A review on reproductivity and time periodicity for incompressible fluids

Bol. Soc. Esp. Ma . Apl. no41(2007), 101–116 A REVIEW ON REPRODUCTIVITY AND TIME PERIODICITY FOR INCOMPRESSIBLE FLUIDS B. CLIMENT-EZQUERRA∗, F. GUILL´ EN-GONZ´ ALEZ∗ AND M. ROJAS-MEDAR† ∗Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Espa˜na. †Depa amen o de Ciencias B´asicas, Uni e sidad del B´ıo-B´ıo, Chile. [email p o ec ed] [email p o ec ed] [email p o ec ed] Abs ac In his a icle, ou aims is o e iew some o he esul s ha a e cu en ly a ailable conce ning he exis ence, uniqueness and egula i y o ep oduc i e and ime pe iodic solu ions o he Na ie -S okes equa ions and some a ian s. By he way, we p esen some open p oblems. Key wo ds: Rep oduc i e and ime pe iodic solu ions, Na ie -S okes ype equa ions, egula i y o solu ions AMS subjec classi ica ions: 35B10 35Q35 76D03 1 In oduc ion We s udy some p oblems ela ed wi h ime pe iodic solu ions o models o incomp essible luids. We s a ecalling he main ideas o p o e he exis ence o ep oduc i e weak solu ions (i.e. weak solu ions de ined in he ime in e al (0, T) aking he same ini ial and inal alues in ime) o he Na ie -S okes equa ions and some a ian s whe e hese ideas a e applicable, such as Boussinesq, mic opola and magne o-mic opola models. This p oo elies on he ob en ion o ime pe iodic Gale kin app oxima ions ia Le ay-Schaude poin ixed a gumen . Mo eo e , in he case o 2Ddomains, using he uniqueness o weak solu ions, he egula izing p ope y o he sys em and he exis ence o global egula solu ions when da a a e egula , one has ha he pe iodic in ime weak solu ions de ined as ex ension o ep oduc i e solu ions o he whole ime in e al (0,+∞) will be egula solu ions. An ex ension o hese esul s o he 3Dcase is possible imposing small enough ex e nal o ce, using he so called “weak/s ong uniqueness” and he global s ong solu ions o small enough da a (see Sec ion 5). Pa ially suppo ed by DGI-MEC (Spain), G an MTM2006–07932 and CGCI MECD- DGU B azil/Spain G an 117/06 101 102 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda Also, we s udy in Sec ion 4 some coupled models o eloci y and p essu e dynamic a iables wi h ano he a iable whe e he maximum p inciple holds, such as he gene alized Boussinesq model (wi h empe a u e-dependen iscosi y) and a nema ic liquid c ys al model wi h a Ginzbu g-Landau penaliza ion. In hese cases one has, hanks o an adequa e e o mula ion o he p oblem by unca ion, exis ence o ep oduc i e weak solu ions as limi o ime pe iodic Gale kin app oxima ions. I is impo an o ema k ha Gale kin app oxima ions do no e i y he maximum p inciple bu hei limi does. Finally, we will see ha , o hese models ela ed wi h he maximum p inciple, he a gumen o p o e egula i y o ep oduc i e solu ions in he Na ie -S okes amewo k (see Sec ion 5 below) a e no alid in gene al. The pa icula case o gene alized Boussinesq model wi h Neumann bounda y condi ion o he empe a u e can be sol ed wi h o he a gumen s, bu he case o nema ic liquid c ys al model emains as an open p oblem. 2 Na ie -S okes equa ions The mode n heo y o he Na ie -S okes equa ions began in he 1930s wi h Le ay’s pionee ing wo k ([10]). Le Ω ⊂IRd(d= 2 o 3) a bounded and egula enough domain illed by he luid, and [0, T] he ime in e al. We deno e Q= (0, T)×Ω and Σ = (0, T)×∂Ω. In he case whe e he luid is subjec o he ac ion o a body o ce , he Na ie -S okes equa ions can be w i en as ollows ∂u ∂ + (u· ∇)u−ν∆u+∇p= ,di u= 0 ,(1) whe e u=u(x, ) is he eloci y ield e alua ed a he poin x∈Ω and a ime ∈[0, T], p=p(x, ) is he p essu e ield and ν > 0 is he coe icien o kinema ical iscosi y (which is aken cons an ). This sys em can be comple ed wi h se e al bounda y condi ions. Fo simplici y, we ix he ollowing non-slip bounda y condi ions: u( , x) = 0,x∈∂Ω, > 0 (2) Finally, supplemen a y condi ions in ime mus be conside ed. The mo e classical is he ini ial condi ion: u(0,x) = u0(x),x∈Ω (3) O he possibili y is o change his ini ial condi ion by he ollowing ime-pe iodic condi ion: u(0,x) = u(T, x),x∈Ω.(4) Ma hema ical p ope ies o sys em (1) ha e been deeply in es iga ed o e he yea s and a e s ill he objec o p o ound esea ches. We in oduce some space unc ions. Le V he ec o ial space o med by all ields ∈C∞ 0(Ω)dsa is ying ∇ · = 0. We conside he Hilbe spaces H Rep oduc i i y and ime pe iodici y o incomp essible luids 103 ( espec i ely V) as he closu e o Vin L2( espec i ely H1). Fu he mo e, one has H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω}, V={u∈H1;∇ · u= 0,u=0on ∂Ω} We deno e L2 0(Ω) = ½p∈L2(Ω) : ZΩ p dx = 0¾. 2.1 Main classical esul s o he ini ial-bounda y p oblem De ini ion 1 Gi en u0∈Hand ∈L2(0, T;H−1(Ω)), i will said ha uis a weak solu ion o he p oblem (1),(2), (3) in (0, T ), i u∈L2(0, T;V)∩L∞(0, T;H), and e i ies (3) and he a ia ional o mula ion ZT 0ZΩn−u( ) ′( ) + ∇u( ) : ∇u( )−(u( )· ∇) ( )u( )− ( ) ( )odxd = 0, o all ∈C1([0, T ]; H)∩C([0, T ]; V), wi h compac suppo con ained in (0, T). In addi ion, i u0∈Vand ∈L2(0, T;L2(Ω)) any weak solu ion will be a s ong solu ion i u∈L2(0, T;H2∩V)∩L∞(0, T;V),u ∈L2(0, T ;H), p ∈L2(0, T;H1∩L2 0(Ω)) and e i ies he sys em (1) poin wise a.e. in (0, T)×Ω. Rema k 1 The p e ious de ini ion can be ex end o he case o inal ime T=∞changing he egula i y L2(0, T)by L2 loc(0,+∞). The ollowing esul s hold. Theo em 1 [22] Fo any u0∈Hand ∈L2(0, T;H−1(Ω)), he p oblem (1)- (2) has (a leas ) a weak solu ion. I Ω⊂IR2, one has uniqueness o weak solu ions. Theo em 2 [22] Fo any u0∈Vand ∈L∞(0,∞;L2(Ω)), he p oblem (1)- (2) has a unique s ong solu ion (u, p)local in ime, de ined in (0, T⋆)wi h T⋆>0small enough. In ac , i a solu ion has he s ong egula i y, i coincides wi h any weak solu ion associa ed wi h he same da a ( his p ope y is called weak/s ong uniqueness). Mo eo e , his s ong solu ion is global in ime, de ined in he whole ime in e al (0,∞)i ei he Ω⊂IR2o Ω⊂IR3and da a (u0, )a e small enough in hei espec i e spaces V×L∞(0,∞;L2(Ω)). 104 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda 2.2 On he ime-pe iodic weak solu ions Theo em 3 [8] Fo any ∈L2(0, T;H−1(Ω)), he e exis s a weak solu ion o (1)-(2) and (4), (i.e. he weak solu ion uhas he so-called ep oduc i e p ope y: u(0, x) = u(T, x)). No ice ha he ime pe iodic ex ension, e u, o any weak ep oduc i e solu ion u o he whole ime in e al (0,+∞) is a pe iodic weak solu ion o (1)-(2) co esponding o he da a, e , de ined as he ime pe iodic ex ension o . Main ideas o he p oo o Theo em 3 Le uk he unique app oxima e solu ion o he Gale kin ini ial-bounda y p oblem o Na ie -S okes in he ini e-dimensional subspace Vk, spanned by he i s kelemen s o he “spec al” basis o V(o hogonal in Vand o hono mal in H), associa ed o a ini ial disc e e da a uk 0∈Vk. Since V֒→H, he e exis s a Poinca ´e cons an c1>0 such ha c1kukk2 L2≤ k∇ukk2 L2, hus, om ene gy inequali y, we ha e d d kukk2 L2+c1kukk2 L2≤Ck k2 H−1,(5) o equi alen ly d d (ec1 kukk2 L2)≤C ec1 k k2 H−1. In eg a ing om 0 o T, we ha e ec1Tkuk(T)k2 L2≤ kuk(0)k2 L2+CZT 0 ec1 k ( )k2 H−1d . (6) Now, we de ine he ope a o Lk: [0, T]→Rkas ollows Lk( ) = (ck 1( ),...,ck k( )) whe e ck i( ), i = 1,...,k, a e he coe icien s o he expansion o uk( ) in Vk. No e ha kLk( )kRk=kukkL2, because we ha e choose he (o hono mal in L2) spec al basis in V. We de ine he ope a o Φk:Rk→Rkas ollows: Gi en Lk 0∈Rk, we de ine Φk(Lk 0) = Lk(T), whe e Lk( ) a e he coe icien s o he Gale kin solu ion wi h ini ial alue wi h coe icien s Lk 0. I is easy o see ha Φkis con inuous and we wan o p o e ha Φkhas a ixed poin . Fo his, hanks o he Le ay-Schaude Theo em, i su ices o show ha o all λ∈[0,1], he possible solu ions o he equa ion Lk 0(λ) = λΦk(Lk 0(λ)),(7) Rep oduc i i y and ime pe iodici y o incomp essible luids 105 a e bounded independen ly o λ. Since Lk 0(0) = 0, i su ices o conside λ∈(0,1]. In his case, (7) is equi alen o Φk(Lk 0(λ)) = 1 λLk 0(λ). Mo eo e , by he de ini ion o Φkand (6), one ob ains ec1T|| 1 λLk 0(λ)||2 Rk≤ kLk 0(λ)k2 Rk+cZT 0 ec1Tk ( )k2 H−1d , which implies kLk 0(λ)k2 Rk≤cRT 0ec1Tk ( )k2 H−1d ec1T−1=M, o each λ∈(0,1]. This bound is independen o λ∈[0,1] and k. Consequen ly, Le ay-Shaude Theo em implies he exis ence o a leas one ixed poin o Φk, ha is he exis ence o ep oduc i e Gale kin solu ion. Thus, since p e ious es ima es a e independen o k, one has he same es ima es o hese ep oduc i e Gale kin solu ions. Finally, he con e gence o a subsequence o a ep oduc i e solu ion o (1),(2), (4) hold. 2.3 Rela ion be ween weak pe iodic solu ions and global solu ions Assume : [0,+∞)→H−1(Ω) and T- ime pe iodic. Na ie -S okes 2D One has (see Theo em 1) uniqueness o weak solu ion o he ini ial-bounda y p oblem (associa ed o any ini ial da a u0). Consequen ly, gi en a ep oduc i e solu ion uassocia ed o u(0) = u(T) := u0, hen uis he (unique) solu ion o he ini ial-bounda y p oblem associa ed o he ini ial da a u0, which is de ined o all ime ∈(0,∞). Mo eo e , his solu ion is T-pe iodic, because in (T, 2T) mus be equal o he ep oduc i e solu ion de ined as ¯ u( ) = u( −T) (which e i ies u(T) = u(2T) = u0) and so on. Finally, using egula i y o solu ion u o s ic ly posi i e imes (see [5]), i is easy o p o e ha e e y pe iodic solu ion is egula . Na ie -S okes 3D Since uniqueness o weak solu ion is no known, i is possible ha he ep oduc i e solu ion uand he global weak solu ion ˜ uassocia ed o he ini ial da a u0:= u(0) = u(T) a e di e en in (0, T), al hough hey coincide locally in ime, nea o he ini ial ime = 0. 2.4 Open p oblems Na ie -S okes wi h la ge Reynolds numbe and a eac ion e m adding ene gy P e ious a gumen s o he p oo o ep oduc i e solu ions a e based on (exponen ial) dec easing o ene gy ( hanks o dissipa i e e ms). Na u ally, he 106 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda same a gumen , is applicable o models wi h ene gy s ic ly dec easing in ini e ime. Bu his is no always possible. Fo ins ance, we conside he ollowing Na ie -S okes sys em wi h la ge Reynolds numbe and a eac ion e m adding ene gy: ∂ u−ε∆u−u+∇p= ,∇ · u= 0, u(0) = u(T),u|Σ= 0.(8) The ene gy inequali y is ∂ kuk2 L2+εk∇uk2 L2≤C(k k2 H−1+kuk2 L2). Assuming εsmall enough such ha kuk2 L26< εk∇uk2 L2, he s ic ly dec easing in ime o kuk2 L2is no clea . Consequen ly, he exis ence o ime-pe iodic weak solu ions o (8) emains as an open p oblem. Ex e io domains Assume Ω is an ex e io domain whe e he Poinca ´e inequali y is no ue. Then, o show he exis ence o ep oduc i e solu ions one could use he “embedding domain echnique” oge he wi h he Gale kin Me hod, ob aining ep oduc i e solu ions in a sequence o (bounded) unca ed domains, see o ins ance [6, 18, 17]. Howe e , since Poinca ´e imbedding is no applicable, i is no clea he con oll o he pass o he limi om unca ed domains o he whole domain. Some pa ial esul s a e known. Fo example, he exis ence o s ong pe iodic solu ions o he Na ie -S okes equa ions in he ollowing unbounded domains, ei he Ω is he whole space Rno he hal -space Rn +has been in es iga ed by Kozono and Nakao [9] and Taniuchi [21] using he semig oup app oach. By using po en ial heo y, Ma emon i [15] p o ed he exis ence o a unique ime pe iodic solu ion on he whole space R3 o small ex e nal o ce. The p oblem, in he hal -space R3 +,was conside ed in [16]. Kozono and Nakao [9], making use o Lp−L es ima es o he semig oup gene a ed by he S okes ope a o , cons uc ed ime-pe iodic solu ions o small ime-pe iodic o ces and he s abili y o hese solu ions was conside ed in [21]. Yamazaki [23] analyzed he same p oblem o [9] in Mo ey spaces. 3 Some a ian s o Na ie -S okes equa ions We can apply he a gumen o ind ep oduc i e solu ions done o Na ie -S okes in he p eceden Sec ion, o some a ian s: 3.1 Boussinesq equa ions The Boussinesq sys em o hyd odynamics equa ions (see Joseph [7]) a ise om ze o o de app oxima ion o he coupling be ween he Na ie -S okes equa ion and he he modynamic equa ion. Such a ma hema ical model eads: Rep oduc i i y and ime pe iodici y o incomp essible luids 107 Find he ield u:Q→R3, he scala unc ions (θ, p) : Q→R2which sa is y he sys em o equa ions: ∂u ∂ −ν∆u+ (u· ∇)u+∇p=αθg+ in Q, ∇ · u= 0 in Q, (9) ∂θ ∂ −χ∆θ+ (u· ∇)θ= 0 in Q. wi h ∂nθ= 0 on ∂Ω and RΩθ= 0 He e u, p, θ deno e he eloci y, he p essu e and he empe a u e, espec i ely. gdeno es he g a i a ional ield, α > 0 is a cons an associa ed o he coe icien o olume expansion and is a ield o ex e nal o ces. Again, ν > 0 is he iscosi y coe icien . Finally, χ > 0 is he he mal conduc i i y coe icien . This sys em is comple ed wi h he bounda y condi ions ( o ins ance) u|Σ= 0, θ|Σ= 0 and he ime-pe iodic condi ions u(0) = u(T), θ(0) = θ(T) in Ω. By aking uand θas es unc ion in he u-sys em and θ-equa ion o (9) espec i ely, adding he esul ing equali ies conside ing an adequa e balance (in o de o elimina e he e m ha con ains g), we ob ain d d kuk2 L2+βd d kθk2 L2+νk∇uk2 L2+βχk∇θk2 L2≤ k k2 H−1,(10) whe e βis a big enough numbe depending on αand kgkL∞. This oge he wi h he Poinca ´e inequali y gi es an inequali y o ype (5). Indeed, i su ices o conside a Gale kin app oxima ion o bo h a iables, eloci y and empe a u e, and o ollow he p oo o Theo em 2.3, changing ukby (uk, θk). Ano he bounda y condi ions a e possible: Neumann, mixed, e c, whene e an inequali y o (u, θ) simila o (5) holds. 3.2 Mic opola equa ions The equa ions ha desc ibes he mo ion o a incomp essible iscous and mic opola luids in Qa e gi en by (see [12]) ∂u ∂ + (u· ∇)u−(ν+ν )∆u+∇p= 2ν o w+ , di u= 0,(11) ∂w ∂ + (u· ∇)w−(ca+cd)∆w−(c0+cd−ca)∇di w+ 4ν w = 2ν o u+g. The unc ions u:Q→R3,w:Q→R3and p:Q→Rdeno e he line eloci y, he angula eloci y (o o a ion o pa icles) and he p essu e o he luid, espec i ely. The unc ions :Q→R3and g:Q→R3deno e ex e nal 108 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda sou ces o linea and angula momen um, espec i ely. The posi i e cons an s ν, ν , c0, caand cda e iscosi ies, such ha c0+cd> ca. This sys em is comple ed wi h he bounda y condi ions u|Σ= 0, w|Σ= 0 ( o ins ance) and he ime-pe iodic condi ions u(0) = u(T), w(0) = w(T) in Ω. By aking uand was es unc ion in he u-sys em and w-sys em o (11) espec i ely, adding he esul ing equali ies, aking in o accoun ha 2ν ( o w,u)+2ν ( o u,w) = 4ν ( o u,w) and |∇u|2=| o u|2, we ob ain d d kuk2 L2+kwk2 L2+νk∇uk2 L2+ (ca+cd)k∇wk2 L2+ (c0+cd−ca)kdi wk2 L2 ≤C(k k2 H−1+kgk2 H−1). S a ing om his inequali y, he a gumen ollows as in p e ious sec ion. 3.3 O he models O he luid models whe e one has exis ence o ep oduc i e solu ions a e: magne ohyd odynamic model [14], Magne o-mic opola luid mo ion [20], a con ec ion-di usion model desc ibing bina y alloy solidi ica ion p ocesses [3], e c. 4 Rep oduc i i y and maximum p inciple Gi en u:Q→R3such ha ∇ · u= 0 in Qand u·n= 0 on ∂Ω, we conside he ( ep oduc i e) di usion-ad ec ion p oblem o he unknown c:Q→R(a concen a ion): ∂ c−∆c+u· ∇c= 0, c|Σ=cΣ, c(0) = c(T), whe e 0 < c ≤cΣ≤con Σ, o some cons an s cand c. In pa icula , ∂ (c−c)−∆(c−c) + (u· ∇)(c−c) = 0 in Q. Mul iplying by (c−c)+and in eg a ing in Ω (no ice ha (c−c)+= 0 on Σ), one has d d ZΩ |(c−c)+|2+ZΩ |∇(c−c)+|2≤0. In eg a ing in ∈(0, T) and using he pe iodic condi ion c(0) = c(T), one a i es a ZT 0 k∇(c−c)+k2 L2= 0. Hence c≤cin Qhold. Simila ly c≥cin Qhold. The e o e, one has he ollowing conclusion: The ep oduc i e solu ion conse e he maximum p inciple. In he ollowing models, he maximum p inciple has an impo an ole. Rep oduc i i y and ime pe iodici y o incomp essible luids 109 4.1 Gene alized Boussinesq sys em, wi h di usion depending on empe a u e When he iscosi y and hea conduc i i y a e empe a u e dependen un ions in he Boussinesq sys em, one has he ollowing sys em:        ∂ u− ∇ · (ν(θ)∇u) + (u· ∇)u+∇p=αθg+ , ∇ · u= 0, ∂ θ− ∇ · (k(θ)∇θ) + (u· ∇)θ= 0, (12) whe e ν: IR →IR+and k: IR →IR+a e s ic ly posi i e con inuous unc ions ( he kinema ic iscosi y and he he mal conduc i i y espec i ely). The p oblem is o ind a egula solu ion {u, θ, p}o (12) in Ω ×[0, T ], oge he he ollowing bounda y Di ichle da a: u= 0, θ =θ∂Ωon ∂Ω×[0, T),(13) and ime-pe iodic condi ions: u(0) = u(T), θ(0) = θ(T) in Ω.(14) We de ine θmin = min θ∂Ωθmax = max θ∂Ω. Thanks o he maximum p inciple, one has θmin ≤θ≤θmax in Q. Then, he e exis s νmin >0, kmin >0, νmax >0 and kmax >0 such ha νmin ≤ν(s)≤νmax and kmin ≤k(s)≤kmax,∀s∈[θmin, θmax]. One can p o es he exis ence o ep oduc i e solu ion in he same way ha in he classical Boussinesq case (see Sec ion 3.1), conside ing he equi alen p oblem ha esul changing νby eνand kby ek, whe e eν(θ) =        ν(θmin) i θ < θmin, ν(θ) i θmin ≤θ≤θmax, ν(θmax) i θ > θmax, ek(θ) =        k(θmin) i θ < θmin, k(θ) i θmin ≤θ≤θmax, k(θmax) i θ > θmax. 4.2 Penalized Nema ic liquid c ys al model We assume he ollowing nema ic liquid c ys al model in (0, T)×Ω, whe e Ω⊂RN o N= 2 o 3 is an open bounded domain: (∂ u+ (u· ∇)u−µ∆u+∇p=−λ∇ · (∇d ∇d),∇ · u= 0, ∂ d+ (u· ∇)d=γ(∆d− ε(d)).(15) 116 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda [17] A.C. Mo e i, M.A. Rojas-Meda and M.D. Rojas Meda . Rep oduc i e weak solu ions o gene alized Boussinesq models in ex e io domains. Ma . Con emp. 23:119–137, 2002. [18] K. ¯ Oeda. Pe iodic solu ions o he hea con ec ion equa ion in ex e io domain. P oc. Japan Acad. Se . A. 73: 49-54, 1997. [19] M.A. Rod ´ıguez-Bellido, M.A. Rojas-Meda and E.J. Villamiza -Roa. Pe iodic solu ions in unbounded domains o he Boussinesq equa ions. Submi ed o publica ion. [20] M.A. Rojas-Meda and J.L. Bold ini. Magne o-mic opola luid mo ion: exis ence o weak solu ions. Re . Ma . Complu . 11(2): 443–460, 1998. [21] Y. Taniuchi. On s abili y o pe iodic solu ions o he Na ie -S okes equa ions in unbounded domains. Hokkaido Ma h. J. 28(1): 147-173, 1999. [22] R. Temam. Na ie -S okes Equa ions, Theo y and nume ical analysis. No h-Holland Plublishing Co., Ams e dam, 1977. [23] M. Yamazaki. Solu ions in Mo ey Spaces o he Na ie -S okes equa ions wi h ime-dependen ex e nal o ce. Funk. Ek ac. 43: 419-460, 2000.