The topological degree method for equations of the Navier-Stokes type.

We obtain results of existence of weak solutions in the Hopf sense 
of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps A−g, where A is invertible and g is 𝒜-condensing, is used.

Various problems for the Navier-Stokes equations describing the motion of the Newton fluid, and its generalizations for nonlinearly-viscous and viscoelastic fluids, have been developed in many papers. We mention here some of the papers which contain surveys on this subject, different approaches, constructions, and methods of investigation: [1], [8], [10]- [16].
Here we consider the problem of the existence of weak solutions, in the Hopf sense, of the initial-boundary value problem for equations of the Navier-Stokes type. These equations include the ones describing the movement of nonlinear-viscous and viscous-elastic fluids. We reduce the above problem to an evolution equation in the space of functionals, and then to the equivalent operator equation. The method of this paper consists of constructing operator equations which approximate the original ones, and then investigating their solvability by means of infinite-dimensional degree theory. As we know, the Galerkin-Faedo method or iteration methods have already been used instead of the degree theory for the classical Navier-Stokes equations and for some their generalizations (see, for example, [1], [10], [12]- [15]). The solution of the original problem may be obtained by passage to the limit in the set of solutions of approximating equations. The results of our paper on the existence of weak solutions generalize the well known ones (see, for example, [2], [10], [13], [15]). This paper consists of four sections. In the first section we introduce the main notations and notions, set up the problem of weak solutions of the initial-boundary value problem for generalized Navier-Stokes equations, and formulate our main results of existence and uniqueness of weak solutions.
In the second section the problem of weak solutions is reduced to the investigation of an equivalent operator equation. Then we construct the approximating equations and investigate the properties of the operators involved.
In the third section a priori estimates of solutions of approximating equations are established and a proposition on the existence of solutions of such equations is obtained.
In the last section the possibility of the limit procedure in the sequence of solutions of approximating equations is established. We present two different approaches to proven the convergence and, as a corollary, we get propositions for the existence of weak solutions of the initial-boundary value problem for some cases of the generalized Navier-Stokes equations. We consider the uniqueness of solutions for dimension n = 2 as well.
It should be noted that our interest in this problem arose when Professor P. E. Sobolevskii posed to one of the authors the question of the applicability of topological methods to the initial-boundary value problems in hydrodynamics. The authors are grateful to P. E. Sobolevskii , and Yu. A. Agranovich for discussions on some problems in hydrodynamics. 1. Introduction. Statement of the problem. Main results

Notations.
Let Ω be a bounded domain in R n with the boundary ∂Ω of class C 2 . For T > 0, we denote by Q T the cylinder (0, T ) × Ω. The bar over Ω, Q T means closure.
We consider different spaces of functions on Ω with values in R n : L 2 (Ω) denotes the space of square integrable functions on Ω. The scalar product of functions u and v from L 2 (Ω) is defined by (u, v) = Ω u(x) · v(x) dx; the norm of the function u in L 2 (Ω) will be denoted by u L 2 (Ω) ; W 1 2 (Ω) denotes the space of functions which belong together with their first order partial derivatives to L 2 (Ω). A norm of the function v from W 1 2 (Ω) is defined by the following equality  H the closure of V with respect to the norm of the space L 2 (Ω); V the closure of V with respect to the norm of the space W 1 2 (Ω). Norms and scalar products in the spaces H and V are defined by the same way as in spaces L 2 (Ω) and W 1 2 (Ω) respectively. Also in the space V the symbol of another scalar product will be used ((u, v) . And the norm generated by this scalar product in the space V is equivalent to the norm induced from the space W 1 2 (Ω). Let V * denote the dual space to V , and h, v means action of the functional h from V * to the element v from V .
Also v(t) X .
The spaces described above are Banach ones. In the case, when the interval [a, b] is clear from a context, the notation [a, b] is omitted: L α (X), L α (X), C(X). A dual space for a space L α ((a, b), X) is the space L α ((a, b), X * ), where 1 α + 1 α = 1. For vector-function v from L α ((0, T ), V ) we denote: by v i the coordinate functions; by ∂v ∂x i , ∂v ∂t the first order partial derivatives; by D 1 v = ∂v i ∂x j . Let us introduce the following notations. Let 1.2. The statement of the problem. The equations with perturbations of retarded type arise in mechanics for visco-elastic materials. By the definition (see [5]), "these materials are such that they have "memory" in sense that at the moment t the tension state depends on all the deformations to which the material have been undergone".
If we reject the proportional dependence D = µE between the stress tensor D and the strain velocity tensor E we obtain the non-Newton or real fluids. We would like to point out some mathematical models describing motion of such fluids.
In the paper [13] Litvinov V.G. investigated equations of fluid motion with relations The Oldroid relation leads to investigation of fluids with "memory". Solving the equation concerning D we obtain Substituting the expression for D into the Cauchy form of the motion equation and transforming the equation we obtain where the vector-function v is connected with the tensor (ε ij ) as follows: It is possible to obtain a model of nonlinear-viscous fluid choosing the nonlinear relation between D and E in the form Expressing D from this relation and substituting it into the motion equation we obtain The existence results for strong solutions in the cases n = 2, 3 can be found in [1].
The phenomenological theory of linear visco-elastic fluids with a finite number of discretely distributed times of relaxation and times of retardation uses the relations For L = M and under additional conditions for coefficients {λ l }, ν and {ae m } (see [8]) the equation of the fluid motion has the following form: In this paper we investigate the above mentioned classes of equations of visco-elastic and nonlinear-viscous fluid motions basing on approximations, using of topological methods for the proof of solvability of approximating problems, and the further limit procedure. It seems that this approach may be useful not only for the solvability but also for the settlement of other questions.
Consider the following initial-boundary value problem for the vectorfunction v :Q T → R n , v = (v 1 , . . . , v n ), and for the scalar function p :Q T → R: Div a(t, s, x, v(s), D 1 v(s))ds (1.4) where µ 0 > 0 is a constant and f : Q T → R n , v 0 : Ω → R n are given functions. Here, and below, E(v) is a matrix function with components Suppose that the scalar functions µ i (s), i = 1, 2, are continuously differentiable on [0, +∞) and satisfy the following conditions: Note that restrictions for µ i , i = 1, 2, mentioned above, may be found in [1], [14].
The essentially bounded function L(t, s) is defined on the set The matrix function a(t, s, x, v, w) is defined for all t ∈ [0, T ], 0 ≤ s ≤ t, x ∈ Ω, v ∈ R n , w ∈ R n 2 and satisfies either the conditions: A 1 ) the functions a ij (components of a) are measurable as functions of t, s, x for all v, w and continuous as functions of v, w for almost all t, s, x; s, x, v and w,w ∈ R n 2 ; or the conditions A 1 ) and is an essentially-boundary function.
Let us point out that the integral equality (1.5) is obtained from (1.1) by scalar multiplication in L 2 (Ω) of each term of (1.1) with h and some simple transformations.
Then for all f ∈ L 2 ((0, T ), H) and v 0 ∈ H there exists at least one weak solution v ∈ W of problem (1.1)-(1.4) satisfying the following inequalities In the case 2 ≤ n ≤ 4 we establish existence of a weak solution for equations of the form: where the elements of the matrix-function a are defined by

Operator and approximating equations
In this section we introduce operator equations which are equivalent to the problem of weak solutions of (1.5)-(1.6), and then we construct a family of approximating equations and investigate properties of the operators involved.
consider each term of (1.5) as the action of some functional on the function h. Thus where f is considered as an element of the space The last equality follows from [15, Lemma 1.1.]. Taking into account the above notations we can rewrite identity (1.5) in the form: 3) the following estimates are valid:

Proof. 1) Consider the function G(t, s, v(s)). By definition
. We rewrite the inequality in the form with some constant C. Note that functions L 1 H and v V are square integrable on T d and, hence, the function G(t, s, v(s)) belongs to L 2 (T d, V * ). Then By assumption A 2 ), the right-hand side of the inequality is square integrable in the variable t. Hence, the function Q(v) = t 0 G(t, s, v(s))ds belongs to the space L 2 ((0, T ), V * ) and where C depends only on L 1 L 2 (Qd) and L 2 L ∞ (Qd) .
2) To prove the continuity of the map it is sufficient to show the continuity of the map It is known [9] that under assumptions A 1 ) − A 3 ) the Nemytskii operator a is continuous. Hence, the map G is continuous too. Thus, Q is continuous as a composition of two continuous maps, namely, G and the integral operator. By similar arguments one can check that the definition is well-defined, prove that the maps A, B 1 , B 2 , C are continuous and obtain the estimates for them.
3) Consider the function K(v). By definition, . By Sobolev's embedding theorem [6], we have the continuous embedding V ⊂ L 4 (Ω) when n ≤ 4 and, hence, X . The continuity of K follows from the continuity of the embedding X ⊂ L 2 ((0, T ), L 4 (Ω)) and the continuity of the Nemytskii operators Hence, applying lemma 2.1, we rewrite the equality (2.1) in the form:
We replace the nonlinear term with ε > 0, and obtain the equation Repeating above arguments for equation (1.1 ε ) instead of (1.1), we obtain that the weak solutions of problem (1.1 ε ) − (1.4) are solutions of the approximating operator equation Since Moreover, the map D ε : X → X * is continuous since it is a Nemytskii operator.
Note that, for v ∈ X, all the terms in (2.3 ε ) (but the first one) belong to the space X * . Therefore, for a solution v of (2.3 ε ) we get v ∈ X * . Hence, any solution belongs to the space

takes values in H and is continuous.
Let us introduce the following notations.
It is easy to see that problem (2.3 ε ), (2.4) is equivalent to the operator equation It follows that the problem of weak solutions of (1.1 ε ) − (1.4) is equivalent to the problem of the solvability of the operator equation (2.6 ε ).
We shall now investigate the properties of the operators A, K ε and g appearing in (2.6 ε ).

Properties of the operator A. W first study the properties of the map A. Then we show that A is an invertible map and its inverse
This statement is well known. For example, it was used in [1]. We give its proof for completeness.
Using the mean value theorem for integrals we write this expression as follows: Observe that if µ i (I 2 (v+s 0 (u−v))) ≥ 0, then the second term is nonnegative.
Since µ i (s) ≥ 0, the first term is also nonnegative. Thus, the integrand is nonnegative.
In the case µ i (I 2 (v + s 0 (u − v))) < 0 we use the inequality and the relation Then This expression is nonnegative since We have actually proved that the integrand is nonnegative. Hence, Using the above relations and inequalities we can similarly get the estimate As we mentioned above, and the equivalent norms Similarly, we define equivalent norms · k,X , · k,X * ×H , · k,L 2 ((0,T ),H) for the spaces X, X * × H and L 2 ((0, T ), H) = L 2 (Q T ), respectively.
where C is independent of u, v and k.
Proof. Let u, v ∈ X. By the definitions of the operators C and B 2 , Using the mean value theorem for integrals we get : By the Cauchy inequality, . Hence, Let us consider the functionsū(t) = e −kt u(t) andv(t) = e −kt v(t). It is obvious that u k,X = ū X and v k,X = v X . By the Hölder inequality we obtain Consider the auxiliary problem (2.11) Letting v(t) = e ktv (t), ϕ = e ktφ (t) and multiplying by e −kt we obtain (2.12) is continuous, monotone and coercive for k large enough.
Proof. The continuity of the operator follows from the continuity of each term.
Let us show the monotonicity of the operator V k . For arbitrary functions u,v ∈ X, we have We evaluate terms at the right hand side of the equation. For k > µ 0 , Applying lemma 2.2 we have Choosing k so that k > µ 0 and C √ 2k < µ 0 2 , we obtain the following estimate: Hence, the operator V k is monotone.
Note that V k (0) = 0. To prove the coercivity property of the operator V k we substituteū = 0 into (2.14) to obtain Using the above propositions we will show that the map A is invertible and expanding with respect to certain special norms.
for k large enough, where the constant C is independent of u, v and the choice of k.
Proof. 1) Let us show the invertibility of the map A. For this purpose it is sufficient to show that the operator equation has a unique solution for each pair ϕ ∈ X * and a ∈ H.
The operator equation (2.16) is equivalent to the problem (2.11) and, hence, the problem (2.12). We can rewrite the problem (2.12) as follows: Since the operator V k is a continuous monotone coercive Volterra operator (according to the terminology of [6]), this problem has a unique solution by [6, Theorem 1.1]. The map a → v is continuous from H to C([0, T ], H). It follows that the map A is invertible.
Subtract the second equality from the first one and consider the actions of the relevant functionals on the functionv(t) −ū(t): Integrating both sides of this equality from 0 to t and using (2.9), (2.13), we obtain By the Cauchy inequality for ε > 0, Choosing k large enough and ε small enough, so that we get the estimate: Proof. By definition, K ε (v) = (D ε (v), 0). Thus, it is sufficient to show that the operator D ε is completely continuous.
The operator D ε is defined by Therefore, the operator D ε is completely continuous if each one of the operators is completely continuous as a composition of the completely continuous embedding W ⊂ L 2 (Q T ) and the continuous map d ij : L 2 (Q T ) → L 2 (Q T ).

Properties of the map g.
In this subsection we show that the map g is A-condensing with respect to the Kuratovskii's measure of noncompactness γ k .
In this work we shall use Kuratovskii's measure of noncompactness in spaces W and X * × H.

Definition 2.2. The number γ k (M ) is called Kuratovskii's measure of noncompactness of the set M ⊂ W if it is equal to the infimum of all numbers d > 0 for which M may be represented as a union of a finite number of subsets M i with diameters less than d.
We mean here a diameter with respect to the norm · k,X .
In the same way we define the measure of noncompactness γ k in the space X * × H. Kuratovskii's measure of noncompactness in the space X * × H has the following properties: The next lemma is an auxiliary one.

Lemma 2.5. If a matrix-function a(t, s, x, v, w) satisfies the conditions
, then for all functions u, v, w ∈ L 2 (Q T ) the following estimate is valid: with C independent of u, v, w and k > 0.

Theorem 2.3. If a matrix-function a(t, s, x, v, w) satisfies the assumptions
, then the map g : W → X * × H is A-condensing with respect to Kuratovskii's measure of noncompactness γ k for all sufficiently large k.
Proof. We define an auxiliary map p : L 2 (Q T ) × X → X * by the equality , s, x, v(s, x), D 1 w(s, x))ds : for h ∈ V . The continuity of the map p is proved in the same way as the continuity of Q in lemma 2.1. Moreover, Let M ⊂ W be an arbitrary bounded set. As we mentioned above, the embedding W ⊂ L 2 (Q T ) is completely continuous. Hence, the set M is completely bounded in L 2 (Q T ). Then for every w ∈ X the set p(M, w) is completely bounded. By lemma 2.5 the map p(v, w) is Lipschitz continuous in its second variable with Lipschitz constant T 2k L 2 L ∞ (Qd) . Hence, by [3, Theorem 1.5.7], the map g is T 2k L 2 L ∞ (Qd) -bounded with respect to Hausdorff's measure of noncompactness χ k , i.e.
Here, χ k (M ) is the Hausdorff's measure of noncompactness in the space X with the norm v k,X , and χ k (g(M )) is the Hausdorff's measure of noncompactness in the space X * with the norm f k,X * (these definitions may be found in [3]). It is known [3, Theorem 1.1.7] that the Kuratovskii and Hausdorff measures of noncompactness satisfy the following inequality Hence, with the constant C from estimate (2.15). Here, γ k (A(M )) is the Kuratovskii measure of noncompactness in the space X * × H with the norm f k,X * + a H , for (f, a) ∈ X * × H. As g(M ) ⊂ X * × {0}, the measures of noncompactness γ k (g(M )) in the space X * and in the space X * ×H coincide. Thus, from (2.18) and (2.19) we obtain Choosing k large enough so that 2T k L 2 L ∞ (Qd) · C < 1, we obtain the inequality γ k (g(M )) < γ k (A(M )).
The choice of k is independent of the set M .

Estimates of solutions and solvability of approximating operator equations
In this section we establish a priori estimates of solutions of approximating operator equations (2.6 ε ) and prove their solvability using the degree theory for A-condensing perturbations of the map A.

Estimates of solutions.
Given ε > 0, consider the auxiliary family of operator equations If τ = 1 this equation coincides with (2.6 ε ). If τ = 0 we get the equation which has a unique solution.
with the constant C depending only on T, n, Ω, ε, and on constants from the assumptions The function v is a solution of equation (3.1 τ ) and, hence, Evaluate the first and second terms at the right-hand side of the inequality. Note that, by (2.5), where the constant C 0 is independent of k and ε. t, s, x, v(s, x), D 1 v(s, x)

Letv(t) = e −kt v(t). Evaluate the second term. By the definition of g and conditions
Repeating the arguments of the proof of lemma 2.5, we obtain From inequality (2.10) and the relation Let k be large enough so that Then the above inequality may be rewritten as follows: with some constant C depending on ε. Taking into account this and the equivalence of norms · k,XC and · XC , as well as the norms · k,X * and · X * , we obtain (3.2). To estimate v L 2 ((0,T ),V * ) , we recall that v is a solution of the equation Now estimate (3.3) follows from estimate (3.2).

The basic facts of the degree theory.
We recall some facts about the degree theory for A-condensing perturbations of different classes of maps. Let E, F be Banach spaces and D a bounded subset of E. Consider the set of maps of the following form: where A is a continuous map, g is continuous and A-condensing with respect to some measure of noncompactness ψ on F , and A(x) = g(x) for x ∈ ∂D.
Two In this work we consider an invertible map A. The construction of the degree of completely continuous perturbations of the invertible map A is assumed to be well known.
We recall some properties of the degree for maps of the form A − g, where A is invertible and g is A-condensing. These properties will be useful in what follows.  Proof. Let f ∈ X * , v 0 ∈ H and ε > 0. Note that (2.6 ε ) coincides with the equation (3.1 1 ). By theorem 3.1, all solutions of the family of equations (3.1 τ ), τ ∈ [0, 1], are contained in the ball B R ⊂ W with radius R = C(2 + f X * + v 0 H ) and center at zero, and there are no solutions on the boundary of the ball. Therefore, for each τ ∈ [0, 1], the degree is defined for where the sum of the completely continuous map K ε and the A-condensing map g is a A-condensing map.
The map A is invertible. Hence, from the choice of R it follows that the equation A(v) = (f, v 0 ) has a solution in the ball B R . Then, by property 3 of the degree, Finally, property 1 implies that (3.1 1 ) has a solution in the ball B R .

Existence theorem for weak solutions
In this section we apply approximate solutions to obtain existence results for weak solutions of the initial-boundary value problem for Navier-Stokestype equations.

A priori estimates for (2.6 ε ).
Here we obtain a priori estimates of solutions of equations (2.6 ε ), ε ≥ 0, which are independent of the parameter ε.

Theorem 4.1. Suppose that the conditions
Then for every solution v ∈ W of the equations (2.6 ε ), for ε > 0, the following estimate is valid: with the constant C independent of ε.
Proof. Let v ∈ W be a solution of the equation (2.6 ε ), for some ε > 0. Then Consider the action on the element v of the above functionals: We show that D ε (v), v = 0. In fact, by the condition of solenoidality of the function v.
where ϕ = f + Q(v). Repeating the arguments of the proof of theorem 2.1 with u = 0, u 0 = 0, ψ = 0, we obtain the estimate for k large enough. Since Choosing k large enough so that C · C 1 · T 2k < 1 2 , we get the estimate which is equivalent to (4.1).

Theorem 4.2. Suppose that assumptions
with the constant C independent of ε.
Proof. Repeating the arguments of the proof of theorem 3.1, we obtain the inequality We evaluate D ε (v) L 1 ((0,T ),V * ) as in the proof of lemma 2.1. By definition, Hence, Using the above estimate, inequalities (2.2) and inequality (4.3), we obtain Now, estimate (4.2) follows from estimate (4.1).

Existence and uniqueness theorems for a weak solution in
The next lemma is an auxiliary proposition which is analogous to theorem 4.2.

Lemma 4.1. Under the conditions of theorem 4.3, every solution
From the above and from estimate (4.4) we have Hence,

Using the embedding W ⊂ C([0, T ], H) and the definition of norm
Hence, repeating the arguments of the proof of theorem 3.1, we get the estimate (4.6). The space V is reflexive and separable, thus the space X = L 2 ((0, T ), V ) is reflexive [6,Remark 1.11], and its dual space X * is also reflexive [6,Theorem 1.14]. In a reflexive Banach space a bounded closed set is weakly compact. By estimates (4.1), (4.6), the set of solutions {v m } is bounded in W . Hence, without loss of generality, passing to subsequences and preserving the notation {v m }, we may assume that Thus, the function v * belongs to W . Using the compactness of the embedding W ⊂ L 2 (Q T ), we may assume that As the space L 1 ((0, T ), H) is separable, by [7,Theorem 6] we may assume that v m v * * −weakly in L ∞ ((0, T ), H). Next we show that the sequence {v m } converges to v * with respect to the norm of the space X and that v * is a solution of the problem (2.3), (2.4).
We subtract from the first equality the second one and consider action of functionals from both sides of the equality on the functionv m (t) −v * (t): Integrating both sides of the equality in t from 0 to T we get (4.7) e −2kt C(e ktvm (t)) − C(e ktv * (t)), e ktvm (t) − e ktv * (t) dt Let k ≥ µ 0 , then

Lemma 2.2 implies
T 0 e −2kt B 1 (e ktvm (t)) − B 1 (e ktv * (t)), e ktvm (t) − e ktv * (t) dt Hence, the left-hand side of (4.7) is no smaller than µ 0 v m −v * X . Denote the terms at the right hand side of (4.7) by I 1 (m), I 2 (m), I 3 (m), I 4 (m), respectively. Then, by (4.7), we obtain the following inequality Estimate for I 4 (m). As v m v * weakly in X,v m v * weakly in X too. Then by the definition of weak convergence Estimate for I 1 (m). From estimate (2.9) with C 1 independent of k and m. Estimate for I 2 (m). By definition, Denote the terms on the right hand-side of the equality by I 2,1 (m) and I 2,2 (m), respectively. We evaluate each one of them. By Hölder's inequality, Hence, by inequality (2.17), we get We now evaluate the second term I 2,2 (m). By Hölder's inequality, The second factor is bounded. We show that the first factor converges to zero as m → ∞. From conditions A 1 ) − A 3 ) it follows that the Nemytskii operator as m → ∞. Hence, I 2,2 (m) → 0 for m → ∞. Thus, we have obtained that where I 2,2 (m) → 0 as m → ∞.
Estimate for I 3 (m). By definition, Let us estimate each term on the right-hand side of the above equality. In the proof of theorem 4.1 it was shown that In the same way we can show that Furthermore, by the definition of the weak convergence v m v * in the space X, We show that the last term in the expression I 3 (m) also converges to zero as m → ∞. By definition, Since for v ∈ W and n = 2, it follows that W ⊂ L 4 (Q T ), and this embedding is As the set of solutions {v m } is bounded in W , the set of functions is bounded in L 2 (Q T ). Therefore, without loss of generality, we may assume This means that (4.9) and, consequently, Thus, we have showed that all the terms in the expression for I 3 (m) are equal to zero or converge to zero as m → ∞, i.e., Summarizing the above investigation of the terms I i (m), i = 1, 2, 3, 4, and applying inequalities, we can rewrite (4.8) as follows: Choosing sufficiently large k such that X ≤ 2 (|I 2,2 (m)| + I 3 (m) + I 4 (m)) . Each term of the right hand side of the above inequality converges to zero as m → ∞, and v m −v * X → 0 as m → ∞. Hence, v m → v * strongly in X.
By lemma 2.1 the maps A, B 1 , C, Q are continuous on X. Hence, Av m → Av * strongly in X * , Passing to the limit, in weak sense, in each term of the equality and applying (4.10), we obtain Thus, we have showed that v * is a solution of equation (2.3) and v * (0) = v 0 .
The uniqueness of weak the solution of the problem (1.1)-(1.4) will be established under assumption that the matrix-function a(t, s, x, v, w) is Lipschitzian in the variables v, w, i.e., for all (t, s, x) ∈ Qd, u,ū ∈ R n , v,v ∈ R n 2 , i, j ∈ 1, n, where L 2 is an essentially bounded function.
Evaluating the functionals from this equality on the functionv(t) −ū(t) and then integrating the result in with respect to t from 0 to τ , we arrive at As in the proof of theorem 4.2, we denote the terms on the right-hand side of this equality by I 1 (τ ), I 2 (τ ), I 3 (τ ), respectively, and suppose that k ≥ µ 0 . Then we obtain (4.10) Let us estimate the terms I 1 (τ ), I 2 (τ ), I 3 (τ ). Estimate for I 1 (τ ). The estimate (2.9) gives with C 1 independent of τ . Estimate for I 2 (τ ). By definition, which implies Estimate for I 3 (τ ). By definition, Hence, taking into account the condition (1.2), we obtain From the inequality v L 4 (Ω) ≤ 2 1/4 v 1/2 L 2 (Ω) · grad v 1/2 L 2 (Ω) ( [15], p. 233), as well as the Schwartz and Hölder inequalities, we obtain Thus, From the estimates for I 1 (τ ) and I 2 (τ ) it follows that Choosing k large enough, so that C 1 +C 2 √ 2k < µ 0 2 , we obtain for all h ∈ V and for all real-valued functions ψ which are continuously differentiable on [0, T ] and have supports in (0, T ). Since Consider a chain of inequalities: .
Under the conditions of the lemma, v ε − v * L 2 (Q T ) → 0 as ε → 0 and v ε L 2 (Q T ) , v * L 2 (Q T ) are uniformly bounded. Hence, to complete the proof it is sufficient to show that the third term in (4.16) also tends to zero as ε → 0.
The convergence v ε → v * a.e. on Q T implies that Thus, → 0 as ε → 0 by the Lebesgue theorem [6, Theorem We are now in the position to prove the weak solvability of the initialboundary value problem for (4.13). All the terms, but the first one, are contained in L 1 ((0, T ), V * ), which implies (v * ) ∈ L 1 ((0, T ), V * ). Therefore, v * is a solution of the equation (4.15). Furthermore, since the sequence {v m } is bounded in the norm of the space L ∞ ((0, T ), H) and {v m (0)} is bounded in H, we may assume, without loss of generality, that v m (0) v * (0) weakly in H, i.e., v * (0) = v 0 . All a priori estimates for solutions v m are also valid for the function v * . Hence, inequalities (4.17) hold.