LOW-CONCENTRATED AQUEOUS POLYMER SOLUTIONS

LOW-CONCENTRATED AQUEOUS POLYMER SOLUTIONS

MIKHAIL V. TURBIN

Received 12 March 2005; Accepted 10 July 2005

The initial-boundary value problem for the mathematical model of low-concentrated aqueous polymer solutions is considered. For this initial-boundary value problem a con- cept of a weak solution is introduced and the existence theorem for such solutions is proved.

Copyright © 2006 Mikhail V. Turbin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The study of fluid motion is a source of a large number of mathematical problems. Mo- tion of an incompressible fluid in a bounded domainΩ⊂Rⁿon a time interval [0,T], (T <∞) is described by the following system of equations in Cauchy’s form [3]:

∂v

∂t+vi∂v

∂xi+ gradp=Divσ+f, (x,t)∈Ω×[0,T], divv=0, (x,t)∈Ω×[0,T].

(1.1) Hereinafter Einstein’s summation convention is supposed. In the given systemv(x,t) is the velocity vector of the particle at a pointxat an instantt;p=p(x,t) is the pressure of the fluid at a pointxat an instantt; f = f(x,t) is the vector of body forces acting on the fluid;σ=(σi j(x)) is the deviator of the stress tensor. By Divσwe denote the vector of divergences of columns of the matrixσ.

The system of (1.1) describes formally the motion of all kinds of fluids. However the number of unknowns in this system is greater than the number of equations. To complete the given system one usually uses various relations between the deviator of the stress tensorσand the strain rate tensorᏱ=(Ᏹi j),

Ᏹi j=Ᏹi j(v)=1 2

∂vi

∂x_j+∂v_j

∂x_i

. (1.2)

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 12497, Pages1–27 DOI10.1155/AAA/2006/12497

Such relations are called constitutive laws. A constitutive law is a hypothesis which should be verified for concrete fluids by experimental data.

The most known constitutive law for viscous incompressible liquid is Newton’s con- stitutive law

σ=2νᏱ. (1.3)

Hereνis the viscosity of the fluid. This model describes the motion of the majority of viscous incompressible fluids appearing in practice at moderate velocities. However this model does not describe the motion of the fluids possessing relaxational properties. These properties consist in the following. After a change of the exterior conditions the equilib- rium state in a fluid does not appear at once but after a while. This while is determined by the processes of interior reorganization. These processes may have various nature (mag- netic, thermal, etc.). The fluid may have some spectrum of relaxation times correspond- ing to these processes.

In papers [16,18] it has been noted, that the account of relaxational properties of a fluid is the account of time connections between the deviator of the stress tensorσ and the strain rate tensorᏱ. The presence of this connection expresses itself in losses of the mechanical energy of a flow, additional to the dissipative losses owing to viscosity. These additional losses of the mechanical energy of the flow are to be taken into account in the general case of motion of a fluid.

The following relation was suggested in paper [16] as a one taking into account the mentioned effects:

σ=2νᏱ+κν⁻¹dᏱ dt

, κ,ν>0. (1.4)

Hereνis the kinematic coefficient of viscosity, andκis the retardation time. Sometimes the coefficientκ is also called the time of strain relaxation. This model describes the motion of viscous non-Newtonian fluid which requires time to begin a motion after an action of a force suddenly applied.

Substitutingσfrom the constitutive law (1.4) into the system of equations of incom- pressible liquid motion in Cauchy’s form (1.1) we obtain the following system of equa- tions:

∂v

∂t ⁻νΔv+vi∂v

∂xi−κ∂Δv

∂t ^−κDiv

vk∂Ᏹ(v)

∂xk

+ gradp=f, x∈Ω,t∈[0,T], (1.5)

divv=0, x∈Ω,t∈[0,T]. (1.6)

The given system of equations differs from the Navier-Stokes system of equations by pres- ence of the additional termκDiv(dᏱ/dt). This term describes relaxational properties of a fluid. In the case of very weak relaxational properties of the fluid (forκclose to zero), and also in the case when the motion of the liquid has steady-state character (the total de- rivative of the strain rate tensor with respect to time is equal to zero) this additional term vanishes. In this case the system of (1.5), (1.6) coincides with the system of Navier-Stokes

equations. But in the case of turbulent behavior of the fluid and on unsteady laminar behavior of the fluid this additional term is nonzero and must play a significant role. Just so low-concentrated aqueous solutions of polymers behave and it has been confirmed by experimental researches of polyethylenoxide and polyacrylamide aqueous solutions [2]

and polyacrylamide and guar gum aqueous solutions [1]. Therefore the given model has also received the title of the mathematical model of motion of low-concentrated aqueous polymer solutions.

Note also that the constitutive law (1.4) is a special case of Noll’s constitutive law [9]:

σi j=ϕI2

Ᏹi j+ψI2

ᏱikᏱk j+ψ1

I2

∂Ᏹ

∂t +vk∂Ᏹi j

∂xk − ∂vi

∂xmᏱm j+ ∂vj

∂xmᏱim

. (1.7) Hereϕ(I2),ψ(I2),ψ1(I2) are material functions depending on the invariantI2=ᏱikᏱki. Noll’s model of fluid motion is a very general model for which the stress tensor at moment tis determined by nonlinear symmetrical tensor functional. This functional is defined on the set of tensor functions determining history of deformation of a medium at all times preceding the considered instantt.

In this paper we consider the system of (1.5), (1.6) in a bounded domainΩ⊂Rⁿ, n=2, 3 with locally-Lipschitz boundary on a time interval [0,T], (T <∞). For (1.5), (1.6) we study the initial-boundary value problem with the initial condition

v(x, 0)=a(x), x∈Ω, (1.8)

and the boundary condition

v|∂Ω×[0,T]=0. (1.9)

We do not know papers in which problem (1.5)–(1.9) would have been investigated.

Since this problem is very complicated for the research. In works [11,16] it was suggested the following simplification:

∂v

∂t ⁻νΔv+vi∂v

∂x_i^−κ ∂Δv

∂t +vk∂Δv

∂x_k

+ gradp= f, x∈Ω,t∈[0,T], divv=0, x∈Ω,t∈[0,T].

(1.10)

This system is obtained from (1.5), (1.6) under the condition (∂v_k/∂x_j)(∂/∂x_k)(∂v/∂x_j+ gradv_j)≡0.

Initial-boundary value problem (1.10), (1.8), (1.9) was studied by Oskolkov in his papers [11,12]. However later he remarked in [13] that these papers contain some er- rors and the obtained results are incorrect. Under the slip boundary condition the sys- tem (1.10), has been investigated by Oskolkov in [15]. Ladyzhenskaya notes in her work [7] that the method of introduction of auxiliary viscosity used for study of the initial- boundary value problem (1.10), (1.8), (1.9) in the above mentioned papers [11,12] is erroneous and the problem of existence of solutions for the given initial-boundary value problem remains open till now.

This work is devoted to the study of the initial-boundary value problem (1.5), (1.6), (1.8), and (1.9). Namely, the problem of existence of weak solutions for this problem is investigated.

2. Principal notations and functional spaces

Let us introduce the following functional spaces used hereinafter:

D(Ω)ⁿ is the space of smooth functions onΩwith values inRⁿ and with compact support contained inΩ;

ᐂ= {v:v∈D(Ω)ⁿ, divv=0}is the set of solenoidal smooth functions;

V is the completion ofᐂin the norm ofW₂¹(Ω)ⁿ; we consider the spaceV with the following norm:

vV =

Ω∇v:∇v dx 1/2

. (2.1)

Here∇u:∇ϕ,u=(u1,. . .,un),ϕ=(ϕ1,. . .,ϕn) denotes the component-wise multiplica- tion of matrixes:

∇u:∇ϕ₌∂ui

∂x_j^·

∂ϕ_i

∂x_j. (2.2)

InVthis norm is equivalent to the norm induced from the spaceW2¹(Ω)ⁿ. Xis the completion ofᐂin the norm ofW₂³(Ω)ⁿ; the norm inXis defined by

vX=

Ω∇(Δv) :∇(Δv)dx^1/2. (2.3)

InXthis norm is equivalent to the norm induced from the spaceW2³(Ω)ⁿ.

ByX^∗we denote the space conjugate to the spaceX. Denote by h,vthe value of a functionalh∈X^∗on a functionv∈X.

Denote byZthe image ofXunder the action of the operator (I−κΔ), that is,Z= (I−κΔ)X. Let us note thatZis a subspace ofVbut does not coincide with it.

Now we can introduce two principal functional spaces used below

E1= v:v∈L_∞0,T;V,v∈L2(0,T;Z^∗) (2.4) with the norm:

vE1= vL∞(0,T;V)+vL2(0,T;Z^∗), (2.5) E2= v:v∈C[0,T],X,v∈L2(0,T;X) (2.6) with the norm:

vE2= vC([0,T],X)+vL2(0,T;X). (2.7)

3. Statement of the problem and the main result

In this section an approximating problem for the investigated initial-boundary value problem (1.5), (1.6), (1.8), and (1.9) is considered. For this problem and for the original problem the concepts of weak solution is introduced. Then the main result is formulated.

3.1. Approximating problem. For study of the initial-boundary value problem (1.5), (1.6), (1.8), and (1.9) we use a modified method of introduction of auxiliary viscosity.

The original method of introduction of auxiliary viscosity for the Navier-Stokes system (see, e.g., Lions [8]) assumes introduction of the additional term (−1)^mεΔ^mv into the system. Hereε >0,m∈N,m(n+ 2)/4. In this work we add the term−εΔ³(∂v/∂t) to (1.5). The obtained equations are called approximating equations. Increase of the order of the equation requires introduction of additional boundary conditions.

Consider the following initial-boundary value problem with small parameterε >0:

∂v

∂t⁻εΔ³∂v

∂t

−νΔv+vi∂v

∂x_i^−κ

∂Δv

∂t ^−κDiv

vk∂Ᏹ(v)

∂x_k

+gradp=f, x∈Ω,t∈[0,T], divv=0, x∈Ω,t∈[0,T],

v(x, 0)=a(x), x∈Ω, v|∂Ω×[0,T]=0,

∂v

∂n

∂Ω×[0,T]=0,

∂²v

∂n²

∂Ω×[0,T]=0.

(3.1) Hereεis some fixed number,ε >0;nis the outer normal to∂Ω.

Suppose that f ∈L2(0,T;V^∗),a∈X.

Definition 3.1. A functionv∈E2is called a weak solution for the initial-boundary value problem (3.1) if for anyϕ∈Xand almost allt∈(0,T) the functionv∈E2satisfies the equality

Ω

∂v

∂tϕ dx−

Ωvivj

∂ϕj

∂xidx+ν

Ω∇v:∇ϕ dx+ε

Ω∇ Δ∂v

∂t

:∇(Δϕ)dx +κ

Ω∇∂v

∂t

:∇ϕ dx−κ 2

Ωv_k∂vi

∂x_j

∂²ϕ_j

∂x_i∂x_kdx

−κ 2

Ωvk

∂vj

∂xi

∂²ϕj

∂xi∂xkdx= f,ϕ

(3.2)

and the initial condition

v(0)=a. (3.3)

3.2. Definition of weak solution for initial-boundary value problem (1.5), (1.6), (1.8), and (1.9). Suppose that f ∈L2(0,T;V^∗),a_∗∈V.

Definition 3.2. A functionv∈E1is a weak solution for the initial-boundary value prob- lem (1.5), (1.6), (1.8), and (1.9) if for anyϕ∈Xand almost allt∈(0,T) the functionv satisfies the equality

(I−κΔ)∂v

∂t,ϕ

−

Ωvivj

∂ϕ_j

∂x_i dx+ν

Ω∇v:∇ϕ dx

−κ 2

Ωv_k∂vi

∂xj

∂²ϕ_j

∂xi∂xkdx−κ 2

Ωv_k∂v_j

∂xi

∂²ϕ_j

∂xi∂xkdx= f,ϕ

(3.4)

and the initial condition

v(0)=a_∗. (3.5)

3.3. The main result. Our main result is the following theorem.

Theorem 3.3. For any f ∈L2(0,T;V^∗),a∈V the initial-boundary value problem (1.5), (1.6), (1.8), and (1.9) has at least one weak solutionv_∗∈E1.

In order to prove this theorem we use the modified method of introduction of auxil- iary viscosity. At the first stage we introduce into (1.5) the term−εΔ³(∂v/∂t). The resolv- ability of the obtained approximating problem is established by topological methods on the basis of a priori estimates of weak solutions. Then it is shown that in a sequence of weak solutions of the approximating problem it is possible to select a subsequence con- verging to a weak solution of the initial-boundary value problem (1.5), (1.6), (1.8), and (1.9) as the parameter of approximation tends to zero.

4. Operator treatment for the approximating problem (3.1)

Let us introduce operators in functional spaces using the following equalities:

A:V−→V, Av,ϕ =

Ω∇v:∇ϕ dx, v,ϕ∈V, N:X−→X^∗, Nv,ϕ =

Ω∇(Δv) :∇(Δϕ)dx, v,ϕ∈X, B1:L4(Ω)ⁿ−→V^∗, B1(v),ϕ=

Ωvivj

∂ϕj

∂xidx, v∈L4(Ω)ⁿ,ϕ∈V, B2:V−→X^∗, B2(v),ϕ=

Ωvk∂v_i

∂xj

∂²ϕj

∂xi∂xkdx, v∈V,ϕ∈X, B3:V−→X^∗, B3(v),ϕ=

Ωvk

∂vj

∂xi

∂²ϕj

∂xi∂xkdx, v∈V,ϕ∈X, J:X−→X^∗, Jv,ϕ =

Ωvϕ dx, v,ϕ∈X.

(4.1)

Since in (3.2) the functionϕ∈Xis arbitrary, this equation is equivalent to the following operator equation:

Jv−B1(v) +νAv+εNv+κAv−κ

2B2(v)−κ

2B3(v)=f . (4.2) Hence the problem of existence of a weak solution for the problem (3.1) is equivalent to the problem of existence of a solutionv∈E2for operator equation (4.2) satisfying the initial condition (3.3).

4.1. Properties of operatorsAandN. In this subsection the properties of operatorsA andNfrom the operator equation (4.2) are investigated.

Lemma 4.1. The operatorAhas the following properties.

(i) The operatorA:V→V^∗is continuous and the following inequality holds:

AuVuV^∗, u∈V. (4.3)

(ii) For anyu∈L2(0,T;V) one hasAu∈L2(0,T;V^∗) and the operatorA:L2(0,T;V)→ L2(0,T;V^∗) is continuous.

(iii) For anyu∈E2one hasAu∈L2(0,T;X^∗) and the operatorA:E2→L2(0,T;X^∗) is compact and the following estimate holds:

AuL2(0,T;X^∗)C0uC([0,T],V). (4.4) Proof. (i) It suffices to show boundedness of the linear operatorA. By definition we have

Au,ϕ=

Ω∇u:∇ϕ dxuVϕV. (4.5) The last inequality implies (4.3). Thus the operatorA:V→V^∗is continuous.

(ii) Letu∈L2(0,T;V). For almost allt∈(0,T) by virtue of (4.3) we have

Au(t)²_V∗u(t)²_V. (4.6)

ThereforeAu∈L2(0,T;V^∗).

Integrating the obtained estimate with respect totfrom 0 toT, we obtain

Au²_L2(0,T;V∗)u²_L2(0,T;V). (4.7)

Since the operatorAis linear and bounded, it is continuous as an operator fromL2(0,T;

V) toL2(0,T;V^∗).

(iii) In order to prove that the operatorA:E2→L2(0,T;X^∗) is compact we use [17, Theorem 2.1, page 184]. We remind here its statement.

Theorem 4.2. LetX0,F,X1be three Banach spaces satisfying the following condition:

X0⊂F⊂X1. (4.8)

Where the embeddings are continuous,X0,X1are reflexive, the embeddingX0→Fis com- pact. LetT >0 be fixed finite number andα0,α1 be two finite numbers such thatαi>1, i=0, 1.

Suppose thatY= {v:v∈L_α0(0,T;X0); v∈L_α1(0,T;X1)}is the space with the norm vY= vLα0(0,T;X0)+vLα1(0,T;X1). Then the embedding ofYintoL_α0(0,T;F) is compact.

In our case

X0=X, F=V, X1=L2(Ω)ⁿ, α0=α1=2, Y= v:v∈L2(0,T;X);v∈L2

0,T;L2(Ω)ⁿ. (4.9)

Since the embeddingX₌V_∩W^◦₂³(Ω)ⁿ⊂V is compact, all conditions of this theorem hold and consequently the embedding ofYintoL2(0,T;V) is compact.

Note thatE2⊂Y and this embedding is continuous. Indeed, it follows from the em- beddingsC([0,T],X)⊂L2(0,T;X) andL2(0,T;X)⊂L2(0,T;L2(Ω)ⁿ) which are continu- ous. From (ii) we have that the operatorA:L2(0,T;V)→L2(0,T;V^∗) is continuous.

Thus we have the following composition E2⊂Y⊂L2(0,T;V)−−−→^A L2

0,T;V^∗⊂L2

0,T;X^∗. (4.10)

Here the first embedding is continuous, the second embedding is compact, and the map Aand the last embedding are continuous. This yields that the mapA:E2→L2(0,T;X^∗) is completely continuous.

The required estimate follows from (4.7) and continuity of embeddingsL2(0,T;V^∗)⊂

L2(0,T;X^∗) andC([0,T],V)⊂L2(0,T;V).

Lemma 4.3. The operatorJ+εN+κAhas the following properties.

(i) The operator (J+εN+κA) :X→X^∗is continuous, invertible and the following in- equalities hold

εuX(J+εN+κA)u_X∗C1+ε+κC2

uX. (4.11)

HereC1,C2 are some constants, which depend fromnand domainΩand do not depend onu.

(ii) For anyu∈L2(0,T;X) one has (J+εN+κA)u∈L2(0,T;X^∗) and the operator (J+εN+κA) :L2(0,T;X)→L2(0,T;X^∗) is continuous, invertible and the following estimate holds

εuL2(0,T;X)(J+εN+κA)u_L2(0,T;X∗). (4.12)

Proof. (i) By linearity of the operatorJ+εN+κAfor the proof of its continuity it suffices to show its boundedness. By definition we have

(J+εN+κA)u,ϕ=

Ωuϕ dx+ε

Ω∇(Δu) :∇(Δϕ)dx+κ

Ω∇u:∇ϕ dx uL2(Ω)ⁿϕL2(Ω)ⁿ+εuXϕX+κuVϕV

C1uXϕX+εuXϕX+κC2uXϕX

=

C1+ε+κC2

uXϕX.

(4.13) We have used two following inequalities:

u²_L2(Ω)nC1u²_X, u²_VC2u²_X. (4.14) Here the constantsC1,C2depend onnand domainΩand do not depend onu.

This implies the right part of estimation (4.11). Thus the operator (J+εN+κA) :X→ X^∗is bounded and hence it is continuous.

In order to prove that the operator (I+εN+κA) :X→X^∗is invertible we use [17, Theorem 2.2, page 17]. We remind here its statement.

Theorem 4.4. LetW be a separable real Hilbert space (norm · W) and leta(u,v) be a linear continuous form onW×W, which is coercive, that is, there existsα >0, such that

a(u,u)αu²W ∀u∈W. (4.15) Then for eachlfromW^∗, the dual space ofW, there exists one and only oneu∈W such that

a(u,v)= l,v ∀v∈W. (4.16)

To apply this theorem it suffices to show that the following continuous bilinear form is coercive.

a(u,v)=

(J+εN+κA)u,v=

Ωuv dx+ε

Ω∇(Δu) :∇(Δv)dx+κ

Ω∇u:∇v dx.

(4.17) In fact, for anyu∈Xwe have

a(u,u)=

(J+εN+κA)u,u

=

Ωu²dx+ε

Ω∇(Δu) :∇(Δu)dx+κ

Ω∇u:∇u dx

= u²_L2(Ω)n+εu²_X+κu²_V≥εu²_X, ε >0.

(4.18)

Thus we have proved that (J+εN+κA) :X→X^∗is an isomorphism and the left part of estimate (4.11) is valid.

Furthermore by virtue of [4, Remark 2.3] the inverse operator (J+εN+κA)⁻¹:X^∗→ Xis Lipschitz-continuous (in the nomenclature of [4]). That is for any f,g∈X^∗:

(J+εN+κA)⁻¹f −(J+εN+κA)⁻¹g_X∗C3f −gX, C3=1

ε. (4.19) (ii) Letu∈L2(0,T;X). Quadrating right part of estimate (4.11) at almost all tand integrating with respect totfrom 0 toT, we get

_T

0

(J+εN+κA)u(t)²_X∗dtC1+ε+κC2

2

u²_L2(0,T;X). (4.20) The right-hand side of this inequality is finite and therefore the left-hand side is also finite. From here it follows that (J+εN+κA)u∈L2(0,T;X^∗) and

(J+εN+κA)u_L2(0,T;X∗)C1+ε+κC2

uL2(0,T;X). (4.21)

The operator (J+εN+κA) :L2(0,T;X)→L2(0,T;X^∗) is linear and bounded and therefore it is continuous.

Now, let us show that it is invertible. First we will prove that the range of the operator (J+εN+κA) coincides withL2(0,T;X^∗). For this purpose it is necessary to show that for anyw∈L2(0,T;X^∗) the equation (J+εN+κA)u=whas a solutionu∈L2(0,T;X).

By (i) the operator (J+εN+κA) :X→X^∗is invertible. This implies that at almost all t∈(0,T) the equation (J+εN+κA)u=whas a solutionu(t)=(J+εN+κA)⁻¹w(t). It is necessary to show that the functionu∈L2(0,T;X). Using the left part of the estimate (4.11) we get

εu(t)_X(J+εN+κA)u(t)_X∗=w(t)_X∗. (4.22) Sincew∈L2(0,T;X^∗) we have from this inequality thatu∈L2(0,T;X). Quadrating this inequality and integrating it along (0,T), we obtain the required inequality (4.12).

Therefore ker(J+εN+κA)= {0}. Thus the operator (J+εN+κA) :L2(0,T;X)→

L2(0,T;X^∗) is invertible.

4.2. Properties of operatorsB1,B2,B3. In this section we consider the properties of op- eratorsB1,B2andB3from (4.2).

Lemma 4.5. The mapB1has the following properties.

(i) The mapB1:L4(Ω)ⁿ→V^∗is continuous and the following inequality holds

B1(v)_V∗C4v²_L4(Ω)ⁿ. (4.23)

(ii) For anyv∈L4(0,T;L4(Ω)ⁿ) one hasB1(v)∈L2(0,T;V^∗) and the mapB1:L4(0,T;

L4(Ω)ⁿ)→L2(0,T;V^∗) is continuous.

(iii) For anyv∈E2one hasB1(v)∈L2(0,T;X^∗), the mapB1:E2→L2(0,T;X^∗) is com- pact and the following estimate holds:

B1(v)_L2(0,T;X∗)C5v²_C([0,T],V). (4.24) Proof. (i) Letv∈L4(Ω)ⁿ,ϕ∈V. Then we obtain

B1(v),ϕ=

Ωvivj

∂ϕj

∂x_idx ⁿ

i,j=1

vi

L4(Ω)vj

L4(Ω)

∂ϕj

∂x_i

L2(Ω)

ⁿ

i,j=1

vL4(Ω)ⁿvL4(Ω)ⁿϕVn²v²_L4(Ω)nϕV

C4v²_L4(Ω)nϕV.

(4.25)

Whence the required estimation (4.23) follows.

Let us prove that the mapB1:L4(Ω)ⁿ→V^∗is continuous. For anyv^m,v⁰∈L4(Ω)ⁿwe obtain

B1

v^m,ϕ− B1

v⁰,ϕ=

Ωv^m_i v^m_j ∂ϕj

∂xidx−

Ωv_i⁰v⁰_j∂ϕj

∂xidx

n i,j=1

v_i^mv^m_j −v⁰_iv⁰_jL2(Ω)∂ϕj

∂xi

L2(Ω)

ϕV^∗

n i,j=1

v^m_i v^m_j −v_i⁰v⁰_jL2(Ω).

(4.26)

We obviously have n

i,j=1

v_i^mv^m_j −v⁰_iv⁰_jL2(Ω)= n i,j=1

v^m_i v^m_j −v_i^mv⁰_j+v^m_i v⁰_j−v_i⁰v⁰_jL2(Ω)

n i,j=1

v_i^mv^m_j −v⁰_jL2(Ω)+ n i,j=1

v⁰_jv^m_i −v⁰_iL2(Ω)

n i,j=1

v_i^m_L4(Ω)v^m_j −v⁰_jL4(Ω)+ n i,j=1

v⁰_jL4(Ω)v_i^m−v_i⁰_L4(Ω) C6v^m_L4(Ω)nv^m−v⁰_L4(Ω)n+C6v⁰_L4(Ω)nv^m−v⁰_L4(Ω)n

=C6

v^m_L4(Ω)n+v⁰_L4(Ω)n

v^m−v⁰_L4(Ω)n.

(4.27) Therefore

B1

v^m−B1

v⁰_V∗≤C6

v^m_L4(Ω)n+v⁰_L4(Ω)n

v^m−v⁰_L4(Ω)n. (4.28) Let a sequence{v^m} ⊂L4(Ω)ⁿconverge to some limit functionv⁰∈L4(Ω)ⁿ. Then the continuity of the mapB1:L4(Ω)ⁿ→V^∗follows from inequality (4.28).

(ii) Letv∈L4(0,T;L4(Ω)ⁿ). Quadrating estimate (4.23) and integrating with respect totfrom 0 toT, we get

_T

0

B1(v)(t)²_V∗dt≤C²_{4 T}

0

v(t)⁴_L4(Ω)ndt=C₄²v⁴_L4(0,T;L4(Ω)ⁿ). (4.29) Sincev∈L4(0,T;L4(Ω)ⁿ), one hasB1(v)(t)V^∗ ∈L2(0,T) and henceB1(v)∈L2(0,T;

V^∗).

Let us prove that the mapB1:L4(0,T;L4(Ω)ⁿ)→L2(0,T;V^∗) is continuous.

Let a sequence{v^m} ⊂L4(0,T;L4(Ω)ⁿ) converge to some limitv⁰∈L4(0,T;L4(Ω)ⁿ).

Quadrating inequality (4.28) and integrating with respect totfrom 0 toT, we get _T

0

B1

v^m(t)−B1

v⁰(t)²_V∗dt

C²_{6 T}

0

v^m(t)_L4(Ω)n+v⁰(t)_L4(Ω)n

2v^m(t)−v⁰(t)²_L4(Ω)ndt

C²_{6 T}

0

v^m(t)_L4(Ω)n+v⁰(t)_L4(Ω)n

4

dt

1/2_T

0

v^m(t)−v⁰(t)⁴_L4(Ω)ndt 1/2

=C²_{6 T}

0

4

i=0

Cⁱ₄v^m(t)ⁱ_L4(Ω)nv⁰(t)⁴_L⁻_4(Ω)ⁱ ndt

1/2_T

0

v^m(t)−v⁰(t)⁴_L4(Ω)ndt 1/2

C²_{6 4}

i=0

Cⁱ₄ T

0

v^m(t)⁴_L4(Ω)ndt i/4T

0

v⁰(t)⁴_L4(Ω)ndt

(4−i)/41/2

× _T

0

v^m(t)−v⁰(t)⁴_L4(Ω)ndt 1/2

.

(4.30) Thus we obtain

B1

v^m−B1

v⁰²_L2(0,T;V∗)

C₆² ₄

i=0

C₄ⁱv^mi/4_{L4(0,T;L4(Ω)}n)v⁰⁽⁴_L4⁻_(0,T;L^i)/4_4(Ω)n)

2

v^m−v⁰²_{L4(0,T;L4(Ω)}n).

(4.31)

HereCⁱ₄=4!/i!(4−i)!. Since the right-hand side of (4.31) tends to zero asm→+∞, the left-hand side tends to zero asm→+∞. This completes the proof of (ii).

(iii) Let us use [17, Theorem 2.1, page 184]. We quoted its statement inLemma 4.1.

In our case

X0=X, F=L4(Ω)ⁿ, X1=L2(Ω)ⁿ, α0=4, α1=2, Y= v:v∈L4(0,T;X);v∈L2

0,T;L2(Ω)ⁿ. (4.32) By Sobolev’s embedding theorem we have compact embeddingX⊂L4(Ω)ⁿ. All con- ditions of [17, Theorem 2.1] hold and we get that the embedding of Y into L4(0,T;

L4(Ω)ⁿ) is compact.