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: