SOLVABILITY OF INITIAL BOUNDARY VALUE PROBLEMS FOR EQUATIONS DESCRIBING MOTIONS OF LINEAR VISCOELASTIC FLUIDS

PROBLEMS FOR EQUATIONS DESCRIBING MOTIONS OF LINEAR VISCOELASTIC FLUIDS

N. A. KARAZEEVA

Received 16 April 2003 and in revised form 28 July 2004

The nonlinear parabolic equations describing motion of incompressible media are inves- tigated. The rheological equations of most general type are considered. The deviator of the stress tensor is expressed as a nonlinear continuous positive definite operator applied to the rate of strain tensor. The global-in-time estimate of solution of initial boundary value problem is obtained. This estimate is valid for systems of equations of any non- Newtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possi- ble to prove the existence of global-in-time solutions of two-dimensional initial boundary value problems for generalized linear viscoelastic liquids, that is, for liquids with linear integral rheological equation, and for third-grade liquids.

1. Introduction

In the paper, we consider rheological equations of general type. We have got the global-in- time estimate of solutions which is true for equations of motion of any non-Newtonian liquid in two-dimensional case. The additional condition imposed on the deviator of the stress tensor permits to obtain the weak solution of periodic initial boundary value problem and of the Cauchy problem. This condition is fulfilled for any linear viscoelastic fluid. Equations of motion of these fluids are approximations of the complete equations of motion of viscoelastic fluids. These equations were investigated in the monographs of Tschoegl [14] and of Golden et al. [6] and they have been of great interest in the recent years. We may point out the papers of Matei et al. [9], Fabrizio et al. [4], Gentili [5].

The theoretical possibility of application of linear viscoelastic equations is proved in the monography of Bird et al. [2]. The experimental verification of these equations is presented in many monographs. Among them I want to mention the monographs of Creus [3], Goodwin et al. [7] and Schwarzl [13].

Creus shows that the concrete structure is described by linear viscoelastic model.

Goodwin and Hughes [7] presented many diagrams of correspondence of experimen- tal data and of data calculated with the help of linear viscoelastic models for different

Copyright©2005 Hindawi Publishing Corporation Journal of Applied Mathematics 2005:1 (2005) 59–80 DOI:10.1155/JAM.2005.59

materials (e.g., for polystyrene latex, polybutadiene, etc.). Schwarzl [13] presents the di- agrams containing intervals of applications of linear viscoelastic models and the critical values for which the linearity is lost.

The solvability of initial boundary value problems for some special linear viscoelastic fluids was investigated by Oskolkov [10,11,12]. He proved the existence of global classi- cal solutions of initial boundary value problems for linear viscoelastic Oldroyd equations in two-dimensional case and for linear viscoelastic Kelvin-Voight equations in three- dimensional case. He did not consider linear viscoelastic equations of general type.

The motion of incompressible media is described by the system of equations

∂

∂tv+v_k∂v

∂x_k+ gradp=divσ+f, divv=0, (1.1) whereσ=(σik) is the deviator of the stress tensor, trσ=0 (by trσ we denote the trace of the tensorσ). The type of the fluid is specified by the so-called rheological equa- tion responding to state equation between σ and the tensor of velocity deformations D=(1/2)(vixk+vkxi). This relation is called the rheological equation or the state equa- tion. The simplest example of the state equation is the equationσ=0 which describes an ideal incompressible fluid. If we have the state equation of the type

σ=2µD, (1.2)

we have a Newtonian fluid, and the motion is described by the Navier-Stokes equations.

In the present paper, we research different types of non-Newtonian fluids with mem- ory. In the first part of the paper, we consider the rheological equation of the type

σ=2µD+KD, (1.3)

where the operatorK is a continuous, positive-definite operator with some additional properties specified below. We consider the two-dimensional initial boundary value prob- lem with nonslip boundary condition and prove the existence of a global strong solution.

The difficulty consists of the fact that the term connected with the memory is nonlocal in time. Therefore, we can not use the estimates on the layersΩ. All the obtained estimates are new.

It is shown that the equations of motion of general linear viscoelastic fluids with rhe- ological equation of the type

σ=2µD+ _t

0K(t−τ)D(x,τ)dτ (1.4)

belong to this class. As a corollary, we apply the obtained results to the equations of mo- tion of linear viscoelastic Oldroyd fluids. These are the equations which are obtained from the equations of motion of the Oldroyd fluids after substitution of the material deriva- tives by the ordinary partial derivatives. This substitution may be done in the case of small displacement gradients, which often occurs in the practice (see [7,13]).

Moreover, the results of this section permit to confirm the solvability of initial bound- ary value problems for equations of motion of third-grade fluids. Flows of these fluids are

described by the rheological equations of the type

σ=2µD+νD³. (1.5)

Ifµandνare positive constants, then the operator in the right-hand side of (1.5) satisfies all the conditions imposed on the operatorK.

In the second part of the paper, we consider three-dimensional flows governed by the rheological equation

σ=2µ1D+ 2KD+ 2µ∂D

∂t (1.6)

with continuous, positive-definite operatorK. For this class of fluids, we investigate an initial boundary value problem with nonslip boundary condition. The existence of a strong solution is proved. Then we show that equations of motion of the linear viscoelas- tic Kelvin-Voight fluids belong to this class. So the existence of a strong global solution for three-dimensional initial boundary value problems for the linear viscoelastic Kelvin- Voight equations is proved.

In the third section, we investigate the most general rheological equation

σ=KD, (1.7)

where the operatorKis continuous and positive-definite. This equations correspond to arbitrary non-Newtonian (or Newtonian) fluid. We obtain the estimate of the solution of periodic initial boundary value problem in two-dimensional case. The requirement of positive definiteness makes a physical sense because the corresponding quadratic form is equal to

−(divσ,v)2,Ω=

Ωσ:∇v=

Ωσ:D, (1.8)

which means that dissipation of energy is positive (see [1, Chapters 2–3]).

The two-dimensional Cauchy problem is investigated for these equations. We prove global solvability of it in the class of weak solutions under some additional hypothesis.

Then we show that the equations of flows of general linear viscoelastic fluids, that is, flows generated by the linear rheological equations of general type

σ= _t

0K(t−τ)D(x,τ)dτ, (1.9)

belong to this class of liquids, and thus we obtain the solvability theorem for the Cauchy problem for fluids of these types. As a corollary of these existence theorems, we prove the global-in-time existence of a weak solution in two-dimensional case for a system of equations of linear viscoelastic Maxwell fluids.

2. Notation

In this section, we state preliminary mathematical notation. The symbolσ:σis used for the summation

σ:σ=tr(σσ)= ⁿ

i,k=1

σikσ_ki. (2.1)

We writevtfor a partial derivative∂v/∂t, and we denote∂v/∂xkbyvxk. The norm in the spaceW2^l(Ω) is denoted byu^l_2,Ω. Hilbert spaceW^0l2is the closure of the setC^0∞ in the normW2^l, whereC^0∞ is the set of infinitely differentiable functions with finite support.

We writeu2,∞for the norm in the spaceL2,∞: u2,∞=sup

t

u(x,t)_2,Ω. (2.2)

The normu2,1in the spaceL2,1is defined as follows:

u2,1= _t

0

u(x,t)_2,Ωdt. (2.3)

Byux2,Ωwe mean

ux²

2,Ω=

Ω

k

u²_xk(x)dx, (2.4)

and (u_x,v_x)_2,Ωis the corresponding scalar product ux,vx

2,Ω=

Ω

k

uxk(x)vxk(x)dx. (2.5)

Scalar product in the spaceL2(Ω) is denoted by (u,v)2,Ω, (u,v)2,Ω=

Ωu(x)v(x)dx. (2.6)

The notationL2,W¹2is used for the spaces of vector functions or tensors with components from the spacesL2orW2¹correspondingly. The space of Lipschitz continuous functions, that is, the space of functions with the finite norm

uC^0,1(Ω)= sup

x,y∈Ω,x=y

f(x)−f(y)

|x−y| (2.7)

is denoted by C^0,1. We use the notation J(Ω) for the set of infinitely differentiable solenoidal vector functions with finite support

J(Ω)=

u∈C^0∞(Ω)|divu=0. (2.8)

The closure of this set in theL2-norm is denoted byJ⁰(Ω), and the closure in theW2¹- norm is denoted byH(Ω).

The subspace of all solenoidal fields in the spaceW¹2(Ω) which are periodic with re- spect toxk is denoted byJ¹₂. The closure of this space in theL2(Ω)-norm is denoted by J⁰2(Ω).

In the paper, we use some well-known inequalities. For the sake of convenience, we list them here.

(1) The Young inequality,

ab≤1

λ^λa^λ+ 1 λ^−λ

b^λ, (2.9)

wherea,b, are arbitrary positive numbers,λandλare more than unit, and (1/λ) + (1/λ)=1.

(2) Inequalities for the normu4,Ω(Ladyzhenskaya [8, Chapter 1]).

For every functionu∈W⁰¹2(Ω), Ω⊂R², it holds that u⁴_4,Ω≤2u²_2,Ωux²

2,Ω, (2.10)

and for every functionu(x)∈W^{0 1}2(Ω),Ω⊂R³, it holds that u⁴4,Ω≤

4 3

3/2

u2,Ωu_x³_2,Ω. (2.11) (3) The Gronwall lemma.

Let a nonnegative absolutely continuous functiony(t) satisfy the inequality dy

dt ^≤c1(t)y(t) +c2(t) (2.12)

for a.e.t∈[0,T], wherec1,c2∈L1[0,T] are nonnegative. Then, y(t)≤exp

_t

0c1(τ)dτ

·

y(0) + _t

0c2(τ)dτ

. (2.13)

3. Generalized equations of motion of viscous fluids

3.1. Statement of the problem. We consider the following system of equations:

∂v

∂t +vk∂v

∂xk−µ∆v−div(KD) + gradp=f, divv=0. (3.1) The system is studied in a bounded domainΩ⊂R²,QT=Ω×[0,T),T∈(0,∞],v= (v1,v2) :Q_T→R²is a velocity field,p:Q_T→Ris a pressure, and f :Q→R²is the field of external forces. The boundary∂Ωis supposed to satisfy some smoothness condition. This condition should be enough to provide the embedding theorems. It is sufficient to assume that∂Ωis Lipschitz continuous. Let the operatorK:L2(Q_T)→L2(Q_T) be continuous and satisfy the following conditions:

(1)Kis bounded, that is,

KD2,QT≤ckD2,QT. (3.2) (2)Kis positive-definite in the following sense:

QT

(KD) :D≥0. (3.3)

(3) the operator (∂/∂t)◦Kis bounded in the spaceL2(QT), that is, ∂

∂t(KD)

2,QT

≤c1D2,QT. (3.4)

We consider the initial boundary value problem

v|t=0=v0(x), v|∂QT=0. (3.5) ByV(QT) we denote the space of vector functionsvequipped with the norm

[v]QT= sup

0_≤t≤T

v(x,t)_2,Ω+vx

2,QT. (3.6)

We introduce the notion of generalized solution of the problem (3.1)–(3.5); the function v∈V(QT) which satisfies the integral identity

QT

−vφ_t−v_kvφ_xk+µv_xφ_x+ (KD)∇φdx dt+

Ωvφ|_t=Tdx−

Ωv0φ|_t=0dx

=

QT

f φ dx dt

(3.7)

for anyφ∈W^01,12 ∩J(QT).

To verify the correctness of the definition, it is necessary to prove the finiteness of the integral_QT(KD)∇φ,

QT

(KD)∇φ≤ KD2,QTφx

2,QT≤ckvx

2,QTφx

2,QT. (3.8) 3.2. A priori estimates. In order to obtain some a priori estimates for solutions, we mul- tiply the system byvand integrate overΩ. After integrating by parts, we arrive at the identity

1 2

d

dtv²2,Ω+µv_x²_2,Ω+

Ω

(KD) :D=

Ωf v. (3.9)

Integration with respect toton [0,t] yields 1

2

v(t)²_2,Ω−v(0)²_2,Ω+µ _t

0

vx²

2,Ω+

QT

(KD) :D=

Qt

f v. (3.10)

By the nonnegativity of terms (3.3), the following estimate holds:

v(t)²_2,Ω≤v(0)²_2,Ω+ 2 _t

0

Ωf v, v(t)²_2,Ω≤v(0)²_2,Ω+ 2

_t

0f2,Ωv2,Ω, v(t)²_2,Ω≤v(0)²_2,Ω+v2,_∞

_t

0f2,Ω.

(3.11)

After maximization of the left-hand side, we obtain the inequality v²2,∞≤v(0)²_2,Ω+ 2v2,∞

_t

0f2,Ω. (3.12)

Solving this quadratic inequality, we get an estimate for the norm ofv:

v2,∞≤ _t

0f2,Ω+ ^t

0f2,Ω

2

+v(0)²_2,Ω:=c2(t). (3.13) Moreover, (3.10) leads to the estimate

1 2

v(t)²_2,Ω−v(0)²_2,Ω+µ _t

0

vx²

2,Ω≤ _t

0

Ωf v. (3.14) Consequently,

v(t)²2,Ω+ 2µ _t

0

vx²

2,Ω≤2 _t

0

(f,v)Ω+v(0)²2,Ω

≤v(0)²_2,Ω+ 2 _t

0f2,Ωv2,Ω, v(t)²2,Ω+ 2µ

_t

0

vx²

2,Ω≤v(0)²2,Ω+ 2v2,_∞

_t

0f2,Ω

≤v(0)²_2,Ω+ 2f2,1

f2,1+

f²2,1+v(0)²_2,Ω :=c3(t).

(3.15)

Now we differentiate the first equation of system (3.1) with respect tot, multiply byv_t, and integrate overΩ. Integration by parts yields

1 2

d dtvt²

2,Ω+µvxt²

2,Ω−

Ω

∂

∂t(KD)∇vt+

Ωvktvxkvt= ft,vt

. (3.16)

By the Cauchy inequality, we have

Ωv_ktv_xkv_t≤v_x2,Ω·v_t²_4,Ω. (3.17)

The terms in the right-hand side may be estimated with the help of (2.10) in two- dimensional case.

Ωv_ktv_xkv_t≤√

2v_x2,Ω·v_t2,Ω·v_xt2,Ω. (3.18) By virtue of the Young inequality, it holds that

Ωvktvxkvt

≤√

2κ2vxt²

2,Ω+cκ2vx²

2,Ωvt²

2,Ω

, (3.19)

wherecκ2=(1/4)κ⁻2¹,κ2is a positive number which will be chosen later.

Due to the Young inequality, the other terms of (3.16) are estimated as follows:

ft,vt

2,Ω≤ft

2,Ωvt

2,Ω≤1 2ft

2,Ω+1 2ft

2,Ωvt²

2,Ω,

Ω

∂

∂t(KD)∇vt

≤ ∂

∂tKD

2,Ω

vxt

2,Ω≤κ1vxt²

2,Ω+cκ1

∂

∂t(KD)

2 2,Ω,

(3.20)

wherec_κ1=(1/4)κ⁻1¹.

We set^√2κ2=µ/4 andκ1=µ/4. Then we get 1

2 d dtvt²

2,Ω+µvxt²

2,Ω≤ 1

2ft

2,Ω+1 2ft

2,Ωvt²

2,Ω

+ µ

4v_xt²_2,Ω+2

µv_x²_2,Ωv_t²_2,Ω

+ µ

4vxt²

2,Ω+1 µ

∂

∂tKD

2 2,Ω

.

(3.21)

Consequently, 1

2 d

dtv_t²_2,Ω≤1

2f_t2,Ω+1

2f_t2,Ωv_t²_2,Ω+2

µv_x²_2,Ωv_t²_2,Ω+1 µ

∂

∂t(KD)

2

2,Ω. (3.22) By the Gronwall lemma (2.13),

vt²

2,Ω≤exp _t

0

ft

2,Ω+4 µvx²

2,Ω

dτ

·

v_t(0)²_2,Ω+ _t

0

f_t2,Ω+1 µ

∂

∂tKD

2 2,Ω

dτ

.

(3.23)

Since₀^t(∂/∂t)KD²_2,Ω≤c²1vx²2,QT (3.4) and 2µvx²2,QT≤c3(3.15), the following esti- mate is true for normv_t²_2,Ω:

v_t²_2,Ω≤exp _t

0

f_t2,Ω

+ 2 µ²c3(t)

·v_t(0)²_2,Ω+ _t

0

f_t2,Ω

+ 1 2µ²c²1c3

:=c4(t).

(3.24)

Moreover, by inequality (3.21), one can estimate₀^tv_xt²2,Ω. Indeed, by integrating (3.21) with respect toton [0,t] we obtain

vt(t)²_2,Ω+µ _t

0

vxt²

2,Ω≤vt(0)²_2,Ω +

_t

0

ft

2,Ω+ _t

0

ft

2,Ω

vt²

2,_∞

+4 µvt²

2,∞

_t

0

vx²

2,Ω+2 µ

_t

0

∂

∂tKD

2

2,Ωdτ,vt(t)²_2,Ω +µ

_t

0

vxt²

2,Ω≤vt(0)²_2,Ω+c4(t) _t

0

ft

2,Ω

+ _t

0

f_t2,Ω+ 2

µ²c3(t)c4(t) + 1

µ²c²1c3=c5(t).

(3.25) 3.3. Existence of solutions. The estimates (3.15), (3.24), and (3.25) obtained inSection 3.2can be used for the proof of the existence theorems for problem (3.1)–(3.5).

Theorem3.1. LetΩbe an arbitrary bounded domain inR²,∂Ω∈C^0,1, f,f_t∈L2,1

Q_T, v0∈W²₂(Ω)∩J⁰(Ω), 0< T≤ ∞. (3.26)

Let the operatorKsatisfy conditions (3.2)–(3.4). Then problem (3.1)–(3.5) has a generalized solution

v∈L_∞0,T;J⁰(Ω), vt∈L_∞0,T;J⁰(Ω). (3.27) Moreover,vx∈L2(QT)andvxt∈L2(QT), and the following estimates hold:

2µ _t

0

vx²

2,Ω+vx²

2,_∞≤c3

f2,1;v(0)2,Ω

,

µ _t

0

v_xt²_2,Ω+v_t²_2,Ω≤c5

f_t2,1;v(0)_2,Ω,

(3.28)

furthermore,c3(t),c5(t)tend to some constantsc3,c5<∞fort→ ∞.

The corollary of this theorem is the existence of smooth global-in-time solution for equations of motion of general linear viscoelastic fluids with rheological equation of the type (1.4) in the case when the inverse Fourier transform of the functionKis positive. The other corollary consists of existence of smooth global-in-time solution of initial boundary value problem for third-grade fluids governed by the rheological equation of the type (1.5).

Proof. To prove this theorem we can use the Galerkin method. Let{ϕ^k},k=1, 2,..., be a complete system of functions inW²₂(Ω)∩H(Ω), which is orthonormal inL2(Ω).

Then for everyv0∈W²2(Ω)∩H(Ω), there is a sequence of functions v⁰(n)(x)=

n k=1

c⁰_knϕ^k(x), (3.29)

such that

v⁰(n)(x)−v0

W2²(Ω)−→0. (3.30)

An approximate solution of problem (3.1)–(3.5) can be represented in the form vⁿ(x,t)=

n k=1

ckn(t)ϕ^k(x), n=1, 2,..., (3.31) where the functionscknsatisfy the following system of integral identities:

v_tⁿ,ϕ^k+µv_xⁿ,ϕ^k_x−

vⁿ_ivⁿ,ϕ^k_xi+K∇vⁿ,∇ϕ^k=

f,ϕ^k, (3.32) and the initial conditions:

c_kn|_t=0=c_kn⁰ , k=1,...,n. (3.33) These identities form the system of ordinary differential equations with respect to the functionscnk,

dckn

dt +µ n i=1

ϕ_kic_in(t) + n i,j=1

ϕ_{ki j}c_in(t)c_jn(t) + n i=1

K∇ϕ_i(x)c_in(t),∇ϕ^k=f_k(t), (3.34)

whereϕ_ki,ϕ_{ki j} are the constants ϕki=

ϕⁱ_x,ϕ^k_x2,Ω, ϕki j=ⁿ

l=1

Ωϕ_l^jϕⁱϕ^k_xl, fk=

f,ϕ^k_2,Ω. (3.35) System (3.34) is a system of ordinary differential equations in the normal form and may be solved, for instance, by the method of Picard iterations.

To prove the theorem, we show that the approximationsvⁿsatisfy estimates (3.15), (3.25). Indeed v_nsatisfy the integral identities (3.9) and (3.16). Identity (3.16) can be obtained in the following way: identities (3.32) are differentiated with respect tot, then multiplied by (d/dt)c_kn, and summarized with respect tok∈ {1,...,n}.

Estimates (3.15), (3.25) and the theorem of weak compactness of bounded sets in Hilbert spaces permit to choose a convergent subsequence{v^nk}from the sequencevⁿ. The functionsv^nk,vx^nk,vⁿ_t^k,v_xt^nkconverge tov,vx,vt,vxt, respectively, in the normL2(QT).

To prove the solvability of the problem, it is necessary to verify that the functionv satisfies the integral identity (3.7). This follows from the possibility of the limit passage

in identities (3.32).

3.4. Equations of motion of the Oldroyd fluids. We now prove that the linear viscoelas- tic Oldroyd flow of orderL=1, 2,..., in a bounded two-dimensional domainΩis a flow of considered type. This flow is described by the system of equations

∂v

∂t+v_k ∂v

∂x_k+ gradp=divσ+f, divv=0, (3.36) where σ is the deviator of the stress tensor. For fluids of Oldroyd type, this tensor is connected with the tensor of velocity deformations by the relation

1 +

L l=1

λl∂^l

∂t^l

σ=2ν

1 + M m=1

κmν⁻¹∂^m

∂t^m

D. (3.37)

Hereλl>0 are the relaxation times, andκm>0 are the retardation times, andL=M.

We apply the Laplace transformᏸto both parts of (3.37) ᏸσ=σ(x,p)=

_∞

0 e^−ptσ(x,t)dt. (3.38) Consider polynomialQ(p)=1 +^L_l=1λ_lp^l. Then it holds

σ(x,p)=2Q⁻¹(p)

ν+ M l=1

κlp^l

D(x,p). (3.39)

To obtainσin explicit form, we apply the inverse Laplace transformᏸ⁻¹to (3.39).

σ(x,t)=2 _t

0G(t−τ)D(x,τ)dτ, (3.40)

where

G(t)=ᏸ⁻¹

Q⁻¹(p)

ν+ M l=1

κlp^l

. (3.41)

By introduction of the polynomialPo(p), Po(p)=ν−µ+

L l=1

κl−µλl

p^l, (3.42)

whereµ=κ_lλ⁻_l¹, we arrive at the expressions

G(t)=ᏸ⁻¹µ+Q⁻¹(p)P_o(p), G(t)=µδ(t) +

L s=1

Po

−αs Q−αse^−αst, G(t)=µδ(t) +

L s=1

β⁽_s^o)e^−αst,

(3.43)

whereδ(t) is the Diracδ-function, (−α_s) are the roots of the polynomialQ(p), and β⁽_s^o)=Po

−αs

Q−αs

₋1

. (3.44)

Suppose that the roots−αsare single, that is,Q(−αs)=0,l=1,...,L, real-valued and negative. The coefficientsβ⁽s^o)are supposed to be positive.

After substituting (3.43) into (3.40), we get σ=µD+

L s=1

β_s^(o) _t

0e^{−αs(t−τ)}D dτ. (3.45) In this way, the equations of motion of the Oldroyd fluids take the form

∂v

∂t +vk ∂v

∂x_k⁻µ∆v− _t

0

L s=1

β⁽_s^o)e^{−αs(t−τ)}∆v(x,τ)dτ+ gradp= f, divv=0. (3.46) This is the system of type (3.1) with the operator

K_ov= _t

0K(t−τ)v(x,τ)dτ (3.47)

with the kernelK

K(t)=^L

s=1

β⁽_s^o)e^−αs(t). (3.48) We should prove that the operatorK_osatisfies conditions (3.2)–(3.4).

Indeed, from the properties of the integral operators, it follows that the operatorK_ois continuous in the spaceL2(Q_T), and its norm does not exceed

c_k=

sup

0≤t≤T

_t

0K(τ)dτ

sup

0≤τ≤T

_T−τ

0 K(t)dt

, c_k= _T

0 K(t)dt. (3.49) Then we prove that the quadratic form_QT(Kv)vis positive definite. This fact follows from the following lemma.

Lemma3.2. For everyu(τ)∈L2([0,t]), it holds that I=

_t

0

_σ

0 K(σ−τ)u(σ)u(τ)dσ dτ≥0. (3.50) Proof. Suppose that the functionK is continued on the negative semiaxis like an even function. ThenIis equal to the half of integral on [0,t]×[0,t]. If we extenduby zero to Rand denote the obtained function by ˜u, we may rewriteIin the form

1 2

_∞

−∞

_∞

−∞K(σ−τ)˜u(σ)˜u(τ)dσ dτ. (3.51)

To prove that the integral (3.51) is positive for everyu∈L2(R), we calculate at first the inverse Fourier transform of the functionK:

Ᏺ⁻¹(K)=L(p)=(2π)^−1/2 L s=1

2β_sα_s

p²+α²_s ^≥0. (3.52) Then we transformI:

I=1 2

_∞

−∞

_∞

−∞K(σ−τ)˜u(σ)˜u(τ)dσ dτ, I=1

2 _∞

−∞

_∞

−∞Ᏺ(L)(σ−τ)˜u(σ)˜u(τ)dσ dτ, I=(2π)^−1/2

R³

e^{−i(σ−τ)p}L(p)dpu(σ)˜˜ u(τ)dσ dτ, I=

R

Re^{−iσ p}u(σ)dσ˜

2

L(p)dp≥0.

(3.53)

The lemma is proved.

We see that condition (3.3) is fulfilled. To verify condition (3.4), transform the corre- sponding relation:

∂

∂t(KD)

2,QT

= ∂

∂t _t

0K(t−τ)D(x,τ)dτ

2,QT

≤K(0)D(x,t)_2,QT+K_oD_2,QT, (3.54) whereK_ois an integral operator of Volterra type with the kernel (d/dt)K:

K_ou(t)= _t

0

d

dtK(t−τ)u(τ)dτ. (3.55)

The operatorK_ois continuous in the spaceL2(Q_t), and _t

0

K_ovx²

2,Ω≤c_{k t}

0

vx²

2,Ω. (3.56)

Thus(∂/∂t)(KD)2,QTcan be estimated in the following way:

∂

∂t(KD)

2,QT

≤K(0)vx

2,QT+c_kvx

2,QT≤ 1 2µ

K(0) +c_kc3:=c1. (3.57)

So we proved that the equation, which describes motion of the Oldroyd fluids, satisfies the condition (3.2)–(3.4). Consequently, the corresponding initial boundary value problem admits a solution.