Existence and Smoothness of Solution of Navier-Stokes Equation on R 3

Transcription

1 Ieaioal Joual of Mode Noliea Theoy ad Applicaio, 5, 4, 7-6 Published Olie Jue 5 i SciRes. hp:// hp://dx.doi.og/.436/ijma.5.48 Exisece ad Smoohess of Soluio of Navie-Sokes Equaio o R 3 Ogje Vukovic Depame fo Fiace, Uivesiy of Liechesei, Vaduz, Liechesei ogje.vukovic@ui.li, oggyvukovich@gmail.com Received 3 May 5; acceped Jue 5; published 5 Jue 5 Copyigh 5 by auho ad Scieific Reseach Publishig Ic. This wok is licesed ude he Ceaive Commos Aibuio Ieaioal Licese (CC BY). hp://ceaivecommos.og/liceses/by/4./ Absac Navie-Sokes equaio has fo a log ime bee cosideed as oe of he geaes usolved poblems i hee ad moe dimesios. This pape poposes a soluio o he afoemeioed equaio o R 3. I ioduces esuls fom he pevious lieaue ad i poves he exisece ad uiqueess of smooh soluio. Fisly, he cocep of ubule soluio is defied. I is poved ha ubule soluios become sog soluios afe some ime i Navie-Sokes se of equaios. Howeve, i ode o defie he ubule soluio, he decay o blow-up ime of soluio mus be examied. Diffeeial iequaliy is defied ad i is poved ha soluio of Navie-Sokes equaio exiss i a fiie ime alhough i exhibis blow-up soluios. The equaio is ioduced ha esablishes he disace bewee he sog soluios of Navie-Sokes equaio ad hea equaio. As i is demosaed, as he ime goes o ifiiy, he disace deceases o zeo ad he soluio of hea equaio is ideical o he soluio of N-S equaio. As he soluio of hea equaio is defied i he hea-sphee, afe is aalysis, i is poved ha as he ime goes o ifiiy, soluio coveges o he saioay sae. The soluio has a fiie ime ad i exiss whe ha implies ha i exiss ad i is peiodic. The afoemeioed saeme poves he exisece ad smoohess of soluio of Navie-Sokes equaio o R 3 ad epeses a majo beakhough i fluid dyamics ad ubulece aalysis. Keywods Navie-Sokes Equaio, Milleium Poblem, Noliea Dyamics, Fluid, Physics. Ioducio I his pape, he followig fom of Navie-Sokes equaios i R 3 is sudied: u p u u y u ( x, ) ( x R, ) () i i j = ν i fi j= xj xi How o cie his pape: Vukovic, O. (5) Exisece ad Smoohess of Soluio of Navie-Sokes Equaio o R 3. Ieaioal Joual of Mode Noliea Theoy ad Applicaio, 4, 7-6. hp://dx.doi.og/.436/ijma.5.48

2 Wih iiial codiios ui div u = = ( x R, ) () x i= i (,) u x = u x x R (3) Hee u = x, C (divegece-fee veco field o x ae he compoes of a give, exeally applied foce, v is a posiive coefficie (he viscosiy) ad = is he Laplacia i space vaiables. If i= xi Eule equaios ae cosideed, he he same se of equaio mus be applied wih he codiio ha viscosiy is equal o zeo. The followig codiios mus be saisfied as i is waed o make sue ha u( x, ) does o gow lage as x : Ad α K K R ), fi (, ) α o u x C x o R fo ay α ad K (4) x K [ [ fo α m f x, C x o Rx, ay α, mk, (5) x αmk The acceped soluio of N-S is physically easoable if i oly saisfies: Ad R ( [ [) pu, C Rx, (6) (, ) d < fo all ( bouded eegy) u x x C (7) A he same ime, i is possible o look a spaially peiodic soluios. We ca assume he followig codiios: Ude he codiio ha ej o ( j) ( j ) o u x e = u x, f x e, = f x, fo j (8) h = j is ui veco i R. I mus be assumed ha K 3 [ [ o u is smooh ad ha α m f x, C o Rx, fo ay α, mk, (9) x αmk The soluio is he acceped if i saisfies: Ad 3 (, ) [, [ u x, = u x e o R x fo j () j ( [ [ ) pu, C Rx, The poblem is o fid ad aalyze whehe a sog, physically easoable soluio exiss fo he Navie- Sokes equaio. The saeme ha will be poved is exisece ad smoohess of Navie-Sokes soluios o R 3. Take v > o ad = 3. Le u ( x ) be ay smooh, divegece-fee veco field saisfyig (.4). Take f( x, ) o be ideically zeo. The hee exis smooh fucios p( x, ), ui ( x, ) o R 3 x[, ] ad he above codiios ad equaios ae saisfied.. Resuls Fisly, he defiiio of ubule soluios (Olive ad Tii) [] is povided. We mus defie he se of all C eal veco fucios ϕ wih compac suppo i R such ha div ϕ =. We defie L as he closue of 8

3 C, wih espec o L om. ; (.,.) is he ie poduc i L. L sads fo he usual L -space ove R,. H, is he closue of C, wih espec o he om φ = φ φ whee H m Whe X is a Baach space,. X deoes he om o X. C ([, ] ; X ) ad ([, ] ; ) ϕ i ϕ =. x j i, j=,, L X ae he Baach spaces, whee m =,,, ad ad ae eal umbes such ha <. C deoes vaious cosas. Def. (Olive ad Tii) [] A ubule soluio of Navie-Sokes equaio is defied as followig: The elaio u L, ; L L, T; H fo all < T < () ) ( ) (, ) T ) ( u, ϕ ) ( u, ϕ) ( u u, ϕ) d = ( a, ϕ( ) ) () (, ) Holds fo almos all T ad all [, [; ϕ C T H L such ha ( T ) Sog eegy iequaliy 3) u( ) u( ) d u( s) s ϕ, = (3) Holds fo almos all s icludig s =, ad all > s. I is ecessay o ioduce he Sokes opeao A i L. The followig Helmholz decomposiio is obaied: whee G { p L ; p L loc } L = L G,< < =. P deoes he pojecio fom L oo L. A defies he Sokes opeao wih domai D( A ) = H L,. A deoes he Sokes opeao A. { E } deoes he specal decomposiio of self-adjoi opeao A. The exisece of ubule soluios fo = 3 ad = 4 is give by Leay ad Kao. I ode o deive he ex esuls, heoem fom Takahio Okabe will be ioduced. Theoem. Le 4 ad le > ad m be Fo =, Fo = 3, 4 < < 4 3, m< 4 3 ad < <, m < K δ fo α { m } δ Suppose ha ˆ m, Km, α = ϕ L ; ϕξ ( αξ fo ξ δ fo αδ>, ad m. If a L L K δ m, α fo some αδ>, he fo evey ubule soluio u( ) hee exis T > ad C m,,, δα,, a > such ha: Eu u C ( m) (4) holds fo all ad fo all > T Def. Le < <, a L. A measuable fucio u defied o R (, ) is called a global sog soluio of Navie-Sokes equaio if: u, Au C, ; L (( ) ) ad u saisfies: ([ ) ) u C, ; L C, ; L (5) 9

4 whee P ( u u) u Au P u u = >, deoes he pojecio fom L oo L of he poduc of he divegece of soluio u ad he soluio iself. Takahio Okabe [], i his pape amed Asympoic eegy coceaio i he phase of he weak soluios o he Navie-Sokes equaio, poves ha ubule soluios of Navie-Sokes equaio become sog soluios afe some defiie ime. So fo he ubule soluio of T > such ha u( ) is a sog soluio of Navie-Sokes equaio o [ ) * iss: (6) u of Navie-Sokes equaio hee exiss T, he he eegy ideiy ex- d / d u A u = (7) Fo T*. Fo ay fixed >, he secod em i (6) is esimaed fom below as: = ρd p ρd p d p (8) *, / A u Eu Eu Eu u Eu Fom (6) o (8), he followig is obaied: d d u u E u χ (9) Afed dividig he boh sides of (9) by u( ), he followig is obaied: d u d u Eu () u χ By (7), he followig is obaied ( dd) u( ) A u( ) u( ) ha: = = i follows fom (7) o () Eu χ u () u u By ioducig he ew heoem ha is poved i Takahio Okabe s pape [], he followig is obaied. Theoem. Le 4. Le ad m be as ) = ) 3 If a L L K δ, 4 4 < <, m< 3 3 < <, m <, evey ubule soluio of u u u of Navie-Sokes equaio saisfies: ( m) O () As. The followig heoem ca be poved by usig well-kow Leay s sucue heoem, evey ubule soluio of N-S becomes he sog soluio afe some ime. Alhough Kao poves ha he sog soluio decays i he same way as he Sokes flow e A, we apply diffee appoach by usig Olive ad Tii s pape [] amed Remak o he Rae of Decay of Highe Ode Deivaives fo soluio o he Navie-Sokes equaio.

5 By ioducig he above meioed heoem, he followig esul is obaied ad i poves Theoem. Eu C ( m) fo all T u( ) This esul poves ha eegy of he molecules of fluid movig is smalle ha some value deemied by C,,, m ad i poves asympoic eegy coceaio. I ode o pove ha ubule soluios ae a he same ime sog soluios, blow-up ime of soluios mus be aalyzed. I is demosaed ha Navie-Sokes equaio ee some class as i was aleady poved ( e A D ; H ) i abiaily sho ime. Foias ad Temam have poved he followig soluio i he case of peiodic bouday codiio ad fo he case of he Navie-Sokes equaio o he wo-dimesioal. Kukavica ad Gujic have obaied he give esuls i L spaces. The followig lemma mus be ioduced ad i is poved i Olive ad p Tii s pape []: Theoem 3. Le, > ad s <. The hee exiss a cosa C= C( s,, ) such ha ay e A D ; H saisfy he iequaliy: wo fucios v ad w i ( s s ) A A s A A A e vw C,, s A e w A e v A e w (3) L H H L The heoem is poved by usig Placheel heoem, he iagle iequaliy, he iequaliy ( x y) ( x y ) ad he covoluio esimae f g f g. These ae he ools used o pove L L L he afoemeioed heoem. Fo fuhe deails, look a he afoemeioed pape. This heoem demosaes ha he blow-up ime is ifiie so ha he soluio is exise. I ode o fid a soluio, i mus be capued i some so of space whee he fucio oscillaes. I ode o ioduce he followig soluio, a few moe esuls will be ioduced. Fisly, we assume he exisece of soluios u L ([, T] ; H ( R )), > is kow fo some T >. I ode o simplify he oaio, he followig is se: whee ( ) = is o be specified lae. The he Gevey om is used o fid he followig esul: R J = Au (4) L L A G = A e u (5) A A G = G vg A e u u A e udx (6) The coibuio of pessue em is zeo because A commues wih he Leay pojecio oo divegece fee veco fields. Noe ha: s A Ae u cg G (7) s H ( s ) By usig Theoem 3 ad Cauchy-Schwaz iequaliy, he followig esul is obaied. ( ) ( ) ( ) (!) A A A A A e u u A e udx A e u u A e u c Gs G G G c Gs G G (8) R I ode o poceed, we ioduce he Theoem 4. Theoem 4. Fo all oegaive p, q ad we have he followig: ( ) p A p q p q A Ae u e Au A e u (9) The poof is simila o ha i Theoem 3, jus i should be oed ha fo evey x, m > oe has x m x e < e x e sice e x x m x < e o [, ] ad e x e fo x. Afe ioducig he heoem ad iepolaig G s by usig Theoem 3 ad Theoem 4 wih p = s, q = s, he simila hig is doe wih G s =. If we apply he Youg iequaliy, he followig esul is obaied.

6 ( ) ( ) ( ) A A A A / / / / s / A e u u A e udx A e u u A e cjs G G c3js G c4 GG (3) R whee 4< s <. Afe seig =, afe iepolaig he fis em o (6) he use he esimae o (3), he followig equaio is obaied: ( ) (,, ) G c u G c s G (3) H This poves ha hee exiss a (,T ] such ha G ( ) = u is fiie fo [, ). This poves ha H if space is fiie, he Gavey space is fiie which demosaes he exisece of saioay soluio. Now he esul of diffeeial iequaliy fo loge ime will be deived. The adius of uifom aalyiciy ρ = iceases like as as he soluios fo hea equaio. Fis he opimal decay ae fo Gevey om is esablished, he opimal decay aes fo oms of fiie ode deivaives will be esablished ad i will be exeded o ifiie ode. If fis wo ems of Equaio (6) ae cosideed ad i is assumed ha oly coibuio fom liea ems is icluded, iepolaio ca be used as well as Youg iequaliy while beakig he secod em i seveal facios. Theoem 3 povides he followig: G J G (3) we all ogehe obai: v v G J G vg G G G (33) v v v v 3v = G G G J G New heoem is ioduced, i is aleady poved by usig Placheel heoem: Theoem 5. Povided ha q p ad >, he followig is obaied: q p q p A, e Au c pq u A u (34) Combiig Theoem 5 wih q = ad he Youg iequaliy, he followig is obaied. J c3 J G (35) 8 If we se = ( α) whee > ad < α v. The followig is immediaely foud. α v = (36) 4 8 So ha he fis wo ems o he igh of equaio (33) ae oposiive ad ca be egleced. The mai ask is ow o aalyze he oliea ems ad if possible pove ha hese oliea soluios do o affec he decay popeies of he soluio o ifiie ode. Applyig he esimae o oliea em ad by iepolaig J s by usig heoem p =, q = s; J s is iepolaed i a aalogous mae. By applicaio of Youg iequaliy, he followig is foud. R ( ) A A Ae u u Ae ux d ( ) s 4 34 s 4 54 s s c J G G c J G c G G c J G 3v c J G C G G 8 ( ) s 4 54 ( s) 6 8 (37) 3. Theoeical Fidigs The followig diffeeial iequaliy is obaied.

7 v v s 3 ( ) s 4 54 ( s) G G c3 J ( ) c7 J G c6 J G c8 G (38) 8 As we ae cosideig global asympoics ad blow-up pofiles, hey ae oly possible i he pesece of a ciical coolled quaiy o he combiaio of a subciical ad a supeciical coolled quaiy. I us ou ha he Navie-Sokes equaio accodig o diffeeial iequaliy eds o coac hese quaiies, i ha way leadig o a useful way o foce fiie ime blow-up. The idea of usig miimal suface aea as coolled quaiies oigiaes fom Hamilo. I ode o discuss he blow-up ime, we ioduce he followig well kow poposiio: Assume ha π ( M ) is o-ivial. Le β : S M be ay immesed sphee o homoopic o a poi. Each such sphee has a eegy E( β, ) : = dβ usig he meic g a ime. If we defie W ( ) o be he ifimum of E(, ) S g β ove all such β. I us ou fom sadad Sacks-Uhlebeck miimal suface heoy ha his ifimum is acually aaied. The diffeeial iequaliy is obaied usig sucue of miimal sufaces ad he Gauss-Boe fomula [3]: W ( ) 4 π mi R W (39) whee R mi is he Ricci scala. I demosaes ha he chage of ifimum of eegy becomes egaive i fiie which is absud. Theefoe his foces blow-up i fiie ime. This meas ha he soluio blows up i a fiie ime, which is why he sugey appoach will be used. If he above meioed sae holds, he he diffeeial iequaliy, i ode o make oliea ems of lowe ode, has o saisfy he followig fom: v s ( ) s 4 4 ( s) > c 7 J G c6 J G c8 G (4) 3 whee s (, ) ( ) is fixed. Fis i mus be oed ha G is a iceasig fucio of, so ha a A he begiig a he iiial ime =, G is bouded bewee Au whe = = ad A e u whe = =. Thus he lef side of equaio (39) diveges fase ha he igh side as, so ha we ca saisfy codiio a = by choosig (, ] small eough. Howeve, wha happes whe does covege o. Imagie, he he lef pa of equaio is ad he igh pa is highe ha zeo, bu ha is o possible, because i is poved above ha he ifimum of eegy becomes egaive, ha is absud. So he soluio mus blow up i some defiie ad he equaio mus hold eve fo as a soluio. This poves ha he soluio is exise ad smooh. I ode o poceed, we will aalyze he oliea ems. Afe havig poved ha he above equaio mus hold eve fo some ha does o covege o, he oly equaio ha mus be solved is he followig: c4 G δ G J ( ) (4) whee δ = v 6. Accodig o assumpio ha hee exis posiive eal umbes M ad γ which may de- ped o ( ) γ γ povided ( α) u such ha u( ) io J M ( ) M fo all whee u( ) is a soluio o he Navie-Sokes equa- = ad J = u, whee L > ad < α v, a fial fom of diffeeial iequaliy is obaied. The iegaig faco fo liea diffeeial iequaliy is: k G δ G (4) ( γ ) δα exp α δ d = α (43) 3

8 So he followig is obaied. d d If we fix α small eough so ha δ α( γ ) δα ( ) ( G δα γ ) k (44) >, he followig is cocluded: δα k k ( ) δ α γ δ α γ G G ( γ ) ( γ ) If he codiio (39) is saisfied fo all, esimae (44) will be global i ime. I is sufficie o show he followig: s ( ) s 4 4 s c7 J G c6 J G c8 G g( ) 3 v (46) fo some o-iceasig fucio g( ). Esimae (44) shows ha his is he case wheeve γ > ad G ( ) ( γ ) (45) k > (47) δ α γ which saisfies he above meioed codiios ad i poves he exisece of a soluio. As, G coveges o zeo heefoe he soluio is exise a he begiig, ad if he equaios exis, he he soluio exiss i he ime. I is obaied ha: The uppe boud of decay is calculaed ad give below: whee (, ) ζ ( q p) c6 δα G ( ) O ( γ ) ( ) (48) γ m m m c γ Au c m J G c m M O 6 δα,, ( ) γ δα (, ) ( O γ m ( )) cc 9 m c m is give above accodig o he followig defiiio ad maximum is aaied a = so he followig defiiio demosaes: (49) q p ( q p) = > (5) c q, p q p e fo q p c q, p =, fo q = p (5) This poves ha soluio is exise eve whe does o covege o. Now i ode o poceed ad aalyze he blow-up ime, v as he soluio of he hea equaio will be ioduced. I should be poved ha he soluio w= u v bewee Navie-Sokes ad hea soluio i A ca m be made sufficiely small so ha u mus decay a he same ae. e A D ; H. Clealy, i saisfies he followig equaio: Fis a esimae o he diffeece w i w v w u u p = (5) w = (53) As he hea equaio peseves he divegece codiio, he followig equaio is obaied w = fo all. Seig: Aw L γ = (54) A ζ = A e w (55) L 4

9 Ad epeaig he seps, he followig esul is obaied: ζ ζ ζ vζ c( Gs G ) G G c( Gs G ) G v v v c3 v = ζ ζ ζ ζ ( ) O 3γ The secod of oliea ems aises fom (47) by usig ad choosig he smalles possible s =. Fo, he followig is obaied: G G G = G G G = G G highe _ ode _ ems s / / The followig diffeeial iequaliy is obaied: Ad he followig is obaied: = O O highe _ ode _ ems γ ( γ ) (56) (57) δ εc8 ζ ( ) ζ O ( γ ) 3γ 5 (58) m A w ( m, ) εc highe _ ode _ ems (59) ( γ m) Afe havig poved ha soluio fo exiss ad if we examie he equaio, as he disace bewee hea equaio soluio ad Navie-Sokes equaio demosaes covegece ad if he followig hea equaio soluio is foud he he soluio fo Navie-Sokes equaios exis ad is i he same age as hea equaio soluio. Now he hea soluio equaio Cao [4] is aalyzed. The soluio of hea equaio: Saisfies a mea-value popey Pecisely if u solves Ad The whee E is a hea ball, u = (6) u =, (6) u = (6) ( x, ) E dom( u) (63) (64) y u( x, ) = u( x y, s) dd, s y 4 E s { } E : = ys, : Φ ys, >, (65) Noice ha ( x) ( ) x Φ, : = 4 π exp. 4 ( E ) o (66) diam = (67) 5

10 So ha demosaes ha equaio is exise ad is capued i he ball if he is fiie. The pevious assumpios ad esuls pove he exisece of smooh ad sog Navie-Sokes soluio of equaio i R 3 ad epese he soluio of milleium poblem i R Coclusio I is poved ha he sog soluio of Navie-Sokes equaio is smooh, exise ad uique. Fisly, ubule soluios ae defied ad i is poved ha hey ae sog soluio, bu as he ubule soluios ae oly possible fo small ime ievals, i is ied o exed he ime ieval by usig he Equaio (39) ad i is poved ha he diffeeial iequaliy (4) holds a he same ime fo some ha does o covege o. The he esul is esablished, i is demosaed ha soluios exhibi possible fiie blow-up ime, which meas ha hey exis ad pesis i he sysem. I ode o esablish if he soluio exiss fo he fiie ime, he hea equaio soluio ad Navie-Sokes soluio ae compaed. I is poved ha wo soluios covege as which poves he exisece of soluio i ifiie ime. If a sugey pocedue is applied, he soluio exiss fo some ime, he blows up, he aises agai ad ha pocess epeas. This saeme poves ha he soluio is eihe exise o peiodic, bu i exiss all he ime. I is possible o ioduce a sochasic pocess i ode o explai he exisece of he dyamical peiodic soluio, bu his is lef fo fuhe eseach. This pape poves he exisece of Navie-Sokes soluio i R 3 ad epeses a beakhough i fluid dyamics aalysis. Ackowledgemes I would like o hak my family, my favouie au, my gadpaes who had a emedous ifluece o my love owads mahemaics. I wa o hak my au Cica, my au Soja ad he husbad Voja, he daughes, my ucle Nemaja ad his family ad all ohe elaives who povided me immese suppo. Love you all. Refeeces [] Olive, M. ad Tii, E.S. () Remak o he Rae of Decay of Highe Ode Deivaives fo Soluios o he Navie- Sokes Equaios i R. Joual of Fucioal Aalysis, 7, -8. hp://dx.doi.og/.6/jfa [] Okabe, T. (9) Asympoic Eegy Coceaio i he Phase Space of he Weak Soluios o he Navie-Sokes Equaios. Joual of Diffeeial Equaios, 46, hp://dx.doi.og/.6/j.jde [3] Tao, T. (6) Peelma s Poof of he Poicaé Cojecue: A Noliea PDE Pespecive. hp://axiv.og/abs/mah/693 [4] Cao, J.R. (984) The Oe-Dimesioal Hea Equaio. Vol. 3. Cambidge Uivesiy Pess, Cambidge. hp://dx.doi.og/.7/cbo