Existence and uniqueness of strong solutions for the incompressible micropolar fluid equations in domains of R3

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–8) Exis ence and Uniqueness o S ong Solu ions o he Incomp essible Mic opola Fluid Equa ions in Domains o R3 J. L. Bold ini 1, M. Du ´ an2, M.A. Rojas-Meda 3 1IMECC-UNICAMP, CP 6065, 13083-859, Campinas-SP, B azil. E-mails: [email p o ec ed]. 2Facul ad de Ingenie ´ıa, Pon i icia Uni e sidad Ca ´olica de Chile, Casilla 306, San iago 22, Chile. E-mail: [email p o ec ed]. 3Uni e sidad del B´ıo-B´ıo Facul ad de Ciencias, Depa amen o de Ciencias B´asicas, Campus Fe nando May, Casilla 447, Chill´an, Chile. E-mails: [email p o ec ed], Palab as cla e: Mic opola luids, unbounded domains, hyd odynamics. exis ence o solu ions Resumen We conside he ini ial bounda y alue p oblem o he sys em o equa ions de- sc ibing he nons a iona y low o an incomp essible mic opola luid in a domain Ω o R3. Unde hypo heses ha a e simila o he Na ie -S okes equa ions ones, by using an i e a i e scheme, we p o e he exis ence and uniqueness o s ong solu ion in Lp(Ω), o p > 3. 1. In oduc ion The objec i e o he p esen wo k is o s udy he exis ence o s ong solu ions o he e olu ion equa ions o he mo ion o incomp essible mic opola (asymme ic) luids in a bounded o unbounded domain Ω ⊂R3ha ing a compac C2-bounda y. Tha is, he domains we a e conside ing include he he so called ex e io domains. To desc ibe hese equa ions, le T > 0 and QT≡Ω×(0, T); hen he sys em we will s udy is he ollowing:              ∂u ∂ + (u· ∇)u−(µ+µ )∆u+∇η= 2µ o w+ in QT, di u= 0 in QT, ∂w ∂ + (u· ∇)w−(ca+cd)∆w+ 4µ w −(c0+cd−ca)∇di w= 2µ o u+gin QT, (1) 1 J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda oge he wi h he ollowing bounda y and ini ial condi ions        u=0on ST, w=0on ST, u(x, 0) = u0(x) in Ω , w(x, 0) = w0(x) in Ω , (2) whe e ST≡∂Ω×(0, T). The ec o - alued unc ions u= (u1, u2, u3),w= (w1, w2, w3) and he scala unc ion ηdeno e espec i ely he eloci y, he angula eloci y o o a ion o pa icles and he p essu e o he luid. The ec o - alued unc ions and gdeno e espec i ely he ex e nal sou ces o linea and angula momen um. The posi i e cons an s µ, µ , c0, caand cda e iscosi ies- ype coe icien s sa is ying he ollowing inequali y c0+ cd> ca. Fo he de i a ion and physical discussion o equa ions (1)-(2) see Pe osyan [15], Condi and Dalhe [1], E ingen [4], [5] and Lukaszewicz [9]. We obse e ha his model o luid include as pa icula case he classical Na ie -S okes equa ions, which has been widely s udied (see o ins ance he books by Ladyzhenskaya [6] o Temam [24], and he e e ences he ein). In his case, since µ = 0, equa ions (1) and (2) decouple. I is app op ia e o ecall ea lie wo ks on he ini ial- alue p oblems closely ela ed o (1)-(2) in o de o cla i y he in ended con ibu ion o he p esen wo k. Le us i s ly conside he si ua ion when Ω is a bounded egula domain. In his case, Lukaszewicz [9] es ablished o a es ic ed class o ini ial da a, he exis ence o weak and s ong global solu ions using, in bo h cases, an i e a i e linea ized scheme oge he wi h a ixed poin esul . Fo ini ial da a simila o he case o he classical Na ie -S okes equa- ions, by applying he spec al Gale kin me hod, Rojas-Meda & Bold ini [17] p o ed he global exis ence and uniqueness o weak solu ions in he wo-dimensional case; exis ence o he local and global in ime s ong solu ion was ob ained espec i ely by Rojas-Meda in [18] and by O ega-To es and Rojas-Meda in [13]. O ega-To es and Rojas-Meda , ollowing he a gumen s gi en by Se in in [21], also conside ed he uniqueness o weak solu ion in [14]. In [19], Rojas-Meda ob ained he con e gence a es associa ed o he app oxima e solu ions cons uc ed by he Gale kin me hod. The exis ence o ep oduc i e solu ion (so called pe iodic weak solu ion) o he p e ious sys em was p o ed in [17]. Recen ly, Res´endiz and Rojas-Meda [16] ha e p o ed he exis ence o weak solu ion in a smoo h ime dependen domain. By using and in e ac i e app oach Rojas-Meda and O ega-To es [20] show he exis ence and uniqueness o he s ong solu ions in bounded domains in he L2-con ex . The exis ence and uniqueness o pe iodic s ong solu ions was done in [10] using he Gale kin me hod. Yamaguchi [25] also s udied he p oblem (1)-(2)in bounded domains using he semig oup app oach in Lp, 1 < p < ∞; he shows he exis ence o global s ong solu ions o small da a. The case o unbounded domains Ω is less s udied. When Ω is an ex e io domain, o he ela ed model o he magne o-mic opola luid, exis ence o a s a iona y weak solu ion was s udied by Du ´an e al. in [2], while he exis ence o ep oduc i e solu ion was es ablished in [3]. Fo wo-dimensional unbounded domains, one can look a he wo d by Lukaszewicz an Sadowski [12]. In he p esen wo k, as we said p e iously, we a e in e es ed in he low o mic opola luids in bounded ou unbounded domains o R3wi h compac C2-bounda ies. By using an i e a i e p ocedu e we will p o e he exis ence and uniqueness o s ong solu ions in 2 Mic opola luids in domains o R3 Lp(Ω), o any p > 3. Speci ically, we will p o e he ollowing (local) exis ence esul o s ong solu ions. Theo em 1.1 Le Ω⊂R3ha e a non- oid egula bounda y ∂Ωin he sense o Solonniko and le p > 3. Assume ha u0(x)∈W2− 2 p(Ω),u0|ST= 0,di u0= 0,w0(x)∈W2− 2 p p(Ω), w0|ST= 0, ,g∈Lp(QT). Then he e exis s T1∈(0, T ]such ha p oblem (1)-(2) has a unique solu ion (u,w, η) sa is ying u∈W2,1 p(QT1),∇η∈Lp(QT1),w∈W2,1 p(QT1). In his s a emen , we used he classical no a ions o he Sobole - ype spaces Wk p(Ω) and W2,1 p(QT). The p esen wo k is o ganized as ollows: in Sec ion 2 we ix he no a ions, and s a e p elimina ies esul s ha will be use ul in he es o he pape . Mo e p ecisely, we s a e he exis ence, he uniqueness and egula i y (a p io i es ima es) o wo linea p oblems closely ela ed o (1)-(2). We also desc ibe in his sec ion he i e a i e scheme ha cons uc he app oxima e solu ions. In Sec ion 3, we ob ain es ima es in se e al no ms o such app oxima e solu ions. Finally, in Sec ion 4, we show ha he app oxima e he solu ions con e ge o a s ong solu ion o ou o iginal p oblem. We ema k ha , as i is usual in his kind o con ex o simpli y he no a ions, we will deno e by c,C0,M0and so on gene ic ini e posi i e cons an s depending only on Ω and he o he ixed pa ame e s o he p oblem (like he ini ial da a). Tha is, hey may ha e di e en alues in di e en exp essions. In a ew poin s o emphasize he ac ha he cons an s a e in ac di e en , we use C1, C2, ..., M1, M2.· · · and so on. 2. P elimina ies and i e a i e scheme Fo any ∈(0, T], we will deno e Q = Ω ×(0, ). As p e iously said, we will use classical no a ions o he Sobole - ype spaces; we will also use eely he s anda d esul s o such spaces. He e we jus ecall ha he es ic ion o a unc ion in W2,1 p(QT) on he hype plane = cons an belongs o ∀ ∈[0, T ] o he Slobode skii-Beso space W2− 2 p p(Ω) and depend con inuously on in he no m o W2− 2 p p(Ω). Mo eo e , i holds ha ku(·, )k W 2− 2 p p(Ω) ≤ ku(·,0)k W 2− 2 p p(Ω) +bckukW2,1 p(QT),(3) whe e he cons an bcdoes no depend on ∈[0, T]. Fo mo e de ails o he Slobode skii- Beso space see [8], o ins ance. Nex , we ecall some esul s associa ed o wo linea p oblems closely ela ed o (1)-(2). The i s esul is p o ed in Solonniko [23] and is he ollowing: Lemma 2.1 Le F(x, )∈Lp(QT)and u0(x)∈W2− 2 p p(Ω) wi h u0|ST= 0 and di u0= 0, hen he ollowing p oblem u −(µ+µ )∆u+∇η=F, di u= 0, u|ST= 0, u(0) = u0(x) 3 J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda has a unique solu ion u∈W2,1 p(QT),η∈W1,0 p(QT)(ηis unique up o a cons an ,) sa is ying kukW2,1 p(QT1)+k∇ηkLp(QT1)≤K1(T1)(ku0k W 2− 2 p p(Ω) +kFkLp(QT1)), whe e K1(·)is an inc easing unc ion o T1∈(0, T ] The ollowing esul is a special case o he esul o pa abolic sys em gi en in [22]. Lemma 2.2 Le G(x, )∈Lp(QT)and w0(x)∈W2− 2 p p(Ω) wi h w0|ST= 0, hen he ollowing p oblem w −(ca+cd)∆w−(c0+cd−ca)∇di w+ 4µ w=G w|ST= 0, w(0) = w0(x) has a unique solu ion w∈W2,1 p(QT), sa is ying kwkW2,1 p(QT1)≤K2(T1)(kw0k W 2− 2 p p(Ω) +kGkLp(QT1)), whe e K2(·)is an inc easing unc ion o T1∈(0, T ]. I e a i e Scheme: Nex , we desc ibe he i e a ion scheme used o cons uc app oxima e solu ions o ou p oblem. Take u(0) =0,w(0) =0 and o k= 1,2,3, . . . ecu si ely ake {u(k), η(k)}and {w(k)} espec i ely as he solu ions o p oblems u(k) −(µ+µ )4u(k)+∇η(k)= + 2µ o w(k−1) −(u(k−1) · ∇)u(k−1), di u(k)= 0, u(k)|ST= 0, u(k)(0) = u0(x) and w(k) −(ca+cd)4w(k)−(c0+cd−ca)∇di w(k)+ 4µ w(k) =g+2µ o u(k−1) −(u(k−1) · ∇)w(k−1), w(k)|ST= 0, w(k)(0) = w0(x). 3. Es ima es o he app oxima e solu ions To ob ain he equi ed es ima es o he sequence (uk, ηk,wk), we s a by de ining: Φ(k)(T1) = ku(k)kW2,1 p(QT1)+kw(k)kW2,1 p(QT1)+k∇η(k)kLp(QT1),(4) o 0 < T1≤T Then, we can p o e he ollowing wo lemmas. 4 Mic opola luids in domains o R3 Lemma 3.1 The elemen s o he sequence {w(k)}sa is y o any T1∈(0, T ] he ollowing es ima e: k∇w(k−1)kLp(QT1)≤C(kw0k W 2− 2 p p(Ω) +aT 1−a ap 1Φ(k−1)(T1) + Tδ1Φ(k−1)(T1)), whe e Cis independen o T1∈(0, T ]and a=p−3 2p−3and δ1= (1 −1 p)(1 −3 p)(1 −a) + 1−a p. Rema k 3.2 Analogous esul is alid o {u(k)}. Lemma 3.3 Le 0< T1≤1. Then, he e is a cons an α > 0such ha k(u(k−1) · ∇)w(k−1)kLp(QT1)≤C[ku0k2 W 2− 2 p 2 p(Ω) +kw0k2 W 2− 2 p 2 p(Ω) +Tα(Φ(k−1)(T1))2]. whe e Cis independen o T1∈(0, T ]. Nex , we p o e he boundness o he sequence {u(k), η(k),w(k)}. Lemma 3.4 Fo su icien ly small T1∈(0, T ], he sequence {u(k), η(k),w(k)}is bounded in W2,1 p(QT1)×Lp(QT)×W2,1 p(QT1). 4. P oo o Theo em 1.1 Se ing u(n,s)( ) = u(n+s)( )−u(n)( ), η(n,s)=η(n+s)−η(n)and w(n,s)=w(n+s)−w(n), we ha e u(n,s) −(µ+µ )4u(n,s)+∇η(n,s)=F(n,s), di u(n,s)= 0, u(n,s)|ST= 0, u(n,s)(0) = 0, (5) whe e F(n,s)= 2µ o w(n−1,s)−(u(n−1,s)· ∇)u(n+s−1) −(u(n−1) · ∇)u(n−1,s).(6) Also w(n,s) −(ca+cd)4w(n,s)−(c0+cd−ca)∇di w(n,s)+ 4µ w(n,s)=G(n,s), w(n,s)|ST= 0, w(n,s)(0) = 0, (7) whe e G(n,s)= 2µ o u(n−1,s)−(u(n+s−1) · ∇)w(n−1,s)−(u(n−1,s)· ∇)w(n−1).(8) 5 J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda We hen a e able o p o e ha kF(n,s)kp Lp(Q )≤cZ 0 ku(n−1,s)kp W2,1 p(Qτ)dτ + (ku0k W 2− 2 p p(Ω) +bcku(n−1+s)(τ)kW2,1 p(Q ))pZ 0bcpku(n−1,s)kp W2,1 p(Qτ)dτ +(ku0k W 2− 2 p p(Ω) (9) +bcku(n−1+s)(τ)kW2,1 p(Q ))pZ 0bcpku(n−1,s)kp W2,1 p(Qτ)dτ. kG(n,s)kp Lp(Q )≤c(k∇u(n−1,s)kp Lp(Q )+k(u(n−1,s)· ∇)w(n−1)kp Lp(Q ) +k(u(n+s−1) · ∇)w(n−1,s)kp Lp(Q ) ≤cZ 0 ku(n−1,s)kp W2,1 p(Qτ)dτ +c(kw0k W 2− 2 p p(Ω) +bckw(n−1)(τ)kW2,1 p(Q ))pZ 0 ku(n−1,s)kp W2,1 p(Qτ)dτ (10) +c(ku0k W 2− 2 p p(Ω) +bcku(n+s−1)kW2,1 p(Q ))pZ 0 kw(n−1,s)kp W2,1 p(Qτ)dτ. F om es ima es (9)-(10) and Lemma 3.4, we conclude ha o ∈[0, T1] and p > 3, i we call Ψ(n,s)( ) = ku(n,s)kW2,1 p(Q )+kw(n,s)kW2,1 p(Q )+k∇η(n,s)kLp(Q ),(11) we hen ha e Ψ(n,s)( )≤cµZ 0 Ψ(n−1,s)(τ)p¶1 p . The e o e, hΨ(n,s)( )ip≤cpZ 0hΨ(n−1,s)(τ)ip dτ, (12) and consequen ly Ψ(n,s)( )→0 as n→ ∞,∀ ∈[0, T1]. In pa icula , since W2,1 p(QT1) and Lp(QT1) a e Banach spaces, he e exis u,w∈ W2,1 p(QT1) and η∈Lp(QT1) such ha un→us ongly in W2,1 p(QT1), wn→ws ongly in W2,1 p(QT1), ηn→ηs ongly in Lp(QT1). The nex s ep is o ake he limi as n→+∞in he app oxima e equa ions in he i e a i e scheme. Howe e , once he abo e con e gences ha e been es ablished, his is s anda d and we ob ain ha u,w, η is a s ong solu ion o he p oblem (1)-(2). 6 Mic opola luids in domains o R3 We need only o conside he uniqueness o he solu ion in o de o comple e he p oo o Theo em. Fo his, suppose ha he e exis s ano he solu ion u1,w1, η1o (1) and (2) wi h he same egula i y as s a ed in he heo em. Then, de ine U=u1−u, W =w1−w, P =η1−η, and obse e ha hese auxilia y unc ions e i y a se o equa ions simila o (5)-(7). Repea ing he a gumen s used o ob ain (12), using he known egula i y o he solu ions, we ge o θ( ) = kUkp W2,1 p(Q )+kWkp W2,1 p(Q )+kPkp Lp(Q )an inequali y o he ollowing ype θ( )≤cZ 0 θ(τ)dτ which by G onwall’s inequali y implies ha U= 0, W= 0, P= 0 and hus he uniqueness o ou s ong solu ions. Acknowledgmen s Du ing his esea ch J.L. Bold ini was pa ially suppo ed by CGCI MECD-DGU B azil/Spain G an 2137-05-4; M.A. Rojas-Meda was pa ially suppo ed by D.G.E.S. and M.C. y T. (Spain) G an BFM2003-06446-C02-01 and CGCI MECD-DGU B azil/Spain G an 2137-05-4. These au ho s a e g a e ul o such suppo . Re e encias [1] D.W. Condi , J.S. Dahle , Fluid mechanics aspec s o an isymme ic s ess, Phys. Fluids, 11, (1964), 842-854. [2] M. Du ´an, E. O ega-To es, M. Rojas-Meda , S a iona y solu ions o magne o-mic opola luids equa ions in ex e io domains. P oyecciones 22 (2003), no. 1, 63–79.. [3] M. Du ´an, J. Fe ei a, M. 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