We consider the initial boundary value problem for the system of equations describing the nonstationary flow of an incompressible micropolar fluid in a domain Ω of R3.Under hypotheses that are similar to the Navier-Stokes equations ones, by using an iterative scheme, we prove the existence and uniqueness of strong solution in Lp(Ω), for p > 3

Boldrini, José Luiz

Durán Toro, Mario Manuel

Rojas Medar, Marko Antonio

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–8)Existence and Uniqueness of Strong Solutions for theIncompressible Micropolar Fluid Equations in Domains ofR3J. L. Boldrini 1, M. Dura´n2, M.A. Rojas-Medar31 IMECC-UNICAMP, CP 6065, 13083-859, Campinas-SP, Brazil. E-mails: boldrini@ime.unicamp.br.2 Facultad de Ingenier´ıa, Pontificia Universidad Cato´lica de Chile, Casilla 306, Santiago 22, Chile.E-mail: mduran@ing.puc.cl.3 Universidad del Bı´o-Bı´o Facultad de Ciencias, Departamento de Ciencias Ba´sicas, Campus FernandoMay, Casilla 447, Chilla´n, Chile. E-mails: marko@ueubiobio.cl,Palabras clave: Micropolar fluids, unbounded domains, hydrodynamics. existence of solutionsResumenWe consider the initial boundary value problem for the system of equations de-scribing the nonstationary flow of an incompressible micropolar fluid in a domain Ω ofR3. Under hypotheses that are similar to the Navier-Stokes equations ones, by usingan iterative scheme, we prove the existence and uniqueness of strong solution in Lp(Ω),for p > 3.1. IntroductionThe objective of the present work is to study the existence of strong solutions of theevolution equations for the motion of incompressible micropolar (asymmetric) fluids ina bounded or unbounded domain Ω ⊂ R3 having a compact C2-boundary. That is, thedomains we are considering include the the so called exterior domains. To describe theseequations, let T > 0 and QT ≡ Ω× (0, T ); then the system we will study is the following:∂u∂t+ (u · ∇)u − (µ+ µr)∆u + ∇η = 2µr rotw + f in QT ,div u = 0 in QT ,∂w∂t+ (u · ∇)w − (ca + cd)∆w + 4µrw− (c0 + cd − ca)∇div w = 2µrrot u + g in QT ,(1)1J.L. Boldrini, M. Dura´n, M.A. Rojas-Medartogether with the following boundary and initial conditionsu = 0 on ST ,w = 0 on ST ,u(x, 0) = u0(x) in Ω ,w(x, 0) = w0(x) in Ω ,(2)where ST ≡ ∂Ω × (0, T ). The vector-valued functions u = (u1, u2, u3), w = (w1, w2, w3)and the scalar function η denote respectively the velocity, the angular velocity of rotationof particles and the pressure of the fluid. The vector-valued functions f and g denoterespectively the external sources of linear and angular momentum. The positive constantsµ, µr, c0, ca and cd are viscosities-type coefficients satisfying the following inequality c0+cd > ca.For the derivation and physical discussion of equations (1)-(2) see Petrosyan [15],Condiff and Dalher [1], Eringen [4], [5] and Lukaszewicz [9]. We observe that this modelof fluid include as particular case the classical Navier-Stokes equations, which has beenwidely studied (see for instance the books by Ladyzhenskaya [6] or Temam [24], and thereferences therein). In this case, since µr = 0, equations (1) and (2) decouple.It is appropriate to recall earlier works on the initial-value problems closely related to(1)-(2) in order to clarify the intended contribution of the present work.Let us firstly consider the situation when Ω is a bounded regular domain. In this case,Lukaszewicz [9] established for a restricted class of initial data, the existence of weak andstrong global solutions using, in both cases, an iterative linearized scheme together with afixed point result. For initial data similar to the case of the classical Navier-Stokes equa-tions, by applying the spectral Galerkin method, Rojas-Medar & Boldrini [17] proved theglobal existence and uniqueness of weak solutions in the two-dimensional case; existenceof the local and global in time strong solution was obtained respectively by Rojas-Medarin [18] and by Ortega-Torres and Rojas-Medar in [13]. Ortega-Torres and Rojas-Medar,following the arguments given by Serrin in [21], also considered the uniqueness of weaksolution in [14]. In [19], Rojas-Medar obtained the convergence rates associated to theapproximate solutions constructed by the Galerkin method. The existence of reproductivesolution (so called periodic weak solution) for the previous system was proved in [17].Recently, Rese´ndiz and Rojas-Medar [16] have proved the existence of weak solution ina smooth time dependent domain. By using and interactive approach Rojas-Medar andOrtega-Torres [20] show the existence and uniqueness of the strong solutions in boundeddomains in the L2-context. The existence and uniqueness of periodic strong solutions wasdone in [10] using the Galerkin method. Yamaguchi [25] also studied the problem (1)-(2)inbounded domains using the semigroup approach in Lp, 1 < p <∞; he shows the existenceof global strong solutions for small data.The case of unbounded domains Ω is less studied. When Ω is an exterior domain, for therelated model of the magneto-micropolar fluid, existence of a stationary weak solution wasstudied by Dura´n et al. in [2], while the existence of reproductive solution was establishedin [3]. For two-dimensional unbounded domains, one can look at the word by Lukaszewiczan Sadowski [12].In the present work, as we said previously, we are interested in the flow of micropolarfluids in bounded ou unbounded domains of R3 with compact C2-boundaries. By usingan iterative procedure we will prove the existence and uniqueness of strong solutions in2Micropolar fluids in domains of R3Lp(Ω), for any p > 3. Specifically, we will prove the following (local) existence result ofstrong solutions.Theorem 1.1 Let Ω ⊂ R3 have a non-void regular boundary ∂Ω in the sense of Solonnikovand let p > 3. Assume that u0(x) ∈W 2−2p (Ω), u0|ST = 0, div u0 = 0, w0(x) ∈W2− 2pp (Ω),w0|ST = 0, f ,g ∈Lp(QT ).Then there exists T1 ∈ (0, T ] such that problem (1)-(2) has a unique solution (u,w, η)satisfying u ∈W 2,1p (QT1), ∇η ∈ Lp(QT1), w ∈W 2,1p (QT1).In this statement, we used the classical notations for the Sobolev-type spaces W kp (Ω) andW 2,1p (QT ).The present work is organized as follows: in Section 2 we fix the notations, and statepreliminaries results that will be useful in the rest of the paper. More precisely, we state theexistence, the uniqueness and regularity (a priori estimates) for two linear problems closelyrelated to (1)-(2). We also describe in this section the iterative scheme that constructthe approximate solutions. In Section 3, we obtain estimates in several norms for suchapproximate solutions. Finally, in Section 4, we show that the approximate the solutionsconverge to a strong solution of our original problem.We remark that, as it is usual in this kind of context to simplify the notations, wewill denote by c, C0, M0 and so on generic finite positive constants depending only on Ωand the other fixed parameters of the problem (like the initial data). That is, they mayhave different values in different expressions. In a few points to emphasize the fact thatthe constants are in fact different, we use C1, C2, ...,M1,M2. · · · and so on.2. Preliminaries and iterative schemeFor any t ∈ (0, T ], we will denote Qt = Ω × (0, t). As previously said, we will useclassical notations for the Sobolev-type spaces; we will also use freely the standard resultsfor such spaces. Here we just recall that the restriction of a function in W 2,1p (QT ) onthe hyperplane t = constant belongs for ∀ t ∈ [0, T ] to the Slobodetskii-Besov spaceW2− 2pp (Ω) and depend continuously on t in the norm of W2− 2pp (Ω). Moreover, it holds that‖u(·, t)‖W2− 2pp (Ω)≤ ‖u(·, 0)‖W2− 2pp (Ω)+ ĉ‖u‖W 2,1p (QT ), (3)where the constant ĉ does not depend on t ∈ [0, T ]. For more details of the Slobodetskii-Besov space see [8], for instance.Next, we recall some results associated to two linear problems closely related to (1)-(2).The first result is proved in Solonnikov [23] and is the following:Lemma 2.1 Let F (x, t) ∈ Lp(QT ) and u0(x) ∈W2− 2pp (Ω) with u0|ST = 0 and div u0 = 0,then the following problemut − (µ+ µr)∆u+∇η = F,div u = 0,u|ST = 0,u(0) = u0(x)3J.L. Boldrini, M. Dura´n, M.A. Rojas-Medarhas a unique solution u ∈ W 2,1p (QT ), η ∈ W 1,0p (QT ) (η is unique up to a constant,)satisfying‖u‖W 2,1p (QT1)+ ‖∇η‖Lp(QT1 ) ≤ K1(T1)(‖u0‖W 2− 2pp (Ω)+ ‖F‖Lp(QT1 )),where K1(·) is an increasing function of T1 ∈ (0, T ]The following result is a special case of the result for parabolic system given in [22].Lemma 2.2 Let G(x, t) ∈ Lp(QT ) and w0(x) ∈ W2− 2pp (Ω) with w0|ST = 0, then thefollowing problemwt − (ca + cd)∆w − (c0 + cd − ca)∇div w + 4µrw = Gw|ST = 0,w(0) = w0(x)has a unique solution w ∈W 2,1p (QT ), satisfying‖w‖W 2,1p (QT1 )≤ K2(T1)(‖w0‖W2− 2pp (Ω)+ ‖G‖Lp(QT1 )),where K2(·) is an increasing function of T1 ∈ (0, T ].Iterative Scheme:Next, we describe the iteration scheme used to construct approximate solutions of ourproblem.Takeu(0) = 0, w(0) = 0and for k = 1, 2, 3, . . . recursively take {u(k), η(k)} and {w(k)} respectively as the solutionsof problemsu(k)t − (µ+ µr)4u(k) +∇η(k) = f + 2µr rot w(k−1) − (u(k−1) · ∇)u(k−1),div u(k) = 0,u(k)|ST = 0,u(k)(0) = u0(x)andw(k)t − (ca + cd)4w(k) − (c0 + cd − ca)∇ div w(k) + 4µrw(k)= g+2µr rot u(k−1) − (u(k−1) · ∇)w(k−1),w(k)|ST = 0,w(k)(0) = w0(x).3. Estimates of the approximate solutionsTo obtain the required estimates for the sequence (uk, ηk,wk), we start by defining:Φ(k)(T1) = ‖u(k)‖W 2,1p (QT1 ) + ‖w(k)‖W 2,1p (QT1 )+ ‖∇η(k)‖Lp(QT1 ), (4)for 0 < T1 ≤ TThen, we can prove the following two lemmas.4Micropolar fluids in domains of R3Lemma 3.1 The elements of the sequence {w(k)} satisfy for any T1 ∈ (0, T ] the followingestimate:‖∇w(k−1)‖Lp(QT1 ) ≤ C(‖w0‖W 2− 2pp (Ω)+ aT1−aap1 Φ(k−1)(T1) + T δ1Φ(k−1)(T1)),where C is independent of T1 ∈ (0, T ] anda =p− 32p− 3 and δ1 = (1−1p)(1− 3p)(1− a) + 1− ap.Remark 3.2 Analogous result is valid for {u(k)}.Lemma 3.3 Let 0 < T1 ≤ 1. Then, there is a constant α > 0 such that‖(u(k−1) · ∇)w(k−1)‖Lp(QT1 ) ≤ C[‖u0‖2W2− 2p2p (Ω)+ ‖w0‖2W2− 2p2p (Ω)+ Tα(Φ(k−1)(T1))2].where C is independent of T1 ∈ (0, T ].Next, we prove the boundness of the sequence {u(k), η(k),w(k)}.Lemma 3.4 For sufficiently small T1 ∈ (0, T ], the sequence {u(k), η(k),w(k)} is boundedin W 2,1p (QT1)× Lp(QT )×W 2,1p (QT1).4. Proof of Theorem 1.1Setting u(n,s)(t) = u(n+s)(t)−u(n)(t), η(n,s) = η(n+s)−η(n) and w(n,s) = w(n+s)−w(n),we haveu(n,s)t− (µ+ µr)4u(n,s) +∇η(n,s) = F (n,s),div u(n,s) = 0,u(n,s)|ST = 0,u(n,s)(0) = 0,(5)whereF (n,s) = 2µr rot w(n−1,s) − (u(n−1,s) · ∇)u(n+s−1) − (u(n−1) · ∇)u(n−1,s). (6)Alsow(n,s)t − (ca + cd)4w(n,s) − (c0 + cd − ca)∇ div w(n,s) + 4µrw(n,s) = G(n,s),w(n,s)|ST = 0,w(n,s)(0) = 0,(7)whereG(n,s) = 2µr rot u(n−1,s) − (u(n+s−1) · ∇)w(n−1,s) − (u(n−1,s) · ∇)w(n−1). (8)5J.L. Boldrini, M. Dura´n, M.A. Rojas-MedarWe then are able to prove that‖F (n,s)‖pLp(Qt) ≤ c∫ t0‖u(n−1,s)‖pW 2,1p (Qτ )dτ + (‖u0‖W2− 2pp (Ω)+ĉ‖u(n−1+s)(τ)‖W 2,1p (Qt))p∫ t0ĉp‖u(n−1,s)‖pW 2,1p (Qτ )dτ+(‖u0‖W2− 2pp (Ω)(9)+ĉ‖u(n−1+s)(τ)‖W 2,1p (Qt))p∫ t0ĉp‖u(n−1,s)‖pW 2,1p (Qτ )dτ.‖G(n,s)‖pLp(Qt) ≤ c(‖∇u(n−1,s)‖pLp(Qt)+ ‖(u(n−1,s) · ∇)w(n−1)‖pLp(Qt)+‖(u(n+s−1) · ∇)w(n−1,s)‖pLp(Qt)≤ c∫ t0‖u(n−1,s)‖pW 2,1p (Qτ )dτ + c(‖w0‖W2− 2pp (Ω)+ĉ‖w(n−1)(τ)‖W 2,1p (Qt))p∫ t0‖u(n−1,s)‖pW 2,1p (Qτ )dτ (10)+c(‖u0‖W2− 2pp (Ω)+ĉ‖u(n+s−1)‖W 2,1p (Qt))p∫ t0‖w(n−1,s)‖pW 2,1p (Qτ )dτ.From estimates (9)-(10) and Lemma 3.4, we conclude that for t ∈ [0, T1] and p > 3, ifwe callΨ(n,s)(t) = ‖u(n,s)‖W 2,1p (Qt)+ ‖w(n,s)‖W 2,1p (Qt)+ ‖∇η(n,s)‖Lp(Qt), (11)we then haveΨ(n,s)(t) ≤ c(∫ t0Ψ(n−1,s)(τ)p) 1p.Therefore, [Ψ(n,s)(t)]p ≤ cp ∫ t0[Ψ(n−1,s)(τ)]pdτ, (12)and consequently Ψ(n,s)(t)→ 0 as n→∞, ∀ t ∈ [0, T1].In particular, since W 2,1p (QT1) and Lp(QT1) are Banach spaces, there exist u,w ∈W 2,1p (QT1) and η ∈ Lp(QT1) such thatun → u strongly in W 2,1p (QT1),wn → w strongly in W 2,1p (QT1),ηn → η strongly in Lp(QT1).The next step is to take the limit as n → +∞ in the approximate equations in theiterative scheme. However, once the above convergences have been established, this isstandard and we obtain that u,w, η is a strong solution of the problem (1)-(2).6Micropolar fluids in domains of R3We need only to consider the uniqueness of the solution in order to complete the proofof Theorem. For this, suppose that there exists another solution u1,w1, η1 of (1) and (2)with the same regularity as stated in the theorem. Then, defineU = u1 − u, W = w1 −w, P = η1 − η,and observe that these auxiliary functions verify a set of equations similar to (5)-(7).Repeating the arguments used to obtain (12), using the known regularity of the solutions,we get for θ(t) = ‖U‖pW 2,1p (Qt)+ ‖W‖pW 2,1p (Qt)+ ‖P‖pLp(Qt) an inequality of the followingtypeθ(t) ≤ c∫ t0θ(τ)dτwhich by Gronwall’s inequality implies that U = 0,W = 0, P = 0 and thus the uniquenessof our strong solutions.AcknowledgmentsDuring this research J.L. Boldrini was partially supported by CGCI MECD-DGUBrazil/Spain Grant 2137-05-4; M.A. Rojas-Medar was partially supported by D.G.E.S. andM.C. y T. (Spain) Grant BFM2003-06446-C02-01 and CGCI MECD-DGU Brazil/SpainGrant 2137-05-4. These authors are grateful for such support.Referencias[1] D.W. Condiff, J.S. Dahler, Fluid mechanics aspects of antisymmetric stress, Phys. Fluids, 11, (1964),842-854.[2] M. Dura´n, E. Ortega-Torres, M. Rojas-Medar, Stationary solutions of magneto-micropolar fluidsequations in exterior domains. Proyecciones 22 (2003), no. 1, 63–79..[3] M. Dura´n, J. Ferreira, M. Rojas-Medar, Reproductive Weak Solutions of Magneto-Micropolar FluidEquations in Exterior Domains, Mathematical and Computer Modelling, 35, (2002), 779-791.[4] A.C. Eringen, Theory of micropolar fluids, J.Math. Mech., 16, (1966), 1-8.[5] A.C. Eringen, Simple microfluids, Int. J. Enging. Sci., 2, (1964), 205-217.[6] O. Ladyszhenskaya, The Mathematical Theory of Viscous Incompressible Flow, second edition, Gor-don and Breach Eds., New York (1969).[7] O. Ladyszhenskaya, V.A. Solonnikov, Unique Solvability of an Initial and Boundary Value Problemfor Viscous Incompressible Nonhomogeneous Fluids, Zap. Naucˆn Sem. - Leningrado Otdel Math.Inst. Steklov, 52, (1976), 52-109; English Transl., J. Soviet Math., 9, (1978), 697-749.[8] O. Ladyszhenskaya, V.A. Solonnikov, N. Ural’ceva, Linear and Quasilinear Equations of ParabolicType, Transl. Math. Monographs, Amer. Math. Soc, 23, (1968).[9] G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Birkha¨user, Berlin (1998).[10] G. Lukaszewicz, E.E. Ortega-Torres, M.A. Rojas-Medar, Strong periodic solutions for a class ofabstract evolution equations. Non. Anal. 54 (2003), no. 6, 1045–1056.[11] G. Lukaszewicz, Asymptotic behavior of micropolar fluid flows. The Eringen anniversary issue (Uni-versity Park, PA, 2002). Internat. J. Engrg. 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Complutense de Madrid, 11 (2), (1998), 443-460.[18] M. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution,Math. Nachr., 188, (1997), 301-319.[19] M. Rojas-Medar, Magneto-micropolar fluid motion: on the convergence rate of the spectral Galerkinapproximations, Z. Angew. Math. Mech., 77, (1997), 723-732.[20] M.A. Rojas-Medar, E.E. Ortega-Torres, E. E., The equations of a viscous asymmetric fluid: aninteractive approach. Z. Angew. Math. Mech. 85 (2005), no. 7, 471–489.[21] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat.Mech. Anal., 9, (1962) 187-195.[22] V.A. Solonnikov, On the boundary-value problems for linear parabolic system of differential equationsof general form, Trudy Mat. Inst. Steklov, 83, (1965), 1-162.[23] V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nacˇn. Sem.Leningrad. Otdel. Mat. Inst. Steklov (LOM), 38, (1973) 153-231.[24] R. 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Existence and uniqueness of strong solutions for the incompressible micropolar fluid equations in domains of R3

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

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idUS. Depósito de Investigación Universidad de Sevilla