Reproductive solution for grade-two fluid model in two dimensions

HRe is a In eg ación Escuela de Ma emá icas Uni e sidad Indus ial de San ande Vol. 27, No. 1, 2009, pág. 15–24 Rep oduc i e solu ion o g ade- wo luid model in wo dimensions L. F iz∗ F. Guillén-González∗∗ M. A. Rojas-Meda ∗∗∗ Abs ac . We ea he exis ence o ep oduc i e solu ion (weak pe iodic solu ion) o a second-g ade luid sys em in wo dimensions, by using he Gale kin app oxima ion me hod and compac ness a gumen s. 1. In oduc ion Fo a gene al incomp essible luid o g ade 2, he Cauchy s ess enso is gi en by T=−pI+µA1+α1A2+α2A2 1, whe e µ≥0is he iscosi y, α1,α2a e ma e ial coe icien s, namely no mal s ess moduli, pis he p essu e and A1,A2a e he i s wo Ri lin-E icksen (see [8] o [9]) enso s de ined by A1=∇u+ (∇u)T, A2=d d A1+A1∇u+ (∇ )TA1. 0Keywo ds: Rep oduc i e solu ion, wo-g ade luid, Gale kin me hod. 0MSC2000: 35Q35, 76D03. 0∗G upo de Ma emá icas Aplicadas, Depa amen o de Ciencias Básicas, Uni e sidad del Bío-Bío, Chillán, Chile. Casilla 447. e-mail:l iz@ oble. do-may.ubiobio.cl 0∗∗ Depa amen o de Ecuaciones Di e enciales y Análisis Numé ico, Facul ad de Ma emá icas, Uni e - sidad de Se illa, Se illa, 41012, España. e-mail:[email protected] 0∗∗∗G upo de Ma emá icas Aplicadas, Depa amen o de Ciencias Básicas, Uni e sidad del Bío-Bío, Chillán,Casilla 447. e-mail:ma [email protected] 15 16 L. F iz, F. Guillén-González, & M. A. Rojas-Meda F om he he modynamical p inciples we ha e ha α1+α2, and he equi emen ha he ee ene gy be a minimum in equilib ium implies ha α1≥0. Wi h all hese condi ions he equa ions o mo ion o an incomp essible luid o g ade wo a e gi en by (∂ ∂ (u−α∆u)−ν∆u+cu l(u−α∆u)×u+∇p= in Ω×]0, T [, di u= 0 in Ω×]0, T [, (1) wi h homogeneous Di ichle bounda y condi ions u= 0,on ∂Ω, and ini ial condi ion u(0) = u0,in Ω. He e, ν > 0 ep esen s he Kinema ic iscosi y and he ex e nal o ces. The s udy o his kind o luids was ini ia ed by Dunn and Fosdick in [4] and by Fosdick and Rajapogal in [5]. The i s success ul ma hema ical analysis o (1) was done by Cio anescu and El Hacène in [1]. Ano he in e es ing wo k is due o Galdi and Sequei a [6], whe e he au ho s ob ain some exis ence esul s. La e Cio anescu and Gi aul in [2] es ablish exis ence, uniqueness and egula i y o a global weak solu ion o (1) wi h small da a and u(0) and he same esul on some in e al o a bi a y da a. The exis ence is ob ained by applying Gale kin’s me hod wi h a special basis. In his pape we seek ep oduc i e solu ions o he wo-g ade luid sys em, i.e. solu ions o he ollowing sys em:          ∂ ∂ (u−α∆u)−ν∆u+cu l(u−α∆u)×u+∇q= in Ω×]0, T [, di u= 0 in Ω×]0, T [, u=0on ∂Ω×]0, T [, u(0) = u(T), (2) by supposing ha depends on he ime (no ice ha i does no depend on , he solu ion o he associa ed s eady-s a e sys em o he second- g ade luid is ac ually a ep oduc i e solu ion). As he eade can see, he usual ini ial condi ion has been changed by a ime pe iodic condi ion. The nex heo em is he main esul o his pape . Theo em 1.1. Fo any ∈L2(0, T ;H(cu l; Ω)) ∩L∞(0, T ;L2(Ω)), he e exis s a weak solu ion o he wo-g ade luid sys em (2). [Re is a In eg ación Rep oduc i e solu ion o g ade- wo luid model in wo dimensions 17 2. P elimina ies Le Ωbe a bounded domain o R2o he class C2,1. To sol e a g ade 2 luid sys em means o ind a ec o alued unc ion u= (u1, u2)and a scala unc ion pde ined on Ω×]0, T [sa is ying (2). Since we a e in wo dimensions (see [7]), he cu l ope a o is de ined by cu l u=∂u2 ∂x1 −∂u1 ∂x2 , and i zis a scala unc ion, we de ine z×u= (−zu2, zu1). In wha ollows, he spaces in bold ace ep esen spaces o bi-dimensional ec o unc- ions. We de ine he Hilbe spaces Hand Vin he ollowing manne : H={Ψ∈L2(Ω) : di Ψ=0,Ψ·n= 0 on ∂Ω}, V={ ∈H1(Ω) : di = 0, =0,on ∂Ω}, H(cu l; Ω) = { ∈L2(Ω) : cu l ∈L2(Ω)}. Fo α∈R+, we in oduce he space (see [1] and [2]) V2= ∈V: cu l ( −α∆ )∈L2(Ω),(3) equipped wi h he scala p oduc (u, )V2= (u, ) + α(∇u,∇ ) + (cu l (u−α∆u),cu l ( −α∆ )),(4) and associa ed no m and semi-no m k kV2= ( , )1/2 V2,| |V2=kcu l ( −α∆ )kL2(Ω).(5) In he ollowing lemma i is p o ed ha he semi-no m | · |V2is a no m in H3. Lemma 2.1 ([1] p 182).Le Ωbe a bounded, simply-connec ed open se o R2o he class C2,1. Then e e y ∈V2belongs o H3(Ω). Mo eo e , he e exis s C > 0such ha k kH3(Ω) ≤Ckcu l ( −α∆ )kL2(Ω). An easy bu edious compu a ion gi es us he ollowing equali y: ZΩ cu l (u−α∆u)×u· dx =b(u;u, )−αb(u; ∆u, ) + αb( ; ∆u,u); Vol. 27, No. 1, 2009] 18 L. F iz, F. Guillén-González, & M. A. Rojas-Meda he e b(u; ,w) = 3 X i,j=1 ZΩ ui ∂ j ∂xi wjdx. F om his, he a ia ional o mula ion o he p ob- lem (1) is he ollowing: Gi en ∈L2(0, T ;H(cu l; Ω) ∩L∞(0, T ;L2(Ω)) and u0∈V2, ind u∈L∞(0, T ;V2)such ha (u′, ) + α(∇u,∇ ) + ν(∇u,∇ ) + b(u;u, ) −αb(u; ∆u, ) + αb( ,∆u,u) = ( , ),∀ ∈V.(6) 3. A p io i es ima es o he Gale kin solu ions By ollowing he ideas gi en in [1] and [2] we conside he basis {wj}j∈N, he eigen- unc ions o he p oblem: Fo j∈N,wj∈V2is he solu ion o (wj, )V2=λj{(wj, ) + α(∇wj,∇ )},∀ ∈V2,(7) whe e (·,·)V2is he scala p oduc in V2. Since he imbedding o V2in o Vis compac , he e exis s a sequence o eigen alues (λj)j≥1and a sequence o eigen unc ions (wj)j≥1 ha cons i u es a basis o V2. Lemma 3.1 ([2] p 326).Le Ωbe a bounded simply-connec ed open se o R3wi h a bounda y Γo class C3,1. Then he eigen unc ions o he p oblem (7) belong o H4(Ω). Fo e e y m∈N, we de ine Vm 2 he ec o space spanned by he i s meigen unc ions {w1,...,wm}, and by Pm he o hogonal p ojec ion on Vm 2 o he scala p oduc in V2. In o de o cons uc a pe iodic solu ion o he p oblem (2) we will use Gale kin’s disc e iza ion. Indeed, o j∈ {1,2,...,m}we ind um( ) = m X j=1 cm j( )wj, solu ion o (u′ m( ),wj) + α(∇u′ m( ),∇wj) + ν(∇um( ),∇wj) + b(um( ); um( ),wj), −αb(um( ); ∆um( ),wj) + αb(wj,∆um( ),um( )) = ( ( ),wj),(8) um(0) = Pm(u0).(9) By mul iplying bo h sides o (8) by cm j( )and summing wi h espec o j, om he an i-symme y o bwe ob ain he equali y 1 2 d d kum( )k2 L2(Ω) +α|um|2 H1(Ω)+ν|um|2 H1(Ω) = ( ( ),um( )). [Re is a In eg ación Rep oduc i e solu ion o g ade- wo luid model in wo dimensions 19 The e o e, by in eg a ing in ime o ∈[0, T ] he abo e equali y we ob ain he ollowing lemma. Lemma 3.2 ([2] p 327).The solu ion um( )o he p oblem (8)-(9) sa is ies he ollowing di e en ial inequali y o each ∈[0, T ]: kum( )k2 L2(Ω) +αk∇um( )k2 L2(Ω) ≤e−νK kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω)+P2 νZ 0 e−νK( −s)k (s)k2 L2(Ω)ds, whe e P>0is he Poinca é cons an and K= (P2+α)−1. In o de o ob ain an es ima ion o he no m kumkV2, we adap he p oo o Theo em 4.4 in [2] and he p oo o he di e en ial inequali y gi en in [1] p 189. A i s , we de ine he ec o - alued unc ion F(um,um)by (F(um( ),um( )), ) = ν(∇um( ),∇ ) + b(um( ); um( ), ) −αb(um( ); ∆um( ), ) + αb( ; ∆um( ),um( )),(10) o e e y ∈H1 0(Ω). Fo 1≤m≤m, by cons uc ion o F(um,um), (u′ m( ),wj) + α(∇u′ m( ),∇wj) + (F(um( ),um( )),wj)−( ( ),wj) = 0.(11) F om Lemma 3.1, F(um( ),um( )) ∈H1(Ω). Nex o each , le m( )∈Vbe solu ion o he S okes equa ion m( )−α∆ m( ) + ∇qm( ) = F(um( ),um( )) − ( ).(12) By classical egula i y esul s, m( )∈H3(Ω) and hen, cu l( m( )−α∆ m( )) belongs o L2(Ω). The e o e, m∈V2. By mul iplying (12) by wj, we ob ain ( m( ),wj) + α(∇ m( ),∇wj) = (F(um( ),um( )) − ( ),wj), hus (11) can be w i en (u′ m( ),wj) + α(∇u′ m( ),∇wj) + ( m( ),wj) + α(∇ m( ),∇wj) = 0.(13) Mul iplying equa ion (13) by λjcm j( )and adding o j= 1,...,m, we ge (u′ m,um)V2+ ( m,um)V2= 0, Vol. 27, No. 1, 2009] 20 L. F iz, F. Guillén-González, & M. A. Rojas-Meda in o he wo ds, (cu l(u′ m−α∆u′ m),cu l(um−α∆um)) + (cu l( m−α∆ m),cu l(um−α∆um)) = 0. By aking cu l in (12), cu l( m−α∆ m) = cu l(F(um,um)− ), and hus (cu l(u′ m−α∆u′ m),cu l(um−α∆um)) + (cu l(F(um,um)− ),cu l(um−α∆um)) = 0. Using de ini ion (10) we ind 1 2 d d kcu l(um−α∆um)k2 L2(Ω) + (cu lF(um,um),cu l(um−α∆um)) = ( ,um) + (cu l ,cu l(um−α∆um)). (14) Now, we will es ima e he e m: T= (cu lF(um,um),cu l(um−α∆um)). Since di um= 0 and Ω⊆R2, i is no so di icul o p o e ha cu l(cu l(um−α∆um)×um) = um· ∇(um−α∆um), and (um· ∇(um−α∆um),cu l(um−α∆um)) = 0, and hus T= (−ν∆cu lum,cu l(um−α∆um)) + (um· ∇(um−α∆um),cu l(um−α∆um)) =ν αkcu l(um−α∆um)k2 L2(Ω) −ν α(cu lum,cu l(um−α∆um)). The e o e, he equa ion (14) can be w i en 1 2 d d kcu l(um−α∆um)k2 L2+ν αkcu l(um−α∆um)k2 L2(Ω) = (cu l ,cu l(um−α∆um)) + ν α(cu lum,cu l(um−α∆um)). [Re is a In eg ación Rep oduc i e solu ion o g ade- wo luid model in wo dimensions 21 Then, we ge he ollowing inequali y: 1 2 d d kcu l(um−α∆um)k2 L2+ν αkcu l(um−α∆um)k2 L2(Ω) ≤ kcu l kL2(Ω)kcu l(um−α∆um))kL2(Ω) +ν αkcu lumkL2(Ω)kcu l(um−α∆um)kL2(Ω) ≤1 2λkcu l k2 L2(Ω) +1 λkcu l(um−α∆um))k2 L2(Ω) +ν 2αεkcu lumk2 L2(Ω) +1 εkcu l(um−α∆um))k2 L2(Ω). I we ake ε= 2 and λ=2α νwe ha e ha 1 2 d d kcu l(um−α∆um)k2 L2+ν αkcu l(um−α∆um)k2 L2(Ω) ≤2ν αkcu lumk2 L2(Ω) +2α νkcu l k2 L2(Ω). Bu kcu lumk2 L2(Ω) ≤2k∇umk2 L2(Ω), hus 1 2 d d kcu l(um−α∆um)k2 L2+ν αkcu l(um−α∆um)k2 L2(Ω) ≤4ν αk∇umk2 L2(Ω) +2α νkcu l k2 L2(Ω). F om Lemma 3.2 1 2 d d kcu l(um( )−α∆um( ))k2 L2+ν αkcu l(um( )−α∆um( ))k2 L2(Ω) ≤4ν α2(kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω)) + 4P2 ανK k kL∞(0,T ;L2(Ω)) +2ν αkcu l ( )k2 L2(Ω). F om all his conside a ions, we ha e p o ed he ollowing lemma. Lemma 3.3. The solu ion umo he p oblems (8) and (9) sa is ies he a p io i es ima e o all ∈[0, T ]: kcu l(um( )−α∆um( ))k2 L2(Ω) ≤e−ν αkcu l(um(0) −α∆um(0))k2 L2(Ω) +2 α(kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω))+ 2P2 ν2Kk kL∞(0,T ;L2(Ω)) +2α νk kL2(0,T ;H(cu l;Ω)). 4. P oo o he Theo em 1.1 In his sec ion, we p o e he Theo em 1.1. To his end, a i s we p o e he exis ence o a sequence o Rep oduc i e Gale kin solu ions, by ollowing he ideas gi en in [3], which con e ges o he ep oduc i e solu ion o he g ade wo sys em luid. Vol. 27, No. 1, 2009] 22 L. F iz, F. Guillén-González, & M. A. Rojas-Meda We de ine he ope a o Lm( ) : [0, T ]→Rmas Lm( ) = (cm 1( ), cm 2( ),...,cm m( )),(15) whe e cm j( )a e he coe icien s o he expansion o um. Fo e e y (ξ1, ξ2,...,ξm)∈Rm, we de ine he ollowing equi alen no ms: k(ξ1, ξ2,...,ξm)k2 a,Rm:= kuk2 L2(Ω) +αk∇uk2 L2(Ω), k(ξ1, ξ2,...,ξm)k2 b,Rm:= kcu l(u−α∆u)k2 L2(Ω), whe e u=ξ1w1+ξ2w2+···+ξmwm. We de ine he ope a o Φm:Rm→Rmin he ollowing manne : Gi en Lm 0∈Rm, we de ine Φm(Lm 0) = Lm(T), whe e Lm( )is de ined in (15). I is clea ha Φmis con inuous and we wan o p o e ha i has a ixed poin . In o de o p o e his esul , we will use he Le ay-Schaude Theo em. Indeed, i su ices o show ha o all λ∈[0,1], he possible solu ion Lm 0(λ)o he equa ion Lm 0(λ) = λΦm(Lm 0(λ)) (16) a e bounded independen ly o λ. Since Lm 0(0) = 0, we will conside λ∈(0,1]. In his case, (16) can be w i en as Φm(Lm 0(λ)) = 1 λLm 0(λ). Thus, by de ini ion o Φmand Lemma 3.2, we ob ain  1 λLm 0(λ) 2 a,Rm ≤e−νKT kLm 0(λ)k2 a,Rm+P2 νZT 0 eνKsk (s)k2 L2(Ω)ds, which implies ha kLm 0(λ)k2 a,Rm≤ P2 νZ 0 eνKsk (s)k2 L2(Ω)ds 1−e−νKT =M0. Now, om Lemma 3.3 and de ini ion o Φm, we ha e ha  1 λLm 0(λ) 2 b,Rm ≤e−νT αkLm 0(λ)k2 b,Rm+2 αkLm 0(λ)k2 a,Rm +2P2 ν2Kk kL∞(0,T ;L2(Ω)) +2α νk kL2(0,T ;H(cu l;Ω)), [Re is a In eg ación Rep oduc i e solu ion o g ade- wo luid model in wo dimensions 23 hen, we deduce ha kLm 0(λ)k2 b,Rm≤ 2M0 α+2P2 ν2Kk kL∞(0,T ;L2(Ω)) +2α νk kL2(0,T ;H(cu l;Ω)) 1−e−νT α =M1, o each λ∈(0,1]. This las es ima e is independen o λ∈[0,1] and m∈ N. Con- sequen ly, Le ay-Shaude Theo em implies he exis ence o al leas one ixed poin o Φm, and hen he exis ence o ep oduc i e Gale kin solu ion um. Mo eo e , since he p e ious es ima es do no depend on m∈ N and by Lemma 3.3, he e exis s M∈R independen o msuch ha kum( )kV2≤M, (17) o each ∈[0, T ], i means ha (um)m≥1is bounded in L∞(0, T ;V2). By Lemma 2.1, we can w i e kum( )kH3(Ω) ≤M, o each ∈[0, T ]. I emains o pass o he limi wi h espec o m. This is a s anda d a gumen and we ha e only o p o e ha (u′ m)m≥1is bounded in L∞(0, T ;V′ 2). In o de o p o e his bound, we use he a gumen s gi en in [1], p 190. A i s , we no e ha |b(um( ),um( ), )| ≤ Ck∇umk2 L2(Ω)k kV2, which implies ha he e exis s T1 m( )∈V′ 2)such ha b(um( ),um( ), ) = hT1 m( ), i,∀ ∈V2, and by es ima e (17), we ha e ha (T1 m)m≥1is bounded in L∞(0, T ;V′ 2). In he same manne , he e exis s a bounded sequence (T2 m)m≥1in L∞(0, T ;V′ 2)such ha b(um( ),∆um( ), ) = hT2 m( ), i,∀ ∈V2. Finally, om equa ion (8), we can conclude ha u′ m=TmPm, whe e (Tm)m≥1is a bounded sequence o L∞(0, T ;V′ 2)and Pmis he p ojec ion o V2on Vm 2. This comple es he p oo . Acknowledgmen s L. F iz was suppo ed by Fondecy (Chile) g an 1090510. F. Guillén-González and M. A. Rojas-Meda we e suppo ed by DGI-MEC (Spain) G an MTM2006-07932 and Fondecy (Chile) G an 1080628. Vol. 27, No. 1, 2009]