On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows

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1 Joural of Applied Mahemaics ad Physics Published Olie Jue i SciRes hp://wwwscirporg/joural/jamp hp://dxdoiorg/6/jamp76 O he Exisece ad Uiqueess of Soluios for oliear Sysem Modelig hree-dimesioal Viscous Sraified Flows Adrei Giiaoullie ovias Casro Deparme of Mahemaics os Ades Uiversiy Bogoa Colombia agiiao@uiadeseduco ecasro7@uiadeseduco Received March ; revised April ; acceped 8 April Copyrigh by auhors ad Scieific Research Publishig Ic his wor is licesed uder he Creaive Commos Aribuio Ieraioal icese (CC BY) hp://creaivecommosorg/liceses/by// Absrac We esablish he uiqueess ad local exisece of wea soluios for a sysem of parial differeial equaios which describes o-liear moios of viscous sraified fluid i a homogeeous graviy field Due o he presece of he sraificaio equaio for he desiy he model ad he problem are ew ad hus differe from he classical avier-soes equaios Keywords Parial Differeial Equaios Sobolev Spaces Fluid Dyamics Sraified Fluid Viscous Fluid Iroducio he objecive of his paper is o sudy he qualiaive properies of he wea soluios of he sysem of parial differeial equaios which describes oliear moios of sraified hree-dimesioal viscous fluid i he graviy field such as exisece uiqueess ad smoohess his model of hree-dimesioal sraified fluid correspods o a saioary disribuio of he iiial desiy i a homogeeous graviaioal field which is of Bolzma ype ad is expoeially decreasig wih he growh of he aliude he resuls may be applied i he mahemaical fluid dyamics modellig real o-liear flows i he Amosphere ad he Ocea he addiioal uow fucio (desiy) as well as he sraificaio equaio iself cosiues he ovely of he problem o cosruc he soluios we will use he Galeri mehod We cosider a bouded domai Ω R wih he boudary of he class Ω C piecewise ad he followig sysem of fluid dyamics How o cie his paper: Giiaoullie A ad Casro () O he Exisece ad Uiqueess of Soluios for oliear Sysem Modelig hree-dimesioal Viscous Sraified Flows Joural of Applied Mahemaics ad Physics hp://dxdoiorg/6/jamp76

2 A Giiaoullie Casro u p ν u+ u u+ = x u p ν u + u u + = x u p ν u + u u + gρ + = () x ρ u = g u u u + + = x x x Here x = ( x x x) is he space variable ( x ) = ( u( x ) u( x ) u( x )) p( x ) is he scalar field of he dyamic pressure ad ( ) u is he velociy field p x is he dyamic desiy I his model he saioary disribuio of desiy is described by he fucio e x where is a posiive cosa he graviaio- al cosa g ad he viscosiy coefficie ν are also assumed as sricly posiive For liear o-viscous case Equaios () are deduced for example i []-[] For liear viscous compressible fluid sysem is deduced for example i [] he liear sysem correspodig o was sudied from various pois of view ad some resuls may be foud i [5]-[] here exiss of course huge bibliography cocerig avier-soes sysem (see for example []-[]) However he o-liear sysem modellig sraified viscous flows have o bee sudied mahemaically ye ad hus our research was moivaed by he ovely of he presece of he erm ρ i he hird equaio of ad also by he presece of he fourh equaio iself Cosrucio ad Exisece of a Wea Soluio We deoe e = ( ) ad observe ha wihou loss of geeraliy we ca cosider g = I his way we wrie Equaios () i he vecor form: We associae sysem wih he iiial codiios ad he Dirichle boudary codiios u ν u+ ( u ) u+ ρe + p = ρ ue = divu = ( ) = ( x) = ρ ( x) u x u x ρ () () u = Ω () ρ = Ω Ω be a fucioal space of smooh soleoidal fucios wih compac suppor i Ω We deoe as Ω he space of soleoidal fuc- e J J ( Ω ) he closure of J ( Ω ) i he orm ( Ω ) We also deoe as J ios from C ( Ω ) which saisfy homogeeous Dirichle codiio Defiiio e < < u( x) ( ) J( Ω ) ( ) J ( Ω) ρ ( x ) ( ) Ω For ay pair of vecor fucios f g ( Ω ) we deoe = ( + + ) Ω f g fg fg fg dx 59

3 A Giiaoullie Casro We will call ( u ρ ) a wea soluio of - if he followig iegral ideiies hold for every pair of fucios ΦΨ such ha { u u ( u ) ν u ρe } u Φ + Φ + Φ + Φ d + Φ = (5) {( ρ ) ( ue ) } ( ρ ) Ψ + Ψ d + Ψ = (6) Φ C ( ) J ( Ω ) Ψ C ( ) ( Ω ) ad Φ = Ψ = We observe ha he relaios 5-6 are obaied i a aural way afer muliplyig by ( ) ad iegraig by pars over Ω for x ad over he ierval ( u ρ ) he fucio p ca be easily foud from e { ϕ } be a complee orhoormal sysem i ΦΨ he sysem for Afer owig he fucios J Ω We observe ha wihou loss of geeraliy i ca be chose as a sysem of eigefucios of Soes operaor (see for example [] [] [5]) We cosruc he soluios of as Galeri approximaios ϕ u x = c x (7) = where c C [ ] are uow coefficies ad ( x ) ρ ( x) ue ρ ( x) ρ( x) Our primary aim is o deermie he exisece of c u If we cosider 5 for he he arbirary elecio of ρ are obaied as soluios of Cauchy problem = = (8) u ad ( x) H ϕ ( x) H C( ) H ad he resulig properies of he approximaios Φ = = H will imply he relaios = u ν u + u u + ρ e ϕ = We rasform ow he sysem 9 io a auoomous sysem of differeial equaios of he firs order wih respec o he variables c where he vecor field is C If we deoe ji j i x = c c β = ϕ ϕ ϕ he i ca be easily see ha he sysem 9 is equivale o (9) where M ( ) x = F x + Gx s ds () xmx ρe ϕ F ( x) = + ν Hx + x Mx ρe ϕ ( ) ( ) β β ϕ e e ϕ ϕ e e ϕ = G = β β ( ϕ e) e ϕ ( ϕ e) e ϕ ϕ ϕ ϕ ϕ H = Φ Ψ = Φi Ψ i= ϕ ϕ ϕ ϕ i 5

4 A Giiaoullie Casro Afer differeiaig we obai ( ) where ( ) ν ( ) K x x = M x x+ M x x + H x + G x j = j j j j j We iroduce he oaio x = z ad hus rewrie as xz is a ifiiely differeiable vecor field he from he heory of ordiary differeial equa- ow we shall deduce some esimaes o prove ha = is idepede o emma he soluios u ( x ) of he approximae sysem 9 are defied uiquely by 7 Addiioally he followig esimaes are valid For all here exiss > such ha sup u x u + ρ Sice ( ) ios we coclude ha admis a maximal soluio i he ierval [ ] x = K xx () x = ( xz ) () z ) u τ x dτ u + ρ ν for all ) a ) u a ab + M for all b a ) u ( x) u + ρ + a ( ab + M ) 5) u ( τ x) ν ν ( ab + M ) sec dτ l for all b sec M b for all 6) u ( τ x) dτ u W ρ + + for all ν where he posiive cosas a b M ad deped oly o he iiial daa he parameer of he sysem ad he domai Ω Remar he values of he cosas a b M ad are give below i 8 ad 9 Proof c For c > we iroduce he followig auxiliary real-valued fucio E = u + ρ We observe ha Afer muliplyig each equaio of 9 by c I his way choosig c = we have ha herefore E E ( ρ ρ) ( ρ) E = u u + c = u u + c u () for all ad hus ad summig hem wih respec o we obai ν ρ u u + u + e u = E = ν u () 5

5 A Giiaoullie Casro I paricular we have j proved From we obai ha j= u + + (5) ρ u ρ = ρ + c u u which will prove he saeme b) Ideed hus he saeme a) of he emma is u τ x dτ u + ρ ν (6) d d ν u τ x τ = E τ τ = E E E + E E = u + ρ ow differeiaig 9 we obai u ν u ( u ) u ( u ) u ρ e ϕ We muliply he las relaios by c = ad sum hem wih respec o : ν ρ u u + u + u u u + u u u + e u = herefore eepig i mid ha ( u ) u u = we have d u + ν u = ( u ) u u ue u (7) d We would lie o esimae he righ-had erms i 7 Evidely from 5 we obai ue u u u u u u u ρ We will eed he followig esimae: ν u = u u ρ e u u u + ρ e u u u + u ρ u + ρ u + u + ρ O he oher had from he geeralized Hölder iequaliy ad he ierpolaio iequaliy 6 f f f we ca esimae he erm ( u ) u u u u u u u u u u 6 (8) (9) = () From he Youg iequaliy wih ε for p = q = ε ( ε) ow usig 9 ad he iclusio Sobolev iequaliy x = u u y = u we obai 6 u u u u u + C u () 6 u C u we have ε u u u u + u + u + u + C C u ν ρ ρ ( ε) If we choose ε such ha C ε C = ν he he las esimae ogeher wih 7 ad 8 implies 5

6 A Giiaoullie Casro ε + u + ρ d ν ε u a+ bu a= b= u ρ d + 8ε ν u + ρ Evidely from we have ν d d from which afer iegraig wih respec o i follows ha u () u () a+ bu a a ab + a b ow o obai a uiform upper esimae wih respec o for u bu ( x) a () we oly eed o prove such esimae for u ( ) x We remid ha he sysem { ϕ } mal i J ( Ω ) ad also i J ( Ω ) wih he scalar produc ( Φ Ψ ) as well as i J( Ω) W ( Ω) scalar produc ( P Φ P Ψ ) where P is a orhogoal projecio of ( Ω ) oo beig formed by eigefucios of Soes operaor is orhoor ρ ν u = e u P u u u u u Proceedig aalogously o 7- ad also usig emma from [] we obai wih he J Ω From 9 we have ρ + ν + ρ + ν + u P u u u P u u u 6 ρ ν P u C u P u u u ν P u C P u u C u ρ ν ρ + + P u + ( C ) ε P u + C( ε) u + C u ( ) If we choose ε = C he we fially have ν u ρ + + P u + ( C + C( ε) ) u ( ) (5) ow from he Bessel iequaliy ad he properies we ca express 5 as c u u P P u ( ) = ϕ ( ) = ϕ ( ) = ϕ ϕ P ϕ (6) ν u ρ + + P u + ( C + C( ε) ) u( ) (7) I his way usig 7 ad he evide iequaliy ( x y z) ( x y z ) we ca esimae as a u a ab + M (8) b 5

7 A Giiaoullie Casro ν b ρ + + P u + ( C( ε) + C ) u where M = a a π M aig < < we assure ha a ( ab + M ) < ad hus we obai he resul ha for all ab here exiss > such ha a u a ab + M for all b (9) which proves he saeme c) of he emma I is easy o see ha he saeme d) is a direc cosequece of 9 ad 9 he saeme e) of he emma is obaied immediaely if we iegrae boh sides of 9 wih respec o I remais o prove he saeme f) For ha we use 5 ad 6 ad herefore obai u ( τ x) d τ = u ( τ x) + u ( τ x) dτ u + ρ + W ν which cocludes he proof of he emma We would lie o obai ow more esimaes for he approximae soluios u ( x ) wih a ieio o show ha heir limi which is a obvious cadidae for a soluio of - would preserve cerai regulariy properies of u emma For all ad for all here exiss a cosa C > which does o deped o u such ha he followig esimae is valid: ( τ ) τ ( ρ Ω) u x d C u () W Proof We coiue usig he oaio of P as a orhogoal projecio of ( Ω ) oo J ( Ω ) e λ be eigevalues correspodig o he eigefucios ϕ of he Soes operaor We muliply 9 by cλ ad sum up he resulig equaios wih respec o I his way we have u ν u + u u + ρ e P ϕ = () We would lie o esimae he followig erm i : ν u P ϕ = u P ϕ + u u P ϕ + ρ e P ϕ () By usig Cauchy ad Hölder iequaliy ogeher wih he iequaliy { } xy x ε y ε + we obai ν u P ϕ ε u u u e + + ρ + 6 P ϕ () ε ow we use Sobolev iclusio iequaliy ad emma from [] which allows us o esimae as We ae ν u P ϕ ε u + C u P u u + u + ρ e + P ϕ ε ε ν = cosider he iequaliy ( x y) ( x y ) + + ad use he propery () 5

8 A Giiaoullie Casro u P ϕ = u P ϕ = P u P ϕ = P u Proceedig i his way we obai { ρ } { u C u P u C u ρ e } ν P u u + C u P u u + u + e ν = ν 6 u + ηc u + P u + C u + ρ e ν η ow if we ae η = we will fially have ν { ρ } 6 P u K u + u + u + e 6 K = C C ν ν max (5) he erm u o he righ side of 5 ca be esimaed by 9 he erms 6 u ad u ca be esimaed aalogously we use he saeme d) of emma ad he iequaliy ( x y) p p ( x p y p ) where we cosider p = ad p = ow aig io accou he properies ( ab τ + M ) τ = ( ab τ + M ) ( ab τ + M ) + + a d a (6) ab ( ab τ + M ) τ = ( ab τ + M ) + ( ab τ + M ) a d a l cos (7) ab we ca proceed esimaig he iegral of 5 as follows: P u ( τ x) dτ ( ) 6 sec ab M cos ab M + a + K l 6 u ρ a ( ab M ) a M l b sec M b cos M a + 6 u + ρ + a ab + M M + u + ρ b b (8) We oe ha he esimae 8 is valid for From 8-9 we have ha he value is chose i such a way ha he righ side of 8 is posiive Fially we use he resul from [5] where here is show J Ω W Ω he orms P f ad f are equivale which cocludes ha i he fucioal space he proof heorem e Ω R be a bouded domai wih he boudary of he class C ad le u J( Ω) W ( Ω) ρ Ω he here exiss a ierval ad here exis he fucios ux ( ) ( x ) px ( ) sysem ad he codiios - i sese of 5-6 such ha W ρ which saisfy he 55

9 A Giiaoullie Casro Proof u J J Ω Ω u C J J W Ω Ω Ω p W Ω ρ Ω u ( ) J( Ω ) From emma ad emma we have ha here exis a fucio u ad a subsequece of { } breviy we will deoe also as { u } ) such ha We oe ha he iclusio J J u (which for u u wealy i ( ) J( Ω) u u wealy i ( ) J( Ω) W ( Ω) (9) u u wealy i ( ) J( Ω) Ω Ω is compac ad coiuous ad ha { } u is bouded i J Ω from emma O he oher had from he saeme c) of emma we have ha for h > ad < < h he followig esimae holds: h h + h ( + ) ( + ) τ ( + ) u h u d a ab M d d h a ab M () I his way from 9 ad emma 5 [] i follows ha { } u u srogly i ( ) J u belogs o a compac se i J Ω herefore he subsequece i 9 ca be chose i such a way ha Ω () I is easy o see ha u saisfies he regulariy properies of he heorem Ideed sice u ( ) J( Ω) W ( Ω) ad u ( ) J( Ω ) he from heorem IV5 [5] we obai ha u C J( Ω) Usig Sobolev iequaliy we have he esimae 6 6 W u u d τ C for all u u u u C u u C u u u C u P u C herefore ; which i ur implies le us show ha u saisfies 5 Evidely m = u ad Sice u u wealy i ( ) J Ω he ρ saisfy 5 for ϕ Ω ow Φ= H H C H = () u vdd x u vdd xv J Ω Aalogously we have ha u v d uv d ρ e Φ d ρe Φ d he limi I remais o prove 56

10 A Giiaoullie Casro o verify we oe firs ha u u Φ d u u Φ d () u u Φ u u Φ = u u u Φ + u u u Φ () We iegrae by he relaio ad use he properies ha u is uiformly bouded i ( ) J( Ω ) ad ha u u srogly i ( ) Ω I his way we obai ha is valid ow we pass o he limi for i ad hus obai { u u ( u ) ν u ρ e } u Φ + Φ + Φ + Φ d + Φ = { u u ( u ) ν u ρe } u Φ + Φ + Φ + Φ d + Φ = (5) for he fucios Φ from Due o he desiy of he se we have ha 5 is valid for all he fucios Φ from Defiiio which complees he proof Uiqueess of he Soluios heorem he wea soluio i sese of Defiiio is uique Proof e us exchage ρ by ρ ad hus rewrie he sysem i a more symmerical way: p x u u u u p u ν u + = u u x u u u u p ρ x () e ( u ρ p ) ad ( w p ) ρ u u u u = + + = x x x ρ be wo soluios of - which saisfy he codiios - ad also he codiios of heorem We deoe U= w up = p p ad hus obai () u u w w U ν U = P+ + u u w w u u w w ( ρ ρ) () From we have ρ ρ = U sx ds We observe ha we ca express as follows: 57

11 A Giiaoullie Casro u U U w U ν U = P+ + u U U w () u U U w ( ρ ρ) We muliply by U iegrae by pars i Ω I his way we have d ( ) d (5) U + ν U = U s x su U wu d By usig he geeralized Hölder iequaliy ad Youg iequaliy ogeher wih he iequiy U CU U we ca esimae he las erm i 5 as If we ae ε ( ε) U w U U w C U U w C U + C U w (6) ν ε = he we will have he esimae C d U U s x d su C U w d (7) + ow we would lie o esimae he erm U s x d su I is easy o see ha U s x d s U U s x d s U x d x U s x ds d x U x dx + Ω Ω Ω I his way from 7 we obai he esimae U ( s x) dxd s U ( x) dx U d s U + + Ω Ω d U U ds+ + C w U d (8) ow le us cosider he followig iiial value problem for he fucio y( ) : Evidely for coiuous y = f ( y) = φ y + y ( s) d s (9) y = φ he uique soluio of he problem 9 is y By compariso priciple herefore for every soluio of he differeial iequaliy z f z z he propery holds: z I his way we coclude from 9 ha Usig we obai also ha ρ = ρ ad hus he heorem is proved Acowledgemes U = which implies ha u = w his research was parially suppored by Fodo de Ivesigacioes Faculad de Ciecias-Uiades 58

12 A Giiaoullie Casro Refereces [] Cushma-Roisi B ad Becers J () Iroducio o Geophysical Fluid Dyamics Academy Press ew Yor [] rio D (99) Physical Fluid Dyamics Oxford UP Oxford [] Kudu P (99) Fluid Mechaics Academy Press ew Yor [] adau ad ifschiz E (959) Fluid Mechaics Pergamo Press ew Yor [5] Maareo Malseva J ad Kazaov A (9) Cojugae Flows ad Ampliude Bouds for Ieral Soliary Waves oliear Processes i Geophysics hp://dxdoiorg/59/pg [6] Maurer B Bolser D ad ide P () Irusive Graviy Curres bewee wo Sably Sraified Fluids Joural of Fluid Mechaics hp://dxdoiorg/7/s99975 [7] Birma V ad Meiburg E (7) O Graviy Curres i Sraified Ambies Physics of Fluids hp://dxdoiorg/6/75655 [8] Masleiova V ad Giiaoullie A (99) O he Irusio Problem i a Viscous Sraified Fluid for hree Space Variables Mahemaical oes hp://dxdoiorg/7/bf558 [9] Giiaoullie A ad Zapaa O (7) O Some Qualiaive Properies of Sraified Flows Coemporary Mahemaics Series AMS 9- [] Giiaoullie A ad Casro () O he Specrum of Operaors of Ier Waves i a Viscous Compressible Sraified Fluid Joural of he Faculy of Sciece Uiversiy of oyo 9 - [] Heywood J (98) he avier-soes Equaios: O he Exisece Regulariy ad Decay of Soluios Idiaa Uiversiy Mahemaics Joural hp://dxdoiorg/5/iumj98998 [] arar (6) A Iroducio o avier-soes Equaios ad Oceaography Spriger Berli [] emam R () avier-soes Equaios: heory ad umerical Aalysis AMS Chelsea Publishig ew Yor [] Sohr H () he avier-soes Equaios: A Elemeary Fucioal Aalyic Approach Birhäuser Zurich [5] Boyer F (5) Elemes d aalyse pour l eude de quelques modeles d ecoulemes de fluides visqueux icompressibles Spriger Berli 59