Micropolar Fluids with Vanishing Viscosity

Abstract

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

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 domain Ω⊂ℝ³ and in a time interval (0,𝑇], 0<𝑇<+∞, this model is given by the following system of differential equations:(𝐮_𝜈)_𝑡−𝜈₁Δ𝐮_𝜈+𝐮_𝜈⋅∇𝐮_𝜈+∇𝑝_𝜈=2𝜇_𝑟𝑟𝑜𝑡𝐰_𝜈+𝐟,in𝑄,(1.1)div𝐮_𝜈=0,in𝑄,(1.2)(𝐰_𝜈)_𝑡−𝜈₂Δ𝐰_𝜈−𝜈₃∇div𝐰_𝜈+𝐮_𝜈⋅∇𝐰_𝜈+4𝜇_𝑟𝐰_𝜈=2𝜇_𝑟𝑟𝑜𝑡𝐮_𝜈+𝐠,in𝑄,(1.3) 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 point (𝐱,𝑡).𝜈₁,𝜈₂, and 𝜈₃ are positive constants which satisfy 𝜈₁=𝜇+𝜇_𝑟,𝜈₂=𝑐_𝑎+𝑐_𝑑,𝜈₃=𝑐₀+𝑐_𝑑−𝑐_𝑎, with 𝑐₀+𝑐_𝑑>𝑐_𝑎, where 𝜇,𝜇_𝑟,𝑐₀,𝑐_𝑎,𝑐_𝑑 represent viscosity coefficients. In particular, 𝜇 is the usual Newtonian viscosity, 𝜇_𝑟 is called the viscosity of microrotation and 𝑐₀,𝑐_𝑎,𝑐_𝑑 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 prescribed𝐮_𝜈(𝐱,0)=𝟎,𝐰_𝜈𝐮(𝐱,0)=𝟎,inΩ,(1.4)_𝜈(𝐱,𝑡)=𝟎,𝐰_𝜈(𝐱,𝑡)=𝟎,on𝜕Ω×[0,𝑇],(1.5) 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 𝜇_𝑟=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 𝐿^𝑝- spaces, including the stability of strong steady solutions, were performed.

This work is concerned with the behavior of the micropolar fluids, in a bounded domain Ω⊂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;𝐇³₀(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:𝐮_𝑡𝐰+𝐮⋅∇𝐮+∇𝑝=𝐟,inΩ×[0,𝑇],(1.6)div𝐮=0,inΩ×[0,𝑇],(1.7)_𝑡+𝐮⋅∇𝐰=𝐠,inΩ×[0,𝑇],(1.8)𝐮(𝐱,0)=𝟎,𝐰(𝐱,0)=𝟎,inΩ,(1.9)𝐮(𝐱,𝑡)=𝟎,𝐰(𝐱,𝑡)=𝟎,on𝜕Ω×[0,𝑇].(1.10) 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 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 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^𝑚(Ω)={𝑓∈𝐿²(Ω)∶‖𝐷^𝑘𝑓‖_𝐿2<∞,|𝑘|≤𝑚},𝑚≥1, with norm denoted by ‖⋅‖_𝐻^𝑚. 𝐻¹₀(Ω) is the closure 𝐶^∞₀(Ω) in the norm ‖⋅‖_𝐻¹. In order to distinguish the scalar-value 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∈𝐋²(Ω)/div𝐯=0inΩ,𝐯⋅𝐧=0on𝜕Ω} and 𝐕={𝐯∈𝐇¹₀(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 𝐿^𝑝-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∈𝐇¹₀∩𝐇³}⊂(𝐿^∞(0,𝑇;𝐇³₀))².(2.1) 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;𝐇¹₀∩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×𝐇¹₀∩𝐇³ and define 𝐮(𝐱,𝑡)=𝑡Φ(𝐱)∈𝐿^∞(0,𝑇;𝐕∩𝐇³),𝐰(𝐱,𝑡)=𝑡Ψ(𝐱)∈𝐿^∞(0,𝑇;𝐇¹₀∩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, 𝐮_𝑡(𝐱,𝑡)=Φ(𝐱) and 𝐰_𝑡(𝐱,𝑡)=Ψ(𝐱). Hence 𝐮_𝑡+𝐮⋅∇𝐮+∇𝑝=Φ+𝑃(𝐮⋅∇𝐮)=Φ+𝑡²𝐰𝑃(Φ⋅∇Φ)=𝐟,_𝑡+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 𝐮=𝐮₁−𝐮₂, 𝐰=𝐰₁−𝐰₂. Then, from (1.6) and (1.8), we have𝐮_𝑡+𝐮₁⋅∇𝐮+𝐮⋅∇𝐮₂+∇(𝑝₁−𝑝₂𝐰)=𝟎,(2.4)_𝑡+𝐮⋅∇𝐰₁+𝐮₂⋅∇𝐰=𝟎.(2.5) Taking the inner product of (2.4) with the function 𝐮 we obtain 12𝑑‖𝑑𝑡𝐮‖²=−(𝐮⋅∇𝐮₂,𝐮)≤𝐶‖𝐮‖²‖∇𝐮₂‖_∞.(2.6) Since 𝐮₂∈𝐇³(Ω) and 𝐇²(Ω)⊂𝐋^∞(Ω), we get 𝑑‖𝑑𝑡𝐮‖²−𝐶₁‖𝐮‖²𝑑≤0⇒𝑑𝑡(exp^−𝐶1𝑡‖𝐮‖²)≤0.(2.7) Integrating the last inequality from 0 to 𝑡, 𝑡≤𝑇, we have exp^−𝐶1𝑡‖𝐮‖²≤0, which implies ‖𝐮‖=0. Consequently 𝐮₁=𝐮₂.
Similarly, by taking the inner product of (2.5) with the function 𝐰 we find12𝑑‖𝑑𝑡𝐰‖²=−(𝐮⋅∇𝐰₁,𝐰)=0.(2.8) Then, by integrating the last equality from 0 to 𝑡, we have ‖𝐰‖=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 (𝐮_𝜈,𝐰_𝜈) of problem (1.1)–(1.5) verifying 𝐮_𝜈∈𝐿^∞(0,𝑇;𝐇)∩𝐿²(0,𝑇;𝐕),𝐰_𝜈∈𝐿^∞(0,𝑇;𝐋²)∩𝐿²(0,𝑇;𝐇¹₀),(2.9) where 𝐮_𝜈 and 𝐰_𝜈 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 ‖𝐮_𝜈−𝐮‖_{𝐿2(0,𝑇;𝐻)}=𝑂((𝜈₁+𝜈₂+𝜈₃)^1/2),‖𝐰_𝜈−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 𝐮_𝜈→𝐮weaklyin𝐿²(0,𝑇;𝐕),𝐰_𝜈→𝐰weaklyin𝐿²(0,𝑇;𝐇¹₀).(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+𝜇_𝑟, if 𝜇_𝑟=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∈𝐇¹₀, and for real constants 𝜉,𝜖>0 consider the operator 𝐵_𝜉 defined by 𝐵_𝜉𝐮=(𝜉+𝜖‖∇𝐮‖²)∇𝐮. Then for all 𝐮,𝐯∈𝐇¹₀, the following inequality holds (𝐵_𝜉𝐮−𝐵_𝜉𝐯,∇(𝐮−𝐯))≥𝜉‖∇(𝐮−𝐯)‖²+𝜖2‖∇𝐯‖²‖∇(𝐮−𝐯)‖².(3.1)

Proof. Using the equality 2(𝐮,𝐯−𝐮)=‖𝐮‖²−‖𝐯‖²+‖𝐮−𝐯‖² and the definition of 𝐵_𝜉𝐮, we obtain (𝐵_𝜉𝐮−𝐵_𝜉𝐯,∇(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.(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 𝜖 with 0<𝜖<𝜈 there is a unique solution (𝐮_𝜈𝜖,𝐰_𝜈𝜖)∈𝐿⁴(0,𝑇;𝐕)∩𝐿^∞(0,𝑇;𝐇)×𝐿⁴(0,𝑇;𝐇¹₀)∩𝐿^∞(0,𝑇;𝐋²) of the problem (𝐮_𝜈𝜖)_𝑡−(𝜈₁+𝜖‖∇𝐮_𝜈𝜖‖²)Δ𝐮_𝜈𝜖+𝐮_𝜈𝜖⋅∇𝐮_𝜈𝜖+∇𝑝_𝜈𝜖=2𝜇_𝑟𝑟𝑜𝑡𝐰_𝜈𝜖+𝐟,in𝑄,(3.3)div𝐮_𝜈𝜖=0,in𝑄,(3.4)(𝐰_𝜈𝜖)_𝑡−(𝜈₂+𝜖‖∇𝐰_𝜈𝜖‖²)Δ𝐰_𝜈𝜖−𝜈₃∇div𝐰_𝜈𝜖+𝐮_𝜈𝜖⋅∇𝐰_𝜈𝜖+4𝜇_𝑟𝐰_𝜈𝜖=2𝜇_𝑟𝑟𝑜𝑡𝐮_𝜈𝜖𝐮+𝐠,in𝑄,(3.5)_𝜈𝜖(𝐱,0)=𝐰_𝜈𝜖𝐮(𝐱,0)=0,inΩ,(3.6)_𝜈𝜖(𝐱,𝑡)=𝐰_𝜈𝜖(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 𝐕_𝑘 the subspace of 𝐕 spanned by {Φ¹(𝐱),…,Φ^𝑘(𝐱)}, and 𝐇_𝑘 be the subspace of 𝐇¹₀ spanned by {Ψ¹(𝑥),…,Ψ^𝑘(𝐱)}. For each 𝑘≥1, the following approximations 𝐮^𝑘_𝜈𝜖 and 𝐰^𝑘_𝜈𝜖, of 𝐮_𝜈𝜖 and 𝐰_𝜈𝜖, are defined: 𝐮^𝑘_𝜈𝜖(𝐱,𝑡)=^𝑘_𝑖=1𝑐_𝑖𝑘(𝑡)Φ^𝑖(𝐱),𝐰^𝑘_𝜈𝜖(𝐱,𝑡)=^𝑘_𝑖=1𝑑_𝑖𝑘(𝑡)Ψ^𝑖(𝐱),(3.8) for 𝑡∈(0,𝑇), where the coefficients 𝑐_𝑖𝑘(𝑡) and 𝑑_𝑖𝑘(𝑡) are calculated such that 𝐮^𝑘_𝜈𝜖 and 𝐰^𝑘_𝜈𝜖 solve the following system: ((𝐮^𝑘_𝜈𝜖)_𝑡,Φ^𝑖)+(𝜈₁+𝜖‖∇𝐮^𝑘_𝜈𝜖‖²)(∇𝐮^𝑘_𝜈𝜖,∇Φ^𝑖)+(𝐮^𝑘_𝜈𝜖⋅∇𝐮^𝑘_𝜈𝜖,Φ^𝑖)=2𝜇_𝑟(𝑟𝑜𝑡𝐰^𝑘_𝜈𝜖,Φ^𝑖)+(𝐟,Φ^𝑖),((𝐰^𝑘_𝜈𝜖)_𝑡,Ψ^𝑖)+(𝜈₂+𝜖‖∇𝐰^𝑘_𝜈𝜖‖²)(∇𝐰^𝑘_𝜈𝜖,∇Ψ^𝑖)+𝜈₃(div𝐰^𝑘_𝜈𝜖,divΨ^𝑖)+(𝐮^𝑘_𝜈𝜖⋅∇𝐰^𝑘_𝜈𝜖,Ψ^𝑖)+4𝜇_𝑟(𝐰^𝑘_𝜈𝜖,Ψ^𝑖)=2𝜇_𝑟(𝑟𝑜𝑡𝐮^𝑘_𝜈𝜖,Ψ^𝑖)+(𝐠,Ψ^𝑖),(3.9) for all Φ^𝑖∈𝐕_𝑘 and Ψ^𝑖∈𝐇_𝑘.
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 have12𝑑𝑑𝑡‖𝐮^𝑘_𝜈𝜖‖²+𝜈₁‖∇𝐮^𝑘_𝜈𝜖‖²+𝜖‖∇𝐮^𝑘_𝜈𝜖‖⁴=2𝜇_𝑟(𝑟𝑜𝑡𝐰^𝑘_𝜈𝜖,𝐮^𝑘_𝜈𝜖)+(𝐟,𝐮^𝑘_𝜈𝜖1),2𝑑𝑑𝑡‖𝐰^𝑘_𝜈𝜖‖²+𝜈₂‖∇𝐰^𝑘_𝜈𝜖‖²+𝜖‖∇𝐰^𝑘_𝜈𝜖‖⁴+𝜈₃‖div𝐰^𝑘_𝜈𝜖‖²+4𝜇_𝑟‖𝐰^𝑘_𝜈𝜖‖²=2𝜇_𝑟(𝑟𝑜𝑡𝐮^𝑘_𝜈𝜖,𝐰^𝑘_𝜈𝜖)+(𝐠,𝐰^𝑘_𝜈𝜖).(3.10) Now, by applying Hölder's and Young's inequalities we get 2𝜇_𝑟((𝑟𝑜𝑡𝐰^𝑘_𝜈𝜖,𝐮^𝑘_𝜈𝜖)+(𝑟𝑜𝑡𝐮^𝑘_𝜈𝜖,𝐰^𝑘_𝜈𝜖𝜈))≤¹2‖∇𝐮^𝑘_𝜈𝜖‖²+𝐶‖𝐰^𝑘_𝜈𝜖‖²,(𝐟,𝐮^𝑘_𝜈𝜖)+(𝐠,𝐰^𝑘_𝜈𝜖)≤𝐶(‖𝐟‖²+‖𝐠‖²+‖𝐮^𝑘_𝜈𝜖‖²+‖𝐰^𝑘_𝜈𝜖‖²).(3.11) Then, summing (3.10), with the help of last inequalities, we obtain 12𝑑𝑑𝑡(‖𝐮^𝑘_𝜈𝜖‖²+‖𝐰^𝑘_𝜈𝜖‖²𝜈)+¹2‖∇𝐮^𝑘_𝜈𝜖‖²+𝜈₂‖∇𝐰^𝑘_𝜈𝜖‖²+𝜖(‖∇𝐮^𝑘_𝜈𝜖‖⁴+‖∇𝐰^𝑘_𝜈𝜖‖⁴)≤𝐶(‖𝐮^𝑘_𝜈𝜖‖²+‖𝐰^𝑘_𝜈𝜖‖²)+𝐶(‖𝐟‖²+‖𝐠‖²),(3.12) and hence, by integrating (3.12) from 0 to 𝑡,𝑡∈(0,𝑇], we find ‖𝐮^𝑘_𝜈𝜖(𝑡)‖²+‖𝐰^𝑘_𝜈𝜖(𝑡)‖²+^𝑡₀(𝜈₁‖∇𝐮^𝑘_𝜈𝜖‖²+2𝜈₂‖∇𝐰^𝑘_𝜈𝜖‖²)𝑑𝑠+2𝜖^𝑡₀(‖∇𝐮^𝑘_𝜈𝜖‖⁴+‖∇𝐰^𝑘_𝜈𝜖‖⁴)𝑑𝑠≤𝐶^𝑡₀(‖𝐮^𝑘_𝜈𝜖‖²+‖𝐰^𝑘_𝜈𝜖‖²)𝑑𝑠+𝐶^𝑡₀(‖𝐟‖²+‖𝐠‖²)𝑑𝑠.(3.13) Applying Gronwall's inequality in (3.13) we get ‖𝐮^𝑘_𝜈𝜖(𝑡)‖²+‖𝐰^𝑘_𝜈𝜖(𝑡)‖²+^𝑡₀(𝜈₁‖∇𝐮^𝑘_𝜈𝜖‖²+2𝜈₂‖∇𝐰^𝑘_𝜈𝜖‖²)𝑑𝑠+2𝜖^𝑡₀(‖∇𝐮^𝑘_𝜈𝜖‖⁴+‖∇𝐰^𝑘_𝜈𝜖‖⁴)𝑑𝑠≤𝐶𝑇𝑒^𝐶𝑇≤𝐶.(3.14) Thus, from (3.14) we conclude that there is 𝐮_𝜈𝜖,𝐰_𝜈𝜖 such that as 𝑘→∞𝐮^𝑘_𝜈𝜖→𝐮_𝜈𝜖weakly-^∗in𝐿^∞(0,𝑇;𝐇)andweaklyin𝐿²𝐰(0,𝑇;𝐕),_𝑘𝜈𝜖→𝐰_𝜈𝜖weakly-^∗in𝐿^∞(0,𝑇;𝐋²)andweaklyin𝐿²(0,𝑇;𝐇¹₀).(3.15) Now, since 𝐕⊂𝐋⁶ and𝐇¹₀⊂𝐋⁶, from (3.14) we have that 𝐮^𝑘_𝜈𝜖,𝐰^𝑘_𝜈𝜖∈𝐿⁴(0,𝑇; 𝐋⁶); consequently (𝐮^𝑘_𝜈𝜖)_𝑡∈𝐿^4/3(0,𝑇;𝐕^∗) and (𝐰^𝑘_𝜈𝜖)_𝑡∈𝐿^4/3(0,𝑇;𝐇⁻¹), with 𝐕^∗ and 𝐇⁻¹ being the topological duals of 𝐕 and 𝐇¹₀, respectively. Therefore (𝐮^𝑘_𝜈𝜖)_𝑡→(𝐮_𝜈𝜖)_𝑡weaklyin𝐿^4/3(0,𝑇;𝐕^∗),(𝐰^𝑘_𝜈𝜖)_𝑡→(𝐰_𝜈𝜖)_𝑡weaklyin𝐿^4/3(0,𝑇;𝐇⁻¹).(3.16) 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 𝐮^𝑘_𝜈𝜖→𝐮_𝜈𝜖in𝐿²(0,𝑇;𝐇),𝐰^𝑘_𝜈𝜖→𝐰_𝜈𝜖in𝐿²(0,𝑇;𝐋²).(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→∞^𝑇₀|(𝐮^𝑘_𝜈𝜖⋅∇𝐮^𝑘_𝜈𝜖−𝐮_𝜈𝜖⋅∇𝐮_𝜈𝜖,Φ^𝑖)𝜙(𝑡)|𝑑𝑡→0,_𝑇0|(𝐮^𝑘_𝜈𝜖⋅∇𝐰^𝑘_𝜈𝜖−𝐮_𝜈𝜖⋅∇𝐰_𝜈𝜖,Ψ^𝑖)𝜙(𝑡)|𝑑𝑡→0,(3.18) for all Φ^𝑖∈𝐕_𝑘,Ψ^𝑖∈𝐇_𝑘 and all 𝜙∈𝐶^∞₀(0,𝑇). Finally, from (3.15) we have (∇𝐮^𝑘_𝜈𝜖,∇Φ^𝑖)→(∇𝐮_𝜈𝜖,∇Φ^𝑖),(∇𝐰^𝑘_𝜈𝜖,∇Ψ^𝑖)→(∇𝐰_𝜈𝜖,∇Ψ^𝑖),as𝑘→∞,(3.19) and hence, by taking ∇Φ^𝑖=∇𝐮^𝑘_𝜈𝜖,∇Ψ^𝑖=∇𝐰^𝑘_𝜈𝜖 in (3.19), and then, ∇Φ^𝑖=∇𝐮_𝜈𝜖,∇Ψ^𝑖=∇𝐰_𝜈𝜖, as 𝑘→∞ we get ‖∇𝐮^𝑘_𝜈𝜖‖²→(∇𝐮_𝜈𝜖,∇𝐮^𝑘_𝜈𝜖)→‖∇𝐮_𝜈𝜖‖²,‖∇𝐰^𝑘_𝜈𝜖‖²→(∇𝐰_𝜈𝜖,∇𝐰^𝑘_𝜈𝜖)→‖∇𝐰_𝜈𝜖‖².(3.20) Moreover, as (‖∇𝐮^𝑘_𝜈𝜖‖²∇𝐮^𝑘_𝜈𝜖−‖∇𝐮_𝜈𝜖‖²∇𝐮_𝜈𝜖,∇Φ^𝑖)=(‖∇𝐮^𝑘_𝜈𝜖‖²−‖∇𝐮_𝜈𝜖‖²)(∇𝐮^𝑘_𝜈𝜖,∇Φ^𝑖)+‖∇𝐮_𝜈𝜖‖²(∇𝐮^𝑘_𝜈𝜖−∇𝐮_𝜈𝜖,∇Φ^𝑖),(‖∇𝐰^𝑘_𝜈𝜖‖²∇𝐰^𝑘_𝜈𝜖−‖∇𝐰_𝜈𝜖‖²∇𝐰_𝜈𝜖,∇Ψ^𝑖)=(‖∇𝐰^𝑘_𝜈𝜖‖²−‖∇𝐰_𝜈𝜖‖²)(∇𝐰^𝑘_𝜈𝜖,∇Ψ^𝑖)+‖∇𝐰_𝜈𝜖‖²(∇𝐰^𝑘_𝜈𝜖−∇𝐰_𝜈𝜖,∇Ψ^𝑖),(3.21) we conclude that ^𝑇₀(‖∇𝐮^𝑘_𝜈𝜖‖²∇𝐮^𝑘_𝜈𝜖−‖∇𝐮_𝜈𝜖‖²∇𝐮_𝜈𝜖,∇Φ^𝑖)𝜙(𝑡)𝑑𝑡→0,_𝑇0(‖∇𝐰^𝑘_𝜈𝜖‖²∇𝐰^𝑘_𝜈𝜖−‖∇𝐰_𝜈𝜖‖²∇𝐰_𝜈𝜖,∇Ψ^𝑖)𝜙(𝑡)𝑑𝑡→0.(3.22)
Now the uniqueness of solution will be analyzed. Let (𝐮¹_𝜈𝜖,𝐰¹_𝜈𝜖,𝑝¹_𝜈𝜖) and (𝐮²_𝜈𝜖,𝐰²_𝜈𝜖,𝑝²_𝜈𝜖) be two solutions of (3.3)–(3.7). We denote 𝐮_𝜈𝜖=𝐮¹_𝜈𝜖−𝐮²_𝜈𝜖,𝐰_𝜈𝜖=𝐰¹_𝜈𝜖−𝐰²_𝜈𝜖, and 𝑝_𝜈𝜖=𝑝¹_𝜈𝜖−𝑝²_𝜈𝜖. Then we have(𝐮_𝜈𝜖)_𝑡−(𝜈₁+𝜖‖∇𝐮¹_𝜈𝜖‖²)Δ𝐮¹_𝜈𝜖+(𝜈₁+𝜖‖∇𝐮²_𝜈𝜖‖²)Δ𝐮²_𝜈𝜖+𝐮_𝜈𝜖⋅∇𝐮¹_𝜈𝜖+∇𝑝_𝜈𝜖+𝐮²_𝜈𝜖⋅∇𝐮_𝜈𝜖=2𝜇_𝑟𝑟𝑜𝑡𝐰_𝜈𝜖,𝐮(3.23)div_𝜈𝜖(𝐰=0,(3.24)_𝜈𝜖)_𝑡−(𝜈₂+𝜖‖∇𝐰¹_𝜈𝜖‖²)Δ𝐰¹_𝜈𝜖+(𝜈₂+𝜖‖∇𝐰²_𝜈𝜖‖²)Δ𝐰²_𝜈𝜖−𝜈₃∇div𝐰_𝜈𝜖+𝐮_𝜈𝜖⋅∇𝐰¹_𝜈𝜖+𝐮²_𝜈𝜖⋅∇𝐰_𝜈𝜖+4𝜇_𝑟𝐰_𝜈𝜖=2𝜇_𝑟𝑟𝑜𝑡𝐮_𝜈𝜖.(3.25) Taking the inner product of (3.23) with 𝐮_𝜈𝜖, of (3.25) with 𝐰_𝜈𝜖, by using (3.24), we obtain 12𝑑‖𝐮𝑑𝑡^𝜈𝜖‖²+(𝐵_𝜈1𝐮¹_𝜈𝜖,∇𝐮_𝜈𝜖)−(𝐵_𝜈1𝐮²_𝜈𝜖,∇𝐮_𝜈𝜖)=−(𝐮_𝜈𝜖⋅∇𝐮¹_𝜈𝜖,𝐮_𝜈𝜖)+2𝜇_𝑟(𝑟𝑜𝑡𝐰_𝜈𝜖,𝐮_𝜈𝜖1),2𝑑‖𝐰𝑑𝑡^𝜈𝜖‖²+(𝐵_𝜈2𝐰¹_𝜈𝜖,∇𝐰_𝜈𝜖)−(𝐵_𝜈2𝐰²_𝜈𝜖,∇𝐰_𝜈𝜖)≤−(𝐮_𝜈𝜖⋅∇𝐰¹_𝜈𝜖,𝐰_𝜈𝜖)+2𝜇_𝑟|(𝑟𝑜𝑡𝐮_𝜈𝜖,𝐰_𝜈𝜖)|.(3.26) Hence, by using Lemma 3.1 we get 12𝑑‖𝐮𝑑𝑡^𝜈𝜖‖²+𝜈₁‖∇𝐮_𝜈𝜖‖²+𝜖2‖∇𝐮²_𝜈𝜖‖²‖∇𝐮_𝜈𝜖‖²≤|(𝐮_𝜈𝜖⋅∇𝐮¹_𝜈𝜖,𝐮_𝜈𝜖)|+2𝜇_𝑟|(𝑟𝑜𝑡𝐰_𝜈𝜖,𝐮_𝜈𝜖1)|,2𝑑‖𝐰𝑑𝑡^𝜈𝜖‖²+𝜈₂‖∇𝐰_𝜈𝜖‖²+𝜖2‖∇𝐰²_𝜈𝜖‖²‖∇𝐰_𝜈𝜖‖²≤|(𝐮_𝜈𝜖⋅∇𝐰¹_𝜈𝜖,𝐰_𝜈𝜖)|+2𝜇_𝑟|(𝑟𝑜𝑡𝐮_𝜈𝜖,𝐰_𝜈𝜖)|.(3.27) Consequently 12𝑑‖𝐮𝑑𝑡^𝜈𝜖‖²+𝜈₁‖∇𝐮_𝜈𝜖‖²≤|(𝐮_𝜈𝜖⋅∇𝐮¹_𝜈𝜖,𝐮_𝜈𝜖)|+2𝜇_𝑟|(𝑟𝑜𝑡𝐰_𝜈𝜖,𝐮_𝜈𝜖1)|,2𝑑‖𝐰𝑑𝑡^𝜈𝜖‖²+𝜈₂‖∇𝐰_𝜈𝜖‖²≤|(𝐮_𝜈𝜖⋅∇𝐰¹_𝜈𝜖,𝐰_𝜈𝜖)|+2𝜇_𝑟|(𝑟𝑜𝑡𝐮_𝜈𝜖,𝐰_𝜈𝜖)|.(3.28) Now, by using Hölder's and Young's inequalities we have |(𝐮_𝜈𝜖⋅∇𝐮¹_𝜈𝜖,𝐮_𝜈𝜖)|≤‖𝐮_𝜈𝜖‖₃‖∇𝐮_𝜈𝜖‖‖𝐮¹_𝜈𝜖‖₆≤𝐶‖𝐮_𝜈𝜖‖^1/2‖∇𝐮_𝜈𝜖‖^3/2‖∇𝐮¹_𝜈𝜖‖≤𝐶‖𝐮_𝜈𝜖‖²‖∇𝐮¹_𝜈𝜖‖⁴+𝜈₁4‖∇𝐮_𝜈𝜖‖²,|(𝐮_𝜈𝜖⋅∇𝐰¹_𝜈𝜖,𝐰_𝜈𝜖)|≤𝐶‖𝐮_𝜈𝜖‖^1/2‖∇𝐮_𝜈𝜖‖^1/2‖∇𝐰_𝜈𝜖‖‖∇𝐰¹_𝜈𝜖‖≤𝐶‖𝐮_𝜈𝜖‖‖∇𝐮_𝜈𝜖‖‖∇𝐰¹_𝜈𝜖‖²+𝜈₂‖∇𝐰_𝜈𝜖‖²≤𝐶‖𝐮_𝜈𝜖‖²‖∇𝐰¹_𝜈𝜖‖⁴+𝜈₁4‖∇𝐮_𝜈𝜖‖²+𝜈₂‖∇𝐰_𝜈𝜖‖²,2𝜇_𝑟|(𝑟𝑜𝑡𝐰_𝜈𝜖,𝐮_𝜈𝜖)|≤𝐶‖𝐰_𝜈𝜖‖‖∇𝐮_𝜈𝜖‖≤𝐶‖𝐰_𝜈𝜖‖²+𝜈₁4‖∇𝐮_𝜈𝜖‖²,2𝜇_𝑟|(𝑟𝑜𝑡𝐮_𝜈𝜖,𝐰_𝜈𝜖)|≤𝐶‖𝐰_𝜈𝜖‖²+𝜈₁4‖∇𝐮_𝜈𝜖‖².(3.29) Then, by summing (3.28), and taking into account the last inequalities, we obtain 12𝑑𝐮𝑑𝑡(‖_𝜈𝜖‖²+‖𝐰_𝜈𝜖‖²)≤𝐶‖𝐰_𝜈𝜖‖²+𝐶(‖∇𝐮¹_𝜈𝜖‖⁴+‖∇𝐰¹_𝜈𝜖‖⁴)‖𝐮_𝜈𝜖‖²≤𝐶𝑀(𝑡)(‖𝐮_𝜈𝜖‖²+‖𝐰_𝜈𝜖‖²),(3.30) where 𝑀(𝑡)=1+‖∇𝐮²_𝜈𝜖(𝑡)‖⁴+‖∇𝐰²_𝜈𝜖(𝑡)‖⁴.
Since 𝑀(𝑡)∈𝐿¹(0,𝑇), by integrating (3.30) from 0 to 𝑡, and then applying Gronwall's inequality, we conclude that‖𝐮_𝜈𝜖‖²+‖𝐰_𝜈𝜖‖²≤0,(3.31) which implies 𝐮_𝜈𝜖=0 and 𝐰_𝜈𝜖=0. Consequently the uniqueness of solution is proved.

Proposition 3.3. Under the assumptions of Lemma 2.1 and Theorem 3.2, if (𝐮_𝜈𝜖,𝐰_𝜈𝜖) and (𝐮,𝐰) are the solutions of problems (3.3)–(3.7) and (1.6)–(1.10), respectively, then ‖𝐮_𝜈𝜖−𝐮‖_{𝐿∞(0,𝑇;𝐻)}=𝑂((𝜈₁+𝜖+𝜈₂+𝜈₃)^1/2𝜈),_11/2‖∇(𝐮_𝜈𝜖−𝐮)‖_{𝐿2(0,𝑇;𝐿2)}=𝑂((𝜈₁+𝜖+𝜈₂+𝜈₃)^1/2),‖𝐰_𝜈𝜖−𝐰‖_{𝐿∞(0,𝑇;𝐿2)}=𝑂((𝜈₁+𝜖+𝜈₂+𝜈₃)^1/2𝜈),_21/2‖∇(𝐰_𝜈𝜖−𝐰)‖_{𝐿2(0,𝑇;𝐿2)}=𝑂((𝜈₁+𝜖+𝜈₂+𝜈₃)^1/2).(3.32)

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_𝜈𝜖−𝐮 and 𝐳=𝐰_𝜈𝜖−𝐰, respectively, we have 12𝑑𝑑𝑡‖𝐯‖²+(𝜈₁+𝜖‖∇𝐮_𝜈𝜖‖²)(∇𝐮_𝜈𝜖,∇𝐯)+(𝐯⋅∇𝐮,𝐯)=2𝜇_𝑟(𝑟𝑜𝑡𝐰_𝜈𝜖1,𝐯),2𝑑𝑑𝑡‖𝐳‖²+(𝜈₂+𝜖‖∇𝐰_𝜈𝜖‖²)(∇𝐰_𝜈𝜖,∇𝐳)+𝜈₃(div𝐰_𝜈𝜖,div𝐳)+(𝐯⋅∇𝐰,𝐳)+4𝜇_𝑟(𝐰_𝜈𝜖,𝐳)=2𝜇_𝑟(𝑟𝑜𝑡𝐮_𝜈𝜖,𝐳).(3.33) Recalling the notation 𝐵_𝜉𝜑=(𝜉+𝜖‖∇𝜑‖²)∇𝜑 we get 12𝑑𝑑𝑡‖𝐯‖²+(𝐵_𝜈1𝐮_𝜈𝜖−𝐵_𝜈1𝐮,∇𝐯)=−(𝐵_𝜈1𝐮,∇𝐯)−(𝐯⋅∇𝐮,𝐯)+2𝜇_𝑟(𝑟𝑜𝑡𝐳,𝐯)+2𝜇_𝑟1(𝑟𝑜𝑡𝐰,𝐯),(3.34)2𝑑𝑑𝑡‖𝐳‖²+(𝐵_𝜈2𝐰_𝜈𝜖−𝐵_𝜈2𝐰,∇𝐳)+𝜈₃‖div𝐳‖²+4𝜇_𝑟‖𝐳‖²=−(𝐵_𝜈2𝐰,∇𝐳)−𝜈₃(div𝐰,div𝐳)−4𝜇_𝑟(𝐰,𝐳)−(𝐯⋅∇𝐰,𝐳)+2𝜇_𝑟(𝑟𝑜𝑡𝐯,𝐳)+2𝜇_𝑟(𝑟𝑜𝑡𝐮,𝐳).(3.35) Using Hölder's and Young's inequalities we bound the right hand of (3.34) and (3.35) as follows: (𝐵_𝜈1𝜈𝐮,∇𝐯)≤¹+𝜖‖∇𝐮‖²2(‖∇𝐮‖²+‖∇𝐯‖²),(3.36)(𝐯⋅∇𝐮,𝐯)≤𝐶‖∇𝐮‖_𝐿^∞‖𝐯‖²≤𝐶‖𝐯‖²,(3.37)2𝜇_𝑟(𝑟𝑜𝑡𝐳,𝐯)≤𝐶𝜈₁𝜈‖∇𝐯‖‖𝐳‖≤²¹8‖∇𝐯‖²+𝐶‖𝐳‖²,(3.38)2𝜇_𝑟(𝑟𝑜𝑡𝐰,𝐯)≤𝜈²₁‖∇𝐰‖²+𝐶‖𝐯‖²,(3.39)(𝐵_𝜈2𝜈𝐰,∇𝐳)≤²+𝜖‖∇𝐰‖²2(‖∇𝐰‖²+‖∇𝐳‖²𝜈),(3.40)₃(div𝐰,div𝐳)≤𝐶𝜈₃‖∇𝐰‖²+𝜈₃‖div𝐳‖²,(3.41)4𝜇_𝑟(𝐰,𝐳)≤𝜈²₁‖𝐰‖²+𝐶‖𝐳‖²,(3.42)(𝐯⋅∇𝐰,𝐳)≤𝐶(‖𝐯‖²+‖𝐳‖²),(3.43)2𝜇_𝑟𝜈(𝑟𝑜𝑡𝐯,𝐳)≤²¹8‖∇𝐯‖²+𝐶‖𝐳‖²,(3.44)2𝜇_𝑟(𝑟𝑜𝑡𝐮,𝐳)≤𝜈²₁‖∇𝐮‖²+𝐶‖𝐳‖².(3.45) Carrying (3.36)–(3.39) in (3.34) and (3.40)–(3.45) in (3.35), we have 12𝑑𝑑𝑡‖𝐯‖²+(𝐵_𝜈1𝐮_𝜈𝜖−𝐵_𝜈1𝜈𝐮,∇𝐯)≤¹+𝜖‖∇𝐮‖²2(‖∇𝐮‖²+‖∇𝐯‖²)+𝐶‖𝐯‖²+𝜈₁8‖∇𝐯‖²+𝐶‖𝐳‖²+𝜈₁‖∇𝐰‖²,12𝑑𝑑𝑡‖𝐳‖²+(𝐵_𝜈2𝐰_𝜈𝜖−𝐵_𝜈2𝜈𝐰,∇𝐳)≤²+𝜖‖∇𝐰‖²2(‖∇𝐰‖²+‖∇𝐳‖²)+𝐶𝜈₃‖∇𝐰‖²+𝜈₁‖𝐰‖²+𝐶(‖𝐯‖²+‖𝐳‖²𝜈)+¹8‖∇𝐯‖²+𝐶𝜈₁‖∇𝐮‖².(3.46)
Now, by using the equality 2(𝐮,𝐯−𝐮)=‖𝐮‖²−‖𝐯‖²+‖𝐮−𝐯‖², the definition of 𝐵_𝜉𝜑 and Lemma 3.1, from (3.46) we get 12𝑑𝑑𝑡(‖𝐯‖²+‖𝐳‖²𝜈)+¹4‖∇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),(3.47) which implies 𝑑𝑑𝑡(‖𝐯‖²+‖𝐳‖²)+𝜈₁‖∇𝐯‖²+𝜈₂‖∇𝐳‖²≤𝐶(𝜖+𝜈₁+𝜈₂+𝜈₃)+𝐶(‖𝐯‖²+‖𝐳‖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‖𝐯(𝑡)‖²+‖𝐳(𝑡)‖²+^𝑡₀(𝜈₁‖∇𝐯(𝑠)‖²+𝜈₂‖∇𝐳(𝑠)‖²)𝑑𝑠≤𝐶(𝜖+𝜈₁+𝜈₂+𝜈₃)𝑇exp(𝐶𝑇),(3.49) and hence the proof of estimates (3.32) is concluded.

Proposition 3.4. Under the assumptions of Theorem 3.2 and considering 𝜈₃<𝜈₂, then as 𝜖→0 the solution (𝐮_𝜈𝜖,𝐰_𝜈𝜖) of (3.3)–(3.7) verifies the following convergences: 𝐮_𝜈𝜖→𝐮_𝜈in𝐿²(0,𝑇;𝐇),𝐰_𝜈𝜖→𝐰_𝜈in𝐿²(0,𝑇;𝐋²),(3.50) where (𝐮_𝜈,𝐰_𝜈) is a solution of problem (1.1)–(1.3).

Proof. Let 𝐮_𝜈𝜖,𝐰_𝜈𝜖 be as in Theorem 3.2. Then from (3.14) we have ‖𝐮_𝜈𝜖‖²_{𝐿∞(0,𝑇;𝐻)}+𝜈₁‖𝐮_𝜈𝜖‖²_{𝐿2(0,𝑇;𝑉)}+𝜖‖𝐮_𝜈𝜖‖⁴_{𝐿4(0,𝑇;𝑉)}≤𝐶,‖𝐰_𝜈𝜖‖²_{𝐿∞(0,𝑇;𝐿2)}+𝜈₂‖𝐰_𝜈𝜖‖²_{𝐿2(0,𝑇;𝐻10)}+𝜖‖𝐰_𝜈𝜖‖⁴_{𝐿4(0,𝑇;𝐻10)}≤𝐶,(3.51) where 𝐶 is a constant which does not depend on 𝜈₁,𝜈₂,𝜈₃, and 𝜖.
Then, since 𝐕⊂𝐋⁶ and 𝐇¹₀⊂𝐋⁶, we have that 𝐮_𝜈𝜖,𝐰_𝜈𝜖∈𝐿⁴(0,𝑇;𝐋⁶). Consequently (𝐮_𝜈𝜖)_𝑡∈𝐿^4/3(0,𝑇;𝐕^∗) and (𝐰_𝜈𝜖)_𝑡∈𝐿^4/3(0,𝑇;𝐇⁻¹), with 𝐕^∗ and 𝐇⁻¹ being the topological dual of 𝐕 and 𝐇¹₀, respectively.
Now, since |(𝐮⋅∇𝐰,𝐳)|≤‖𝐮‖₃‖∇𝐰‖‖𝐳‖₆≤𝐶‖𝐮‖^1/2‖𝐮‖_𝑉^1/2‖∇𝐰‖‖∇𝐳‖, from (3.3) and (3.5) we obtain ‖(𝐮_𝜈𝜖)_𝑡‖_𝑉∗≤𝐶(𝜈₁‖𝐮_𝜈𝜖‖_𝑉+𝜖‖𝐮_𝜈𝜖‖³_𝑉+‖𝐮_𝜈𝜖‖^1/2‖𝐮_𝜈𝜖‖_𝑉^3/2+𝜈₁‖𝐰_𝜈𝜖‖+‖𝐟‖),‖(𝐰_𝜈𝜖)_𝑡‖_𝐻−1≤𝐶(𝜈₂‖𝐰_𝜈𝜖‖_𝐻10+𝜖‖𝐰_𝜈𝜖‖³_𝐻10+𝜈₃‖𝐰_𝜈𝜖‖_𝐻10+‖𝐮_𝜈𝜖‖^3/2‖𝐮_𝜈𝜖‖_𝑉^3/2)+𝐶(‖𝐰_𝜈𝜖‖_𝐻^3/2₁₀+𝜈₁‖𝐰_𝜈𝜖‖+𝜈₁‖𝐮_𝜈𝜖‖+‖𝐠‖).(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 ‖(𝐮_𝜈𝜖)_𝑡‖_𝑉^4/3_∗≤𝐶(𝜈₁^4/3‖𝐮_𝜈𝜖‖_𝑉^4/3+𝜖^4/3‖𝐮_𝜈𝜖‖⁴_𝑉+‖𝐮_𝜈𝜖‖²_𝑉+𝜈₁^4/3+‖𝐟‖^4/3),‖(𝐰_𝜈𝜖)_𝑡‖_𝐻^4/3₋₁≤𝐶(𝜈₂^4/3‖𝐰_𝜈𝜖‖_𝐻^4/3₁₀+𝜖^4/3‖𝐰_𝜈𝜖‖⁴_𝐻10+𝜈₃^4/3‖𝐰_𝜈𝜖‖_𝐻^4/3₁₀)+𝐶(‖𝐮_𝜈𝜖‖²_𝑉+‖𝐰_𝜈𝜖‖²_𝐻10+𝜈₁^4/3+‖𝐠‖^4/3).(3.53) Integrating the last inequalities from 0 to 𝑇 we conclude ^𝑇₀‖(𝐮_𝜈𝜖)_𝑡‖_𝑉^4/3_∗𝑑𝑡≤𝐶(𝜈₁^4/3𝑇₀‖𝐮_𝜈𝜖‖_𝑉^4/3𝑑𝑡+𝜈₁^4/3𝑇+^𝑇₀‖𝐟‖^4/3𝑑𝑡)+𝐶(𝜖^4/3‖𝐮_𝜈𝜖‖⁴_{𝐿4(0,𝑇;𝑉)}+‖𝐮_𝜈𝜖‖²_{𝐿2(0,𝑇;𝑉)}),_𝑇0‖(𝐰_𝜈𝜖)_𝑡‖_𝐻^4/3₋₁𝑑𝑡≤𝐶((𝜈₂^4/3+𝜈₃^4/3)^𝑇₀‖𝐰_𝜈𝜖‖_𝐻^4/3₁₀𝑑𝑡+𝜈₁^4/3𝑇+^𝑇₀‖𝐠‖^4/3𝑑𝑡)+𝐶(𝜖^4/3‖𝐰_𝜈𝜖‖⁴_{𝐿4(0,𝑇;𝐻10)}+‖𝐮_𝜈𝜖‖²_{𝐿2(0,𝑇;𝑉)}+‖𝐰_𝜈𝜖‖²_{𝐿2(0,𝑇;𝐻10)}).(3.54) Using Hölder's inequality and (3.51), since 𝜈₃<𝜈₂ we obtain 𝜈₁^4/3𝑇₀‖𝐮_𝜈𝜖‖_𝑉^4/3𝑑𝑡≤𝐶𝜈₁^4/3‖𝐮_𝜈𝜖‖_𝐿^4/3_{2(0,𝑇;𝑉)}≤𝐶𝜈₁^2/3,(𝜈₂^4/3+𝜈₃^4/3)^𝑇₀‖𝐰_𝜈𝜖‖_𝐻^4/3₁₀𝑑𝑡≤𝜈₂^4/3𝐶‖𝐰_𝜈𝜖‖_𝐿^4/3_{2(0,𝑇;𝐻10)}≤𝐶𝜈₂^2/3.(3.55) Hence, from (3.54)–(3.55) and (3.51) we get ‖(𝐮_𝜈𝜖)_𝑡‖_𝐿^4/3_{4/3(0,𝑇;𝑉∗)}≤𝐶(𝜈₁^2/3+𝜈₁^4/3+1+𝜖^1/3+𝜈₁⁻¹),‖(𝐰_𝜈𝜖)_𝑡‖_𝐿^4/3_{4/3(0,𝑇;𝐻−1)}≤𝐶(𝜈₂^2/3+𝜈₁^4/3+1+𝜖^1/3+𝜈₁⁻¹+𝜈₂⁻¹).(3.56) Thus, since 𝜖<𝜈₁, from last two inequalities we have ‖(𝐮_𝜈𝜖)_𝑡‖_{𝐿4/3(0,𝑇;𝑉∗)}≤𝐶(𝜈₁^1/2+𝜈₁+𝜈₁^1/4+𝜈₁^−3/4+1),‖(𝐰_𝜈𝜖)_𝑡‖_{𝐿4/3(0,𝑇;𝐻−1)}≤𝐶(𝜈₂^1/2+𝜈₁+𝜈₁^1/4+𝜈₁^−3/4+𝜈₂^−3/4+1),(3.57) 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→0𝐮_𝜈𝜖⇀𝐮_𝜈weak-^∗in𝐿^∞𝜈(0,𝑇;𝐇),_11/2𝐮_𝜈𝜖⇀𝜈₁^1/2𝐮_𝜈weaklyin𝐿²𝜖(0,𝑇;𝐕),_1/4𝐮_𝜈𝜖⇀0weaklyin𝐿⁴(0,𝑇;𝐕),(𝐮_𝜈𝜖)_𝑡⇀(𝐮_𝜈)_𝑡weaklyin𝐿^4/3(0,𝑇;𝐕^∗𝐮),_𝜈𝜖(0)⇀𝐮_𝜈(0)weaklyin𝐕^∗.(3.58) Similarly, as 𝜖→0𝐰_𝜈𝜖⇀𝐰_𝜈weak-^∗in𝐿^∞(0,𝑇;𝐋²𝜈),_21/2𝐰_𝜈𝜖⇀𝜈₂^1/2𝐰_𝜈weaklyin𝐿²(0,𝑇;𝐇¹₀𝜖),_1/4𝐰_𝜈𝜖⇀0weaklyin𝐿⁴(0,𝑇;𝐇¹₀),(𝐰_𝜈𝜖)_𝑡⇀(𝐰_𝜈)_𝑡weaklyin𝐿^4/3(0,𝑇;𝐇⁻¹𝐰),_𝜈𝜖(0)⇀𝐰_𝜈(0)weaklyin𝐇⁻¹.(3.59) 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 𝐮_𝜈𝜖→𝐮_𝜈in𝐿²(0,𝑇;𝐇),𝐰_𝜈𝜖→𝐰_𝜈in𝐿²(0,𝑇;𝐋²).(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:𝑑𝑑𝑡(𝐮_𝜈,Φ)+𝜈₁(∇𝐮_𝜈,∇Φ)+(𝐮_𝜈⋅∇𝐮_𝜈,Φ)=2𝜇_𝑟(𝑟𝑜𝑡𝐰_𝜈,𝑑Φ)+(𝐟,Φ),𝑑𝑡(𝐰_𝜈,Ψ)+𝜈₂(∇𝐰_𝜈,∇Ψ)+𝜈₃(div𝐰_𝜈,divΨ)+(𝐮_𝜈⋅∇𝐰_𝜈,Ψ)+4𝜇_𝑟(𝐰_𝜈,Ψ)=2𝜇_𝑟(𝑟𝑜𝑡𝐮_𝜈,Ψ)+(𝐠,Ψ),(3.61) for all Φ∈𝐕,Ψ∈𝐇¹₀.
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_𝜈𝜖⋅∇𝐮_𝜈𝜖,Φ)→(𝐮_𝜈⋅∇𝐮_𝜈,Φ),(𝐮_𝜈𝜖⋅∇𝐰_𝜈𝜖,Ψ)→(𝐰_𝜈⋅∇𝐰_𝜈,Ψ).(3.62) Moreover it is not difficult to check that (𝐮_𝜈(𝐱,0),Φ)=0,∀Φ∈𝐕,(𝐰_𝜈(𝐱,0),Ψ)=0,∀Ψ∈𝐇¹₀.(3.63) Finally, it is clear that for all Φ∈𝐕,Ψ∈𝐇¹₀, as 𝜖→0 it holds (𝜖‖∇𝐮_𝜈𝜖‖²)(∇𝐮_𝜈𝜖,∇Φ)→0,(𝜖‖∇𝐰_𝜈𝜖‖²)(∇𝐰_𝜈𝜖,∇Ψ)→0.(3.64)

Proof. 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_𝜈𝜖,𝐰_𝜈𝜖).
Now the second part of the Theorem 2.2 will be proved. Let (𝐮,𝐰) be solution of problem (1.6)–(1.10). From (3.58)-(3.59) we have ∇(𝐮_𝜈𝜖−𝐮)⇀∇(𝐮_𝜈−𝐮)weaklyin𝐿²(0,𝑇;𝐋²),∇(𝐰_𝜈𝜖−𝐰)⇀∇(𝐰_𝜈−𝐰)weaklyin𝐿²(0,𝑇;𝐋²).(3.65) Consequently 𝜈₁^1/2‖∇(𝐮_𝜈−𝐮)‖_{𝐿2(0,𝑇;𝐿2)}≤liminf_𝜖→0𝜈₁^1/2‖∇(𝐮_𝜈𝜖−𝐮)‖_{𝐿2(0,𝑇;𝐿2)},𝜈₂^1/2‖∇(𝐰_𝜈−𝐰)‖_{𝐿2(0,𝑇;𝐿2)}≤liminf_𝜖→0𝜈₂^1/2‖∇(𝐰_𝜈𝜖−𝐰)‖_{𝐿2(0,𝑇;𝐿2)}.(3.66) Hence, from the last inequalities and taking into account (3.32) we conclude that 𝜈₁^1/2‖∇(𝐮_𝜈−𝐮)‖_{𝐿2(0,𝑇;𝐿2)}≤𝐶(𝜈₁+𝜈₂+𝜈₃)^1/2,𝜈₂^1/2‖∇(𝐰_𝜈−𝐰)‖_{𝐿2(0,𝑇;𝐿2)}≤𝐶(𝜈₁+𝜈₂+𝜈₃)^1/2.(3.67) 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 ‖∇(𝐮_𝜈−𝐮)‖_{𝐿2(0,𝑇;𝐿2)}≤𝑀₁,‖∇(𝐰_𝜈−𝐰)‖_{𝐿2(0,𝑇;𝐿2)}≤𝑀₂,(3.68) with 𝑀₁,𝑀₂ positive constants independent of 𝜈₁,𝜈₂,𝜈₃. Moreover, by using (3.32) we obtain ‖𝐮_𝜈−𝐮‖_{𝐿2(0,𝑇;𝐻)}≤‖𝐮_𝜈𝜖−𝐮_𝜈‖_{𝐿2(0,𝑇;𝐻)}+‖𝐮_𝜈𝜖−𝐮‖_{𝐿2(0,𝑇;𝐻)}≤‖𝐮_𝜈𝜖−𝐮_𝜈‖_{𝐿2(0,𝑇;𝐻)}+𝐶(𝜈₁+𝜈₂+𝜈₃+𝜖)^1/2,‖𝐰_𝜈−𝐰‖_{𝐿2(0,𝑇;𝐿2)}≤‖𝐰_𝜈𝜖−𝐰_𝜈‖_{𝐿2(0,𝑇;𝐿2)}+‖𝐰_𝜈𝜖−𝐰‖_{𝐿2(0,𝑇;𝐿2)}≤‖𝐰_𝜈𝜖−𝐰_𝜈‖_{𝐿2(0,𝑇;𝐿2)}+𝐶(𝜈₁+𝜈₂+𝜈₃+𝜖)^1/2.(3.69) Thus, by taking into account (3.60), as 𝜖→0 we find ‖𝐮_𝜈−𝐮‖_{𝐿2(0,𝑇;𝐻)}≤𝐶(𝜈₁+𝜈₂+𝜈₃)^1/2,‖𝐰_𝜈−𝐰‖_{𝐿2(0,𝑇;𝐿2)}≤𝐶(𝜈₁+𝜈₂+𝜈₃)^1/2,(3.70) with the constant 𝐶 independent of 𝜈₁,𝜈₂,𝜈₃. Hence, the proof of theorem is finished.

Acknowledgment

The first and third authors were partially supported by Fondecyt-Chile, Grant 1040205, 7060025.