On the Incompressible Navier-Stokes Equations with Damping *

Transcription

1 Apple Maheacs hp://xoorg/436/a34489 Publshe Ole Aprl 3 (hp://wwwscrporg/oural/a) O he Icopressble Naver-Sokes Equaos wh Dapg * Weya Zhao Zhbo Zheg # Depare of Maheacs Baosha Uversy Baosha Cha Eal: # zhegzhbo34569@6co Receve Noveber 3 ; revse February 7 3; accepe March 3 3 Copyrgh 3 Weya Zhao Zhbo Zheg Ths s a ope access arcle srbue uer he Creave Coos Arbuo Lcese whch pers uresrce use srbuo a reproco ay e prove he orgal work s properly ce ABSTRACT We coser yacs syse wh apg whch are obae by soe rasforaos fro he syse of copressble Naver-Sokes equaos These have slar properes o orgal Naver-Sokes equaos he scalg varace Due o he presece of he apg er coclusos are ffere wh provg he org of he copressble Naver-Sokes equaos a ge soe ew coclusos For oe for of yacs syse wh apg we prove he exsece of soluo a ge he exsece of he aracors Moreover we scuss wh l-behavor he eforaos of he Naver-Sokes equao Keywors: Icopressble Naver-Sokes Equao; Soluo; Maxal Aracor; L-Behavor Iroco Cocere wh he perurbe Naver-Sokes equaos: We ee he followg prelares: Equaos () are suppleee wh a bouary coo Two cases wll be cosere: The oslp bouary coo The bouary s sol a a res; hus u () The space-peroc case ere L L a u u u up f u x () u where R s a sooh boue oa wh bouary u u u a p u s he velocy vecor p x s he pressure a x a e a s he keac vscosy a f represes volue forces ha are Reark u p he frs ervaves of u are -peroc (3) apple o he flu a where s he Reark If s sol bu o res he he oslp 4 bouary coo s u o where frs egevalue of A (see Reark 4) The Equao () x s he gve velocy of s Naver-Sokes equaos as whch show he Reark Tha s u a p ake he sae values a exsece of absorbg ses a he exsece of a axal aracor he uversal aracor aracor u- correspog pos of Furherore we assue hs case ha he average boue oa (see [-5]) ACTA Maheacal Applcao Sca I [6] where hey soe eresg re- flow vashes suls as I [78] Bab shk a Abergel coser axal aracors of segroups correspog o ux (4) evoluo ffereal equaos exsece a fe esoaly of global aracor for evoluo equaos plee hese equaos wh Whe a al-value proble s cosere we sup- o uboue oas I [9] A Pazy coser Segroups of lear operaor a applcao o paral ffereal equao u x u x x (5) For he aheacal seg of hs proble we coser a lber space (see [8]) whch s a close subspace of L ( here) # Correspog auhor I he oslp * The research s suppore by he Scece Fouao of Baosha Uversy (No3BY33) case Copyrgh 3 ScRes

2 W Y ZAO Z B ZENG 653 u L u u v v o (6) a he peroc case Reark 3 ul u u Λ (7) We refer he reaer o R Tea [] for ore eals o hese spaces a parcular a race heore showg ha he race of uv o exss a belog o whe ul a vul The space s eowe wh he scalar proc a he or of L eoe by a Reark 3 a are he faces x a x L of The coo u expresses he u perocy of u v sasfyg (4) Aoher useful space s ; L s he space of u L a close subspace of he oslp case a he space-peroc case u per vu (8) (9) where per s efe [] I boh case v s eowe wh he scalar proc u u a he or v u v uv x x u u u We eoe by A he lear uboue operaor whch s assocae wh a he scalar proc uv Auv uv The oa of A s eoe by DA; A s self-ao posve operaor Also A s a soorphs fro DA oo The space DA ca be fully characerze by usg he regulary heory of lear ellpc syses (see [3]) D A Ι a D A per he oslp a peroc cases; furherore Au s o DA a or equvale o ha ce by Le be he al of ; he ca be efe o a subspace of a we have DA () where he clusos are couous a each space s ese he followg oe Reark 4 I he space-peroc case we have Au u u DA whle he oslp case we have Au P u uda where P s he orhogoal pro L o he space We ca also say ecor ha ud A ha here exss p Au f f s equvale o sayg such ha ugra p f vu u The operaor a sce he ebeg of L s copac he ebeg of s copac Thus A s a self-ao couous copac operaor a by he classcal specral heores here exss a sequece A s couous fro o D A Λ a a faly of elees w hooral a such ha of D A whch s or- Aw w () We ee he followg a Resul: Lea (see [4]) (Ufor Growall Lea) Le g hy be hree posve locally egrable fuco o such ha y s locally egrable o a sasfy for y gy h for x r r r g s s a h s s a y s s a where ra a a 3 3 a 3 are posve cosa The y r a exp a r The evoluo of he yacal syse s escrbe by a faly of operaors S ha ap o self a eoy he usual segroup properes (see [8]): Iefy S s S S s s S I s acouous S T fro o self olear operaor () (3) The operaor S are uforly copac for large By hs we ea ha for every boue se X here exss whch ay epe o X such ha s relavely copac see [] (4) S X for every boue se C r sup S as (5) c C Of course f s Baach space ay faly of operaors sasfyg (4) also sasfes (5) wh S Theore (see [4]) We assue ha s a erc Copyrgh 3 ScRes

3 654 W Y ZAO Z B ZENG space a ha he operaors S are gve a sasfy () (3) a eher (4) or ( 5) We also assue ha here exss a ope se a aboue se X of such ha X s absorbg The w-l se of X wx s a copac aracor whch aracs he boue se of I s he axal boue aracor (for he cluso relao) Furherore f s a Baach space f U s covex a he appg S u s couous fro R o for every u ; he s coece oo The res of h s paper s orgaze such ha Seco coas a skech of exsece a uqueess of soluo of he equaos; Seco 3 we show he exsece of absorbg se a he exsece of a axal aracor; Seco 4 coa he proof of exsece a uqueess of soluo of he equaos Seco 5 scusse he perurbao coeffces Exsece a Uqueess of Soluo of he Equaos The weak fro of he Naver-Sokes equaos e o J Leray [-3] volves oly u as I s obae by ulply () by a es fuco v a egrag over Usg he Gree forula () a he bouary coo we f ha he er volvg p sappears a here reas uv uv buuv f v uv () where v buvw u w x () x wheever he egrals ake sese Acually he fro b s rlear couous o a parcular o We have he followg equales gvg varous couy properes of b: buvw u Au v w uda v w c u u v w w uvw [] (3) where c s a approprae cosa A alerave fro of () ca be gve usg he operaor A a he blear operaor B fro o efe by we also se B u v w b u v w u v w (4) B u B u u u a we easly see ha () s equvale o he equao whle (5) ca be rewre AuBu f u (5) u u (6) We assue ha f s epee of so ha he yacal syse assocae wh (5) s auooous f f (7) Exsece a uqueess resuls for (5) (6) are well kow as (see [3]) The followg heore collecs several classcal resuls Theore 3 Uer he above assupo for f a u gve here exss a uque soluo u of (4) (5) sasfyg u L T; Ι L ; ; Furherore u s aalyc wh values D A for a he appg u α u s couous fro o D A ; Fally f u he u L T; Ι L T; DA T Soe caos for he proof of Theore 3 wll be gve Seco 4 Ths heore allows us o efe he operaors S : u α u These operaor eoy he segroup properes () a he are couous fro o self a eve fro D A o 3 Absorb g Ses a Aracor The par proof abou global aracor s slar o he Tea s book bu he exss of perurbao er s ffere fro he Tea s book so we reprove for egraly Theore 4 The yacal syse assocae wh he ow-esoal ofe Naver-Sokes equaos suppleee by bouary () or (3) (4) possesses a aracor ha s copac coecea axal aracs he boue ses of a s also axal aog he fucoal vara se boue Proof We frs prove he exsece of a absorbg se A frs eergy-ype equaly s obae by akg he scalar proc of (5) wh u ece We see ha buvv uv (3) Bu u a here reas u u f u u u (3) We kow ha u u where s he frs egevalue of A ece we ca aorze he rgh-ha Copyrgh 3 ScRes

4 W Y ZAO Z B ZENG 655 se of (3) by f u f u u f 4 he esaes uu : uu u u u 4 ece we oba u u f (33) u u f Usg he classcal Growall Lea we oba exp u u f exp Thus (34) (35) l sup u f (36) We fe r (35) ha he ball B of wh are posvely vara s for he segroup S a hese balls are absorbg for ay We choose a clue a ball B R of I s easy o ece fro (35) ha S X XB for X where log R We he fer fro (33) afer egrao ha (37) r u s f u r (38) Wh he use of (36) we coclue ha r r l sup r u s f u (39) 3 a f u X B R a X he r r u s f (3) ) Absorbg se A coue a show he exsece of a absorbg se For ha purpose we oba aoher eergy-ype equao by akg he scalar proc of (5) wh Au Sce Au u uu u we f u Au B u Au f Au u Au we wrer uau u Au Au u 4 f Au f Au Au f 8 a usg he seco equaly (3) ece a sce We also have 3 B u Au c u u Au c Au u u u Au 4 c f u u 3 A L L u 4 (3) D A (33) u Au 4 c f u u u 4 3 (34) We a pror esae of ul T; T follows easly fro (34) by he classcal Growall lea usg he prevous esaes o u We are ore erese a esae val for large Assug ha u belog o a boue se X of a ha X as (37) we apply he ufor Growall lea o (34) wh g hy replace by c 4 3 u u f u u Thaks o (4) (8) we esae he quaes a a a Lea by 3 a we oba c a 4r a r a 3 a3 f 3 f (35) Copyrgh 3 ScRes

5 656 W Y ZAO Z B ZENG a3 u a exp afor r r (36) as (37) Le us fx r a eoe by he rgh-ha se of (4) We he coclue ha he ball B of eoe by X s a absorbg se for he segroup Furherore f X s ay boue se of he S X X for X r Ths shows he exsece of a absorbg se aely X a also ha he operaors S are uforly copac e Theore s sasfe ) Maxal aracor All he assupo of Theore are sasfe a we ece fro hs heore he exsece of a axal aracor for ofe Naver-Sokes equaos 4 Proof of Theore 3 The exsece of a soluo of ( 4) (5) ha belog o L T; L T; T s frs oba by he Fa eo-gakerk (see [3]) eho We plee hs approxao procere wh he fuco w represeg he egevalues of A (see Reark 4) For each we look for a approxae soluo u of he for sasfyg u g w w a u w b u u w f w u w u P u (4) (4) where P s proecor (or ) o he space spae by w w Sce A a P coue he relao (3) s also equvale o We prove hece Au P B u P f P u u w o g hu g ww s Lp couous h g h g g ww g ww M (43) here s such ha g M w Whe w w w s esablshe a whe w w s esablshe he h g h g C g g O boh ses he egral he we wrer h g h g x C g g x Cg g u w u w C g g u w o g ece s Lp couous The exsece a uqueess of u o soe erval T s eleeary a he T because of he a pror esaes ha we oba for u A eergy equaly s obae by ulplyg (4) by g a sug hese relaos for We ob a (3) exacly wh u replace by u a we ece fro hs relao ha u reas boue L T; L ; T Due o (3) a he las equaly (3) B Therefor ; T (44) (45) B u a Pu rea boue L T a by (43) u reas boue L T; (46) By weak copacess follows fro (43) ha here exss u L ; T; L T; a a subsequece sll eoe such ha ; weakly a ; u ul T L T weak-sar L T; weakly (47) Due o (46) a a classcal copacess heore (see []) we also have u u L T ; srogly (48) Ths s suffce o pass o he l (4)-(43) a we f (4) (5) a he l For (5) we sply observe ha (47) ples ha u u weakly or eve T By (4) u belog o L T; a u s L T; The uqueess a couous epee of u o u ( ) follow by saar usg [] The fac ha u L T; L T; DA T s prove by ervg furher a pror esaes o u They are obae by ulplyg (4) by g a sug hese relaos for Λ Usg () we f a relao ha s exacly (3) wh u replace by we ece for hs relao ha u Copyrgh 3 ScRes

6 W Y ZAO Z B ZENG 657 T T DA u reas boue L ; L ; (49) T A he l we he f ha u s L T : D The fac ha u L T he follows fro a approprae applcao of Lea 3 [] Fally he fa c ha u s aalyc wh values DA resuls fro oally ffere ehos for whch he reaer s referre o C Foas a R Tea [] or R Tea [3] owever hs propery was gve for he sake of copleess a s ever use here a esseal aer 5 The L-Behavor of Naver-Sokes Equao wh Nolear Perurbao We coser he l-behavor of Naver-Sokes equao wh olear perur bao o he wo esoal space we use he space whch s gve (6) (8) The a avaage we see s ha applyg he Growall lea o he soluo of proble () approaches a soluo of Naver-Sokes equao o L a as Theore 6 Uer assupo (6) he he soluo of ux of () s approxae soluo of Naver-Sokes equaos a hs soluo s sable as Proof Le u x u s a soluo of () as : u u uup f u x u u u Le u x u (5) s a soluo of () as : u uuup f u x u (5) u u Ulzg (5)-(5) a le v u u ece v u u u u v u u I s obae by ulply (54) by a fuco a egrag over v v uuuu vx u u v x Usg he seco equaly (3) a (53) v buvw (54) s rlear couous: a u u u u vx bu u v bu u vbu u v bu u v bu u v b u u v b v u v b u v v c v v u v v c u u v v v c v u u v x u v u v u v u v u v u v u v u u v ulzg equaly (**) (36) (346) we esae Du u v : u we wre hece v u v x u u u x u x u u x u u u c u u v 3 vv v x v xc v c3 v c where c c c3 s a approprae cosa ece v v c c c3 v e (55) 3 v c c c v (56) Usg he classcal Growall Lea we oba Copyrgh 3 ScRes

7 658 W Y ZAO Z B ZENG where C v v expc exp C C Noce ha hece C c C c c 3 v Thaks o l C l v ( 57) ece C C as accorg o sable coo hus hs sol uo s sable REFERENCES [] C Foas a R Tea Aracor Represeg Tule Flows Meors of Apple Maheacal Sceces ol 53 No [] C Foas a R Tea O he Deso of he Aracors Two-Deesoal Turbulece Physca D ol 3 No pp o: 6/67-789(88)9-X [3] C Foas a R Tea O he Large-Te Galerk Approxao of he Naver-Sokes Equaos SI Joural o Nuercal Aalyss ol No pp o:37/743 [4] J E Marse L Srovch a F Joh Ife-Desoal Dyacal Syses Mechacs a Physcs Apple Maheacal Sceces ol Sprger- erlag New York pp 5-5 [5] F Abergel Aracor for a Naver-Sokes Flow U- o he Whole Twoboue Doa Maheacal Moellg a Nuercal Aalyss ol 3 No pp [6] C S Zhao a K T L The Global Aracor of N-S Equao wh Lear Dapess Desoal Space a Esaes of Is Deesos ACTA Maheacal Applcao Sca ol 3 No pp 9-96 [7] A Bab a M I shk Maxal Aracors of Segroups Correspog o Evoluo Dffereal Equaos Maheacs of he USSR-Sbork ol 54 No 986 pp o:7/sm986v54abe976 [8] F Abergel Exsece a Fe Desoaly of Global Aracor for Evoluo Equaos o Uboue Doas Joural of Dffereal Equaos ol 83 No 99 pp 85-8 o:6/-396(9)97-6 [9] R S As Sobolve Space Acaec Press New York 975 [] A Pazy Segroups of Lear Operaor a Applcao o Paral Dffereal Equao Apple Maheacal Sceces Sprger-erlag New York 6 pp -38 Copyrgh 3 ScRes