Micropolar Fluids with Vanishing Viscosity

and Applied Analysis 3 a bounded domain of R3. Indeed, the analysis of our situation is still more difficult. The difficulties arise from the lack of smoothness of the weak solution. To overcome this difficulty a penalization argument is needed. This argument generalizes the penalization method given in 10 , for the Navier-Stokes equations, to this case of micropolar fluids. In fact, if we take the viscosity of microrotation μr 0, our results imply the other ones in 10 , where the analysis of the convergence in an appropriate sense, of solutions of Navier-Stokes equations to the solutions of the Euler equations on a small time interval, is given. It is worthwhile to remark that 10 has been the unique work where the convergence of nonstationary Navier-Stokes equations, with vanishing viscosity, to the Euler equations, in a bounded domain of R3, has been considered. In the whole space R3, the authors of 11–13 analyzed the convergence, as the viscosity tends to zero, of the Navier-Stokes equations to the solution of the Euler equations on a small time interval. The two-dimensional case is more usual in the literature. In fact, the book 14 presents a result where the fundamental argument involves the stream formulation for the Navier-Stokes equations, which is not applicable in the three-dimensional case. This paper is organized as follows. In Section 2 the basic notation is stated and the main results are formulated. In Section 3, the analysis of convergence of solutions of the initial value problem 1.1 – 1.5 , when the viscosities ν1, ν2, ν3 tend to zero, is done. This analysis is based on the ideas of 10 for Navier-Stokes equations in bounded domains. 2. Statements and Notations Let Ω be a bounded domain of R3 with smooth enough boundary ∂Ω. We consider the usual Sobolev spaces H Ω {f ∈ L2 Ω : ‖Df‖L2 < ∞, |k| ≤ m}, m ≥ 1, with norm denoted by ‖ · ‖Hm . H1 0 Ω is the closure C∞ 0 Ω in the norm ‖ · ‖H1 . In order to distinguish the scalarvalue functions to vector-value functions, bold characters will be used; for instance, H H Ω 3 and so on. The solenoidal functional spaces H {v ∈ L2 Ω /divv 0 in Ω, v · n 0 on ∂Ω} and V {v ∈ H0 Ω /divv 0 in Ω}, will be also used. Here the Helmholtz decomposition of the space L2 Ω H ⊕G, where G {φ : φ ∇p, p ∈ H1 Ω }, is recalled. Throughout the paper, P denotes the orthogonal projection from L2 onto H. The norm in the L-spaces will be denoted by ‖ · ‖p. In particular, the norm in L2 and its scalar product will be denoted by ‖ · ‖ and ·, · , respectively. Moreover 〈·, ·〉 will denote some duality products. We remark that, in the rest of this paper, the letter C denotes inessential positive constants which may vary from line to line. In order to study the behavior of system 1.1 – 1.5 , when the viscosities ν1, ν2, ν3 tend to zero, the initial value problem 1.6 – 1.10 is required to study. An immediate question related to the system 1.6 – 1.10 is to know about the existence of its solution. In the following lemma a partial result about the existence and uniqueness of solution of problem 1.6 – 1.10 is given. For that, let us consider the following functional space: F0 {( Φ t2P Φ · ∇Φ ,Ψ t2Φ · ∇Ψ ) : Φ ∈ V ∩H3,Ψ ∈ H0 ∩H3 } ⊂ ( L∞ ( 0, T ;H0 ))2 . 2.1 Thus we have the following lemma. 4 Abstract and Applied Analysis Lemma 2.1. Let f,g ∈ F0. Then there is a unique solution u ∈ L∞ 0, T ;V ∩ H3 , w ∈ L∞ 0, T ;H0 ∩ H3 , p ∈ L∞ 0, T ;H2/R of problem 1.6 – 1.10 . Proof. The proof follows by using the arguments of 10, Lemma 3.1 . Indeed, with f,g being an element of F0, we consider Φ,Ψ ∈ V ∩H3 ×H0 ∩H3 and define u x, t t Φ x ∈ L∞ ( 0, T ;V ∩H3 ) , w x, t tΨ x ∈ L∞ ( 0, T ;H0 ∩H3 ) . 2.2 Note that the pair u,w satisfies conditions 1.4 and 1.5 . Moreover, u · ∇u ∈ L∞ 0, T ;L2 and thus, u · ∇u I −P u · ∇u P u · ∇u . Then, ut x, t Φ x andwt x, t Ψ x . Hence ut u · ∇u ∇p Φ P u · ∇u Φ t2P Φ · ∇Φ f, wt u · ∇w Ψ u · ∇w Ψ t2Φ · ∇Ψ g, 2.3 with ∇p − I − P u · ∇u ∈ L∞ 0, T ;H1 . Therefore the proof of the existence is finished. In order to prove the uniqueness, we consider u1,w1, p1 and u2,w2, p2 two solutions of 1.6 – 1.10 and define ũ u1 − u2, w̃ w1 − w2. Then, from 1.6 and 1.8 , we have ũt u1 · ∇ũ ũ · ∇u2 ∇ ( p1 − p2 ) 0, 2.4 w̃t ũ · ∇w1 u2 · ∇w̃ 0. 2.5 Taking the inner product of 2.4 with the function ũwe obtain 1 2 d dt ‖ũ‖ − ũ · ∇u2, ũ ≤ C‖ũ‖‖∇u2‖∞. 2.6 Since u2 ∈ H3 Ω and H2 Ω ⊂ L∞ Ω , we get d dt ‖ũ‖ − C1‖ũ‖ ≤ 0 ⇒ d dt ( exp−C1t‖ũ‖2 ) ≤ 0. 2.7 Integrating the last inequality from 0 to t, t ≤ T, we have exp−C1t‖ũ‖2 ≤ 0, which implies ‖ũ‖ 0. Consequently u1 u2. Similarly, by taking the inner product of 2.5 with the function w̃ we find 1 2 d dt ‖w̃‖ − ũ · ∇w1, w̃ 0. 2.8 Then, by integrating the last equality from 0 to t, we have ‖w̃‖ 0 and thus w1 w2. In the next theorem our main result is stated. Abstract and Applied Analysis 5 Theorem 2.2. Let f,g be in F0. Then one has the following.and Applied Analysis 5 Theorem 2.2. Let f,g be in F0. Then one has the following. (1) Existence There is a weak solution uν,wν of problem 1.1 – 1.5 verifying uν ∈ L∞ 0, T ;H ∩ L2 0, T ;V , wν ∈ L∞ ( 0, T ;L2 ) ∩ L2 ( 0, T ;H0 ) , 2.9 where uν and wν are dependent on ν1, ν2, ν3. (2) Convergence If u,w is the unique solution of problem 1.6 – 1.10 given by Lemma 2.1, then ‖uν − u‖L2 0,T ;H O ( ν1 ν2 ν3 1/2 ) , ‖wν −w‖L2 0,T ;L2 O ( ν1 ν2 ν3 1/2 ) . 2.10 Moreover, if ν3 < ν1 < ν2 < kν1 for some constant k, as ν1, ν2, ν3 → 0 one has uν −→ u weakly in L2 0, T ;V , wν −→ w weakly in L2 ( 0, T ;H0 ) . 2.11 Remark 2.3. 1 Due to that we are interested in the convergence of system 1.1 – 1.5 when ν1, ν2, ν3 go to zero, the assumptions in item 2 of Theorem 2.2 are verified. Moreover, since ν1 μ μr, if μr 0, system 1.1 – 1.5 decouples and therefore, if ν1 tends to zero, the known results for the Navier-Stokes equations are recovered. 2 Note that although in Theorem 2.2 the external sources f and g are assumed in the class F0, the case of constant external sources is covered. 3. Vanishing Viscosity: Proof of Theorem 2.2 The aim of this section is to prove Theorem 2.2. For this the following auxiliary result is needed. Lemma 3.1. Let u ∈ H0, and for real constants ξ, > 0 consider the operator Bξ defined by Bξu ξ ‖∇u‖2 ∇u. Then for all u,v ∈ H0, the following inequality holds ( Bξu − Bξv,∇ u − v ) ≥ ξ ‖∇ u − v ‖ 2 ‖∇v‖‖∇ u − v ‖. 3.1 6 Abstract and Applied Analysis Proof. Using the equality 2 u,v − u ‖u‖2 − ‖v‖2 ‖u − v‖2 and the definition of Bξu, we obtain ( Bξu − Bξv,∇ u − v ) ( ξ ‖∇v‖ ) ‖∇ u − v ‖ ( ‖∇u‖ − ‖∇v‖ ) ∇u,∇ u − v ( ξ ‖∇v‖ ) ‖∇ u − v ‖ 2 ( ‖∇u‖ − ‖∇v‖ )2 2 ( ‖∇u‖ − ‖∇v‖ ) ‖∇ u − v ‖ ≥ ξ ‖∇ u − v ‖ 2 ‖∇v‖‖∇ u − v ‖. 3.2 Hence the proof of lemma is finished. The next theorem is crucial in the proof of our main result. Theorem 3.2. Let f,g be in F0 and ν min{ν1, ν2, ν3}. Then, for each with 0 < < ν there is a unique solution uν ,wν ∈ L4 0, T ;V ∩ L∞ 0, T ;H × L4 0, T ;H0 ∩ L∞ 0, T ;L2 of the problem uν t − ( ν1 ‖∇uν ‖ ) Δuν uν · ∇uν ∇pν 2μrrotwν f, in Q, 3.3 divuν 0, in Q, 3.4 wν t− ( ν2 ‖∇wν ‖ ) Δwν −ν3∇divwν uν · ∇wν 4μrwν 2μrrotuν g, in Q, 3.5 uν x, 0 wν x, 0 0, in Ω, 3.6 uν x, t wν x, t 0, on ∂Ω × 0, T . 3.7 Proof. In order to prove the existence of solutions of system 3.3 – 3.7 , the Galerkin method is used. Let Vk the subspace of V spanned by {Φ1 x , . . . ,Φ x }, and Hk be the subspace of H0 spanned by {Ψ1 x , . . . ,Ψ x }. For each k ≥ 1, the following approximations uν andwν , of uν andwν , are defined:


Introduction
The aim of this work is to analyze the convergence of the evolution equations for the motion of incompressible micropolar fluids, when the viscosities related to the physical properties of the fluid tend to zero.The equations that describe the motion of a viscous incompressible micropolar fluid express the balance of mass, momentum, and angular momentum.In a bounded domain Ω ⊂ R 3 and in a time interval 0, T , 0 < T < ∞, this model is given by the following system of differential equations: with Q Ω × 0, T , where the unknowns are u ν , w ν , and p ν , which denote, respectively, the velocity of the fluid, the microrotational velocity, and the hydrostatic pressure of the fluid, at a bounded domain of R 3 .Indeed, the analysis of our situation is still more difficult.The difficulties arise from the lack of smoothness of the weak solution.To overcome this difficulty a penalization argument is needed.This argument generalizes the penalization method given in 10 , for the Navier-Stokes equations, to this case of micropolar fluids.In fact, if we take the viscosity of microrotation μ r 0, our results imply the other ones in 10 , where the analysis of the convergence in an appropriate sense, of solutions of Navier-Stokes equations to the solutions of the Euler equations on a small time interval, is given.It is worthwhile to remark that 10 has been the unique work where the convergence of nonstationary Navier-Stokes equations, with vanishing viscosity, to the Euler equations, in a bounded domain of R 3 , has been considered.In the whole space R 3 , the authors of 11-13 analyzed the convergence, as the viscosity tends to zero, of the Navier-Stokes equations to the solution of the Euler equations on a small time interval.The two-dimensional case is more usual in the literature.
In fact, the book 14 presents a result where the fundamental argument involves the stream formulation for the Navier-Stokes equations, which is not applicable in the three-dimensional case.This paper is organized as follows.In Section 2 the basic notation is stated and the main results are formulated.In Section 3, the analysis of convergence of solutions of the initial value problem 1.1 -1.5 , when the viscosities ν 1 , ν 2 , ν 3 tend to zero, is done.This analysis is based on the ideas of 10 for Navier-Stokes equations in bounded domains.

Statements and Notations
Let Ω be a bounded domain of R 3 with smooth enough boundary ∂Ω.We consider the usual Sobolev spaces In order to distinguish the scalarvalue functions to vector-value functions, bold characters will be used; for instance, H m H m Ω 3 and so on.The solenoidal functional spaces H {v ∈ L 2 Ω /div v 0 in Ω, v • n 0 on ∂Ω} and V {v ∈ H 1 0 Ω /div v 0 in Ω}, will be also used.Here the Helmholtz decomposition of the space L 2 Ω H ⊕ G, where G {ϕ : ϕ ∇p, p ∈ H 1 Ω }, is recalled.Throughout the paper, P denotes the orthogonal projection from L 2 onto H.The norm in the L p -spaces will be denoted by • p .In particular, the norm in L 2 and its scalar product will be denoted by • and •, • , respectively.Moreover •, • will denote some duality products.We remark that, in the rest of this paper, the letter C denotes inessential positive constants which may vary from line to line.
In order to study the behavior of system 1.1 -1.5 , when the viscosities ν 1 , ν 2 , ν 3 tend to zero, the initial value problem 1.6 -1.10 is required to study.An immediate question related to the system 1.6 -1.10 is to know about the existence of its solution.In the following lemma a partial result about the existence and uniqueness of solution of problem 1.6 -1.10 is given.For that, let us consider the following functional space:

2.1
Thus we have the following lemma.
Lemma 2.1.Let f, g ∈ F 0 .Then there is a unique solution u ∈ L ∞ 0, T; Proof.The proof follows by using the arguments of 10, Lemma 3.1 .Indeed, with f, g being an element of F 0 , we consider Note that the pair u, w satisfies conditions 1.4 and 1.5 .Moreover, u • ∇u ∈ L ∞ 0, T; L 2 and thus, u • ∇u Therefore the proof of the existence is finished.
In order to prove the uniqueness, we consider u 1 , w 1 , p 1 and u 2 , w 2 , p 2 two solutions of 1.6 -1.10 and define u u 1 − u 2 , w w 1 − w 2 .Then, from 1.6 and 1.8 , we have Taking the inner product of 2.4 with the function u we obtain 1 2 Integrating the last inequality from 0 to t, t ≤ T, we have exp Similarly, by taking the inner product of 2.5 with the function w we find 1 2 Then, by integrating the last equality from 0 to t, we have w 0 and thus w 1 w 2 .
In the next theorem our main result is stated.
Abstract and Applied Analysis 5 Theorem 2.2.Let f, g be in F 0 .Then one has the following. (

2) Convergence
If u, w is the unique solution of problem 1.6 -1.10 given by Lemma 2.1, then

2.11
Remark 2.3. 1 Due to that we are interested in the convergence of system 1.1 -1.5 when ν 1 , ν 2 , ν 3 go to zero, the assumptions in item 2 of Theorem 2.2 are verified.Moreover, since ν 1 μ μ r , if μ r 0, system 1.1 -1.5 decouples and therefore, if ν 1 tends to zero, the known results for the Navier-Stokes equations are recovered.2 Note that although in Theorem 2.2 the external sources f and g are assumed in the class F 0 , the case of constant external sources is covered.

Vanishing Viscosity: Proof of Theorem 2.2
The aim of this section is to prove Theorem 2.2.For this the following auxiliary result is needed.
Lemma 3.1.Let u ∈ H 1 0 , and for real constants ξ, > 0 consider the operator B ξ defined by B ξ u ξ ∇u 2 ∇u.Then for all u, v ∈ H 1 0 , the following inequality holds

Abstract and Applied Analysis
Proof.Using the equality 2 u, v − u u 2 − v 2 u − v 2 and the definition of B ξ u, we obtain

3.2
Hence the proof of lemma is finished.
The next theorem is crucial in the proof of our main result.
Theorem 3.2.Let f, g be in F 0 and ν min{ν 1 , ν 2 , ν 3 }.Then, for each with 0 < < ν there is a unique solution u ν , Proof.In order to prove the existence of solutions of system 3.3 -3.7 , the Galerkin method is used.Let V k the subspace of V spanned by {Φ 1 x , . . ., Φ k x }, and H k be the subspace of H 1 0 spanned by {Ψ 1 x , . . ., Ψ k x }.For each k ≥ 1, the following approximations u k ν and w k ν , of u ν and w ν , are defined: for t ∈ 0, T , where the coefficients c ik t and d ik t are calculated such that u k ν and w k ν solve the following system: Then, by multiplying 3.9 by c ik and d ik , respectively, summing over i from 0 to k and taking into account 3.8 , we have 1 2

3.10
Now, by applying H ölder's and Young's inequalities we get

3.11
Then, summing 3.10 , with the help of last inequalities, we obtain 1 2 3.12 and hence, by integrating 3.12 from 0 to t, t ∈ 0, T , we find

3.13
Applying Gronwall's inequality in 3.13 we get ds ≤ CT e CT ≤ C.

3.16
Since V ⊂ H is compact and H ⊂ V * is continuous, as well as V ⊂ L 2 is compact and L 2 ⊂ H −1 is continuous, then as k → ∞, we obtain

3.17
In order to pass to the limit in 3.9 we take into account 3.15 and 3.17 .Indeed, the convergence in the linear terms follows directly.Moreover, as in 2, page 289 , we can prove that as k → ∞ Finally, from 3.15 we have and hence, by taking ∇Φ i ∇u k ν , ∇Ψ i ∇w k ν in 3.19 , and then,

3.22
Proof.Considering the differences between 1.6 and 3.3 , as well as between 1.8 and 3.5 and then by taking the inner product with v u ν − u and z w ν − w, respectively, we have 1 2

3.33
Recalling the notation

3.35
Using H ölder's and Young's inequalities we bound the right hand of 3.34 and 3.35 as follows:

3.46
Now, by using the equality 2 u, v − u u 2 − v 2 u − v 2 , the definition of B ξ ϕ and Lemma 3.1, from 3.46 we get 1 2

3.48
Since v 0 0 and z 0 0, by integrating 3.48 from 0 to t, t ∈ 0, T , and then applying Gronwall's inequality, we obtain and hence the proof of estimates 3.32 is concluded.
Proof.Let u ν , w ν be as in Theorem 3.2.Then from 3.14 we have where C is a constant which does not depend on ν 1 , ν 2 , ν 3 , and .

3.52
Then, by using n i 1 |a i | 4/3 ≤ C n i 1 |a i | 4/3 and 3.51 , from the last inequalities we get

3.53
Integrating the last inequalities from 0 to T we conclude

3.54
Since V ⊂ H is compact and H ⊂ V * is continuous, as well as V ⊂ L 2 is compact and L 2 ⊂ H −1 is continuous, then as → 0 we have u ν −→ u ν in L 2 0, T; H , w ν −→ w ν in L 2 0, T; L 2 .

3.60
We can verify that u ν , w ν is a weak solution of 1.1 -1.3 .Indeed, we need to verify that u ν , w ν satisfies the following variational system: 3.61 for all Φ ∈ V, Ψ ∈ H 1 0 .Note that the before convergence results enable us to pass to the limit in the linear terms of 3.3 -3.7 , obtaining the linear term in 3.61 .Furthermore, through standard arguments one can obtain u ν • ∇u ν , Φ −→ u ν • ∇u ν , Φ , u ν • ∇w ν , Ψ −→ w ν • ∇w ν , Ψ .

3.63
Finally, it is clear that for all Φ ∈ V, Ψ ∈ H 1 0 , as → 0 it holds

3.64
Proof of the Theorem 2.2.The existence of a solution of 1.1 -1.5 is given by using Proposition 3.4 as the limit u ν , w ν of the sequence u ν , w ν .Now the second part of the Theorem 2.2 will be proved.Let u, w be solution of problem 1.6 -1.10 .