# Micropolar Fluids with Vanishing Viscosity

Authors: E. Ortega-Torres; E. J. Villamizar-Roa; M. A. Rojas-Medar
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

A study of the convergence of weak solutions of the nonstationary micropolar fluids, in
bounded domains ofℝn, when the viscosities tend to zero, is established. In the limit, a fluid governed by an Euler-like system is found.


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## Body

## 1. 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 domainΩ⊂ℝ3 and in a time interval (0,T], 0<T<+∞, this model is given by the following system of differential equations:(1.1)(uν)t-ν1Δuν+uν·∇uν+∇pν=2μrrotwν+f,inQ,(1.2)divuν=0,inQ,(1.3)(wν)t-ν2Δwν-ν3∇divwν+uν·∇wν+4μrwν=2μrrotuν+g,inQ,
with Q=Ω×(0,T], where the unknowns are uν,wν, and pν, which denote, respectively, the velocity of the fluid, the microrotational velocity, and the hydrostatic pressure of the fluid, at a point (x,t).ν1,ν2, and ν3 are positive constants which satisfy ν1=μ+μr,ν2=ca+cd,ν3=c0+cd-ca, with c0+cd>ca, where μ,μr,c0,ca,cd represent viscosity coefficients. In particular, μ is the usual Newtonian viscosity, μr is called the viscosity of microrotation and c0,ca,cd are new viscosities related to the asymmetry of the stress tensor. The fields f and g are given and denote external sources of linear and angular momentum, respectively.With (1.1)–(1.3) the following initial and boundary conditions are prescribed(1.4)uν(x,0)=0,wν(x,0)=0,inΩ,(1.5)uν(x,t)=0,wν(x,t)=0,on∂Ω×[0,T],
where, for the simplicity in this exposition, homogeneous boundary conditions have been taken. The initial data is also assumed to be equal to zero due to the nature of the solutions of the Euler-like system (1.6)–(1.10) below.Theory of micropolar fluids was proposed by Eringen [1] and describes flows of fluids whose particles undergo translations androtations as well. In this sense, micropolar fluids permit to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for viscous incompressible fluids. Indeed, if μr=0 in system (1.1)–(1.3), the equations are decoupled and (1.1) reduces to the incompressible Navier-Stokes equations (see [2]). For the derivation and physical discussion of system (1.1)–(1.3), see the references [1, 3, 4].There is extensive literature related to the solutions of micropolar fluids. In a hilbertian context, in [4–6] and some references therein, results of existence, uniqueness and regularity of weak solutions were found. On the other hand, in [7, 8], by using semigroups approach, some recent results related to the initial value problem (1.1)–(1.5) with initial data in Lp- spaces, including the stability of strong steady solutions, were performed.This work is concerned with the behavior of the micropolar fluids, in a bounded domainΩ⊂ℝ3, with boundary ∂Ω smooth enough, when the viscosities ν1,ν2,ν3 tend to zero. We will prove that there is a subspace ℱ0 of (L∞(0,T;H03(Ω)))2 such that, for external sources (f,g) in ℱ0, the weak solutions of the micropolar fluid system (1.1)–(1.3) converge in L2(Ω)×L2(Ω), when the viscosities ν1,ν2,ν3 in (1.1)–(1.3) tend to zero, to the solution (u,w) of the following Euler-like system:(1.6)ut+u·∇u+∇p=f,inΩ×[0,T],(1.7)divu=0,inΩ×[0,T],(1.8)wt+u·∇w=g,inΩ×[0,T],(1.9)u(x,0)=0,w(x,0)=0,inΩ,(1.10)u(x,t)=0,w(x,t)=0,on∂Ω×[0,T].
As far as it is known, the analysis of convergence of the evolution equations for the motion of incompressible micropolar fluids, when the viscosities tend to zero, in an open set Ω×(0,T) with Ω being a bounded domain of ℝ3, is still unknown. In [9] a nonhomogeneous, viscous incompressible asymmetric fluid in Ω=ℝ3 was considered, and the existence of a small time interval where the fluid variables converge uniformly as the viscosities tend to zero was proved. However, the results of [9] are not applicable in our case, that is, when Ω is 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 Section2 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

LetΩ be a bounded domain of ℝ3 with smooth enough boundary ∂Ω. We consider the usual Sobolev spaces Hm(Ω)={f∈L2(Ω):∥Dkf∥L2<∞,|k|≤m},m≥1, with norm denoted by ∥·∥Hm. H01(Ω) is the closure C0∞(Ω) in the norm ∥·∥H1. In order to distinguish the scalar-value functions to vector-value functions, bold characters will be used; for instance, Hm=(Hm(Ω))3 and so on. The solenoidal functional spaces H={v∈L2(Ω)/divv=0inΩ,v·n=0on∂Ω} and V={v∈H01(Ω)/divv=0inΩ}, 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 Lp-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: (2.1)ℱ0={(Φ+t2P(Φ·∇Φ),Ψ+t2Φ·∇Ψ):Φ∈V∩H3,Ψ∈H01∩H3}⊂(L∞(0,T;H03))2.
Thus we have the following lemma.Lemma 2.1.
Let(f,g)∈ℱ0. Then there is a unique solution u∈L∞(0,T;V∩H3),w∈L∞(0,T;H01∩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×H01∩H3 and define
(2.2)u(x,t)=tΦ(x)∈L∞(0,T;V∩H3),w(x,t)=tΨ(x)∈L∞(0,T;H01∩H3).
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) and wt(x,t)=Ψ(x). Hence
(2.3)ut+u·∇u+∇p=Φ+P(u·∇u)=Φ+t2P(Φ·∇Φ)=f,wt+u·∇w=Ψ+u·∇w=Ψ+t2Φ·∇Ψ=g,
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(2.4)ũt+u1·∇ũ+ũ·∇u2+∇(p1-p2)=0,(2.5)w̃t+ũ·∇w1+u2·∇w̃=0.
Taking the inner product of (2.4) with the function ũ we obtain
(2.6)12ddt∥ũ∥2=-(ũ·∇u2,ũ)≤C∥ũ∥2∥∇u2∥∞.
Since u2∈H3(Ω) and H2(Ω)⊂L∞(Ω), we get
(2.7)ddt∥ũ∥2-C1∥ũ∥2≤0⇒ddt(exp-C1t∥ũ∥2)≤0.
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(2.8)12ddt∥w̃∥2=-(ũ·∇w1,w̃)=0.
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.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
(2.9)uν∈L∞(0,T;H)∩L2(0,T;V),wν∈L∞(0,T;L2)∩L2(0,T;H01),
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
(2.10)∥uν-u∥L2(0,T;H)=O((ν1+ν2+ν3)1/2),∥wν-w∥L2(0,T;L2)=O((ν1+ν2+ν3)1/2).
Moreover, if ν3<ν1<ν2<kν1 for some constant k, as ν1,ν2,ν3→0 one has
(2.11)uν→uweaklyinL2(0,T;V),wν→wweaklyinL2(0,T;H01).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 Theorem2.2

The aim of this section is to prove Theorem2.2. For this the following auxiliary result is needed.Lemma 3.1.
Letu∈H01, and for real constants ξ,ϵ>0 consider the operator Bξ defined by Bξu=(ξ+ϵ∥∇u∥2)∇u. Then for all u,v∈H01, the following inequality holds
(3.1)(Bξu-Bξv,∇(u-v))≥ξ∥∇(u-v)∥2+ϵ2∥∇v∥2∥∇(u-v)∥2.Proof.
Using the equality2(u,v-u)=∥u∥2-∥v∥2+∥u-v∥2 and the definition of Bξu, we obtain
(3.2)(Bξu-Bξv,∇(u-v))=(ξ+ϵ∥∇v∥2)∥∇(u-v)∥2+ϵ(∥∇u∥2-∥∇v∥2)(∇u,∇(u-v))=(ξ+ϵ∥∇v∥2)∥∇(u-v)∥2+ϵ2(∥∇u∥2-∥∇v∥2)2+ϵ2(∥∇u∥2-∥∇v∥2)∥∇(u-v)∥2≥ξ∥∇(u-v)∥2+ϵ2∥∇v∥2∥∇(u-v)∥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 ϵ with 0<ϵ<ν there is a unique solution (uνϵ,wνϵ)∈L4(0,T;V)∩L∞(0,T;H)×L4(0,T;H01)∩L∞(0,T;L2) of the problem
(3.3)(uνϵ)t-(ν1+ϵ∥∇uνϵ∥2)Δuνϵ+uνϵ·∇uνϵ+∇pνϵ=2μrrotwνϵ+f,inQ,(3.4)divuνϵ=0,inQ,(3.5)(wνϵ)t-(ν2+ϵ∥∇wνϵ∥2)Δwνϵ-ν3∇divwνϵ+uνϵ·∇wνϵ+4μrwνϵ=2μrrotuνϵ+g,inQ,(3.6)uνϵ(x,0)=wνϵ(x,0)=0,inΩ,(3.7)uνϵ(x,t)=wνϵ(x,t)=0,on∂Ω×[0,T].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),…,Φk(x)}, and Hk be the subspace of H01 spanned by {Ψ1(x),…,Ψk(x)}. For each k≥1, the following approximations uνϵk and wνϵk, of uνϵ and wνϵ, are defined:
(3.8)uνϵk(x,t)=∑i=1kcik(t)Φi(x),wνϵk(x,t)=∑i=1kdik(t)Ψi(x),
for t∈(0,T), where the coefficients cik(t) and dik(t) are calculated such that uνϵk and wνϵk solve the following system:
(3.9)((uνϵk)t,Φi)+(ν1+ϵ∥∇uνϵk∥2)(∇uνϵk,∇Φi)+(uνϵk·∇uνϵk,Φi)=2μr(rotwνϵk,Φi)+(f,Φi),((wνϵk)t,Ψi)+(ν2+ϵ∥∇wνϵk∥2)(∇wνϵk,∇Ψi)+ν3(divwνϵk,divΨi)+(uνϵk·∇wνϵk,Ψi)+4μr(wνϵk,Ψi)=2μr(rotuνϵk,Ψi)+(g,Ψi),
for all Φi∈Vk and Ψi∈Hk.
Then, by multiplying (3.9) by cik and dik, respectively, summing over i from 0 to k and taking into account (3.8), we have(3.10)12ddt∥uνϵk∥2+ν1∥∇uνϵk∥2+ϵ∥∇uνϵk∥4=2μr(rotwνϵk,uνϵk)+(f,uνϵk),12ddt∥wνϵk∥2+ν2∥∇wνϵk∥2+ϵ∥∇wνϵk∥4+ν3∥divwνϵk∥2+4μr∥wνϵk∥2=2μr(rotuνϵk,wνϵk)+(g,wνϵk).
Now, by applying Hölder's and Young's inequalities we get
(3.11)2μr((rotwνϵk,uνϵk)+(rotuνϵk,wνϵk))≤ν12∥∇uνϵk∥2+C∥wνϵk∥2,(f,uνϵk)+(g,wνϵk)≤C(∥f∥2+∥g∥2+∥uνϵk∥2+∥wνϵk∥2).
Then, summing (3.10), with the help of last inequalities, we obtain
(3.12)12ddt(∥uνϵk∥2+∥wνϵk∥2)+ν12∥∇uνϵk∥2+ν2∥∇wνϵk∥2+ϵ(∥∇uνϵk∥4+∥∇wνϵk∥4)≤C(∥uνϵk∥2+∥wνϵk∥2)+C(∥f∥2+∥g∥2),
and hence, by integrating (3.12) from 0 to t,t∈(0,T], we find
(3.13)∥uνϵk(t)∥2+∥wνϵk(t)∥2+∫0t(ν1∥∇uνϵk∥2+2ν2∥∇wνϵk∥2)ds+2ϵ∫0t(∥∇uνϵk∥4+∥∇wνϵk∥4)ds≤C∫0t(∥uνϵk∥2+∥wνϵk∥2)ds+C∫0t(∥f∥2+∥g∥2)ds.
Applying Gronwall's inequality in (3.13) we get
(3.14)∥uνϵk(t)∥2+∥wνϵk(t)∥2+∫0t(ν1∥∇uνϵk∥2+2ν2∥∇wνϵk∥2)ds+2ϵ∫0t(∥∇uνϵk∥4+∥∇wνϵk∥4)ds≤CTeCT≤C.
Thus, from (3.14) we conclude that there is uνϵ,wνϵ such that as k→∞(3.15)uνϵk→uνϵweakly-*inL∞(0,T;H)andweaklyinL2(0,T;V),wνϵk→wνϵweakly-*inL∞(0,T;L2)andweaklyinL2(0,T;H01).
Now, since V⊂L6 andH01⊂L6, from (3.14) we have that uνϵk,wνϵk∈L4(0,T;L6); consequently (uνϵk)t∈L4/3(0,T;V*) and (wνϵk)t∈L4/3(0,T;H-1), with V* and H-1 being the topological duals of V and H01, respectively. Therefore
(3.16)(uνϵk)t→(uνϵ)tweaklyinL4/3(0,T;V*),(wνϵk)t→(wνϵ)tweaklyinL4/3(0,T;H-1).
Since V⊂H is compact and H⊂V* is continuous, as well as V⊂L2 is compact and L2⊂H-1 is continuous, then as k→∞, we obtain
(3.17)uνϵk→uνϵinL2(0,T;H),wνϵk→wνϵinL2(0,T;L2).
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→∞(3.18)∫0T|(uνϵk·∇uνϵk-uνϵ·∇uνϵ,Φi)ϕ(t)|dt→0,∫0T|(uνϵk·∇wνϵk-uνϵ·∇wνϵ,Ψi)ϕ(t)|dt→0,
for all Φi∈Vk,Ψi∈Hk and all ϕ∈C0∞(0,T). Finally, from (3.15) we have
(3.19)(∇uνϵk,∇Φi)→(∇uνϵ,∇Φi),(∇wνϵk,∇Ψi)→(∇wνϵ,∇Ψi),ask→∞,
and hence, by taking ∇Φi=∇uνϵk,∇Ψi=∇wνϵk in (3.19), and then, ∇Φi=∇uνϵ,∇Ψi=∇wνϵ, as k→∞ we get
(3.20)∥∇uνϵk∥2→(∇uνϵ,∇uνϵk)→∥∇uνϵ∥2,∥∇wνϵk∥2→(∇wνϵ,∇wνϵk)→∥∇wνϵ∥2.
Moreover, as
(3.21)(∥∇uνϵk∥2∇uνϵk-∥∇uνϵ∥2∇uνϵ,∇Φi)=(∥∇uνϵk∥2-∥∇uνϵ∥2)(∇uνϵk,∇Φi)+∥∇uνϵ∥2(∇uνϵk-∇uνϵ,∇Φi),(∥∇wνϵk∥2∇wνϵk-∥∇wνϵ∥2∇wνϵ,∇Ψi)=(∥∇wνϵk∥2-∥∇wνϵ∥2)(∇wνϵk,∇Ψi)+∥∇wνϵ∥2(∇wνϵk-∇wνϵ,∇Ψi),
we conclude that
(3.22)∫0T(∥∇uνϵk∥2∇uνϵk-∥∇uνϵ∥2∇uνϵ,∇Φi)ϕ(t)dt→0,∫0T(∥∇wνϵk∥2∇wνϵk-∥∇wνϵ∥2∇wνϵ,∇Ψi)ϕ(t)dt→0.
Now the uniqueness of solution will be analyzed. Let(uνϵ1,wνϵ1,pνϵ1) and (uνϵ2,wνϵ2,pνϵ2) be two solutions of (3.3)–(3.7). We denote ũνϵ=uνϵ1-uνϵ2,w̃νϵ=wνϵ1-wνϵ2, and p̃νϵ=pνϵ1-pνϵ2. Then we have(3.23)(ũνϵ)t-(ν1+ϵ∥∇uνϵ1∥2)Δuνϵ1+(ν1+ϵ∥∇uνϵ2∥2)Δuνϵ2+ũνϵ·∇uνϵ1+∇p̃νϵ+uνϵ2·∇ũνϵ=2μrrotw̃νϵ,(3.24)divũνϵ=0,(3.25)(w̃νϵ)t-(ν2+ϵ∥∇wνϵ1∥2)Δwνϵ1+(ν2+ϵ∥∇wνϵ2∥2)Δwνϵ2-ν3∇divw̃νϵ+ũνϵ·∇wνϵ1+uνϵ2·∇w̃νϵ+4μrw̃νϵ=2μrrotũνϵ.
Taking the inner product of (3.23) with ũνϵ, of (3.25) with w̃νϵ, by using (3.24), we obtain
(3.26)12ddt∥ũνϵ∥2+(Bν1uνϵ1,∇ũνϵ)-(Bν1uνϵ2,∇ũνϵ)=-(ũνϵ·∇uνϵ1,ũνϵ)+2μr(rotw̃νϵ,ũνϵ),12ddt∥w̃νϵ∥2+(Bν2wνϵ1,∇w̃νϵ)-(Bν2wνϵ2,∇w̃νϵ)≤-(ũνϵ·∇wνϵ1,w̃νϵ)+2μr|(rotũνϵ,w̃νϵ)|.
Hence, by using Lemma 3.1 we get
(3.27)12ddt∥ũνϵ∥2+ν1∥∇ũνϵ∥2+ϵ2∥∇uνϵ2∥2∥∇ũνϵ∥2≤|(ũνϵ·∇uνϵ1,ũνϵ)|+2μr|(rotw̃νϵ,ũνϵ)|,12ddt∥w̃νϵ∥2+ν2∥∇w̃νϵ∥2+ϵ2∥∇wνϵ2∥2∥∇w̃νϵ∥2≤|(ũνϵ·∇wνϵ1,w̃νϵ)|+2μr|(rotũνϵ,w̃νϵ)|.
Consequently
(3.28)12ddt∥ũνϵ∥2+ν1∥∇ũνϵ∥2≤|(ũνϵ·∇uνϵ1,ũνϵ)|+2μr|(rotw̃νϵ,ũνϵ)|,12ddt∥w̃νϵ∥2+ν2∥∇w̃νϵ∥2≤|(ũνϵ·∇wνϵ1,w̃νϵ)|+2μr|(rotũνϵ,w̃νϵ)|.
Now, by using Hölder's and Young's inequalities we have
(3.29)|(ũνϵ·∇uνϵ1,ũνϵ)|≤∥ũνϵ∥3∥∇ũνϵ∥∥uνϵ1∥6≤C∥ũνϵ∥1/2∥∇ũνϵ∥3/2∥∇uνϵ1∥≤C∥ũνϵ∥2∥∇uνϵ1∥4+ν14∥∇ũνϵ∥2,|(ũνϵ·∇wνϵ1,w̃νϵ)|≤C∥ũνϵ∥1/2∥∇ũνϵ∥1/2∥∇w̃νϵ∥∥∇wνϵ1∥≤C∥ũνϵ∥∥∇ũνϵ∥∥∇wνϵ1∥2+ν2∥∇w̃νϵ∥2≤C∥ũνϵ∥2∥∇wνϵ1∥4+ν14∥∇ũνϵ∥2+ν2∥∇w̃νϵ∥2,2μr|(rotw̃νϵ,ũνϵ)|≤C∥w̃νϵ∥∥∇ũνϵ∥≤C∥w̃νϵ∥2+ν14∥∇ũνϵ∥2,2μr|(rotũνϵ,w̃νϵ)|≤C∥w̃νϵ∥2+ν14∥∇ũνϵ∥2.
Then, by summing (3.28), and taking into account the last inequalities, we obtain
(3.30)12ddt(∥ũνϵ∥2+∥w̃νϵ∥2)≤C∥w̃νϵ∥2+C(∥∇uνϵ1∥4+∥∇wνϵ1∥4)∥ũνϵ∥2≤CM(t)(∥ũνϵ∥2+∥w̃νϵ∥2),
where M(t)=1+∥∇uνϵ2(t)∥4+∥∇wνϵ2(t)∥4.
SinceM(t)∈L1(0,T), by integrating (3.30) from 0 to t, and then applying Gronwall's inequality, we conclude that(3.31)∥ũνϵ∥2+∥w̃νϵ∥2≤0,
which implies ũνϵ=0 and w̃νϵ=0. Consequently the uniqueness of solution is proved.Proposition 3.3.
Under the assumptions of Lemma2.1 and Theorem 3.2, if (uνϵ,wνϵ) and (u,w) are the solutions of problems (3.3)–(3.7) and (1.6)–(1.10), respectively, then
(3.32)∥uνϵ-u∥L∞(0,T;H)=O((ν1+ϵ+ν2+ν3)1/2),ν11/2∥∇(uνϵ-u)∥L2(0,T;L2)=O((ν1+ϵ+ν2+ν3)1/2),∥wνϵ-w∥L∞(0,T;L2)=O((ν1+ϵ+ν2+ν3)1/2),ν21/2∥∇(wνϵ-w)∥L2(0,T;L2)=O((ν1+ϵ+ν2+ν3)1/2).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=uνϵ-u and z=wνϵ-w, respectively, we have
(3.33)12ddt∥v∥2+(ν1+ϵ∥∇uνϵ∥2)(∇uνϵ,∇v)+(v·∇u,v)=2μr(rotwνϵ,v),12ddt∥z∥2+(ν2+ϵ∥∇wνϵ∥2)(∇wνϵ,∇z)+ν3(divwνϵ,divz)+(v·∇w,z)+4μr(wνϵ,z)=2μr(rotuνϵ,z).
Recalling the notation Bξφ̃=(ξ+ϵ∥∇φ̃∥2)∇φ̃ we get
(3.34)12ddt∥v∥2+(Bν1uνϵ-Bν1u,∇v)=-(Bν1u,∇v)-(v·∇u,v)+2μr(rotz,v)+2μr(rotw,v),(3.35)12ddt∥z∥2+(Bν2wνϵ-Bν2w,∇z)+ν3∥divz∥2+4μr∥z∥2=-(Bν2w,∇z)-ν3(divw,divz)-4μr(w,z)-(v·∇w,z)+2μr(rotv,z)+2μr(rotu,z).
Using Hölder's and Young's inequalities we bound the right hand of (3.34) and (3.35) as follows:
(3.36)(Bν1u,∇v)≤ν1+ϵ∥∇u∥22(∥∇u∥2+∥∇v∥2),(3.37)(v·∇u,v)≤C∥∇u∥L∞∥v∥2≤C∥v∥2,(3.38)2μr(rotz,v)≤Cν1∥∇v∥∥z∥≤ν128∥∇v∥2+C∥z∥2,(3.39)2μr(rotw,v)≤ν12∥∇w∥2+C∥v∥2,(3.40)(Bν2w,∇z)≤ν2+ϵ∥∇w∥22(∥∇w∥2+∥∇z∥2),(3.41)ν3(divw,divz)≤Cν3∥∇w∥2+ν3∥divz∥2,(3.42)4μr(w,z)≤ν12∥w∥2+C∥z∥2,(3.43)(v·∇w,z)≤C(∥v∥2+∥z∥2),(3.44)2μr(rotv,z)≤ν128∥∇v∥2+C∥z∥2,(3.45)2μr(rotu,z)≤ν12∥∇u∥2+C∥z∥2.
Carrying (3.36)–(3.39) in (3.34) and (3.40)–(3.45) in (3.35), we have
(3.46)12ddt∥v∥2+(Bν1uνϵ-Bν1u,∇v)≤ν1+ϵ∥∇u∥22(∥∇u∥2+∥∇v∥2)+C∥v∥2+ν18∥∇v∥2+C∥z∥2+ν1∥∇w∥2,12ddt∥z∥2+(Bν2wνϵ-Bν2w,∇z)≤ν2+ϵ∥∇w∥22(∥∇w∥2+∥∇z∥2)+Cν3∥∇w∥2+ν1∥w∥2+C(∥v∥2+∥z∥2)+ν18∥∇v∥2+Cν1∥∇u∥2.
Now, by using the equality2(u,v-u)=∥u∥2-∥v∥2+∥u-v∥2, the definition of Bξφ and Lemma 3.1, from (3.46) we get (3.47)12ddt(∥v∥2+∥z∥2)+ν14∥∇v∥2+ν22∥∇z∥2≤ν1+ϵ∥∇u∥22∥∇u∥2+ν2+ϵ∥∇w∥22∥∇w∥2+C(∥v∥2+∥z∥2)+C(ν1+ν3),
which implies
(3.48)ddt(∥v∥2+∥z∥2)+ν1∥∇v∥2+ν2∥∇z∥2≤C(ϵ+ν1+ν2+ν3)+C(∥v∥2+∥z∥2).
Sincev(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(3.49)∥v(t)∥2+∥z(t)∥2+∫0t(ν1∥∇v(s)∥2+ν2∥∇z(s)∥2)ds≤C(ϵ+ν1+ν2+ν3)Texp(CT),
and hence the proof of estimates (3.32) is concluded.Proposition 3.4.
Under the assumptions of Theorem3.2 and considering ν3<ν2, then as ϵ→0 the solution (uνϵ,wνϵ) of (3.3)–(3.7) verifies the following convergences:
(3.50)uνϵ→uνinL2(0,T;H),wνϵ→wνinL2(0,T;L2),
where (uν,wν) is a solution of problem (1.1)–(1.3).Proof.
Letuνϵ,wνϵ be as in Theorem 3.2. Then from (3.14) we have
(3.51)∥uνϵ∥L∞(0,T;H)2+ν1∥uνϵ∥L2(0,T;V)2+ϵ∥uνϵ∥L4(0,T;V)4≤C,∥wνϵ∥L∞(0,T;L2)2+ν2∥wνϵ∥L2(0,T;H01)2+ϵ∥wνϵ∥L4(0,T;H01)4≤C,
where C is a constant which does not depend on ν1,ν2,ν3, and ϵ.
Then, sinceV⊂L6 and H01⊂L6, we have that uνϵ,wνϵ∈L4(0,T;L6). Consequently (uνϵ)t∈L4/3(0,T;V*) and (wνϵ)t∈L4/3(0,T;H-1), with V* and H-1 being  the topological dual of V and H01, respectively.
Now, since|(u·∇w,z)|≤∥u∥3∥∇w∥∥z∥6≤C∥u∥1/2∥u∥V1/2∥∇w∥∥∇z∥, from (3.3) and (3.5) we obtain (3.52)∥(uνϵ)t∥V*≤C(ν1∥uνϵ∥V+ϵ∥uνϵ∥V3+∥uνϵ∥1/2∥uνϵ∥V3/2+ν1∥wνϵ∥+∥f∥),∥(wνϵ)t∥H-1≤C(ν2∥wνϵ∥H01+ϵ∥wνϵ∥H013+ν3∥wνϵ∥H01+∥uνϵ∥3/2∥uνϵ∥V3/2)+C(∥wνϵ∥H013/2+ν1∥wνϵ∥+ν1∥uνϵ∥+∥g∥).
Then, by using (∑i=1n|ai|)4/3≤C∑i=1n|ai|4/3 and (3.51), from the last inequalities we get
(3.53)∥(uνϵ)t∥V*4/3≤C(ν14/3∥uνϵ∥V4/3+ϵ4/3∥uνϵ∥V4+∥uνϵ∥V2+ν14/3+∥f∥4/3),∥(wνϵ)t∥H-14/3≤C(ν24/3∥wνϵ∥H014/3+ϵ4/3∥wνϵ∥H014+ν34/3∥wνϵ∥H014/3)+C(∥uνϵ∥V2+∥wνϵ∥H012+ν14/3+∥g∥4/3).
Integrating the last inequalities from 0 to T we conclude
(3.54)∫0T∥(uνϵ)t∥V*4/3dt≤C(ν14/3∫0T∥uνϵ∥V4/3dt+ν14/3T+∫0T∥f∥4/3dt)+C(ϵ4/3∥uνϵ∥L4(0,T;V)4+∥uνϵ∥L2(0,T;V)2),∫0T∥(wνϵ)t∥H-14/3dt≤C((ν24/3+ν34/3)∫0T∥wνϵ∥H014/3dt+ν14/3T+∫0T∥g∥4/3dt)+C(ϵ4/3∥wνϵ∥L4(0,T;H01)4+∥uνϵ∥L2(0,T;V)2+∥wνϵ∥L2(0,T;H01)2).
Using Hölder's inequality and (3.51), since ν3<ν2 we obtain
(3.55)ν14/3∫0T∥uνϵ∥V4/3dt≤Cν14/3∥uνϵ∥L2(0,T;V)4/3≤Cν12/3,(ν24/3+ν34/3)∫0T∥wνϵ∥H014/3dt≤ν24/3C∥wνϵ∥L2(0,T;H01)4/3≤Cν22/3.
Hence, from (3.54)–(3.55) and (3.51) we get
(3.56)∥(uνϵ)t∥L4/3(0,T;V*)4/3≤C(ν12/3+ν14/3+1+ϵ1/3+ν1-1),∥(wνϵ)t∥L4/3(0,T;H-1)4/3≤C(ν22/3+ν14/3+1+ϵ1/3+ν1-1+ν2-1).
Thus, since ϵ<ν1, from last two inequalities we have
(3.57)∥(uνϵ)t∥L4/3(0,T;V*)≤C(ν11/2+ν1+ν11/4+ν1-3/4+1),∥(wνϵ)t∥L4/3(0,T;H-1)≤C(ν21/2+ν1+ν11/4+ν1-3/4+ν2-3/4+1),
where the constant C is independent of ν1,ν2,ν3, and ϵ.
From (3.51) and (3.57), taking subsequences if necessary, we deduce that, as ϵ→0(3.58)uνϵ⇀uνweak-*inL∞(0,T;H),ν11/2uνϵ⇀ν11/2uνweaklyinL2(0,T;V),ϵ1/4uνϵ⇀0weaklyinL4(0,T;V),(uνϵ)t⇀(uν)tweaklyinL4/3(0,T;V*),uνϵ(0)⇀uν(0)weaklyinV*.
Similarly, as ϵ→0(3.59)wνϵ⇀wνweak-*inL∞(0,T;L2),ν21/2wνϵ⇀ν21/2wνweaklyinL2(0,T;H01),ϵ1/4wνϵ⇀0weaklyinL4(0,T;H01),(wνϵ)t⇀(wν)tweaklyinL4/3(0,T;H-1),wνϵ(0)⇀wν(0)weaklyinH-1.
Since V⊂H is compact and H⊂V* is continuous, as well as V⊂L2 is compact and L2⊂H-1 is continuous, then as ϵ→0 we have
(3.60)uνϵ→uνinL2(0,T;H),wνϵ→wνinL2(0,T;L2).
We can verify that(uν,wν) is a weak solution of (1.1)–(1.3). Indeed, we need to verify that (uν,wν) satisfies the following variational system:(3.61)ddt(uν,Φ̃)+ν1(∇uν,∇Φ̃)+(uν·∇uν,Φ̃)=2μr(rotwν,Φ̃)+(f,Φ̃),ddt(wν,Ψ̃)+ν2(∇wν,∇Ψ̃)+ν3(divwν,divΨ̃)+(uν·∇wν,Ψ̃)+4μr(wν,Ψ̃)=2μr(rotuν,Ψ̃)+(g,Ψ̃),
for all Φ̃∈V,Ψ̃∈H01.
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(3.62)(uνϵ·∇uνϵ,Φ̃)→(uν·∇uν,Φ̃),(uνϵ·∇wνϵ,Ψ̃)→(wν·∇wν,Ψ̃).
Moreover it is not difficult to check that
(3.63)(uν(x,0),Φ̃)=0,∀Φ̃∈V,(wν(x,0),Ψ̃)=0,∀Ψ̃∈H01.
Finally, it is clear that for all Φ̃∈V,Ψ̃∈H01, as ϵ→0 it holds
(3.64)(ϵ∥∇uνϵ∥2)(∇uνϵ,∇Φ̃)→0,(ϵ∥∇wνϵ∥2)(∇wνϵ,∇Ψ̃)→0.Proof.
The existence of a solution of (1.1)–(1.5) is given by using Proposition 3.4 as the limit (uν,wν) of the sequence (uνϵ,wνϵ).
Now the second part of the Theorem2.2 will be proved. Let (u,w) be solution of problem (1.6)–(1.10). From (3.58)-(3.59) we have (3.65)∇(uνϵ-u)⇀∇(uν-u)weaklyinL2(0,T;L2),∇(wνϵ-w)⇀∇(wν-w)weaklyinL2(0,T;L2).
Consequently
(3.66)ν11/2∥∇(uν-u)∥L2(0,T;L2)≤lim infϵ→0ν11/2∥∇(uνϵ-u)∥L2(0,T;L2),ν21/2∥∇(wν-w)∥L2(0,T;L2)≤lim infϵ→0ν21/2∥∇(wνϵ-w)∥L2(0,T;L2).
Hence, from the last inequalities and taking into account (3.32) we conclude that
(3.67)ν11/2∥∇(uν-u)∥L2(0,T;L2)≤C(ν1+ν2+ν3)1/2,ν21/2∥∇(wν-w)∥L2(0,T;L2)≤C(ν1+ν2+ν3)1/2.
Therefore, since max{ν1,ν2,ν3}<1, with the additional condition ν3<ν1<ν2<kν1 for some positive constant k, from (3.67) we get
(3.68)∥∇(uν-u)∥L2(0,T;L2)≤M1,∥∇(wν-w)∥L2(0,T;L2)≤M2,
with M1,M2 positive constants independent of ν1,ν2,ν3. Moreover, by using (3.32) we obtain
(3.69)∥uν-u∥L2(0,T;H)≤∥uνϵ-uν∥L2(0,T;H)+∥uνϵ-u∥L2(0,T;H)≤∥uνϵ-uν∥L2(0,T;H)+C(ν1+ν2+ν3+ϵ)1/2,∥wν-w∥L2(0,T;L2)≤∥wνϵ-wν∥L2(0,T;L2)+∥wνϵ-w∥L2(0,T;L2)≤∥wνϵ-wν∥L2(0,T;L2)+C(ν1+ν2+ν3+ϵ)1/2.
Thus, by taking into account (3.60), as ϵ→0 we find
(3.70)∥uν-u∥L2(0,T;H)≤C(ν1+ν2+ν3)1/2,∥wν-w∥L2(0,T;L2)≤C(ν1+ν2+ν3)1/2,
with the constant C independent of ν1,ν2,ν3. Hence, the proof of theorem is finished.



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*Source: 843692-2010-05-13.xml*