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

The nonlinear parabolic equations describing motion of incompressible media are investigated. 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 nonNewtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possible 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.

materials (e.g., for polystyrene latex, polybutadiene, etc.).Schwarzl [13] presents the diagrams 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 classical solutions of initial boundary value problems for linear viscoelastic Oldroyd equations in two-dimensional case and for linear viscoelastic Kelvin-Voight equations in threedimensional case.He did not consider linear viscoelastic equations of general type.
The motion of incompressible media is described by the system of equations where σ = (σ ik ) is the deviator of the stress tensor, trσ = 0 (by trσ we denote the trace of the tensor σ).The type of the fluid is specified by the so-called rheological equation responding to state equation between σ and the tensor of velocity deformations D = (1/2)(v ixk + v kxi ).This relation is called the rheological equation or the state equation.The simplest example of the state equation is the equation σ = 0 which describes an ideal incompressible fluid.If we have the state equation of the type 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 memory.In the first part of the paper, we consider the rheological equation of the type σ = 2µD + KD, (1.3) where the operator K is a continuous, positive-definite operator with some additional properties specified below.We consider the two-dimensional initial boundary value problem 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 layers Ω.All the obtained estimates are new.
It is shown that the equations of motion of general linear viscoelastic fluids with rheological equation of the type σ = 2µD + t 0 K(t − τ)D(x,τ)dτ (1.4) belong to this class.As a corollary, we apply the obtained results to the equations of motion 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 derivatives 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 boundary value problems for equations of motion of third-grade fluids.Flows of these fluids are N. A. Karazeeva 61 described by the rheological equations of the type σ = 2µD + νD 3 . (1.5) If µ and ν are positive constants, then the operator in the right-hand side of (1.5) satisfies all the conditions imposed on the operator K.
In the second part of the paper, we consider three-dimensional flows governed by the rheological equation with continuous, positive-definite operator K.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 viscoelastic 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 where the operator K is 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 which means that dissipation of energy is positive (see [1,).
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 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.

Notation
In this section, we state preliminary mathematical notation.The symbol σ : σ is used for the summation σ ik σ ki . (2.1) We write v t for a partial derivative ∂v/∂t, and we denote ∂v/∂x k by v xk .The norm in the space where 0 C ∞ is the set of infinitely differentiable functions with finite support.We write u 2,∞ for the norm in the space L 2,∞ : (2. 2) The norm u 2,1 in the space L 2,1 is defined as follows: By u x 2,Ω we mean and (u x ,v x ) 2,Ω is the corresponding scalar product The notation L 2 , W 1 2 is used for the spaces of vector functions or tensors with components from the spaces L 2 or W 1  2 correspondingly.The space of Lipschitz continuous functions, that is, the space of functions with the finite norm is denoted by C 0,1 .We use the notation J(Ω) for the set of infinitely differentiable solenoidal vector functions with finite support The closure of this set in the L 2 -norm is denoted by J 0 (Ω), and the closure in the W 1 2norm is denoted by H(Ω).
The subspace of all solenoidal fields in the space W 1 2 (Ω) which are periodic with respect to x k is denoted by J 1  2 .The closure of this space in the L 2 (Ω)-norm is denoted by J 0 2 (Ω).
In the paper, we use some well-known inequalities.For the sake of convenience, we list them here.

For every function u
and for every function u(x)

Generalized equations of motion of viscous fluids
3.1.Statement of the problem.We consider the following system of equations: The system is studied in a bounded domain is a pressure, and f : Q → R 2 is the field of external forces.The boundary ∂Ω is supposed to satisfy some smoothness condition.This condition should be enough to provide the embedding theorems.It is sufficient to assume that ∂Ω is Lipschitz continuous.Let the operator K : L 2 (Q T ) → L 2 (Q T ) be continuous and satisfy the following conditions: (1) K is bounded, that is, (2) K is positive-definite in the following sense: We consider the initial boundary value problem By V (Q T ) we denote the space of vector functions v equipped with the norm We introduce the notion of generalized solution of the problem (3.1)-(3.5); the function To verify the correctness of the definition, it is necessary to prove the finiteness of the integral QT (KD)∇φ, (3.8)

A priori estimates.
In order to obtain some a priori estimates for solutions, we multiply the system by v and integrate over Ω.After integrating by parts, we arrive at the identity 1 2 Integration with respect to t on [0, t] yields By the nonnegativity of terms (3.3), the following estimate holds: After maximization of the left-hand side, we obtain the inequality Solving this quadratic inequality, we get an estimate for the norm of v: Moreover, (3.10) leads to the estimate Consequently, (3.15) Now we differentiate the first equation of system (3.1) with respect to t, multiply by v t , and integrate over Ω. Integration by parts yields 1 2 By the Cauchy inequality, we have The terms in the right-hand side may be estimated with the help of (2.10) in twodimensional case.
By virtue of the Young inequality, it holds that where c κ2 = (1/4)κ −1 2 , κ 2 is a positive number which will be chosen later.Due to the Young inequality, the other terms of (3.16) are estimated as follows: where Consequently, 1 2 By the Gronwall lemma (2.13),Let Ω be an arbitrary bounded domain in

Let the operator K satisfy conditions (3.2)-(3.4). Then problem (3.1)-(3.5) has a generalized solution
v ∈ L ∞ 0,T;J 0 (Ω) , v t ∈ L ∞ 0,T;J 0 (Ω) . (3.27) , and the following estimates hold: 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 function K is 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).
Then for every v 0 ∈ W 2 2 (Ω) ∩ H(Ω), there is a sequence of functions An approximate solution of problem (3.1)-(3.5)can be represented in the form where the functions c kn satisfy the following system of integral identities: and the initial conditions: These identities form the system of ordinary differential equations with respect to the functions c nk , where ϕ ki , ϕ ki j are the constants 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 approximations v n satisfy estimates (3.15), (3.25).Indeed v n satisfy 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 to t, then multiplied by (d/dt)c kn , and summarized with respect to k ∈ {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 sequence v n .The functions v nk , v nk x , v nk t , v nk xt converge to v, v x , v t , v xt , respectively, in the norm L 2 (Q T ).To prove the solvability of the problem, it is necessary to verify that the function v satisfies the integral identity (3.7).This follows from the possibility of the limit passage in identities (3.32).N. A. Karazeeva 69

Equations of motion of the Oldroyd fluids.
We now prove that the linear viscoelastic Oldroyd flow of order L = 1,2,..., in a bounded two-dimensional domain Ω is a flow of considered type.This flow is described by the system of equations 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 Here λ l > 0 are the relaxation times, and κ m > 0 are the retardation times, and L = M.We apply the Laplace transform ᏸ to both parts of (3.37) To obtain σ in explicit form, we apply the inverse Laplace transform ᏸ −1 to (3.39). where By introduction of the polynomial P o (p), where µ = κ l λ −1 l , we arrive at the expressions where δ(t) is the Dirac δ-function, (−α s ) are the roots of the polynomial Q(p), and Suppose that the roots −α s are single, that is, Q (−α s ) = 0, l = 1,..., L, real-valued and negative.The coefficients β (o)  s are supposed to be positive.After substituting (3.43) into (3.40),we get (3.45) In this way, the equations of motion of the Oldroyd fluids take the form This is the system of type (3.1) with the operator

.48)
We should prove that the operator K o satisfies conditions (3.2)-(3.4).Indeed, from the properties of the integral operators, it follows that the operator K o is continuous in the space L 2 (Q T ), and its norm does not exceed Then we prove that the quadratic form QT (Kv)v is positive definite.This fact follows from the following lemma.Lemma 3.2.For every u(τ) ∈ L 2 ([0,t]), it holds that (3.50) Proof.Suppose that the function K is continued on the negative semiaxis like an even function.Then I is equal to the half of integral on [0,t] × [0,t].If we extend u by zero to R and denote the obtained function by ũ, we may rewrite I in the form To prove that the integral (3.51) is positive for every u ∈ L 2 (R), we calculate at first the inverse Fourier transform of the function K: Then we transform I: The lemma is proved.
We see that condition (3.3) is fulfilled.To verify condition (3.4), transform the corresponding relation: where K o is an integral operator of Volterra type with the kernel (d/dt)K: The operator K o is continuous in the space L 2 (Q t ), and Thus (∂/∂t)(KD) 2,QT can be estimated in the following way: The system is considered in the cylinder Q T = Ω × [0,T), 0 < T ≤ ∞, Ω is a bounded three-dimensional domain.Suppose that the operator K satisfies conditions (3.2)- (3.4) and that the solution of the system satisfies the following initial boundary conditions: We introduce the notion of generalized solution.A function v ∈ V(Q T ) is called the generalized solution if it satisfies the integral identity for any t ∈ [0,T] and for any test function φ

A priori estimates.
To obtain some a priori estimates for solutions of initial boundary value problem (4.1)-(4.2),we multiply the first equation of system (4.1) by v and integrate with respect to x.
Integrating system (4.4) again with respect to t over the interval [0, t), we get Consequently, Maximization of the left-hand side implies Thus the norm v 2,Ω is estimated by From (4.6) and (4.8), we deduce that ,Ω := c 7 (t). (4.9) Due to (4.5), we can moreover estimate the norm ,Ω : Now we differentiate the first equation of system (4.1) with respect to t, multiply by v t , and integrate over Ω: Using similar estimates for Ω v kt v xk v t , Ω (∂/∂t)(KD)(∇v) t and ( f t ,v t ) Ω as in (3.21) and applying the Young inequality (2.9), we arrive at the following relation: Define the constants θ, κ i : We obtain the estimate 1 2 We abbreviate ,Ω , c µ1 = 2(4 7/4 3 1/4 /µ 3 1 ).Then, By the Gronwall lemma Thus the following estimate is true: Let Ω be an arbitrary bounded domain in
The proof of this theorem is similar to the proof of Theorem 3.1.

Equations of motion of the linear viscoelastic
Kelvin-Voight fluids.Now we show that the system of equations describing the motion of a linear viscoelastic Kelvin-Voight fluid belongs to the considered class.
The system of equations describing the Kelvin-Voight flow can be obtained from the system of equations of incompressible media (3.36) when the deviator of the stress tensor σ and the tensor of velocity deformations D satisfy the relation (3.37).We assume, that M = L + 1, L = 0,1,.... Transforming this relation in a similar way as in the previous section, we obtain an expression for the tensor σ where (−α s ) are the roots of polynomial Q(p) and the coefficients β (k)  s are supposed to be positive

.22)
P k are the polynomials We assume again that the coefficients −α s are single, that is, Q (−α s ) = 0, l = 1,2,...,L, real valued and negative.Thus the equations of motion of the Kelvin-Voight fluids take the form Obviously, this system is of type (4.1) with the operator where the kernel K has the form The operator K k has a similar form as the operator K o from the previous section.So it possesses the properties (3.2)- (3.4).Thus the existence theorem is valid also for the systems describing motions of the Kelvin-Voight fluids.

Equations of general type
5.1.Formulation of the problem.In this section, we consider rheological equations of more general type (1.7).Suppose that the operator K maps L 2 (Q T ) into L 2 (Q T ) and satisfies conditions (3.2), (3.3).These conditions are fulfilled for rheological equations of any non-Newtonian (and Newtonian) fluid.Substituting (1.7) into the equation of motion of incompressible fluid (1.1), we obtain the system At first, we consider an initial boundary value problem with periodic boundary conditions. Let Suppose that the functions v and p satisfy periodic boundary conditions on the sides of the square.We formulate them for the function v: Moreover, the function v satisfies the initial condition (5.6) We prove the existence of at least a one weak solution from the space L ∞ (0,T; J 1 2 (Ω)) which satisfies the integral identity for any solenoidal periodic φ such that φ x ,φ xt ∈ L 2 (Q T ).
We project system (5.1)onto the space of divergence-free vector fields and annihilate the term gradp, find a weak solution of obtained boundary value problem.When the velocity is found, the pressure may be found from system (5.1).N. A. Karazeeva 77

A priori estimates of the solutions. Now we assume that the operator K possesses the following property:
− rot div(KD),rotv ≥ 0. (5.8) The operation rot is understood as the scalar function in the space (5.9) If the operators K and ∂/∂x k , k = 1,2 commute then condition (5.8) is a corollary of (3.3).This condition (5.8) is fulfilled, for instance, in the case when the operator K is integral operator of the form (1.9).Then the condition of positiveness (3.3) may be improved in the following manner.We may suppose that the kernel K(t) of the operator K is a positive definite function.It means that the matrix {K (t i − t j )} N i, j=1 is positive in C N for every N and for any t 1 ,..., t N ∈ R N .
We estimate the space derivative of the function v.
Consider the stream function v = (−ψ x2 ,ψ x1 ).To obtain the following relation, we apply the operator rot to both parts of the first equation of system (5.1).Then it holds where ϕ(x,t) = (∂ f 2 /∂x 1 ) − (∂ f 1 /∂x 2 ).Moreover, we suppose that the function ψ satisfies periodic boundary conditions.Let all the first-order and the second-order space derivatives of the function ψ be periodic functions in Ω R .The initial condition takes the form (5.12) This relation is similar to relation (3.9).Thus we find an estimate similar to estimate (3.13) for the function ∆ψ, namely, (5.14)

Existence of the solution.
To prove the existence of a weak solution, we apply Galerkin's method with the system of eigenfunctions of the Stokes problem (5.15) as basis functions.By [8] the functions {ϕ l } ∞ l=0 form an orthonormal system in the space J 0 2 (Ω), and so it is the case in the spaces J 1 2 (Ω) and J 2 2 (Ω).The Galerkin approximations are defined as v n (x,t) = n l=0 c ln (t)ϕ l (x), (5.16)where the coefficients c ln (t) are determined from the system d dt v n ,ϕ l + Kv n x ,ϕ l x + v n k v n xk ,ϕ l = f ,ϕ l (5.17) with initial conditions c ln (0) = v 0 ,ϕ l , l = 0,...,n. (5.18) The approximations v n satisfy the same identities as the exact solutions.So estimates (3.13) and (5.14) are true for them.And a majorant is independent of n.This permits to pass to the limit and prove the next theorem.
The corollary of this theorem consists of the existence of a solution (5.21) for the systems of equations of general linear viscoelastic liquids with rheological equation of the type (1.9).All the conditions (3.2), (3.3), and (5.8) are fulfilled for these equations.where J 1 2 (R 2 ) is a space of solenoidal vector fields with finite norm u 2 + u x 2 .Then there exists a generalized solution of system (5.1) which satisfies the initial Cauchy condition v(x,0) = v 0 (x), x ∈ R 2 . (5.23) This solution belongs to L ∞ (0,T;J 1 2 (R 2 )) and satisfies estimates (3.13) and (5.14).This theorem is also true for general linear viscoelastic liquids.
Proof.The theorem can be proved in the following way.We choose the sequence of extending domains Ω Rn , R n → ∞, and construct the solution v n of periodic boundary value problem in every domain.The norms v n 2,∞ , v n x 2,∞ of all solutions v n are bounded by a majorant, which is independent of the domain.If we use compactly supported test functions, then we can pass to the limit in the identities (5.7) for functions v n .

4 . 4 . 1 .
.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.Equations of motion of generalized Kelvin-Voight fluids The abstract problem.Consider a PDE system of the following type:

. 3 ) 5 )
Multiply the system (5.1) by v and integrate over Ω.After integrating by parts, Proceeding as in Section 3.2 and using the nonnegativity of the term QT ((KD) : D), we get a priori estimate for the norm of

e
Maxwell fluids.The motion of the Maxwell fluid is determined by the relation (3.37), together with M = L − 1, L = 1,2,.... Proceeding exactly as in the previous sections, we can represent the deviator of the stress tensor σ in the form σ = −αs(t−τ) D(x,τ)dτ, P m is given by P m (p) = ν + the Maxwell fluid is described by the system (5.1) with the operator K m that the operator K m possesses properties (3.2), (3.3), and (5.8), and therefore the existence theorem is also valid for the Maxwell fluids.