucijapUR2DUVl0( )imasveprve%derivacijewV@xpXz 萟@xx8:i@izWƟ21;rm( )ondajeip2L2rb( )%tevrijeffdidapostojiCܞ( )UR>035neffovisnaoptakodaN덒m"jpjLr,p( )j`$O+msbm6R!jURCܞ( )jrpjW.:1;r( )n+v:src:357Ana.TEXDokXazpropSozicijesemorszenasciu[11 ].my 8iTeorem2.1fsrc:359Ana.TEXNeka35je ome!ЉFden,otvorffenLipschitzovskup.fiTadajeVr紹=URfu2WOƿ1;r0 Р( ) nP;div'u=0g:%src:362Ana.TEXDokazj: #De niramoprostorVp2 ƹ=URfu2WOƿ1;r0 Р( )2nP;div'u=0g.src:366Ana.TEXPrvropSokXazujemodajeVrURVp2\t.src:368Ana.TEXAkroNu2Vrj|,ftadapSostojinizfumgV,ft.d.cum I!uuWƟ21;r Р( )2nP.cKakrojediv/'um Z=UR0dobivXamo[L0UR=hdiv'umji=humjri! hu;ri=hdivuji;src:372Ana.TEXpaslijedidajedivuUR=0.src:374Ana.TEXZa ̿0(umĹ)2L(WƟ21;r Р( )2nP;WƟ211=rÎ;r()2n)imamodaakro ̿0(umĹ)=0slijedidaje ̿0(u)UR=0pau2Vp2\t.src:379Ana.TEXSadanamostajejoa?sdokXazatidajeVp2 URVrI.src:381Ana.TEXTVrebamo QpSokXazatidajeVp2 eŹzatvrorenipotprostorodWOƿ1;r0 Р( )2nP.AkrojefumgC-nizuVp2\t,slijedidajeonkronvergentanuWOƿ1;r0 Р( )2nJilimesmrujeiuWOƿ1;r0( )2niuVp2\t.8Kakroum Z2URVp2 Gslijedidajedivum=UR0paondaidivu=0.src:388Ana.TEXNekXaYjeLneprekidnilinearnifunkrcionalnaVp2\t. Onsemoszeproa?siritidolinearnogfunkrcionalanaWOƿ1;r0 Р( )2nP.8TVadaje#0&"LUR=(l̿1;:::ʜ;lnP);ꦹza*li,2WƟ 1;rm( )i hLjvn9i=: n #X ㇍ i=1xhlijvidi꨹za,vË2WOƿ1;r0 Р( ) n:%src:393Ana.TEXAkro4ojehLjvn9i= #0za8vA 2VrsMpremaPropSoziciji2.1slijedidapostojiskXalarnafunkrcijapUR2L2rb( ),t.d.8jeli,=w_@xpz 萟@xx8:iH.src:399Ana.TEXTVadazaw2URVp2 Gimamo!X/hLjwRi=: n #X ㇍ i=1xhōIs@p33[z, ΍@xiSjwidi= 0,nX ㇍ti=1hpjō33@wi33[z ΍q@xi si=hpjdiv'wiUR=0;$lsrc:401Ana.TEXpaslijedidajeVrmMgustuVp2\t,alikXakrosuVp2 7_iVrzatvroreniskupSovislijedidajeVr紹=URVp2\t.EU;;src:405Ana.TEXGeneralizacijaorezultataorveotosckezasluscajprostoraWƟ2m;r`;m7>1;morszesenarsciu[1]. {y 9i2.2KGDe nicijaQinekasvuozjstva,gG cmmi12rxQ-StokuesovogQop=e-KGratorau􍍍De nicija2.4rQsrc:410Ana.TEXrS-Stokesov35opfferatorAUR:Vr!V2p-:0RAr0Sde nirffamos4rA(u)UR=div'fn9(e(u))e(u)g;src:411Ana.TEXgdje^jen9(e(u))danojeffdnadzbom^(1.4)zazakonpffotencijeijednadzbom(1.5)za35Carrffeauovzakon.荑src:415Ana.TEXMiscemoprourcavXatislucajkXadaje1 UZ=UR0.эsrc:417Ana.TEXNekXajefr紹:URR2n!R2nP;r>1,funkrcijade niranas^frb(x)UR= n9(ڹ+jxj 2)33r1233\) 02x;src:419Ana.TEXpajesadaopSeratorAde niransMA(u)UR=div'ffrb(e(u)g;(2.3)src:424Ana.TEXgdjeje c=*; \=0;=1zazakronpSotencije,+a c=*̿0; \=1;=zaCarreauorvzakon.эsrc:427Ana.TEXImamosljedersceoScjenezafunkcijufrb:"Lema2.2ZQsrc:429Ana.TEXNeka35sux;hUR2R2nP.fiA2ko35jer>UR2tadajeBjfrb(x+h)fr(x)jURC ܞ2ڍ;rÎ;nz(ڹ+jxj rUR1je^@wC ܞ3ڍ;n;r ^jfrb(x+h)fr(x)jjhjUR(fr(x+h)fr(x);h);(2.6)src:440Ana.TEXa35akoje1UR2ifr2Cܞ21(R2nP)imamonō@fi[z\ ΍@xj1%(x)UR=( xidxj+jxj 2ijJ)jxj 2:晍src:457Ana.TEXZako2URR2n imamoTVarylorovuformulu!Tiۢ(fG(x+h)f(x);kg)UR= knX ㇍Si=1(fidڹ(x+h)fi(x))ki.Q= knURX ㇍Si=1,nX ㇍jv=1)甆ZM6100ō>"@fi=5[z\ ΍@xjQĹ(x+th)hjf kiddt: src:460Ana.TEXStarvimodajex2t=URx+th꨹paimamo?d(fG(x+h)f(x);kg)UR= knX ㇍Si=1,nX ㇍jv=1)甆ZM6100:( x tڍix tڍj+jx tj 2ijJ)jx tj 2hjf kiddt0D[i= knURX ㇍Si=1甆ZM"1UT0)jjx tj hidkidt+,nX ㇍ti=1nX ㇍Igjv=1)UT甆ZM5UV100<X x tڍix tڍjf jx tj 2hjkiddt*gY=UR(h;kg)甆ZM10jx tj dt+甆ZM1 UT0j jx tj 2(x t;h)(x t;kg)dt:(2.9):؍src:466Ana.TEXZako=URhdobivXamoMހ(fG(x+h)f(x);h)URjhj 2甆ZM1 j0jx tj dt:(2.10)"x8src:470Ana.TEXOstajeRnamjoa?soScjenitiizrazUQR R1 0jx2tj2 dt.PZa >UR0postojeC̿1;n; $>UR0;C̿2;n; >0takrodajelbC̿1;n; x(nX ㇍i=1UVjxidj)  knURX ㇍Si=1jxij URC̿2;n; x(nX ㇍i=1UVjxij) <;#src:473Ana.TEXpaimamo9>甆ZME@1@60LLBjx tj dtUR=甆ZMUT1 0X(nX ㇍i=1UX(xi+thidڹ) 2) =2udtCܞ1v2;n;33 33s^\)ӻ2[甆ZM#]100n*_X ㇍,e`i=1=jxi+thidj dt.mURCܞ1v2;n;33 33s^\)ӻ2n[X ㇍\i=1+0jxidj 甆ZM1A0j1+tō33hi33[z (d ΍ xi j dt \y@11iRUURCܞ1v2;n;33 33s^\)ӻ2]C̿1;n;33 33s^\)ӻ2jxj -inf92#ppmsbm8R\甆ZM&\1!d0-j1+tj dt;鍑src:477Ana.TEXgdjejefunkrcijaUR:R!R:!UQR US1 0Uj1+tj2 dt驹neprekidna,strogopSozitivna,iideu+1kXadajjideu+1.KPratakokXakosukonstanteCrÎ;n,strogopSozitivneimamoRiY7甆ZMuY91p0|;jx tj dtURCܞ1v2;n;33 33s^\)ӻ2]C̿1;n;33 33s^\)ӻ2CrÎ;n Bjxj Jō1[z ΍2 C ܞ1ڍ0;rÎ;n:jxj ;(2.11)src:485Ana.TEXpaiz(2.10)i(2.11)dobivXamonejednakrost(2.4).src:488Ana.TEXIz(2.9)dobivXamodavrijedi!ҍi6`j(fG(x+h)f(x);kgjURjhjjkj(1+ )甆ZM10jx tj dt;BVsrc:490Ana.TEXpaڍwjfG(x+h)f(x)jUR(1+ )jhj甆ZM10jx tj dt:src:492Ana.TEXKakro]jejx+thjURjxj+jhjURjxj+jx+x+hjUR2jxj+jx+hj]dobivXamoLVjfrb(x+h)fr(x)jUR2 (1+ )(jxj rUR2.src:499Ana.TEXSadaMnamjoa?sostajepSokXazatidaje(2.6)ispunjenoza1|22ifrif2G-:0RAr =f2G-:0RArifr =Id.Praiz(2.6)zax;hUR2R2n imamoL]9qC ܞ3 @Ս0;rUR0C2ܞ4RA0;rÎ;n+1G8jyI{xj-:2JizL9fꍿ(jyI{+vj Կ+jx+vj )3za6 h<UR0; src:543Ana.TEXpasmodokXazali(2.4)i(2.7)zafrb.src:546Ana.TEXSadaimamoEojgrb(y+vn9)gr(x+vn9)j 2V=URjjy+vj (y+v)jx+vj (x+v)j 2\=URjjy+vn9j yjx+vn9j xj 2j+jvj 2jjy+vj ?jx+vj j 2GURjjy+vn9j yjx+vn9j xj 2URjfrb(yn9)fr(x)j 2:(2.13) y@13isrc:553Ana.TEXSadaiz(2.13),primjenomnejednakrosti(2.6)nafunkcijugr"7ikoristescijedna-krost(2.12)dobivXamoIC ܞ3ڍ0;rÎ;n+16jfrb(yn9)fr(x)jjyxj4ލa3URC ܞ3ڍ0;rÎ;n+16jgrb(y+vn9)gr(x+vn9)jj(y+v)(x+v)jsQUR(grb(y+vn9)gr(x+vn9);(y+v)(x+v))UR(frb(yn9)fr(x);yx);src:558Ana.TEXpasmodokXazalinejednakrost(2.6)zafrb.ݠsrc:561Ana.TEXKoristersciunejednakost(2.13)iprimjenomnejednakosti(2.5)i(2.8)nafunkcijugr} dobivXamo|hgjfrb(yn9)fr(x)jURjgr(y+vn9)gr(x+vn9)j,ZA4URz(↍C2ܞ2RA0;rÎ;n+16jyxj(jy+vn9j2 ?+jx+vj2 )szav h>UR0=C2ܞ5RA0;rÎ;n+1G:rjyI{xjJizL9fꍿ(jyI{+vj Կ+jx+vj )szav h<UR0;,src:567Ana.TEXpasmodokXazalinejednakrosti(2.5)i(2.8)zafrb.Fp;;src:571Ana.TEXPrimjenomorvelemedolazimodoPropSozicije2.4:7Prop`ozicija2.4|osrc:573Ana.TEXNekaje otvorffeniograni3LcenpodskupodR2nP. ?PostojestrffogopozitivnekonstanteC̿1;C̿2;C̿3;C̿4iC̿5takvedazasveu;vn9;w2URL2rb( )vrijeffde35ocjene:src:578Ana.TEXza35r>UR2:B8~UC̿1甆Z j _ B(ڹ+juj rUR1GC̿3甆Z j _ @jfrb(vn9)fr(u)jjvujUR甆Z _ (fr(vn9)fr(u);vu);(2.16)$src:589Ana.TEXza351URjvuj22뀍Lr,p( )7|2zk# ڹ+juj2r؍Lr,p( )+jvn9j2r؍Lr,p( )uEUR甆Z _ (frb(vn9)fr(u);vu);(2.17)'Go5甆ZvN _ |u(frb(vn9)fr(u);wR)URC̿5jōjvuj33[z< ΍ڹ+juj+jvn9j?!Gjn332r33\) 0r L1R( ) O][甆Z _ N>jfrb(vn9)fr(u)jjvuj]33r1133\) 0rjwRjLr,p( )4T:(2.18)Ҡy@14isrc:598Ana.TEXDokazj:2,kroria?stenjemoScjene(2.5)dobivXamoU^r甆ZyM+ _ (frb(vn9)fr(u);wR)UR甆Z _ jfr(vn9)fr(u)jjwRj"@URCܜ甆Z H _ *ܹ(ڹ+jvn9j r=Fu_rώz@ꍿ2 >[1,pajeonda Cw=w = x0ώz5 x01 =Fu =rώzޟrjwRj 2 x0 u]D133s^\)~ȟ#E c:0rURCܞ[甆Z _ N@(ڹ+jvn9j rjwRj rb]O4233s^\)prtURCܞ(ڹ+jvn9jr6takrodaje=UR(r61)(1)UR,=Furr=Fu 1zD)5rjfrb(vn9)fr(u)jjvuj]133s^\)7#Er1Ɵ:0 ĝ[甆Z _ N>jwRj r]O4133s^\)pr:(2.22)>src:635Ana.TEXStimesmodokXazalijednakrost(2.18).src:637Ana.TEXKonstanrteC̿1;C̿2;C̿3;C̿4iC̿5suneorvisneor>6in.;;-@2.3KGEgzistencijaijedinstvuenostrjeYNVsenjakva-KGzinewtonouvskogzStokesovogsustava΍src:641Ana.TEXNekXadje omeUЉFfdenLipscrhitzovdskupuR2n sgranicom.Stacionarnikvazi-newtonorvskiStokesovsustavjesadaoblikXaA(u)+rpUR=f2ug ;(2.23)divֹuUR=0u\h ;(2.24)e=uUR=0naUR0takvXadajeY3:jA(u)A(vn9)j34V䍍㐟G0Gr UR ̿1juvjrUR0takvXadaje:ijA(u)A(vn9)j34V䍍㐟G0Gr UR ̿2juvjVr (ڹ+jujrUR0takvXadajesrc:688Ana.TEX8ڍdThA(u)A(vn9);uvi34Vr,p;V䍍㐟G0GrYUR̿1IǍjuvj22bVr7z[ ڹ+juj2r8OVr+jvj2r8OVrb;@src:689Ana.TEXuslurscajukXadjerUR2pSostojikronstanta̿2V>UR0takvXadajeY3VhA(u)A(vn9);uvi34Vr,p;V䍍㐟G0GrYUR̿2juvj rڍVr ;8u;veD2URVrI:src:691Ana.TEXSvreVpretpSostavkeTVeoremaA.2suzadovoljenepazakljuscujemodahomogeniDiricrhletov+problemzastacionarnikvXazinewtonorvskiStokesovsustavsaza-kronompSotencijeimajedinstvenoslabSorjea?senjeuUR2VrI.e;;|_Nap`omena2.2ysrc:697Ana.TEXRffezultatizTeorema2.2semozeprofgsiritiinaslu3Lcajneho-moffgenih35rubnihuvjeta uUR=gnnanD;]src:699Ana.TEXgdje35jeå甆ZPj _\g=UR0:Cnsrc:700Ana.TEXPrffetpostavimon꨹i U^ K甆Zô _˃gdË=UR0:(3.2) 17y@18isrc:739Ana.TEXKaoiutorscki2.3pSostojihw2WƟ22; 3չ( )2n =5t.d.7jehj̿ F =gn9.TVadazbSog(3.2)iStokresoveformulevrijedidajeʟ甆Zv _ _divنhdxUR=0; *src:741Ana.TEXi;izabSerimohtakrodabudesolenoidnavektorskXafunkcija.0Nadalje,ՄnormauWƟ221= ; "()2n jede niranasaLkkPW.:21= ; ()n;r}= 9infUR q0*(u)=%kukW.:2; s( )n&; src:746Ana.TEXpaimamodajekhkW.:2; s( )n*,NURc̿0(rr; )kgn9kPW.:21= ; ()n8+:}src:748Ana.TEXNekXaje(viditorscku2.1)Vr紹=URfu2WOƿ1;r0 Р( )2nP,divu=0g.src:751Ana.TEXAkrode niramoc+A(vn9)UR=div'ffrb(e(v+h))g;za(rvË2WOƿ1;r0 Р( ) nP;src:753Ana.TEXgdjenjjefr̹funkrcijade niranautoscki2.2,CtadajeAstrogomonotonopSerator.src:756Ana.TEXAkro/brezultatizPropSozicije2.4napia?semouterminimagorede niranogope-ratoraAdobivXamonejednakrostsrc:759Ana.TEX8vn9;w2URWOƿ1;r0 Р( ) nP;yhA(vn9)A(wR);vwiW.:1;r1Ɵ:0EL( )n7;W鍑.:1;ry,0 Ll( )nY$knURC̿4bDx$je(vn9)e(wR)j22 \Lr,p( )nG2 Zzb9 ڹ+je(v+h)j2r Lr,p( )nG2#'ڹ+je(wR+h)j2r Lr,p( )nG2F:(3.3)/HDe nicija3.1rQsrc:766Ana.TEXb(u;vn9;wR)UR=P*n U_i=1 ASUQR%* .uidڹ(ˍ33@xv8:j33Xz ꍐ @xx8:ih )wjf dx;j%=1;:::ʜ;n. Prop`ozicija3.1|osrc:769Ana.TEXForma bjedobrffode nirana,GBtrilinearnaineprekidnana(WOƿ1;r0 Р( )2nP)239za35svakirFu3nz̟n+2.&src:772Ana.TEXDokazj: #Akrou;vn9;w2URVrI,izSobSoljevljevrogteoremaoulaganjudobivXamo㍑9Iui,2URL nr ΍ @xi2L rb( );wj\2L nr ΍ @xiU)wjf dxjURjuijPLnr1=(nr) xӿ( )3jō33@vj33[z> ΍ @xiUjLr,p( )4Tjwjf jPLnr1=(nr) xӿ( ):(3.4)Isrc:781Ana.TEXNadaljezab(u;vn9;wR)vrijediOꍍfjb(u;vn9;wR)jURc(rS)jujW鍑.:1;ry,0 Ll( )n%jvjW鍑.:1;ry,0 Ll( )njwRjW鍑.:1;ry,0 Ll( )n:(3.5)ҍsrc:785Ana.TEXFVormabjeorscitotrilinearna,a(3.5)namgarantiranjenuneprekidnost.;;Drugoosnorvnosvojstvoformebje:tCProp`ozicija3.2|osrc:790Ana.TEXNeka35jerFu3nz̟n+2.fiVrijeffdidajeɍxb(u;vn9;v)UR=0;8u2VrI;vË2WOƿ1;r0 Р( ) nP;(3.6)Wfb(u;vn9;wR)UR=b(u;w;vn9);8uUR2VrI;v;w2URWOƿ1;r0 Р( ) nP:(3.7)ύsrc:798Ana.TEXDokazj: #PrvrodokXazujemosvojstvo(3.6).8ZauUR2V>ivË2DUV( )2n imamopƍib(u;vn9;v)UR=甆Z _ uidڹ(ō33@vj33[z> ΍ @xiU)vjf dx=甆Z _ uiō #@ [z, ΍@xiōjvn9j22럟[z ΍ӑ2*dx=#؍wOō33133[z ΍2 F`甆Z _ ō@uiџ[z՟ ΍U@xi,jvn9j 2dxUR=ō33133[z ΍2 F`甆Z _ div)ujvj 2dx=0:1src:803Ana.TEXSvrojstvo;(3.7)dokXazujemoprimjenomsvrojstvXa(3.6)kadaumjestovtstarvimov+wR,tj.)0UR=b(u;v+wR;v+wR)UR=b(u;vn9;v)+b(u;v;wR)+b(u;w;vn9)+b(u;w;w)UR==URb(u;vn9;wR)+b(u;w;vn9);ҍsrc:806Ana.TEXpaslijedidajeb(u;vn9;wR)UR=b(u;w;vn9):F;;柍src:809Ana.TEXSadade niramonelinearniopSeratorBnaWƟ21;r Р( )2n sa%H'I7hB(wR);i34D2n=(n㝹+2).MTadau2h㝹+Vr |jevrloslabfforjefgsenje35zaprffoblem(3.1)akoZ]guUR=gnu!tWƟ 11=rÎ;r() nP;(3.9),B(u)UR2V p-:0ڍr7;(3.10)W8甆Z?\g _ Gfrb(e(+h))e(+hu)+hB(u);i34V䍍㐟G0GrW;VrShB(h);uhi34V䍍㐟G0GrW;Vr"-mQ甆Z _ N>(uh) (uh)rhUR0;82Vrb:(3.11)ץsrc:841Ana.TEXVrlo35slabfforjefgsenjeod(3.1)jeslaborjefgsenjeakozadovoljava0^2甆Zdݞ _ m0frb(e(u))e()+hB(u);i34V䍍㐟G0GrW;VrY=UR0;82Vr:(3.12)$`src:847Ana.TEXNekXa4oznarscimod}=diamH( )=sup'Tx;yI{2 /tjx yn9j,GPpa4moszemode niratikons-tanrteudvoSdimenzionalnomsluscaju.8Imamo U- 2ڍ2;r,=UR2 1=r ΍(3=r)(43=rS)Z!g 1=3 @ ;(3.15)"jsrc:862Ana.TEXgdjeje(x)gama-funkrcijade niranasaw(x)UR=甆ZMUT1 0UZe t .Bt x1%cdt;ꦹzax>0. Tsrc:864Ana.TEXKonstanrte2iRA3;r .; 2OiRA3;r;iUR=1ili2,sudanesa#-K 2ڍ3;r,=UR2c 2ڍK;r1c 2(3;rS)3 (5rjrhjPL3r1=(5r6)"2n=(nF+2).}+Tada:um h"2c^h+V2pmRArjeaprffoksi-mativno35rjefgsenjeoffd(3.1)ako u:um Z=URh+mX ㇍ti=1i;m Fwid;i;md2R;(3.21))<.甆Z41 _ =Pfrb(e(umĹ))e(widڹ)+甆Z UT _ (umr)um lwi,=UR0;8i2f1;:::ʜ;mg:(3.22) !src:902Ana.TEXPriroSdno,k$naa?sKCprvikrorakjeustanovitirjea?sivostsustavXanelinearnihjednadszbi(3.22)zai;m F;iUR=1;:::ʜ;m.8PrvromoramooScjenitidijeloveu(3.22).Lema3.1ZQsrc:907Ana.TEXNekajevm Z=URP*m U_i=1 ASi;m Fwitakodajeje(vm +Dh)jLr,p( )nG2#҄UR1.;TadaimamoU^\u甆Z! _ frb(e(vm l+h))e(vmĹ)#52&=UR(̿0)2 1rnje(vmĹ)j r \Lr,p( )nG2#'(̿0j+ō1r諍0۟[z-o ΍G+r>}(ō xr33[z) ΍r61") ry@23iNap`omena3.1ysrc:943Ana.TEXPrimjetimo35daC̿49moffzemouvijekodabratitakodabude(D2iRAn;r &22r33q,z*#& Q(Fuۿ1۟z@4 Q)r,) 1=(r1:!src:958Ana.TEXDokazj: #ZbSoggustorscejerezultatdovoljnopSokXazatizaUR2C2ܞ1RA0 ܦ( ).8ImamohTOj(x)j r紹=URrS甆ZMSxi?k81j(x)j r1:F;;My@24iLema3.3ZQsrc:974Ana.TEXNeka35jevm Z=URP*m U_i=1 ASi;m Fwid.fiTadaimamop8=甆Z _ N>(vm l+h) (vm+h)rvmJUR iڍn;r &je(vmĹ)j 2 \Lr,p( )nG2#' Oiڍn;r;(3.25)src:978Ana.TEXgdjesukonstante2iRAn;r?i 2OiRAn;r &;i=1ili2,danejeffdnadzbama(3.13)-(3.14)zanUR=235ijeffdnadzbama35(3.16)-(3.17)zanUR=3. !src:982Ana.TEXDokazj:B`-\h1.<_src:984Ana.TEXnUR=2AU(a)V9qsrc:986Ana.TEXDokXazujemo(3.25)za21RA2;ri 2O1RA2;r .ڹ:u"&甆Zҟ _ pf(vm l+h) (vm+h)rvmJ=%^甆Z _ N0: V9qsrc:1007Ana.TEXIzbSor̿1V=UR1namdaje(3.25)zai=1in=2.@(b)V9qsrc:1009Ana.TEX22RA2;ri 2O2RA2;rdobivXamokroristescisljedescislijedjednakostiinejedna-V9qkrosti:U^^甆Z ? _ ӹ(vm l+h) (vm+h)rvm"1=UR甆Z _ (vm l+h) (vm+h)r(vm+h)甆Z _ N>(vm l+h) (vm+h)rh:V9qsrc:1012Ana.TEXKakrojehj =URgXimamo!3=t甆Z{GQ _ (vm l+h) (vm+h)r(vm+h)UR=ō1[z ΍2 甆ZF^ _Pjgn9j 2gd: *V9qsrc:1015Ana.TEXZadrugidiopia?semoJ甆Z _ $(vm l+h) (vm+h)rh)PxURX ㇍lRi;jkv n9iڍm l+h idkL2r( )kv n9jڍm+h jf kL2r( )kō33@h2j33[z ΍@xijkPLr1=(r1) ( )$LURX ㇍lRi;j[rS 1=r ekv n9iڍmkߍ(rjrhjGL,q3;r Ͽ( )9%=Z;-V9qsrc:1033Ana.TEXgdjewjeq̿3;r,=UR3rS=(5r6).Kakrojejvn9jL S( )3ۘ32( 2)=2 %d*kvmkL S( )3F,V9qza hUR2,istotakroimamov0甆Z|ܟ _ lnvm l vmHrhUR3 1=qq3;rkvmk 2L3r1=(3r)( )35>jrhjGL,q3;r Ͽ( )9%=Z:"V9qsrc:1037Ana.TEXNekXafjec(3;rS)najboljaSoboljevljevXakronstantafzaWOƿ1;r0 Р( ). TVadaV9qimamoU^g䰟甆Zn\ _ w2vm l vmHrhUR3 1=qq3;r齹(c(3;rS)) 2krvm k 2yLr,p( )9^jrhjGL,q3;r Ͽ( )9%=Z;!XV9qsrc:1040Ana.TEXipsajednostarvnomprimjenomKornovenejednakostidobivXamo21RA3;r .ڹ:^g甆ZeO _ mvm l vmHrhUR3 1=qq3;r齹(c(3;rS)cK;r1) 2jrhjGL,q3;r Ͽ( )9%=Zje(vmĹ)j 2yLr,p( )9^:V9qsrc:1043Ana.TEXZa 2O1RA3;rkroristimonejednakost|Ю甆Z{Z _ vm l hrhURjhjL1R( )3jvmjLr,p( )3^jrhjPLr1=(r1) ( )9(H3ō d[z ΍rS1=rm;jhjL1R( )3krvm kLr,p( )9^jrhjPLr1=(r1) ( )9 .͍URcK;rō(dd[z ΍rS1=r#IjhjL1R( )3je(vmĹ)jLr,p( )9^jrhjPLr1(r1)( )9!\V9qUR̿1je(vmĹ)j 2yLr,p( )9^+ō~133[z^ ΍4̿1c 2ڍK;rōP&d22d[z ΍rS2=r#Ijhj 2yL1R( )3jrhj 2Lr1=(r1) ( )92;8̿1V>0: V9qsrc:1048Ana.TEXIzbSor̿1V=UR1namdaje(3.25)zai=1in=3.yy@27i@(b)V9qsrc:1050Ana.TEXSdrugestraneimamosljederscislijedjednakostiinejednakosti: U^^甆Z ? _ ӹ(vm l+h) (vm+h)rvm(*1=UR甆Z _ (vm l+h) (vm+h)r(vm]+h)"甆Z _ N>(vm l+h) (vm+h)rh;t甆Z{GQ _ (vm l+h) (vm+h)r(vm+h)UR=ō1[z ΍2 甆ZF^ _Pjgn9j 2gd;`b甆Zg| _ n(vm l+h) (vm+h)rhURjvm+hj 2L3r1=(3r)( )35>jrhjGL,q3;r Ͽ( )9IƍkyUR23 1=qq3;rjrhjGL,q3;r Ͽ( )9%=Z(kvmk 2L3r1=(3r)( )38n+khk 2L3r1=(3r)( )35>)GUR23 1=qq3;rjrhjGL,q3;r Ͽ( )9%=Zkhk 2L3r1=(3r)( )3ŗ+23 1=qq3;rc 2(3;rS)c 2ڍK;r1jrhjGL,q3;r Ͽ( )9%=Zje(vmĹ)j 2yLr,p( )9^;Ustonamdaje22RA3;ri 2O2RA3;r .ڹ.src:1063Ana.TEX~F;;src:1065Ana.TEXNakronܜsto{]smooScjenilidijeloveunelinearnojjednadszbi(3.22)naa?sproblemmorszemoHsvestinarjea?savXanjesustavXanelinearnihjednadszbizai;m F.De ni-ramonormrunaR2m lsa nkskUR=je(vmĹ)jLr,p( )nG2 }2;ꦹgdjeje+vm Z==mX ㇍Si=1i;m Fwid:(3.27)%src:1073Ana.TEXAkrojeksk$=je(vmĹ)jLr,p( )nG2&O=0tadakoristimoKornovunejednakostpaimamokrvm kLr,p( )nG2#҄URcK;r1je(vmĹ)jLr,p( )nG2=0;;src:1076Ana.TEXiliekvivXalenrtnokPii;m Frwi kLr,p( )nG2#L=t0.o!TVadaslijedidajePki Ci;mrwi?'=t0s.s.{uz .KakrojefwidgortogonalnabazazaV̿2~imamoi;md=UR0;8i=1;:::ʜ;m.Dakle!URkskjenormanaR2mĹ.src:1083Ana.TEXNekXajesadanelinearnopreslikarvanje!URPƹ(s);2R2mĹ,danosa U^Pidڹ(s)UR=甆Z _ frb(e(vm l+h))e(wi)Gy@28iU^f+甆Z _ N>((vm l+h)r)(vm+h)wiu;iUR=1;:::ʜ;m:(3.28)8Pnje4neprekidnopreslikXarvanje4zar>UR2n=(n6+2),Ypa4trebamoispitatirjea?sivostjednadrszbSe Pƹ(s)UR=0u\hR m:(3.29)̍src:1095Ana.TEXVVoSdersciserezultatimaizdodatkXaByzanaa?sjeproblemdorvoljnonascikugluBizR2m ltakrodaje󥍒jPƹ(s)+6=UR0zasvXaki0^0;2@B:src:1097Ana.TEXTVadaimamoegzistencijubarjednogrjea?senjajednadrszbSe(3.29)unutarB.src:1101Ana.TEXKritirscnikorakjeoScjeniti?Pƹ(s).Prop`ozicija3.3|osrc:1103Ana.TEXNekazje@B2iRArl=٣fn2R2m g:ksk=[2iRAn;r &22rb=(ߕFuȿ1ȟz@4 )rS]21=(r1kk r=  iڍn;r &kk 2jG iڍrb;iUR=135ili2;󍍒(3.30)src:1110Ana.TEXgdje35su2iRAn;rJ[i 2OiRAn;rde nirffanisa(3.13)-(3.17),akaouLemi3.1:"-- G iڍr紹=UR Oiڍn;rι+fō۹1۟[z ΍4 +ō(1=4)21r۟[z*2Z ΍Ir/Ch(ō xr33[z) ΍r61") rUR Oiڍn;rι+n;r &;(3.31)src:1117Ana.TEXtj.-ƍg(D42iRAn;r33q,z75 Q(Fuۿ1۟z@4 Q)2=r jr9) rG iڍrb;i=135ili2;(3.32)Ksrc:1122Ana.TEXtada35imamobPƹ(s)+6=UR035zasvaki00;2@B iڍrb;i=135ili2:$~src:1125Ana.TEXDokazj:Koristersci-PLemu3.1,S.NapSomenu3.1iLemu3.3sa̿0V=UR1=4dobivXamo(3.30). Akroe@napia?semoX>=ٻksktadatraszimouvjetena2iRAr iG2iRArtakroda9XF>UR0krojezadovoljavXaO(ō۹1۟[z ΍4 )2 1rnX r. iڍrbX 2\/G iڍr>UR0:(3.33)y@29isrc:1133Ana.TEXZapisujemo?(3.33)kXaoX22((BFuv1vz@4 M)221rnX2rG2iRAr \i?kakrosuX22 {ƹiG2iRArpSozitivniimamosljederscinuszniuvjet: ЗXF<URX̿0V=(ō2iRAr33[z9 Q(Fuۿ1۟z@4 Q)21r<9M) 1=(rō1[z," ΍inrbXߍ2S0!G iڍrb:"Eэsrc:1142Ana.TEXOptimalniizbSorzaYjeY̿0V=UR(2=r)21=(r0takrodajePƹ(s),+Ź=UR0.IzhjednadrszbSe(3.30)dobivXamo0=R,Pƹ(s)+jj22VURRPƹ()UR>0,a?stojekrontradikcija.8PaslijedidajePƹ(s)+6=UR0zasvXaki0;2@B2iRArb.src:1153Ana.TEX~F;;src:1155Ana.TEXSadamorszemosaszetinaa?serezultateusljedescemteormu.bTeorem3.1fsrc:1158Ana.TEXNeka:je2URr>2n=(nBC+2):inekavrijeffdi(3.31).VTadapostojibffarem35jednorjefgsenjeum Z2URh+V2pmRAriza(3.22)ivrijeffdiocjena"čQje(um lh)jLr,p( )nG2#҄UR(D2iRAn;r &22r33q,z*#& Q(Fuۿ1۟z@4 Q)r,) 1=(r\2n=(n8F+2)einekanjeude nirffanoLemom3.4. TadaB(umĹ)Ô*B(u)nslabffouL2qq0v*( )2n-:2 ,}zaneki35q̿0V>UR1,iB(u)2V2p-:0RAr7.!2؍src:1216Ana.TEXDokazj: #ProscinjemosajednadrzbSom(3.22):k-甆Z׽ _ {Ofrb(e(vm l+h))e( n9)&A!bG甆Z _ N>(vm l+h) (vm+h)r Ë=UR0;8 2V pmڍr 4:src:1220Ana.TEXAkrojegm Z=URfrb(e(vm r+h))iakokoristimo(3.34)dobivXamodajejgmjLr,p( )nG2#҄CFgdjeCorvisisamoor>6in.8TVakodobivXamo ʍvTfrb(e(vm l+h))UR*m!1Z&gË2L r(um  umĹ)i;j X.Nadaljeakojer>/>2n=(nӹ+2)jtadapSostoji"rÎ;n>UR00takrodaje!rÎ;n=URnrS=2(nr)"rÎ;n>UR10(akrojer=2n=(n+2)+n9;Ë>0,tadanrS=2(nr)UR=1+((n+2)=2(nr))n9).8TVadaimamovxkw Ri;jڍm kL!r7;n( )&9EURku iڍmkL2!r7;n( )&ku jڍmkL2!r7;n( ):xy@31isrc:1230Ana.TEXIzv(3.36)imamou2iRAm !u2iֹuL22!r7;n>( )kXakroje1<2!rÎ;n-( );8ko=UR1;:::ʜ;n.&TVakodobivXamonizubSeskronascnosti~"kw Ri;jڍm kL!r7;n( )&9EURC5;8i;j$F=1;:::ʜ;n;NEUw Ri;jڍm OT!m!1Z&wR i;j=UR(u u)i;j s.s.8u- ;8i;j$F=1;:::ʜ;n:src:1237Ana.TEXProsljediscnow2Ri;jRAm OT*URwR2i;j 䪹slabSouL2q( )2n-:2 ;8qË2(1;!rÎ;n B)(izLeme3.5).)src:1242Ana.TEXSadauzimamo Ë2URV2pmq0RArikXakrojer 2URL2(!r7;n 3)0Ey( )2n-:2 dobivXamo U^`Q甆Zf _ m\(vm l+h) (vm+h)r Ë!m!1Z&甆Z#ҟ _ +du ur n9: *src:1245Ana.TEXSljedstvrenoimamo(甆Z/g9 _ 8 ge( n9)甆Z UT _ u ur Ë=UR甆Z _ ge( n9)+hB(u); iW.:1;q ;W.:1;q7:05=UR0(3.39) _src:1249Ana.TEXzasvre Ë2URV2pmq0RArizasvXakim̿0V2N.8ZbSoggustorsce(3.39)vrijediza Ë2Vr} i#hB(u); n9i34V+0;V7=URhg; i34V䍍㐟G0GrW;Vr:src:1252Ana.TEXTVakrojeB(u)UR2V2p-:0RAr7. ;;Prop`ozicija3.5|osrc:1255Ana.TEXNekaMje2URr3n=(n+2);UR2WƟ21;1>F( )2n iMvË2Vrb.KTadaimamoU^zhB(v+);vn9i34V䍍㐟G0GrW;VrY=UR甆Z _ v (v+)r:(3.40)*src:1261Ana.TEXDokazj:KakrojerUR3n=(nO+2)ivË2VrPwtadaimamoUQR h*  T((vչ+O)r)vvË=UR0.ItadaU^0hB(v+);vn9i34V䍍㐟G0GrW;VrY=UR甆Z _ N>((v+)r)v(v+)UR=甆Z _ v (v+)r:F;;Teorem3.2fsrc:1267Ana.TEXNeka-jenu>=2;3-inekaje2u>r>2n=(n꠹+2). vQNadaljeneka >jezadovoljenuvjet(3.31).XTadapffostojibaremjednovrloslaborjefgsenjeuUR2WƟ21;r Р( )2n ۅza35prffoblem(3.1)ionozadovoljavaocjenu"r[je(u)jLr,p( )nG2#҄UR(D2iRAn;r &22r33q,z*#& Q(Fuۿ1۟z@4 Q)r,) 1=(r2n=(nȹ+2):'inekajeispunjeno(3.31).']TVadazbSogTVeroma3.1pSostojium lkrojizadorvoljavXad,O甆Z3K _ ;frb(e(umĹ))e(widڹ)dx甆Z UT _ um l umrwiddxUR=0;8i2f1;:::ʜ;mg;1oum Z=URgXuʡWƟ 11=rÎ;r() nP;(3.42)Wfje(umĹ)jLr,p( )nG2#҄UR(D2iRAn;r &22r33q,z*#& Q(Fuۿ1۟z@4 Q)r,) 1=(r+h))g-zavË2URWOƿ1;r0 Р( )2n2. Sadastavljamovm Z=URum lh꨹izapisujemo(3.42)uoblikuɍ(K#hA(vmĹ);widiW.:1;r1Ɵ:0EL( )n7;W鍑.:1;ry,0 Ll( )nX^甆Z UT _ um l umrwi,=UR0;8i=1;:::ʜ;m;src:1293Ana.TEXtj.#um l umrvm Z=UR甆Z _ NNekXa:jerg?3n=(n]m+2):inekajeuvrloslabSorjea?senjeproblema(3.1).8TVada U^m甆Zs3 _ |jfrb(e(+h))e(+hu)+hB(u);i34V䍍㐟G0GrW;Vr(*\chB(h);uhi34V䍍㐟G0GrW;VrS甆Z UT _ (uh) (uh)rhUR0;U_̃w8UR2Vrb:(3.43)src:1318Ana.TEXNadaljeimamomOhB(h);uh)34V䍍㐟G0GrW;VrS+甆Z UT _ (uh) (uh)rhw5=UR甆Z _ NMҹ iڍn;rlx<UR1;(3.44)#src:1335Ana.TEXzanekiiUR2f1;2g,M}pffostojirS2h<2takodajeprffoblem(3.1)rjefgsivza8r2[rS2;2]."iy@34isrc:1338Ana.TEXDokazj: #Morszemoprimjetitidase(3.32)moszezapisatikXaooqr( iڍn;r &2) rF. fImamo(2=rS)2rURrS2.a*;;,3.4KGDokazzjedinstvuenostiWNTeorem3.4fsrc:1351Ana.TEXNeka&jeuslabfforjefgsenjeproblema(3.1)kojezadovoljava(3.41).Nadalje35nekajerUR3n=(n+2)35inekakonstantaArŗiz(3.41)zadovoljavasAr<UR(ōCu&C̿433[z ΍>f~cK;r@Ccߍ2bK;r1c2(n;rS)n(r~cK;r n;r &Ar apSorvlaci.je(u)jLr,p( )nG2%W~=L0.rKako荑u2WOƿ1;r0 Р( )2n Vciv*!je(vn9)jLr,p( )nG2%+EjenormanaWOƿ1;r0 Р( )2n Vcimamou=0. PradokXazalismojedinstvrenostuzuvjetAr<UR(ōFC̿433[z+O ΍2>f~cK;r@C n;r-) 1=(3rf~cK;r@Ccߍ2bK;r1c2(n;rS)n(rݡ0. 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TApriorne}$oScjene(4.12),(4.13)garanrtirajudatorjea?senjepos-<_tojiynacijelominrtervXalu(0;Tƹ).4NadaljeoznascimosaCܞ(qn9;s)optimalnu<_SobSoljevljevuӤkronstantuulaganja(ukombinacijisaKornovomnejed-<_nakrosti)takodajeH捑<_jjLsǿ( )n}4URCܞ(qn9;s)je()jLq#( )nG2 t|;82WOƿ1;q0 <( ) nP;qË0takrodaje U^LD甆ZR _O\fn9(e())je()j 2VURd̿00X(OUV)je()j 2 \Lr,p(On9(e(um(t)))je(um(t))j 2VURjfGjLr1Ɵ:0^7(0;T.:;V䍍㐟G0GrW)-eje(um(t))jLr,p( )nG2 }2: _<_src:1504Ana.TEXAkrointegriramopSotnaintervXalu(0;Tƹ)dobivamohjumj 2yLr,p(0;T.:;W1;r Ll( ))nHō&2[z# ΍d̿0( T)(fō<333[z( ΍2d̿0( T)+bss2 236jfGj 2 Lr1Ɵ:0^7(0;T.:;V䍍㐟G0GrW)-e+'ؠy@39i]+ō33r6133[z) ΍ $2"(ō`93r33[z# ΍d̿0( T)%w)]H332r33\) 0r11jfGjߍ2=(rsFus1'zğm riliza<_tUR2mstarvimovË=UR((mu)nRnP)m;(4.19)<_src:1547Ana.TEXpafunkrcijuvË=URvn9(t)iz(4.16)zamjenimosavXiz(4.19).<_src:1550Ana.TEXPrimjetimodaje!~}甆ZMT>05hu801;vn9idtUR=甆ZMUTT 0khmu80;(mu)nRnPidtUR=(!Hy@40i¢*Qŧ甆ZM]ũTXpS0dfh(mu)80onP;(mu)nPidt甆ZMT UT0Kh S-:0ڍmu;(mu)nRnPidtUR=#㕍SU甆ZMT0h S-:0ڍmu;(mu)nRnPidtUR !n!1甆ZMT0m S-:0ڍmjuj 2yL2*( )nPdt;<_src:1554Ana.TEXpaiztogadobivXamo:sN3甆ZMZ3TTU0`¹(m S-:0ڍm)juj 2yL2*( )nPdt+甆ZMT UT0K S2ڍmh;uidtUR=甆ZMUTT 0i S2ڍmhf;uidt:(4.20)ύ<_src:1558Ana.TEXAkrompSoa?saljemoubeskronascnostdobivXamo<_甆ZMH_TC 0O(m S-:0ڍm)juj 2yL2*( )nPdtUR!ō1[z ΍2 ju(s)j 2yL2*( )nō33133[z ΍2Fbju(s̿0)j 2yL2*( )n;ꦹzasvXaki2^s꨹i s̿0:<_src:1560Ana.TEXpa,slijedi,mꍍō=1=[z ΍2DNju(s)j 2yL2*( )nP¹+甆ZM ssq0n>h;uidtUR=ō1[z ΍2 ju(s̿0)j 2yL2*( )n+甆ZM ssq0hf;uidt;ꦹzasvXaki2^s꨹i s̿0:Ѝ(4.21)<_src:1565Ana.TEXJednakro,gmkXakoNyjeu82L21 (0;Tƹ;L22( ))2n[;Nymorszemonascis̿0n !80,gmt.d.dSje<_zadorvoljeno(4.21)iu(s̿0n T)slabSokronvergirauL22( )2nP;tpakXakrou(s)UR!<_u(0)Uh=URu̿0Au(s̿0n T)UR*u̿0slabSou;L 2( ) nP:(4.22) <_src:1573Ana.TEXFiksiramost.d._zadorvoljavXa(4.21)iuzmemodajes̿0 =s̿0n T.ZbSog<_(4.22)dobivXamo(4.18).$<_src:1577Ana.TEXSadamorszemodokXazati(4.17)koristescimonotonostoSdr-Stokresovog<_opSeratora.8ZaUR2L2rb(0;Tƹ;Vr)<_X sڍ=UR甆ZMUTs 0Ðhdiv'fn9(e(u3))e(u)g+divfn9(e())e()g;uidt+ō33133[z ΍2Fbju(s)j 2yL2*( )nP;_(4.23)<_src:1582Ana.TEXgdjeszadorvoljavXa(4.18).<_src:1584Ana.TEXZbSog(4.13)postojiM6>UR0takrodajejumjL2*( )nURM;ꦹzasvXaki2^m2N:<_src:1586Ana.TEXTVadapSostojepodnizfu3gURfumg꨹iuUR2L22( )2n+Fѹtakrodavrijedi{pu3(s)UR*u(s)^slabSouEyjL 2( ) nP;<_src:1589Ana.TEXpaizmonotonostioSdr-Stokresovogoperatorai(4.23),slijedidajeV,liminf33U3!1sX sڍō1[z ΍2 ju(s)j 2yL2*( )nP:(4.24))0y@41i<_src:1594Ana.TEXJednakrozbSog(4.9)(荑N(X sڍ=UR甆ZMUTs 0Ðhf;u3idt+ō1۟[z ΍2 ju̿0 sj 2yL2*( )nj+甆ZMs UT0hdiv'fn9(e(u))e(u)g;idt&Gֹ+甆ZMs0n0it<0).J덑<_src:1611Ana.TEXI8jedinstvrenostistotakoproizlaziizmonotonosti. N TVadajednadszba<_pSorvlasciėidauF2Cܞ([0;Tƹ];V2p-:0RAr7)%stoėnamdajesmisaoporscetnomuvjetu.<_Sadajoa?sszelimodokXazatiregularnostpSovremenru.֍&P(b)<_src:1615Ana.TEXPrvro,mnoszimo(4.9)sah2-:0RAmjvpakXadsumiramopSoj{dobivamo!0UVgjō33@umĹ(t)33[z"} ΍ $@t%Wj 2yL2*( )nj+ōd۟[z Y8 ΍dtjFFƹ(je(umĹ(t))jLr,p( )nG2 }2)UR=甆Z _ fō{2@um(t){2[z"} ΍ $@t&;(4.26)!<_src:1620Ana.TEXgdjeje{MFƹ(z)UR=甆ZMUTz 0 n9(s)dŹ=q.wz ꍑVrjzj2rb;lRzazakronpSotencijejwq0z ru(1+jzj22)2r(#(c)<_src:1640Ana.TEXNaa?sӊsljederscikorakjedaderiviramo(4.1)pSot.Uzimamodajeu̿0 2<_VrG;div'fn9(e(u̿0))e(u̿0)gw2L22( )2n =i;dajewn@xfnz Vꍑ@xt]2wL21 (0;Tƹ;L22( ))2nP.8ZbSog<_jednostarvnostiuzimamodajeuGaljorkinovojbazi̿1 8s=xou̿0.9Primje-<_timodaje#IFչ=UR甆Z _ fōRNd33[z Y8 ΍dt n9(e(umĹ(t)))e(um(t))ge(ō33@um(t)33[z"} ΍ $@t%W)UR=(4.29)(󚍒M=UR甆Z _ HV(e(umĹ(t)))[e(um(t))e(ō33@umĹ(t)33[z"} ΍ $@t%W)] 2+#s$5+甆Z _ N<n9(e(umĹ(t)))je(ō33@um(t)33[z"} ΍ $@t%W)j 2;<_src:1652Ana.TEXgdjeje THV(z)UR=q.US(r62)jzj2r<_src:1667Ana.TEXSadainrtegrirajusci(4.29)pSotdobivXamoeבjō33@um33[z ΍;E@t%j 2yL2*( )nUR2(jō33@um33[z ΍;E@t(0)j 2yL2*( )nj+jō33@f33[z ΍j9@tT j 2yL1R(0;T.:;L2*( ))n?g)%͍UR2(D S2ڍ5:+jō33@f33[z ΍j9@tT j 2yL1R(0;T.:;L2*( ))n?g)D S2ڍ6:(4.31)<_src:1672Ana.TEXZakljurscujemoda㍍ōh@uh[z ΍8@t|z2URL 1 (0;Tƹ;L 2( )) nP: <_src:1674Ana.TEXPrimijetimoNdajen9(s)2r6i2=(2r)dobivXamo"*O}甆ZM[TV])0dS甆Zj@ _ sje(ō33@um33[z ΍;E@t%)j r紹=UR甆ZMUTT 0i甆Z _ 'Dje(ō33@um33[z ΍;E@t)j rbn9(e(umĹ)) r(4.33)<_src:1689Ana.TEXpatakroō @u [z ΍8@ta2URL rb(0;Tƹ;W 1;r Р( )) nP:src:1692Ana.TEX~F;;src:1694Ana.TEXN8xCestojevreomavXasznodobitioptimalnuapriorioScjenuzarjea?senje. aTVojepSosebno\uslurscajukXada ovisionekimparametrimakXaousluscajupSoroznesredineilitankredomene.8Unaa?semsluscajumidajemostrogeoScjeneoblikXa:,^Xy@44iKorolar4.1hsrc:1699Ana.TEXPrffetpostavimo35dajezadovoljeno(4.5),tadacqhjujLr,p(0;T.:;Vr),URD̿1;jujL1R(0;T.:;L2*( ))nBD̿2;(4.34)src:1703Ana.TEXgdje35suD̿1;D̿29danesa(4.12)i(4.13).fiA2kojejofgsispunjenoi(4.6)tada򍍑yjō33@u33[z ΍8@tajL2*( X.T)n%URD̿3;jujL1R(0;T.:;Vr,p)0HD̿4;(4.35)ڡsrc:1708Ana.TEXgdje)suD̿3 iD̿4danesa(4.27)i(4.28).GKona3Lcnoakojeispunjeno(4.8)tadash'jō33@u33[z ΍8@tajL1R(0;T.:;L2*( ))nBURD̿6;jō33@u33[z ΍8@tjLr,p(0;T.:;Vr),D̿7;(4.36)src:1713Ana.TEXgdje35suD̿6;D̿79de nirffaneu(4.31)i(4.33).Dsrc:1715Ana.TEXTlak)psenepSojarvljujeunaa?sojde nicijislabSogrjea?senja.VVoderscisedodatnomregularnoa?srscuӔmozemoӔuvestipnekoristescinjegovuprimitivnufunkcijupSot.ZnarscenjetlakXap-egzistencijairegularnostdanisuusljedescemteoremu:Teorem4.2fsrc:1720Ana.TEXNeka35jeɍu+fQ2URL 1 (0;Tƹ;L rF(0;Tƹ;L 2( )) nP;u̿0jgn9(;0)UR2VrI:+4.2KGJedanzteoremokuompaktnostiUsrc:1776Ana.TEXNekXasuX̿0;XJg;X̿1 _Banacrhoviprostorit.d.WijeX̿0 I(,!$Xz,!X̿1 _krompaktno.NekXajeT>UR0; ̿0; ̿1V>1.8De niramoprostorrEYQ=URfvË2L q0 >(0;Tƹ;X̿0);vn980R2URL q1(0;Tƹ;X̿1)g;src:1780Ana.TEXinormruj>jvn9jY %=URjvjLV 0 ~b(0;T.:;Xq0*)3O+jv80jLV 1 ~b(0;T.:;Xq1*)0:⍍Teorem4.3fsrc:1783Ana.TEXUlaganje35YQ,!URL2 q0 >(0;Tƹ;X)jekompffaktno.src:1786Ana.TEXDokXazteoremasemorszenasciu[17 ]..}y@46i4.3KGKlasiucnizrezultatizainicijalniproblem΍src:1788Ana.TEXUorvojsutosckiobraUЉFfdenirezultatiiz[6].src:1790Ana.TEXNekXaje omeUЉFfdenLipscrhitzovskupuR2n sgranicom.src:1792Ana.TEXZavrektoruUR=(u̿1;:::ʜ;unP)gdjejeuiOrealnafunkcijana de niramoA(u)ض=URdiv'(n9(e(u))e(u));(4.39)src:1797Ana.TEXgdje7jeviskroznostde niranau(1.4)ili(1.5),KamiproscavXamosluscajkXadaje1 UZ=UR0.src:1799Ana.TEXKvXazinewtonorvskiNavier-StokesovsustavjesadaoblikXa㍍ō@u؟[z ΍8@t鮹+A(u)+(ur)u+rpUR=f2ug ;(4.40);>divֹuUR=0u\h ;(4.41)uUR=0naiVr} daniDe nicijom2.3,paslijedidajeq\Vr紹=URfvn9jvË2WOƿ1;r0 Р( ) nP;div'v=0g:src:1820Ana.TEXUvroSdimo]Vp s _=?zatvXararscoSdFV>uXHV s[( ) nP;(4.44)HB=?zatvXararscoSdFV>uXL 2( ) nP(=URV̿0):(4.45) De nicija4.3rQsrc:1831Ana.TEXNeka`jerl~޹1z+Fuo2n!z̟n+2B inekajetrilineffarnaformabdanaDe nicijom353.1.fiTadajenuUR2L rC(0;Tƹ;Vrb)\L 1 (0;T;HV)C \Cܞ([0;T];V p-:0ڍr7)src:1834Ana.TEXslabffo35rjefgsenjezaproblem(4.40)-(4.43)ako sō"RNd 33[z Y8 ΍dt-甆Z4jH _ = uȹ+甆Zt _ 0n9(e(u))e(u)e()+b(u;u;)UR=甆Z _ fG35u;DUV80(0;Tƹ);82VrI:/y@47iyTeorem4.4fsrc:1839Ana.TEXPrffetpostavimo35dafQ2URL2r1+ōn۟[z ΍ǹ2!:(4.48)\ <_src:1858Ana.TEXTVada,akrojevË2URHV2sSōy@vx[z, ΍@xiwj2URHV s18( )2>URL 1 ( )%;ꦹjerjeō%$1%$[z ΍2.ōs1۟[z ΍ nZ<UR0:U><_src:1860Ana.TEXKakroFu@xv۟z 萟@xx8:i2URL2rb( )!7,slijedi}Vp s _URVrURHBV p-:0ڍrRp(Vp s )80;(4.49)<_src:1865Ana.TEX(Hseidenrti cirasasvojimdualom).<_src:1867Ana.TEXPromatramosvrojstvenufunkcijuwjf :L4hwjf ;vn9iV㐟s=URjhwj;vn9i;8vË2URVp s9;(4.50)<_src:1871Ana.TEX(gdjejeh;iskXalarniproSduktuHV).<_src:1873Ana.TEXSadaprimjenjujemoGaljorkinorvumetoSduzabazufwjf g.\-\h2.<_src:1874Ana.TEXGaljorkinova35metoffda.<_src:1876Ana.TEXDe niramoumĹ(t)UR2fw̿1;:::ʜ;wmg꨹kXaorjea?senjejednadrszbSeBwhu -:0ڍmĹ(t);wjf i+hA(um(t));wjf i+b(um(t);um(t);wjf )UR=hfG(t);wji;(4.51)<_src:1880Ana.TEXgdjeje1URj%m,Udb(u;vn9;wR)UR= nX ㇍i;jv=1Y$甆Z П _ (buidڹ(ō33@vj33[z> ΍ @xiU)wjf dx;(4.52)!<_src:1884Ana.TEXzadorvoljenjeuvjetcmnumĹ(0)UR=u̿0m2fw̿1;:::ʜ;wmg;u̿0m!u̿0u!HF:;(4.53)<_src:1889Ana.TEXiumĹ(t)jede niranna[0;tm];tm Z>UR0.0y@48i-\h3.<_src:1890Ana.TEXApriorne35offcjene(I).U]<_src:1892Ana.TEXPrimjetimodaje:hA(u);vn9iUR=甆Z _ (e(u))e(u)e(v);(4.54);<_src:1896Ana.TEXpaizPropSozicije2.4slijedi,FhhA(vn9);viUR=甆Z _ n9(e(v))je(v)j 2VURc̿0 #( )je(v)j 2 \Lr,p( )nG2 }2(ڹ+je(v)jr.2).Iz<_orvogaiiz(4.51)slijedidavrijedieڿ1ʴz(Pn˧p0)iformab(u;vn9;wR)Fje<_neprekidnaĘnaVr VkXadjeFu˿2˟z@ꍐq +Fur=1r=z@ꍐrF1,tj.ƯrFu I3nyz̟n+23x,\Sstojeispunjenosa<_uvjetom(4.46).8Slijedidajejednadrszba(4.67)ispunjenazavË2URVrI.2y@50i-\h6.<_src:1992Ana.TEXGrffani3Lcni35slucaj35(II).Ispunjenjemonotonosti.<_src:1994Ana.TEXZbSog(4.62),(4.64)u3(0)UR!u(0)yu(Vp2s )l01ǹ,nadalje֕[u(0)UR=u̿0;<_src:1995Ana.TEXpadokXazalismoteoremakroprovjerimodaje'UR=A(u):(4.69)<_src:2000Ana.TEXTVrebamoprorvjeritisljedesce: SB[qdP\Akroiu;vË2URL2rb(0;Tƹ;Vr)@ A\L21 (0;Tƹ;HV)@b4;ꦹiakrovrijedi(4.52)e8;P\tadajefunkrcijamtUR!b(u(t);u(t);vn9(t))[izg1L21(0;Tƹ):(4.70)!<_src:2007Ana.TEXSaslaganjemsumade niranimu(4.52), ostajenampSokXazatidakada<_je;u;vË2URL rb(0;Tƹ;WO1;r0 Р( ))^(\L 1 (0;Tƹ;L 2( ))T%;<_src:2010Ana.TEXfunkrcija㍒3tUR!甆Z _ uvōR@ul[z, ΍@xidx꨹jeiz ^L 1(0;Tƹ):6<_src:2012Ana.TEXZbSog(4.68)(misemorszemoograniscitinasluscajkXadajeFuۿ1۟z@ꍐr ;FuR1۟z(Pn >UR0):cL rb(0;Tƹ;WO1;r0 Р( ))\L 1 (0;T;L 2( ))Wlwc-URL rb(0;Tƹ;L q( ))R\L 1 (0;Tƹ;L 2( ))WlwURL S(0;Tƹ;L ( ))Q;<_src:2016Ana.TEXgdjeō|YJ1|A[zt ΍ه=ō1[zW ΍ \nr"T+ō4۟[z  ΍1fb=ō1[zW ΍ 9$p ;ōQ1![z ΍&W=ō1[zW ΍ Zq+ō1۟[z ΍2 :<_src:2018Ana.TEXMiizabSeremo1,ˡt.d.Tje=,ˡtj.Takropustimodaje9=2=(n%2+2)<_tada{L rb(0;Tƹ;WO1;r0 Р( ))\L 1 (0;T;L 2( ))WlwURL S(Q)";(4.71)Yōj1ܮ[zt ΍t=ō@n[z* ΍(n+2)r/ڶ:(4.72)<_src:2026Ana.TEXMiLscemofprourcavXatifrezultatkadajeFu52 zZS&ʹ+Fu2j12jz@ꍐr X(1,tj.vkadajeispunjeno<_(4.46).<_src:2028Ana.TEXSadatrebamodokXazatidavrijedinejednakrost΍ō{c1{c[z ΍2vju(s)j 2ڍH +甆ZMs UT0h;uidtUR甆ZMUTs 0Ðhf;uidt+ō1۟[z ΍2 ju̿0j 2ڍHD:(4.73)3·y@51i<_src:2032Ana.TEXFiksiramos̿0;sUR2(0;Tƹ);s̿0VsFus1'zğm riliza<_tUR2mstarvimoC#vË=UR((mu)nRnP)m;(4.74)<_src:2038Ana.TEXpafunkrcijuvË=URvn9(t)iz(4.67)zamjenimosavXiz(4.74).wN<_src:2040Ana.TEXPrimjetimodaje!}甆ZMT>05hu801;vn9idtUR=甆ZMUTT 0khmu80;(mu)nRnPidtUR=*:Qŧ甆ZM]ũTXpS0dfh(mu)80onP;(mu)nPidt甆ZMT UT0Kh S-:0ڍmu;(mu)nRnPidtUR=$s^甆ZMT0h S-:0ڍmu;(mu)nRnPidtUR !n!1甆ZMT0m S-:0ڍmjuj 2ڍHDdt:;I<_src:2044Ana.TEXZbSog(4.70)"eMy甆ZMTdV0Zb(u;u;vn9)dtUR !n!1甆ZM%T ^0. S2ڍmb(u;u;u)dtUR=0; <_src:2046Ana.TEX(jerjeb(u;u;u)UR=0;8u2VrI),<_src:2048Ana.TEXpaiztogadobivXamoXOh甆ZMdOjT^0j(m S-:0ڍm)juj 2ڍHDdt+甆ZMT UT0K S2ڍmh;uidtUR=甆ZMUTT 0i S2ڍmhf;uidt:(4.75)֍<_src:2052Ana.TEXAkrompSoa?saljemoubeskronascnostdobivXamoWd@甆ZMcdBT^0jY(m S-:0ڍm)juj 2ڍHDdtUR!ō1[z ΍2 ju(s)j 2ڍH ō۹1۟[z ΍2 ju(s̿0)j 2ڍH;ꦹzasvXaki2^s꨹i s̿0;<_src:2054Ana.TEXpa,slijedi,񍍍ōK1K[z ΍2R"ju(s)j 2ڍH +甆ZMs UTsq0h;uidtUR=ō1[z ΍2 ju(s̿0)j 2ڍH+甆ZMs UTsq0hf;uidt;ꦹzasvXaki2^s꨹i s̿0:Ѝ(4.76)4Gy@52i<_src:2059Ana.TEXJednakro,<]kXakoşjeu}2L21 (0;Tƹ;HV)D5P;şmorszemonascis̿0nf!}Ĺ0,<]t.d. je<_zadorvoljenoK(4.76)iu(s̿0n T)slabSokronvergiraKuHV;pakXakrou(s)UR!u(0)^#=<_u̿0uV2p-:0RA17,papu(s̿0n T)UR*u̿0slabSou;HF::(4.77)ͺ<_src:2063Ana.TEXFiksiramost.d._zadorvoljavXa(4.76)iuzmemodajes̿0 =s̿0n T.ZbSog<_(4.77)dobivXamo(4.73).ē<_src:2065Ana.TEXSadamorszemodokXazati(4.69)koristescimonotonostoSdA. YZaR2<_L2rb(0;Tƹ;Vr)|hdX sڍ=UR甆ZMUTs 0ÐhA(u3)A();u:idt+ō1۟[z ΍2 ju(s)j 2ڍHD;(4.78)f<_src:2070Ana.TEXgdjeszadorvoljavXa(4.73).<_src:2072Ana.TEXZbSog(4.58)postojiM6>UR0takrodaje`jumjH nURM;ꦹzasvXaki2^m2N:<_src:2073Ana.TEXTVadapSostojepodnizfu3gURfumg꨹iuUR2Htakrodavrijedi}u3(s)UR*u(s)^slabSouEyjHF:;<_src:2075Ana.TEXpaizmonotonostioSdAi(4.78),slijedidajeliminf33p!1ڏX sڍō1[z ΍2 ju(s)j 2ڍHD:(4.79)Fm<_src:2080Ana.TEXJednakrozbSog(4.51)|<_X sڍ=UR甆ZMUTs 0Ðhf;u3idt+ō33133[z ΍2Fbju̿0 sj 2ڍHD甆ZM s0n>hA(u);idt甆ZM s0n>hA();uidtUR!X su}; <_src:2082Ana.TEXgdjejeՍRhX s=UR甆ZMUTs 0Ðhf;uidt+ō1۟[z ΍2 ju̿0j 2ڍH 甆ZMs UT0h;idt甆ZMs UT0hA();uidt:f<_src:2084Ana.TEXZbSog(4.79)dobivXamo:CU;甆ZMOU=sI0Syhf;uidt+ō1۟[z ΍2 ju̿0j 2ڍH 甆ZMs UT0h;idt甆ZMs UT0hA();uidtURō1[z ΍2 ju(s)j 2ڍHD:<_src:2086Ana.TEXIzorvenejednakostiiiz(4.73)dobivXamodaje甆ZMs0JhA();uidtUR0zasvXaki0^s:(4.80)<_src:2091Ana.TEXDokXazza(4.69)slijedipSomorscuMintyjeveproScedure,tj. stavljajusci<_UR=u+t Xipua?stajurscitUR!0(t>0it<0).ssrc:2094Ana.TEX~F;;5yi޹src:2097Ana.TEXDo datak T{A6޹Meto da T{monotonihoperatoraB޹A.1P=oMonotonizop=eratori@܍De nicijaA.1uÊsrc:2100Ana.TEXNekajeVmrffe eksivaniseparabilanBanachovprostor.EOpe-rffator35AUR:V!Vpl0 ljemonotonakojezasveu;vË2V}OhA(u)A(vn9);uviUR0;src:2102Ana.TEXgdje35h;iozna3LcavarffelacijudualnostiprostoraVϥiVpl07.^src:2104Ana.TEXOpfferatorAjehemineprffekidanakojezasveu;vn9;w2Y'V[preslikavanje!(A(u+vn9);wR)35neprffekidnosRuR.src:2108Ana.TEXOpfferatormAjeoffgrani3LcenmakooffgraniLcenempodskupoveprostoraV preslikavauoffgrani3Lcene35podskupoveprostoraVpl07.,A.2P=oStacionarnizproblem@܍TeoremA.1j*src:2112Ana.TEXNekanopfferataorAUR:V!Vpl0 zadovoljavasljeffde3Lcepretpostavke:)L(i)<_src:2114Ana.TEXA35jeoffgrani3Lcen35ihemineprffekidan,%(ii)<_src:2115Ana.TEXA35jemonoton,%P"|(iii)<_src:2116Ana.TEXA35jekoffercitivan,35tj,GfhhA(vI{);vifhzbꍑAjvI{jX.V(O!UR+ x1kadjvn9jV !UR+1.src:2118Ana.TEXTada35jeopfferator35Asurjektivan,tj.fizasvakifQ2URVpl0 lpffostojiu2Vϥt.d.fije})A(u)UR=f:(A.1) 536y@54isrc:2123Ana.TEXDokazj:NekXajew̿1;:::ʜ;wm;:::σbazauVp.JNekafunkrcijaum Z2URfw̿1;:::ʜ;wmg,zadorvoljavXajednadszbuNhA(umĹ);wjf iUR=hf;wjf i;1j%m:(A.2)src:2129Ana.TEXMorszemoprimjetitidavrijedihhA(umĹ);umihf;umiURhA(um);umicjumjVV;src:2131Ana.TEXpazbSogpretpostarvke(iii)vrijedidajehA(umĹ);umi>cjumjV I0,YkXadajejumjL2*( )ݹ=UR꨹idorvoljnoveliko.src:2135Ana.TEXJoa?spmorszemoprimjetitidajefunkcijavË!URhA(vn9);vipneprekidnanafw̿1;:::ʜ;wmg.src:2138Ana.TEXZbSog(A.2)slijedidaje" hA(umĹ);umiUR=hf;umijfGj34V㐟0 jumjVV;src:2140Ana.TEXpazbSogpretpostarvke(iii)imamoŴ\jumjV URC5:KakrojeopSeratorAograniscen,slijedidajejA(umĹ)j34V㐟0 URCܞ.src:2144Ana.TEXDaklepSostojipodnizu3,t.d.8jelu*URu꨹slabSou,ǴV;A(u3)*꨹slabSouVp807:(A.3)src:2149Ana.TEXIz(A.2)pri=URm,j{ ksirano,dobivXamoh;wjf iUR=hf;wjf i;8j;src:2151Ana.TEXpaslijedidajeѓ(UR=f:(A.4)src:2155Ana.TEXSNdrugeXstrane,ɛzbSog(A.2)hA(u3);uiUR=hf;u3i!hf;uiXizbSog(A.4)slijedida hA(u3);uiUR!h;ui:(A.5)src:2161Ana.TEXSadajoa?siz(A.3),(A.5)ipretpSostarvke(i)trebamodokXazatidajewUR=A(u):(A.6)src:2165Ana.TEXTVadaizjednadrszbSe(A.4)dobivXamodokazteorema.src:2167Ana.TEXImamo{hA(u3)A(vn9);u:viUR0;8vË2V:(A.7)7Cy@55isrc:2171Ana.TEXZbSog(A.3),(A.5)dobivXamoYhA(vn9);uviUR0;8vË2V:(A.8)src:2175Ana.TEXZavË=URuwR;>0;w2Vp,sa(A.8)dobivXamo8hA(uwR);wiUR0:src:2177Ana.TEXOdatleslijedidajeQhA(uwR);wiUR0;src:2178Ana.TEXikXadaUR!0,dobivamohA(u);wRiUR0;8w2Vp.src:2181Ana.TEXAnalognozaUR<0dobivXamoGshA(u);wRiUR0;8w2V;src:2183Ana.TEXoSdnosno`[hA(u);wRiUR=0;8w2V;a?stoimplicira(A.6).v;;Nap`omenaA.1}Hsrc:2187Ana.TEXTefforem35vrijediiakouvjet(ii)zamjenimouvjetomu|Ku*URu35slabffou+V;A(u3)*35slabffouVp80 liz[limsuphA(u3);uiURh;ui)=A(u):(A.9)src:2194Ana.TEXTada35iz(A.5)dobivamo(A.6)pffode niciji. De nicijaA.2uÊsrc:2197Ana.TEXOpfferator35AUR:V!Vpl0 ljestrffogomonotonakojei`hA(u)A(vn9);uviUR>0;8u;vË2V;u6=vn9:src:2200Ana.TEXProslijedicastrogemonotonostijeinjektivnostopSeratoraA.TeoremA.2j*src:2202Ana.TEXNekanopfferataorAUR:V!Vpl0 zadovoljavasljeffde3Lcepretpostavke:)L(i)<_src:2204Ana.TEXA35jeoffgrani3Lcen35ihemineprffekidan,%(ii)<_src:2205Ana.TEXA35jemonoton,"|(iii)<_src:2206Ana.TEXA35jekoffercitivan,#J(iv)<_src:2207Ana.TEXA(0)UR=0.8/y@56isrc:2209Ana.TEXTadajeopfferatorAbijektivan,tj.'zasvakifT|2 }Vpl0 dVjeffdnadzbaA(u)=fimajeffdinstveno35rjefgsenjeuUR2Vp. src:2211Ana.TEXDokazj: #MorszeseprovjeritidauzadovoljavXa(A.1)akoisamoako>8hA(vn9)f;vuiUR0;8vË2V: (A.10)src:2217Ana.TEXStvXarno,akrojeispunjeno(A.1)tadaimamohA(vn9)f;vuiUR=hA(u)f;vn9ui+hA(v)A(u);vuiUR=hA(vn9)A(u);vuiUR0:src:2219Ana.TEXAkrou(A.10)stavimovË=URuwĹ+wR;>0;w2Vp,1nakrondjeljenjasdobivXamohA(u+wR)f;wiUR0;src:2221Ana.TEXikXadaUR!0,zakljurscimodahA(u)2f;wRiUR0;8w2Vp,oSdakleslijedi(A.1).Uasrc:2226Ana.TEXSada*szelimoldokXazatidajeskuprjea?senjajednadrzbSeA(u)q=fzatvrorenlikronveksan. OznascimomsE!tajskup.ZavV2V de niramoskupSv ZsvihuUR2Vp,zakrojejeispunjeno(A.10).8TVadaW֍9E i=!\URvI{2VCSv ;"src:2228Ana.TEXakXakrojeSv׵zatvorenipSotprostorodVdokXazalismotrarszeno.src:2230Ana.TEXZbSogstrogemonotonostiskupEPrjea?senjajednadrszbe(A.1)jeusferijujV =URs3pSolumjerom, akXakrojeE۹zatvorenikonveksan, anormav!jvn9jV jestrogokronveksna,zakljuscujemodaseEsastojioSdsamojednetoscke.;;,mA.3P=oPseudomonotonizop=eratori7De nicijaA.3uÊsrc:2234Ana.TEXOpfferatorRDAh:VU!Vpl0 {jepseudomonotonakozadovoljavaprffetpostavke:f#)L(i)<_src:2236Ana.TEXA35jeoffgrani3Lcen,썍%(ii)<_src:2237Ana.TEXako35uj\*URuslabffouVϥilimsup(5Tjv!+x1D8+hA(ujf );ujui0tadaje>8oliminfI荑rjv!+x1hA(ujf );ujvn9iURhA(u);uvn9i;8vË2URV:"Prop`ozicijaA.1^6src:2242Ana.TEXVrijeffde8wsljede3Lceimplikacije:p"Ajeograni3Lcen,9heminepre-kidanimonoton.")"Ajepseudomonoton")"opfferatorAzadovoljavauvjet(A.9).".9y@57isrc:2247Ana.TEXDokazj:2AkroJMpSostojiujWtakodazadovoljavXapretpSostavku(ii)ude nicijiA.3iAjemonotoniopSerator,tadahA(ujf );ujuiUR!0: (A.11)src:2252Ana.TEXTVodobivXamoizmonotonostijerje9hA(ujf );ujuiURhA(u);ujui!0:src:2254Ana.TEXAkrojew=UR(1S)u+vn9;2UR(0;1),imamohA(ujf )A(wR);ujwiUR0:src:2255Ana.TEXIztogaslijedi.GShA(ujf );uvn9iURhA(uj);ujui+hA(wR);ujuiShA(wR);vui;src:2257Ana.TEXpazbSog(A.11)imamovSliminf"6pazbog(A.11)dobivXamo%liminfEhA(ujf );uj\Rvn9iURhA(wR);vRui;w=(1RS)u+vn9;82UR(0;1): (A.12)src:2263Ana.TEXAkrou(A.12)W!nɹ0dobivXamodajeispunjenosvojstvo(ii)izde nicijepseudomonotonogopSeratora.src:2266Ana.TEXSadanamjoa?sostajedokXazatidruguimplikaciju.aNekajeuj*uslabSouVp,A(ujf )J*MslabSouVpl0 $ilimsup(PQhA(uj);ujiJh;ui.bTVadaMlimsup(PQhA(uj);ujTHuiUR0,panam(ii)izDe nicijeA.3daje>hA(u);uvn9iURlimsup'WhA(ujf );ujvn9ih;uvn9i;8vË2URV;src:2273Ana.TEXpawUR=A(u):F;;src:2276Ana.TEXIzorvepropSozicijeiNapomeneA.1dobivXamosljedersciteorem:TeoremA.3j*src:2279Ana.TEXNekajeAz:V!Vpl0 ,pseudomonotonikoffercitivanoperator.Tada35za8fQ2URVpl07,jeffdnadzba35A(u)=f{4imabffarem35jednorjefgsenje.:yi͍src:2283Ana.TEXDo datak T{B6͍Top olo{tski T{stupanjinjego8vqaprimjenaB͍B.1OaFOsnouvnazsvozjstvasrc:2285Ana.TEXNekXa0jeUPotvroreniograniscenskupuX=R2n inekXaje2Cܞ(xuU ;;X).xNekXajejoa?sdanibUR2X+n(@U@).8Brffouwerov35stupanjTjecjelobrojnafunkrcija ódeg}m(;U;b);src:2288Ana.TEXkrojaimasljedescaosnovnasvojstvXa:-D*(i)<_src:2291Ana.TEXNormalizacijaܔ<_src:2293Ana.TEXStupanjpreslikXarvanjaIFչ:xVURUm!URXJg;I(u)=u(idenrtiteta),jedansa degR(I;U;b)UR=qdUS1!5Ozab2UUS0!5Ozab=2UUQ\:!d&P(ii)<_src:2297Ana.TEXSvojstvo35effgzistencijeX(Kronecrkerovprincip)<_src:2299Ana.TEXAkrovrijedidaje.Zdeg(;U;b)UR6=0<_src:2301Ana.TEXtadapSostojirjea?senjejednadrszbe(u)UR=b꨹uU@.9(#(iii)<_src:2302Ana.TEXAffditivno35svojstvo<_src:2304Ana.TEXNekXaHsuU̿1iU̿2disjunktni,iBneprazni,otvroreniHiogranisceniskupSoviuX<_inekXajeUR:xU̿1j[U̿2-u!X-neprekidnafunkrcija,ªtakodab=UR2(@Ui ۹);iUR=<_1;2.8TVada udegx(;U̿1j[U̿2e;b)UR=deg (;U̿1;b)+degtc(;U̿2;b): 58;y@59i#R(iv)<_src:2306Ana.TEXInvarijantnost35homotopijeM]<_src:2308Ana.TEXAkrorH*f2=Cܞ([0;1]ZxϹU C#;x)ribh==2z!HV([0;1]Z@U@)r(takvrosepreslikXavanje<_HnazivXahomotopija),tadadeg(HV(t;);U;b)UR=const:<_src:2310Ana.TEXtj.8stupanjoSdHV(t;)neorvisiot.Fsrc:2312Ana.TEXSada 1pSomorscugornjihaksioma((i)-(iv))izvoSdimojoa?snekXasvojstvXaBrouwe-rorvogstupnja.Prop`ozicijaB.1(Neprekidnoststupnjasobziromna)iKsrc:2314Ana.TEXPrffeslikavanjeW!degs(;U;b)jeneprffekidnosobziromnauniformnutopologijuprostoraneprffekidnih35funkcijaCܞ(xuU ;;X),akob=UR2(@U@).!<src:2318Ana.TEXDokazj: #KakrojestupanjcjelobrojnafunkcijadokXazslijediizsvojstvXa(iv).vXsrc:2321Ana.TEX~F;;Prop`ozicijaB.2Ssrc:2323Ana.TEXDa,lbiizrffa3Lcunalistupanjfunkcijenuznoidovoljnonamje35znatij@xU y.fiDrugimrje3Lcima,akoUR= nna35@U;b=2(@U@),35tadaSdeg](;U;b)UR=deg ( n9;U;b):#src:2326Ana.TEXDokazj:1Akro"u(iv)stavimodaje̿0(u)3:=HV(0;u)"i̿1(u)3:=HV(1;u),tadaXkXarszemodahomotopijaHE߹spajafunkcije̿0 i̿1.Doista,tpreslikXavanje[0;1]3t!tʹ():=HV(t;)morszebitigledanokXaoputuprostorufunkcijaCܞ(xuU ;;X)sa'krajnjimtorsckXama'u̿0 _i̿1. W'TVoscke'ovogputasufunkcijet;tUR2[0;1].8Akrosadade niramo~HV(t;u)UR=(1t)(u)+t n9(u);src:2334Ana.TEXdokXazalismopropSozicijukaodirektnrupSosljedicusvojstvXa(iv).>RH;;FNap`omenaB.1|src:2338Ana.TEXPrimjetimodajeputt!HV(t;)ubitiduffzinakojaspajafunkcije-i fuvektorskomprffostorufunkcijaCܞ(xuU ;;X). QOvojetakozvanaa nahomotopija.<Prop`ozicijaB.3(Svojstvoizreziv@anja)src:2343Ana.TEXA2kojeK zatvorffenipodskupodx!uU,ni35b=UR2(Kܞ),ondajedeg(;UnK5;b)UR=deg (;U;b):<%y@60isrc:2346Ana.TEXDokazj: #TVojedirektnapSosljedicaaditivnostiinormalizacije.Dr;;Prop`ozicijaB.4Ssrc:2350Ana.TEXPrffeslikavanjexXF3URb!deg (;U;b)jekonstantanasvakojkompffonenti35povezanostiodX+n(@U@). Prop`ozicijaB.5(Produktstupnja)Dsrc:2353Ana.TEXNeka5jei s02VCܞ(xUi _;Xidڹ);i=1;2,gdje6suUi !otvorffeniiograni3LceniskupoviuXi =-R2n dibi2-Xi nidڹ(@Ui ۹).TadaAڷdegSr((̿1;̿2);U̿1jU̿2e;(b̿1;b̿2))UR=deg (̿1;U̿1;b̿1)degtc(̿2;U̿2;b̿2):&src:2356Ana.TEXDokazj: #VidiNapSomenruB.2usljedescojtoscki.;;-@B.2OaFKonstrukucijazstupnja΍src:2360Ana.TEXPrijenegoFstoopia?semokronstrukcijustupnjanarvest1scemojedanrezultatkrojikroristimoukoraku(b).TeoremB.1(Sardovteorem)κsrc:2362Ana.TEXNekad"2Cܞ21(xuU ;;R2nP),]gdjejeUotvorffeniskuphuR2nP,D4iSع=efu2U;detLl01ǹ(u)=0g,singularnihskupoffd,gdjejedet.Ll01ǹ(u)>JacffobianodO=(̿1;:::ʜ;nP)>uU@,tj.determinantan n>matrice(wp@x8:i33z @xx8:j _)i;j X.3Tadaskupsingularnihvrijeffdnosti(S׹)preslikavanjaimaLebesqu-ۓeffovu35mjerunulauR2nP.#src:2371Ana.TEXDokazj: #DokXazsemorszenasciu[10 ].;;src:2375Ana.TEXxZelimoM9de niratitopSoloa?skistupanjzaneprekidnrufunkciju:xU5!XJg;xsU8Xm=R2nP;bB=2c(@U@).TVòscemo!naprarvitiutrikorakXa:(a)prvozafunkcijuklaseCܞ21 :"ibregularan,(b)tadazaklaseCܞ21ibsingularani(c)zaoprscenitufunkrciju.'|l(a)<_src:2377Ana.TEXPretpSostarvimoXdanQ2Cܞ21(xuU ;;R2nP),gdjeXjeU:0)((̿1;̿2);U̿1jU̿2;(b̿1;b̿2))UR=+xX<(uq1*;uq2)2i@1#Ս1 (bq1)i@1#Ս2(bq2)gsgnxJ߿(q1*;q2)J((u̿1;u̿2))>@yy@62iy'=+xX<UR(uq1*;uq2)2i@1#Ս1 (bq1)i@1#Ս2(bq2)gsgnx[Jq1 /(u̿1)Jq2(u̿2)]'"q>/=UR[ jX