ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

Kucher Nikolay A; Zhalnina Aleksandra A

doi:doi:10.21603/2500-1418-2017-2-2-54-71

ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

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ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

Журнал: SCIENCE EVOLUTION Том 2 № 2 , 2017

Рубрики: MATHEMATICAL SCIENCES

Kucher Nikolay A ¹

Zhalnina Aleksandra A ²

Информация об авторах и публикации

Авторы:

1. Kemerovo State University

Россия

2. Kemerovo State University

Россия

Тип:

Статья

DOI:

https://doi.org/10.21603/2500-1418-2017-2-2-54-71

Страницы:

с 54 по 71

Статус:

Опубликован

Получено:

30.12.2017

Одобрено:

30.12.2017

Опубликовано:

30.12.2017

Классификаторы:

ГРНТИ 29.01 Общие вопросы физики
ГРНТИ 27.01 Общие вопросы математики
ГРНТИ 31.01 Общие вопросы химии
ГРНТИ 34.01 Общие вопросы биологии

Язык материала:

английский

Ключевые слова:

Mixture of viscous compressible fluids, boundary value problem

Аннотация и ключевые слова

Аннотация (русский):
Mathematical models of multi-velocity continua, through which the motion of multicomponent mixtures are described, represent a rather extensive area of modern mechanics and mathematics. Mathematical results (statements of problems, theorems on the existence and uniqueness, properties of solutions, etc.) for such models are rather modest in comparison with the results for classical single-phase media. The present paper aims to fill this gap in some extent and is devoted to investigating the global correctness of the boundary value problem for a nonlinear system of differential equations, which is some regularity of the mathematical model of nonstationary spatial flows of a mixture of viscous compressible fluids. Construction of the solution of the problem considered in this article is a key step for the mathematical analysis of the initial model of the mixture, since it allows to obtain globally defined solutions of the latter by means of a limiting transition and, in addition, the proposed algorithm for constructing solutions to the regularized problem is practical. This algorithm is based on the finite-dimensional approximation procedure for an infinite-dimensional problem, and therefore a mathematically grounded algorithm for the numerical solution of the boundary value problem of the motion of a mixture of viscous compressible fluids in a region bounded by solid walls can be constructed on this basis. The local in time solvability of finite- dimensional problems is proved by applying the principle of contracting mappings and the local solution can be extended to an arbitrary time interval with the help of a priori estimates.

Ключевые слова:
Mixture of viscous compressible fluids, boundary value problem

Текст

INTRODUCTION > 0, 4 ( )2 > 0, = 2 . (4) Let us assume that a binary mixture of viscous gases 11 22  12  21  ij ij  ij fills a bounded region Ω  R3 of the Euclidean space of 2ν It is assumed that the pressure pi and density i in  points x = (x1, x2 , x3 ) with the class C boundary and the i -th component are linked by the relation pi = i i , its state is characterized by the distributions of densities  where  i > 1 is the adiabatic exponent. The summands i (t, x) , pressures pi (t, x) , and velocity fields u(i) (t, x) J (i) characterizing the intensity of the momentum of the constituent components ( i = 1,2 ). They satisfy the following equations [1] exchange between the components of the mixture are determined by the formula [2,3] t i  (ρ u (i) i )  div(ρ u (i)  u (i) )  pi= (i) J = (1) i1  a(u(2)  u(1) ), a = const > 0, i = 1,2. (5) =divσ (i) J  (i) , in QT=( 0,T)  Ω, (i) (1) Equations (1) and (2) are supplemented by the initial conditions t(ρi )  div(ρi u )=0, in QT i=1,2. (2) i t =0 i =  0 , i  u(i) t =0 = q (i) t =0 0 = q (i) in , i = 1,2, (6) Here the tensors of viscous stresses given by equalities  (i) , i = 1,2 are and boundary conditions σ(i)(u( 1 ),u( 2 ) )= 2 2 μ D(u(j) )  λ divu(j)  I , u(i) = 0 on (0,T )  , i = 1,2, (7)    ij  ij  j=1 D(u)= 1 u  (u)T , I is unit tenso , (3) 2 meaning that the flow region boundary is a fixed solid wall. Global existence theorems and results for the stabilization of solutions for nonstationary equations of multispeed continua of the form of (1)-(5) have been in which the (constant) viscosity coefficients i, j = 1,2, satisfy the inequalities 2 ij , ij , obtained only in the case of a one-dimensional motion with plane waves, when the solution depends on only one spatial variable [3,4,5]. The first results for the 11 > 0, 41122  (12  21) > 0,11 > 0,22 > model of a mixture in the Stokes approximation in the Please cite this article in press as: Kucher N.A. and Zhalnina A.A. On the regularization of equations of the mechanics of mixtures of viscous compressible fluids. Science Evolution, 2017, vol. 2, no. 2, pp. 54-71. DOI: 10.21603/2500-1418-2017-2-2-54-71. Copyright © 2017, Kucher et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license. This article is published with open access at http:// science- evolution.ru/. Science Evolution, 2017, vol. 2, no. 2 case of more than one spatial variable were obtained in [6, 7]. In [8], a so-called quasi-stationary model of a i  n = 0, on (0,T )  , i = 1,2, (12) mixture of compressible liquids in a limited region where  > 0 ,  > 0 are small parameters, and the with special boundary conditions was studied. In [9], a values i ,i = 1,2 are chosen sufficiently large. stationary solution of the first boundary value problem for the complete equations (1)-(5) was constructed for three spatial variables, and in [10, 11, 12], the existence of weak stationary solutions of the system of equations for mixtures of viscous liquids taking into account thermal conductivity is proved. STATEMENT OF THE PROBLEM AND MAIN RESULTS Equations (9) are supplemented by homogeneous Neumann boundary conditions (12). The aim of this paper is to construct a solution of the regularized problem by the finite-dimensional approximation method. This paper uses standard notations for spaces of continuous functions, Lebesgue spaces, and Sobolev spaces (see, for example, [10, 13-15]). The main results for the correctness of the regularized problem are given by the following theorem. In order to construct a generalized solution of the problem (1)-(7), let us consider its regularization. Theorem 1. Let the coefficients ij and ij satisfy  ( u (i ))  div( u (i ) u (i ))  ( i )  ( i )  the conditions (4) and the adiabatic exponents 3 t i i i i   i > ,i = 1,2. Let the parameters  ,  , i be chosen so 2 i  u(i )   = div (i )  J (i ) , in QT = (0, T )  , (8) that  > 0,  > 0, i  15,i = 1,2. . Suppose that  is a 2,  ( )  div( u(i) ) =  , in Q , (9) bounded domain of the class C ,   (0,1] and t i i i T 0 <    0   < ,  0 W 1, (), q (i) L2 () .  =  0 ,  (i) =  (i), , (10) i i 0 (i) i t =0 i iu t =0 q 0 in  Then there exist pairs (i, ,u ) , i, = i, , , u(i ) = 0, on (0, T )  , (11) u(i) = u(i) ,i = 1,2, with the following properties:   ,   C 0(I , Li ()),   C 0(I , Lp ())  Li 1 (Q ), 1  p <  . 1   0 a.e in Q ,  2 i  L2 (I ,W 1,2 ()), i, 5i 3 4 weak  i,   5i 3 4 33   T (i) 2 i 1,2 i,  (i) T i, (i)  2i  0 i 1  t i,  L i (QT ), D i,  L i (QT )  ,|  |= 2. u  L (I ,W0 ()), q  = i, u  C  I , Lweak (),  (i )    2  , 6i i 6 (    ), (i ) 2  1      6i  2  4i 3  q   L  I L     i, | u  |  L (I , L ())  L  I , L  () ,   5i 3 5i 3  10i 6 ,5i 3  (i) 4 (i) 4  3 3 4  0 u i,  L i (QT ), i, , i, u  L i  I , E0 i   i () , i, dx = i dx.  Equations (8) are satisfied in the space of distributions D(QT )          2        t (i, u (i))  div(i, u (i) u (i))  (i,i  i, i ) = 2 j=1 ij u ( j)   (   )divu ( j)   (  )u (i)  (1)i1 a(u (2) u (1)),i = 1,2.  ij ij  j=1 i,    d Equations   dx   u(i) dx      dxdt = 0,  C ( 3 ), i = 1,2 , are satisfied in D(I ) . dt i, i  i,  i  i, i i  The following relations are satisfied: 0   (i)  (i)  (i)  (i)  (i)  lim ρi,ε(t)ηi dx=ρi ηi dx, ηi  C 0 (Ω), lim ρi,εu ε (t) dx=q 0  dx,   C 0 (Ω) t 0 Ω Ω t 0 Ω Ω There are energy inequalities in the differential form d Eˆ (ρ u (i) )  c (|u ( 1 )|2  |u ( 2 )|2 )dx  ε γ 2 β 2 2 (γ ρ i  δβ ρ i )|ρ |2dx  a |u ( 2 ) u ( 1 )|2dx  0 â D(I), (13) dt δ i,ε ε 0  ε ε Ω  i Ω i=1 i,ε i i,ε i,ε  ε ε Ω as well as in the integral form  (i) t 2  (i) 2 t 2 γ  2 β  2 2 t  ( 2 )  ( 1 ) 2 (14) δ i,ε ε 0  ε L2   i i,ε ε   0 Eˆ (ρ u )(t)  c 0 i= 1 П  u П (Ω ) dτ  ε 0 Ω i= 1 (γ ρ i δβ i ρi, i )|  ρi,ε| dxd τ  a |u ε 0 Ω u ε | dxd τ  Eˆ δ, , for almost all t  I. 2  2 γk βk   Here Eˆ (ρ u (i) )= 1 ρ u(k) ρk  δρk k  dx . The value E  E (ρ0u (i) ) is determined by the initial data and δ i,ε ε   k k=1 Ω 2 γk  1 β  1 ˆ ˆ δ,0 δ i 0 does not depend on the number n . If   (0,1) , then the following estimates are valid, uniform with respect to the parameter  : 2  (i) 1/βi  ˆ 12 (15) Пu ε П 2 1,2  L(Eˆδ,0 ), Пρi,ε П  γ  L(γi ,Eˆδ,0 ), δ Пρi,ε П β L(βi ,Eδ,0 ), i= , , i=1 L (I,W ε Пρ ()) П L (I,L i (Ω ))  L(δ(δ ,ρ0 ,q (i) ), i=1,2, Пρ |u (i)|2 П L(I,L i ())  L(Eˆ ), i=1,2, (16) i,ε  (i) 2 L2(QT ) i i 0 i,ε ε  (i) L(I,L1()) δ,0 (17) Пρi,ε|u ε | П  (i) 6 βi L2(I,L4 βi 3())  L(Eˆ δ,0 ), i=1,2, Пρi,εu ε П  (i) 2 βi L(I,L βi 1())  L(Eˆδ,0 ), i=1,2, (18) Пρi,εu ε П 6 βi L2(I,L βi  6())  L(Eˆδ,0 ), i=1,2, Пρi,εu ε П 10 βi 6 i 3 β  3 L (QT )  L(Eˆ δ,0 ), i=1,2, i εПρi,ε П 10 β 6 3 β  3 δ,0 i,ε ε  L(Eˆ ), i=1,2, ε, Пρ  u (i) П 5 βi 3 4 β  L(Eˆδ,0 ), i=1,2. (19) L i (QT ) L i (QT ) Here L is a positive constant, independent of  . Moreover, if the parameter  is not specified in the (L2 ())3 to properties. X n by Pn and recall some of its argument L , then L does not depend on  . Lemma 2. The following relations are valid: APPROXIMATION OF THE FAEDO-GALERKIN P (u)  vdx = u  P (v)dx, v  (L2 ())3 , n n AUXILIARY PROBLEM (8)-(12)   In this section, we construct a scheme for approximating the regularized problem (8)-(12) by n 2 2 || P || = 1, L( L (  ),L ( )) means of finite-difference problems. Let us study the local and then time-global solvability of these problems. Further, using a priori estimates of solutions of the lim || (Pn  n I )z || L2 () = 0, z  (L2 ( ))3 ,   n k,2  k,2   ( ))  ( )) 1 2 Galerkin equations, we will prove the possibility of a limit transition, as a result of which we obtain a strong ||P z|| 0 c||z|| , z W 1,2(Ω 3 W k,2(Ω 3 , k= , , generalized solution of the problem (8)-(12). lim || ( )  || = 0,  ( 1,2 ( ))3 , n  Pn  I z 1,2 z  W0  PRELIMINARY ASSUMPTIONS ||  || ||  || ,  2 ( ), Let us choose a system of sufficiently smooth Pn z 1,2  c z 1,2 z  L  functions   that forms an orthonormal basis in    || ( I  Pn )z  ||1,2,   i i=1 lim n  sup 2 || z ||  = 0. (L2 ())3 , and also an orthonormal basis in 0 (W 1,2 ())3  zL () 0,2,  (with suitably chosen scalar product) Let us consider a sequence of finite-dimensional Let us note that all norms on k , p X n , and, in particular, Euclidean spaces X n with a scalar product X (,) = (,) W () - the norms k = 0,1, , 1  p   are defined as n equivalent on X n .  n       Given a function Xn = span i =1, (u,v ) = u  vdx, u,v  Xn . 0 1 1 i  g C (I , L ()),  g  L (Q ),  Let us denote an orthogonal projection from t T essinf g(t, x)  a > 0. (t ,x )QT (20) Since the mapping w  l(w ) = g(t)v  w dx is a  (0) = 0 in  (26)  bounded linear functional on X n and and the boundary condition L () L ()     n  = 0 on I  . (27) | l(w) ||| v ||  || u ||  g(t)dx, then, by the Riesz  Here,  (t, x) , t  I , x   is the desired function, theorem, it can be represented as a scalar product  is a bounded domain,  > 0  is a given constant, 0 (M g v,w ) , M g(t)  Xn . Thus for all t  I the linear is a given function and u(t, x) is a given vector field mapping M g(t): X n  X n , (M g v,w )=g(t)v  w dx, v,w  X n . Ω vanishing on the boundary of the domain  . Lemma 4. Let 0 <   1 ,  be a bounded domain of class, C 2, , 0 <    <  and 1, is defined. Let us note the following properties of the 0 W (),   0  . operator M g(t) : Then there exists a single-valued mapping (21)  1, 3 0 1,2 ||M || g(t) L(X n ,X n )  c(n) g(t)dx, t  I. Ω 0 S : L (I,(W0 ()) )  C (I ,W ()), such that:  2 2, p 0 1, p The inverse operator exists for all t  I , and the 0 S (u)  RT = { :   L (I ,W ())  C (I ,W ()), following estimate is valid t    L2 (I , Lp ()) 1 < p < }; ||M 1 || g(t) L(X n,X n )  1 . a (22) The function   = S 0 (u) satisfies the equation (25) From (21) and (22) it follows that e. in QT , the initial condition (26) a. e. in  and the boundary condition (27) in the sense of traces a. e. in I ; ||M 1 M M 1 ||  c(n)||g (t)|| , t  I. t t g(t) g1(t) g(t) L(X n ,X n ) a2 1 0 ,1  ||u(τ)|| 0 dτ 1,  ||u(τ)|| 0 dτ 1, ||M  M || , )  ||(g2  g1 )(t)||0,1, t  I. (23) ρe ρ  [S (u)](t,x)  ρe , 0 g2(t) g1(t) L(X n X n t  I, x  Ω; From the identity 1  1 1  1 we obtain g g g g g g M M =M (M 2 1 2 1 M )M 2 1  1, If || u ||  K , where K > 0 , then L (I ,W ())  1   1  2 ||M M || , ) c K  K t g 2(t) g1(t) L(X n X n  || S 0 (u) || L ( I t ,W 1,2 ( ))  c || 0 ||1,2 ( ) e 2 ,  c(n)||(g  g )(t)|| , t  I. (24) a2 2 1 0 ,1 It = (0,t), t  I , (28) for all g1, g2 that satisfy conditions (20). Further, we 2  c c (K  K 2 )t 2 ε have the relations || S ρ (u)|| 2  0 L (QT ) ε t||ρ0||1,2  K  e , 1 1 (M g v,w )=(v,M g w ), (M g v,w )=(v,M g w ), t  I, (29) t(M g v,w )=t gv  w dx, v,w  Xn. Ω ||t S (u)||  c ρ0 L2(QT ) t||ρ0||1,2 c (K  K 2 )t K  e 2 ε , T Lemma 3. Let us assume that g W 1,1 (Q ) , t  I; (30) essinf g(t, x)  a > 0 . Then the following relations are ||[S (u )  S (u )](t)||  c(K,ε(K,ε (t,x)QT ρ0 1 ρ0 2 L2(Ω ) valid:  (31) 1   1 1     , ||ρ0||1,2||u1  u2||  1, L (I t ,W () , t  I t t(M g v,w)=  (M g M  g M g v,w), in D(I), v,w  X n The constant c in the inequalities (28)-(30) depends t(M g v,w )=  (M g M g Mg v,w )  (M g t v,w ), 1 1 1 1  t only on  (in particular, it does not depend on  , K ,T , 0 ,u ). n n in D(I), v C 1(I,X ), w  X . CONTINUITY EQUATION WITH DISSIPATION Let us consider the equation GALERKIN APPROXIMATIONS For an arbitrary chosen T   (0,T ] , let us find vector functions  (i) 0 t   div(u) =  in QT , (25) u  C (I ', X n ), I  = (0,T ), i = 1,2, (32) supplemented by the initial condition atisfying the equations  (i )   (i )  t  2 2  ( j )  ( j )   i (t)u dx  q 0  dx = ij u (ij  ij )divu  i i  i i    0  j =1 j =1  div(ρ u (i) u (i) )  ε(ρ  )u (i) (  1 )i 1  a(u ( 2 ) u ( 1 ) )dxdτ, i=1,2, t  I ,   X , (33) i i n where i (t) = [S 0 (u (i))](t) is the solution of the problem (25)-(27), constructed in Lemma 4. i Equations (33) can be represented in the form  (i)  t  (i)    (i)  ( 1 )  ( 2 )  , (M u ρ (t) ,)X  (Pq 0 ,)X = (P[N i(S 0 (u X ),u ,u )],) dτ n where i n n ρi 0 2 N (ρ ,u ( 1 ) ,u ( 2 ) )=μ 2 Δu (j)  (λ  μ )divu (j)  (ργi )  δ(ρβi )  div(ρ u (i) u (i) )  i i ij j=1 ij ij j=1 i i i i  ε(ρ )u (i)  (  1)i 1  a(u ( 2 )  u ( 1 ) ), i=1,2, and, consequently, we can write them in the operator form    t        u (i)(t) = M 1  Pq (i)  P  (S (u (i)),u (1),u (2)) d , i = 1,2. (34) 0  [S (u (i))](t) i  0  i  0  0 i v =M 0 Pn q 0    EXISTENCE OF A LOCAL SOLUTION OF EQUATIONS (34) * (i) 1 (i) , and, consequently, ρi Lemma 5. There exists such T  > 0 that on the  2  (i)  (i) 2 1/ 2  interval 0 < t < T  , there exists a unique solution  u v *   C 0(I,X n ) u (i) C((0,T ), X (34). n ), i = 1,2 of the system of equations  i=1  2 1/ 2 The proof of Lemma 5 is based on the use of the C  q (i)   2  1  0 2     =K~ . principle of contracting mappings.  i=1  1 ρ L (Ω )       EXISTENCE OF A TIME-GLOBAL Thus, the solution of the system of equations (34) a SOLUTION OF EQUATIONS (34) priori belongs to a ball  ~ (v ) , v = (v (1) , v (2) ) , and Theorem 6. In any finite interval 0 < t < T , the K1,T * * * * ~ system of equations (34) has a unique solution in the therefore, choosing a number K1  K1 as the radius K1 class C 0((0,T ), X n ) . of the ball 1 0 K , , we continue the local solution of Proof. Let us note that the possibility of extending the local solution constructed in Lemma 5 to an arbitrary finite interval (0,T ) follows from the boundedness in the equations (34) for an arbitrary finite time interval [0,T ] in a finite number of steps. n space C 0((0,T ), L2 ()) of the family of solutions of Let u (i)= u (i)(t, x) = c(i) (t) (x)  C 0(I , X ) be equations (34). Indeed, having an estimate n  j j n j=1 Пu (i)П C 0(I,L2(Ω ))  C=const, i=1,2, (35) the solution of equations (34), which in this case is more convenient to be represented in the form (33). For each we obtain П u (i) v (i)П  C  1 П q (i) П , basis function k (x) , k = 1,n from (33) (after * C 0(I,X n ) ρ 0 L2() differentiation according to t ) we obtain the identities d  i  u (i ) k ( x)dx = div (i )  ( i )  ( i )  div u (i ) u (i )  ( )u (i )  (1)i 1 a  u (2)  u (1)  dt   i i k  ( x)dx, i i k = 1,2,, n, div (i) = 2 { u ( j)  (   )divu ( j)},  ij j=1 ij ij k Multiplying these equations, respectively, by C(i) (t) and summing over k , we obtain    (i) d u i ρ |u (i)|2 dx  ρ u (i) dx=divσ (i)  (ργi )  δ(ρβi )  Ω dt Ω Ω i t i i  divρ u (i) u (i)  ε(ρ )u (i)  (  1)i 1a  u ( 2 )  u ( 1 )  u (i)dx. (36) i i Since i (t) = S 0 (u (i)) , i = 1,2 , the following identities are valid i   (i) d (( i ),u )dx =    2  i i dx    i |  |2 dx, i = 1,2, (37) i i dt  1 i i i    i  i  (( i ),u )dx =     (i) d   2 i dx    |  |2 dx, (38) i    (i)  i dt   1 i i i      u (i), u dx = d 1  | u (i)|2 dx  (div u (i) u (i) ,u (i))dx   ((  )u (i),u (i))dx. (39)    i t   dt 2 i i i    From the relations (36)-(39) and the inequality 2 0  (div (i) ,u (i))dx  c | u (1)|2  | u (2)|2 dx we obtain the i=1   following energy inequality on the solutions of the Galerkin equations (34): d 1 ρ δ 2  γi i  ρ |u (i)|2  i  β  i 0 ρ i dx  c |u ( 1 )|2 |u ( 2 )|2 dx  dt Ω i=1  2 γi 1 βi 1  Ω  2 ε γ ργi  2 β  2 δβ ρ i | ρ|2dx a |u ( 1 ) u ( 2 )|2dx 0.  i i Ω i=1 i i  i     Ω (40) Integrating inequality (40), we obtain that t 2 t γk 2 2 ˆ  (i)    ( 1 ) ) 2   ( 2 ) ) 2   k k 2  k  E [ρ ,u ](t) c  ||u (τ || ||u (τ || , (Ω dτ εγ  ρ | ρ | dτ δ i 0 0 2 t βk 2 2 W 1,2(Ω ) t W 1 2 ) k=1 0 L2(Ω ) εδ β ρ 2 |ρ | dτ  a ||u ( 1 )(τ)  u ( 2 )(τ)||2 dτ  Eˆ [ρ ,u (i) ]( 0 ), (41) where  k  k k=1 0 k  L2(Ω ) 0 L2(Ω ) δ i 2 Eˆ [ρ ,u (i) ](t)=  (i) 2 ρ 1 γk δ  ρ |u |  k  β  k ρ k dx (42) δ i  k k=1 Ω  2 γk 1 βk 1  Since u n = u  X n , then u n t=0 = Pnu 0 by that || u || || u || . In addition, i = i,n = S 0 (u n ) and (i) (i)  (i)  (i) (i ) 0, n 0,2 0 0,2 (i ) 0  (i) i therefore (according to the definition of the operator S 0 ) i,n t =0 = i i . Given these facts, we obtain 2 2 0 γk 1 (ρ ) Eˆ [ρ0 ,u (i) ]( 0 )=   ρ0|u (k) |2 dx   k δ (ρ0 )βk dx  δ i,n 0,n  k 0,n   k  k=1 Ω 2 k=1 Ω  γk 1 βk 1  2 2  0 γk   1 ρ||u (k)||2 dx  (ρk ) δ (ρ0 )βk dx=Eˆ , (43) i. e. 2  k=1 Ω 0 L2(Ω ) γ  k=1 Ω  k 1 k  βk 1  δ,0 ˆ 0  (i) ˆ δ,0 where Eˆ Eδ[ρi,n,u 0,n ]( 0 )  Eδ,0, is a known constant independent of n . (44) From the inequalities (40)-(43), in particular, we have the estimates ρ |u | dx  t Eˆ , k=1,2, П u (τ)П П u (τ)П , dx 0 Eˆδ, , 0 2 , then taking into account PASSAGE TO THE LIMIT IN THE CONTINUITY EQUATIONS WITH DISSIPATION inequalities (64) and (50), the estimate follows Lemma 8. From the sequence n u (i ) , || t i,n || 2i  c(ˆ  ,0 , i ,T ). (65) i,n S 0 u n , n = 1,2,, of solutions of equations 2 i 1 * = ( (i)) i 1, L ( I ,(W ()) ) (34) constructed in Theorem 6, we can select a subsequence (which retains the previous notation), From (49) with i > 3 it follows that which converges as n   in the following sense || i,n || 1, 2i  c,  i   1 i,n  i * weakly in L (I , L ()), L (I ,(W i ())*) and therefore the sequences { i,n }  n=1 are equicontinuous T  (i )  (i )  (1) (2) (69) in W 1, 2i i 1 () = (W 1, 2i i 1 ())* .  i (u n  u 0  )  dxdt = I n  In .  In view of Lemma 4, for each n functions For any vector-function  (t, x)  L2 (I , L i ()) , i,n S 0 u n are continuous on I with values in 6  = ( (i)) i i  i = i 5i  6 we have that    L2 (I , L6/5 ()) (by W 1, p () , 3 < p <  , and moreover (according to the virtue of relation (59)). Since embedding theorem) in C 0 () .  n u (i )  u (i ) weakly in L2 (I , L6 ()) (due to continuity of Thus, for each n i,n weak (t) C 0(I, L i ()) . Hence it the embedding 0 W 1,2 () in L6 () ), then follows that for some subsequence (which retains the previous notation), I (2)  0 as n  . i,n  i i = 1,2 в C (I, L i 0  weak ()). With i 2 > 3 , n (70) Let us consider the first integral in (69). The following estimate is obvious 1,2 1, 2 i  1 |I ( 1 )|  ||ρ  ρ || ||u (i)|| . the embeddings W () L () W i (). n i,n β i L i (Ω ) n L2(I,L6(Ω )) are valid The sequences { (estimates (49) and (50)) i,n }  n=1 , i = 1,2 , are bounded ù σ |||| 2 β ,ùi= i . (71) in L (I , L2 ())  L2 (I ,W 1,2 ()) , and the sequences of  L i (I,L i (Ω ))   β  2 i  i,n  Since L i (I , L i ()) L2 (I , L i ()) ( 2 <  i <  ), the derivatives 2   t  n=1 are bounded in then on the basis of formulas (59), (62) from (70) and (71) we can assert the weak convergence of the sequence L2 (I ,W 1, i i 1 ()) . Then, according to the Lions-Aubin n i,nu (i)  to iu (i) in the space L  i (I , L 6i i 6 ()) , theorem, for some subsequence 1 1 i, n  i strongly in T L2 (Q ). (66)   i  i = 1 ,  i < 2 , “containing” the space From relation (66), estimate (56), and the interpolation inequality L2 (I , L 6i i 6 ()) . Taking into account the fact that the 6i Пρ  ρ П  cП ρ ρ Пθ  П ρ ρ П 1θ ,  (i) 2  6 i,n i Lp(QT ) i,n i L2(QT ) i,n i i 4 β L3 (QT ) limit element is iu  L (I , L i 6i ()) and  (i)  i 6 1 = θ  3( 1θ) , 0  θ  1, i,nu n  wi weakly in L2 (I , L ()) , we conclude p 2 4 βi that wi i =  u (i) . The formula (60) is proved. we obtain that On the basis of the estimate (64) we obtain (passing to a subsequence if required) that    strongly in Lp (Q ), 1  p < 4  . (67) 2i i,n i T 3 i  i   i 1 n i,nu ( ) wi * weakly in L (I , L ()). Due to The relation (62) is proved. By virtue of inequality  6i   2i   2 i 6    i 1  uniqueness of the limit in topological space and property  (i)   (i)   w  L (I , L ())   w  L (I , L ())  L n 6 i,nu n dx || i,n || i  ()  || u ||  ||  || L () i 6         5 6 and estimates (49), we obtain L i () (60), we obtain (61). The properties (59)-(61) allow to make the passage to the limit in the weak sense in equations (46) and ||  , u (i)|| 6  c(ˆ  ,0 ). (68) prove that the limit functions i  , u (i ) satisfy the i n n i 2 i  6 L (I ,L ()) continuity equations in the sense of identity (63). PASSAGE TO THE LIMIT IN Let us prove the relations (60), (61). Let us turn to THE MOMENTUM BALANCE EQUATIONS equality T T T Let us prove that the limit functions i , u (i ) satisfy  u (i )dxdt   u (i )dxdt = (   )u (i )dxdt  (9) almost everywhere in QT and the boundary i ,n n i 0  0  i ,n i n 0  conditions (12) in the sense of traces.  17 , 5  From (49) and (78) we easily obtain the estimate  Lemma 9. There exist values  34 10  ti  , 16 4  i,n n t εП ρ u (i) П  c(Eˆδ,0 ), (79) ri  15 , 3  , i = 1,2 such that the sequences {t i,n} , L i (QT )  { 2 i,n} ,  i = 1,2 t are bounded in, the sequences are εП div(ρi,n u  (i) n t )П L i (QT )  c(Eˆ δ,0 ),where bounded in Li (QT ) , the sequences {i,n} are bounded in ti (I , ri ()) , the sequences { u (i)} are t = t ( ) = 2ri = 5i  3 L L t ri ,t i,n n i i i ri  2 4i bounded in Li (I , E0 i ()) . Consequently, the limit 17 5 functions i , u (i ) belong to the same functional (with i  4 we have  ti < ). (80) classes, that is, in particular, 16 4 t t r Then the estimates (80) imply the inequalities t i  Li (QT ), i  Li (I , L i ()), i = 1,2, (72) i Пt ρi,n П t  c(Eˆδ,0 ,ε),  (i) t ri ,t L (QT ) i ,u  Li (I , E0 i ()), i = 1,2. (73) П  2 ρ П t  c(Eˆδ,0 ,ε). (81) and satisfy (9) almost everywhere in QT . The initial i,n L i (QT ) conditions (10) are satisfied in the sense that p The estimates (78), (80), (81) prove the statements i  C 0(I , L i ()), i = 1,2, 1  pi < i . (74) (72), (73), (75), and also the postulations about the boundedness of the corresponding sequences. From the The boundary conditions (12) are satisfied in the sense of the trace, i. e. boundedness of the sequence {t  2 i,n }  n=1 , i = 1,2 in the j ( ) = 0, j ( u (i)) = 0, a.e. in I. (75) 1, i i 1 n i n i space L2 (I ,(W ())* ) proved in Lemma 8 it 2 roof. Let us use the interpolation inequality follows that t i  L2 (I ,(W 1, i i 1 ())* ) . Пρ u (i) П  Пρ u (i) П 1θi  Пρ u (i) Пθi , (76) According to what has been proved above i,n n L ri (Ω ) i,n n 2 βi i β 1 L (Ω ) i,n n 6 βi i β  6 L ()  ri ( , 1,ri ()) i L I W , ri > 2 . Choosing numbers i so i  where i  4 , 1 = (1   ) i  1 i i  6 , ri i = 2 , that ri (i ) = i 2 5  3   2i (for example, i > 21 ) ri 2i 6i 3 i  1 i 1 4 i = 1,2 , and, consequently, i r = 2(5i  3) , 2 1, i i  = 3(i  1) (note that 3(i  1) i 34  r < 10 ). Due to we obtain that 1, 2i i   L2 (I ,W i 1 ()) . Since the 5i  3 15 3 embedding i  1 W () in a Hilbert space L2 () is estimates (64), (68) from (32) we obtain that  (i) (77) bounded, then П ρi,nu n П r L i (QT )  c(Eˆδ,0 ). i   C 0(I , L2 ()), i = 1,2. (82) Using the results for the regularity of solutions of the parabolic Neumann problem from equation (9) we obtain From (82) and the estimate (49), using the interpolation inequality, we obtain property (74). Since  2  2 εПρi,n П r L i (QT )  c(Eˆδ,0 ), for every vector a  L (QT ) ,  t a  L (QT ) the equality t (Pn a) = Pnt a is valid a. e. in QT , then from П ρ П  c(β ,Ω)Пρ0 П . (78) equations (33) there follow the identities i,n (I,L L ri (Ω)) i L i ri (Ω) 2 2  P ( u (i)) dx =  u ( j): (P )dx  (   )divu ( j) div(P )dx  ( u (i) u (i)) : (Pn)dx  t n i,n n    ij n  j=1  n ij ij n n  j=1     i,n n n     ( i   i )div(P  )dx   ( )u (i) P ( )dx  (1)i1 a(u (2)  u (1) )P dx,t  I ,   D(), i = 1,2. (83) i,n  i,n n i,n n n   n n n Lemma 10. There is a uniform estimate hand part of equality (83) 2 П  P(ρ u (i) )П t  c(Eˆδ,0 , δ, ε, ), i=1,2 (84) ( ) =  ( j ): ( ) . t n i,n n L i (I,W 2,2(Ω )) J1  ij u n  j=1  Pn dx Obviously, the Proof. Let us consider the summand in the rightequality |J (i)()| 2 c (μ ) || u (j)|| W ПPП 2 || J (i) || 2 ( , 1,2 ( 1 )  c (ij ,ij )ˆ  ,0 1  1 ij   j=1 n L2(Ω )  n 1,2 0  (Ω ) L I W  ) 2  c П u (j) П  ПП . || J (i) ||  c(ˆ ,  , ). (87) 1  j=1 n L2(Ω ) 0 W 1,2(Ω ) t 2 L i (I ,W 2,2 ())  ,0 ij ij Let us consider the functional 2 Thus, ПJ (i)П  c Пu (j)П , and due to J (i) () = ( u (i) u (i)) : (P )dx. By virtue of 1 W 1,2(Ω) 1 j=1 n L2(Ω ) 3 i,n n n n  the estimate (49) (i) ПJ1 П 2 1,2  c1Eˆδ,0. inequality ρ u (i) u(i) dx  П ρ u (i) П  L (I,W  (Ω ) (85)  i,n n n i,n n 2 β Since 2  3 t = i < 2 i 4i and the embedding W 1,2 () in Ω n  П u (i) 0 ,6 П  ПП 0, 6 βi 0 , i βi 1 W 2,2 () is also bounded, (85) implies the estimate 2 βi  3 (i) П J1 П t  c(Eˆδ,0 ,μij ,T). (86) and the estimates (49), (64), we obtain L i (I,W  2 ,2(Ω )  (i)  (i) (88) Similarly, the following summand is estimated 2 П ρi,nu n  u n П 2 6 βi 4 βi 3  c(Eˆδ,0 ), i=1,2. J (i) () =  (   )divu ( j ) div(P )dx, i. e. L (I,L () 2  ij ij n n  j=1 Consequently, |J (i)()|  П ρ u (i) u (i) П  П (P )П  c(Eˆ )|||| (89) 3 i,n n i n 6 β 0, 4 βi 3 i n 6 β 0 , 2 βi 3 δ,0 W 2 ,2(Ω ) (since 6i 2i 3  6 when i  3 , then, by virtue of the boundedness of the embedding W 1,2 () in L6 () , the inequality П (P )П  c |||| is valid). From inequality (89), these estimates follow. i n 6 β 0, 2 βi 3 0 W 2 ,2(Ω ) || J (i) 3 ||L2 (I ,W 2,2 ())  c(ˆ  ,0 ), || J (i) 3 t || L i (I ,W 2,2 ())  c(ˆ  ,0 ). (90) To estimate the functional J (i) 4 () =  i,n n  i div(P )dx , we use the inequality  | J (i) () |||  i ||  || div(P ) || ||  i || c ||  || , 4 i,n L4/3 () n L4 () i,n L4/3 () 0 W 2,2 () from which it follows that J 4 || (i) 4/3 2,2 ||  L ( I ,W ()) 3 () ||  || c 4 i 0 i,n 4/3 L . i (QT ) (91) Since t < 5 < 4 , then it follows from (91) and (57) that i 4 3 ||J || (i) 4 t 2 2  c(Eˆδ,0 ,ε,). )) (92) Estimation of the functional J (i) 5 () =   i,n n i div(P )dx is similar to the previous one and, as a result, we arrive at inequalities (i) П J5 П 4 / 3  2,2 (i)   c(ε, , Eˆδ,0 ), П J5 П t  c(ε, , Eˆδ,0 ) . (93) L (I,W (Ω )) L i (I,W 2 ,2(Ω )) Let us consider the summand J (i) () =  (  )u (i) P ()dx. By virtue of the embedding W 2,2 () C (()) we 6 i,n n n  6 obtain that | J (i) () |  ||  i,n n  u (i)|| L ti () c0 2,2 ||  || W () , and from this and the estimate (79) it follows that (i) || J4 i || t L (I ,W 2,2())  c(ˆ  ,0 ). (94) For the functional J (i) () = (1)i1 a(u (2)  u (1))P dx, it is obvious that the inequalities 7 n n n  | J (i) () | a || u (2) u (1)||  || P  || , || J (i) ||  a || u (2) u (1)|| are valid. Due to the estimate (49), it follows 7 n n L2 () n L2 () 7 L2 () n n L2 () (i) that || J7 2 ||L (I ,L2 ())  c(ˆ  ,0 ), and therefore the following estimate is true || J (i) 7 t || L i (I ,W 2,2())  c(ˆ  ,0 ). (95) From the equality iu i,nu n i u Pn i,nu n Pn i,nu n i,nu n Summing the relations (86), (87), (90), (92)-(95), we (i) (i)= [ ( (i))] [ ( (i)) (i)] arrive at the estimates (84). The Lemma is proved. Lemma 11. The following relations are valid: and properties (100), (101) it follows that (96). Property (97) follows from the estimate (88) and relations  u (i)  u (i) strongly in L2 (I ,W 1,2 ()), i = 1,2, (96) (59), (61). i,n n i 6i Indeed, due to weak convergence 2 1,2 n u (i)  u (i) in  (i )  (i )  (i )  (i ) 2 ( , 4 i 3 ( )), Ł (I ,W ()) and the embedding theorem, we have i,nu n  u n  iu  u weakly in L I L  u n  u strongly in L (I , L ()) for any q < 6 . i = 1,2. (97) that (i) 2 q  Proof. From the inequalities Let the function   L2 (I , L i ()) , where  (i)  (i) 1  i 1  1 , i > 1 . Let us consider the integral W ПPn(ρi,n u n )П 1,2 (Ω ) L (Ω )  П Pn(ρi,n u n )П 2  i 2i q T  П ρ u (i)П  c(Ω)Пρ u (i)П I (1) =  u (i) (u (i)  u (i)) dxdt . The sequence i,n n L2(Ω) i,n n L ri (Ω) n i,n n n 0  (the numbers ri > 2 are defined in (76)) and estimates  ,  (i) (77) we obtain that the uniform inequalities are valid i nu n is bounded in space L2 (I , Lq ()) , 1  1 = 1 and, therefore I (1) ( )  0 as n   . Since n i,n n r П P (ρ u (i) )П L i (I,W 1,2Ω)  c(Eˆδ,0 ), i=1,2. (98) q q n 1  4i  3 , then the space In addition, it follows easily from the estimates (77) and  i >  i , where i = 1 6i the projection properties Pn  (i) that (99)  L i () is everywhere dense in   L i () . Let П Pn(ρi,nu n )П r  c(Eˆδ, ), i=1,2.  2 L i (I,L2Ω) 0   L (I , L i ()) be an arbitrary element. For an   Since the chain of embeddings arbitrary  > 0 , there exists  0  L2 (I , L i ()) such L2 () W 1,2 () W 2,2 (), takes place, we 0 that Пψ  ψ П o <ε . Then L2(I,L i (Ω ) conclude from the estimates (84), (98), (99) and the Aubin-Lions theorem that (after a transition, if required | (1) ( ) | | (1) () (1) ( ) | | (1) ( ) | . to a subsequence), the sequence In  In In 0  In 0) n P ( u (i)) converges strongly to u (i) in L2 (I ,W 1,2 ()). (100) n i,n i The first term on the right-hand part of this inequality, by virtue of (88), can be estimated as follows Due to the assertion of Lemma 2, for n  N ( ) , we ( 1 ) 0 n  ( 1 )  (i) n (i) (i)   (i) |In (ψ) I (ψ )| П ρi,nu n (u u )П 6 βi obtain sup W || (Pn  I )i,nu n ||  (i) 1,2 () <  (77) L2(I,L4 βi  3(Ω ))    , u (i)L2 () L () || i,nu n || 2  Пψ  ψ0 П σ  c(Eˆδ,0 )  ε. i n n L2(I,L i (Ω ) implies the boundedness of the sequence { , u (i)} in (1) i n n According to In (0 )  0 that has been proved L2 (I , L2 ()) and, consequently, above with n    and, thus,  n I (1) ( )  0 with n   || (Pn  I ) i,n n u (i)|| L2 ( I ,W 1.2 ())  0 as n  . (101) for any   L2 (I , L i ()) . Let us consider the integral T Proof. The sequences { } and {  } , I (2) () = ( , u (i)  iu (i))  u (i) dxdt. For any i,n n=1 t i,n n=1 n i n n 0  1, 2i     2i i = 1,2 , are bounded in the spaces L2 (I ,W i 1 ()) and   L2 (I , L i ()) element u (i)  L1 (I , Li 1 ()) . By 1, 2i 2 i 1 * , respectively (see (78), where virtue of the * -weak convergence  , u (i) iu (i) in L (I ,(W ()) ) 2i i n n ri (i )  2i , ri > 2 and (65)). L (I , L  i 1 ()) , we obtain that  n I (2) ( )  0 as n   i  1 In particular, we have i,n  C(I , L2 ()), i. e. the    L2 (I , L i ()) . From the equality functions t  Пρi,n(t)П L2(Ω) are continuous on the T ( , u (i) u (i)   u (i) u (i)) dxdt = I (1) ( )  I (2) ( ) segment [0,T ] . On the other hand, estimates (78), (81) i n n n i 0  n n imply that the sequences { 1,t i,n }  n=1 , i = 1,2 are bounded, and what we have proved above, property (97) follows. in space W i (QT ) (recall that ti < ri ), and therefore, by Lemma 11 is proved. Proposition 12. The following properties are valid: T virtue of the compactness of the embedding W 1,ti (Q ) in qi () , 1 < q < q* = 3ti i ( q*  1,085 ) we can assume ||  (t)   (t) ||  0, ; L i i 4  ti i,n i L2 () that, if necessary, passing to subsequences, {i,n }, n  , t uniformly in t t  [0, T ]; (102) strongly converging in L qi ()  t [0,T ] .  2 divu (i) dxd   2 divu (i) dxd . (103) By virtue of the interpolation inequality i,n n i 0  0  Пρ (t)  ρ (t)П  cП ρ (t)  ρ (t)П θ  Пρ β (t)  ρ (t)П1θ , 1 =   1   , 0 <  < 1 i,n i L2(Ω ) i,n L i qi (Ω ) i,n i L i (Ω ) 2 qi i and the estimate (49), we obtain (102), from which, in particular, Let us prove the formula (103) i,n L2 ()  i L2 () , n   uniformly in t [0,T ]. t t  2 divu (i) dxd   2 divu (i) dxd  ( 2   2 )divu (i) dxd   2 (divu (i) u (i))dxd = I (1)  I (2) . i,n n 0  i 0  Qt i,n i n i n n n Qt The first summand I (1) n is estimated as | I (1) n L (Q ) ||| i,n  i || 4 T L (Q )  || i,n  i || 4 T (i)  || divu n || 2 . L (QT ) From the n inequalities (49), (51), (67) it follows that I (1)  0 при n n  . The second summand I (2) tends to zero, due to the weak convergence (59) and the fact that  2  L2 (Q ) . i T Proposition 13. The following property is valid i,n  i is strong in T L2 (Q ), i = 1,2. (104) Proof. From equations (46), conditions (48) and u (i)= 0 on  , we obtain the identity t i,n 2 2  2 1 0 2 t 1  2  (i) . П ρ П L (Ω ) ε |ρi,n| dxdτ  0 Ω П ρ П 2 L (Ω ) i 2  2 0 Ω ρi,n divu n dxdτ (105) On the other hand, by Lemma 9, equations (9) and conditions (10), (11) imply similar identities for the limit functions i , u (i ) : t i 2 2  2 1 0 2 t 1  2  (i) . П ρ П L (Ω ) ε |ρi| dxdτ  0 Ω П ρ П 2 L (Ω ) i 2  2 0 Ω ρi divu dxdτ (106) From Proposition 11 and the identities (105), (106) we obtain that П ρi,n П L2(QT )  П ρi П L2(QT ) , since i,n  i T is weak in L2 (Q ) (on the basis of the estimate (78), since ri > 2 ), the Proposition 13 is proved. Corollary 14 The following relation is valid i,n n u (i)i u (i) weakly in L1(QT ), i = 1,2. (107) Proof. This property is easily obtained from equality  ( u (i)   u (i))dxdt =  (   )u (i)dxdt   (u (i)  u (i))dxdt,   L (Q ), i,n n i QT i,n i n i n T QT QT n since each of the integrals in the right-hand part tends to zero by virtue of the properties of the sequences {i,n} and {u (i)} . Proposition 15. The following formula is valid: lim  (  )u (i) P dxdt =  (  )u (i) dxdt,   D(). n QT Proof. Indeed, the equality i,n n n i QT  i,nu n  iu  dxdt (108) lim ( n QT (i) (i)) = 0 holds according to property (107). Let us prove the formula lim  ( u (i))  (P   )dxdt = 0. (109) According to Holder's inequality n QT i,n n n (ρ u (i) ) (P   )dxdt  П ρ u (i) П  П(P  I)П ;  i,n n n QT i,n n L6 / 5(QT ) n L6(QT ) П(P I)П =T1/ 6 П(P E)П T 1/ 6c (Ω)П(P E)П 0 with n   in view of the properties of the  n L6(QT ) n  L6(Ω )  0 n  5  3 0 W 1,2(Ω )   (i) projection operator. Under the condition i  15 , ti = i  6 4i 5 and, thus, the norms L || i,n u n || 6/5 (QT ) are uniformly bounded due to the estimate (79). The formula (109) is proved, and together with it, with the help of (108), the Proposition 14 is proved. PROOF OF PROPERTY (II) OF THEOREM 1  Let us prove that the limit functions i = i, , u (i)= u (i) satisfy equations (8) as a result of the passage to the limit in Galerkin equations (33). On the basis of the identities (83) we have 2 2   P ( u (i))  dxd =   ij u nj): (Pn)dxd   (ij  ij )divu nj) div(Pn)dxd  t n i,n QT  (i) n  (i) ( QT j=1    QT j=1  (  (i)    ( u  u ) : (P )dxd   ( i   i )div(Pn )dxd   (i n  )u n  Pn ( )dxd  i,n n n n QT i,n QT i,n , QT 6  (1)i1 a(u (2)  u (1))P dxd = (i) ,   D(), i = 1,2. (110)  n n n QT  k =1 k ,n Proposition 16. For each   D() , the following lim (i)    =  ( i   i )div dxd , (114) formulas are valid 2 n 4,n i i QT lim  = (i) 1,n   ij u ( j ): dxd , (111) lim (i) = ( )  (i)  , (115) n QT j=1 n 5,n   QT i   u   dxd 2 (i)  i1   (2)  (1)  (116) lim (i) =  (   )divu ( j ) divdxd , (112) lim 6,n = ( 1) a(u u ) dxd . n 2,n   ij ij QT j=1 n QT lim n  = (i) 3,n  (iu (i) QT  u (i)) : dxd , (113) Proof. Formula (111) follows from the inequality (i) 2  ( j):  2 (  ( j) 2  ( j)) :   ( j): (  ) , 1,n   iju QT j=1 dxd   ij u n QT j=1 u dxd   iju n QT j=1  Pn   dxd n weak convergence of sequences {u ( j )} in T L2 (Q ) and the projection properties Pn (see Lemma 2). Formula (112) is proved on the basis of the same considerations. To prove (113), let us consider the difference (i)   ( u (i) u (i)) : dxd =  ( u (i) u (i))  ( u (i) u (i)): dxd   i  i   3,n i i,n n n i ( u ( ) u ( )) : [(P  )   ]dxd . QT QT  i ,n n n n QT The first integral in the right-hand part tends to zero by (97), and the second integral is represented in the form  (i,nu n  u n ) : (Pn  )dxd =   (i,n  u n )  (u n  (Pn  ))dxd  (i) (i) (i) (i) QT   ( u (i) u (i))  (P   )dxd    QT divu (i)(u (i) (P   ))dxd = R(i)  R(i)  R(i) . i,n n n n QT i,n n n QT 1,n 2,n 3,n 1,n The summand R(i) under the condition i  15 can be estimated as follows |R(i)|  Пρ П  Пu (i) П 2  ПP   П  1,n i,n L3(QT ) n L4(QT ) n L6(QT )  (i) 2  1/ 6 1/ 6 1  (117) r 0  c (Ω)П ρ П i  Пu П 1 2 )  П(P  I)П 1 2  T  c0T  ε c(Eˆ 0 )П (P  I )П 1 2 i,n L (QT ) n W0 , (QT ) n W0 , (Ω ) δ, n W0 , (Ω (here we must take into account the limited nature of the embeddings T W 1,2 (Q ) T Lp (Q ) for all p  4 and W 1,2 () Lq () for all q  6 , and also ri (i )  3 with i  15 , in addition, (49) and (78) are taken into account. From the inequality |R(i) |  П u (i) П  П ρ u (i)П  ПP   П and the estimates (49), (77) we obtain 2 ,n n L2(QT ) i,n n L3(QT ) n L6(QT ) | R(i) | c(ˆ )T 1/6 || P   || . (118) 2,n  ,0 0 n W1,2 () 3,n Obviously, the integral R(i) can be estimated similarly: | R(i) | c(ˆ )T 1/6 || P   || . (119) 3,n  ,0 0 n W1,2() Formula (113) now follows from the inequalities (117)-(119) and the properties of the projection operator. Let us prove the formula (114). Let us consider the difference t t t        S =  i div(P  )dxd   i div dxd = ( i  i i )div(Pn )dxd  i ,n i ,n n i 0  0  i ,n 0  t   i div(P   )dxd = S (1)  S (2) . (120)  i n 0  i,n i,n By virtue of Holder's inequality, | S (1) ||| i  i ||  || div(P ) || . The multiplier || div(P ) || is i,n uniformly bounded by virtue of inequality i,n i L6/5(QT ) n L6 (QT ) n L6 (QT ) || div(P ) || = T 1/6 || div(P ) ||  c T 1/6 || div(P ) ||  c T 1/6 || P  ||  c T 1/6 ||  || . n L6 (QT ) p n L6 () 0 4 n W1,2 () 0  n W 2,2 () 0  p/ W 2,2 () Since i,n  i in L (QT ) , 1  p < i , 3 i = 1,2 (see Lemma 8), then n i, i  i i in L i (QT ) . In particular, the  convergence of the sequence { , i } takes place in L6/5 (Q ) <  6 4    . Thus we have proved the relation i n (1) T  5 3  0 . Si,n  as n   (121) t 1  1  Let us rewrite the summand S (2) in the form of S (2) = 2 2 i  ( 2 i )(P   )dxd . In view of the estimate i,n 1  i,n i i n 0  (53), the function ( 2 i )  L2 (0,T ; L2 ()) = L2 (Q ) , and according to estimate (55)   i  L4/3 (0,T ; L2 ()) , and i T i i,n therefore the integral S (2) is estimated as T 1 ( 2 ) βi 1 βi   1   βi  T β 1/ 2  |S |  2 (П ρ 2 П П(ρ 2 )П ПP   П )dt  c(Ω(Ω P   П П (ρ 2 )П  П ρ i П dt  . i,n  i L3(Ω ) 0 i L2(Ω ) n L6(Ω ) n W 1,2(ΩΩ i  L2(QT )   i 0 L3 / 2(Ω )   i,n Hence, S(2)  0 as n  , and together with relations (120), (121) this proves the formula Similarly, using estimates (57), we prove that Si,n  0 as n  . (122) t    t    n lim n i,i div(Pn )dxd = i i div dxd . (123) 0  0  The formula (114) is a collorary of (122), (123). The formula (115) is proved in the Proposition 15. The proof of (116) n is elementary in view of the convergence of the sequences {u (i )} , i = 1,2 in T Lq (Q ) , q < 4 and the properties of the projection operator Pn . The Proposition 16 is proved. Due to Proposition 16, it follows from (110) that t t 2 t 2 lim  P ( u (i))  dxd = ij u ( j ): dxd  (ij  ij )divu ( j ) divdxd  t n n 0  i,n n 0  j=1 0  j=1 t t t t  ( u (i) u (i)) : dxd  (i  i )divdxd   ( )u (i)dxd  (1)i1 a(u (2) u (1))dxd . (124) i i i i 0  0  0  0  Hence, taking into account the properties (61) and (100), we see that 2i  u (i) L (I , Li 1 ()), i = 1,2,   u (i)   L1 (I ),   D(), i t i   (i) 2i 0 i 1 and therefore, due to Proposition 2.18, there exist q  C (I , Lweak ()) , i = 1,2 , such that for t  I a. e. we have the equality i q (i)(t) =  (t)u (i)(t) п. в. в . Moreover, as a result of the change in the velocity field 2i u (i ) on the set of measure zero in I , we obtain that  u (i) C 0(I , Li 1 ()). i weak This implies the formulas lim  u (i)dx = q (i) dx , i = 1,2. . From (124), by virtue of the arbitrariness t  (0,T ) , the following equalities follow i n 0 t0   2 2 t iu (i) dx = ij u ( j): dx  (ij  ij )divu ( j) divdx  (iu (i) u (i)) : dx       j=1  j=1   ( i   i )div dx   (  )u (i) dx  (1)i1 a(u (2)  u (1))dx, i i i      D() a. e. in I . Hence, for each function g(t)  D(I ) we obtain the identities T T  ρ u (i) gdxdt=  2 T μ u (j):(g)dxdt  2 (λ  μ T )divu (j) div(g)dxdt  (ρ u (i) u (i) ):  i t 0 Ω  ij 0 Ω j=1  ij ij 0 Ω j=1  i 0 Ω T :(g)dxdt  γ β  T (ρ i  δρ i )div(g)dxdt  ε T (ρ )u (i)(g)dxdt  (  1 )i 1 a(u ( 2 ) u ( 1 ) )g dxdt. (125)  i i 0 Ω  i  0 Ω 0 Ω Since the family of functions {(x)  g(t)}, i 2 | |2 i 2 2 . ,   D() , g  D(I ) is everywhere dense in D(QT ) , i,n i,n  i i a e in  the identities (125) prove that the limit functions  2 n | 2  2 | | |2 . , i = i, , , u = u  (i)  (i)  , satisfy the regularized i,i i,n  i i i a e in  momentum balance equations (8). and then, according to Fatou's lemma, we obtain that T  2 T  2 PASSAGE TO THE LIMIT IN THE ENERGY  (t) i |  |2 dxdt  lim  (t) , i | i,n |2 dxdt, INEQUALITY AND THE ESTIMATES THAT ARE INDEPENDENT OF THE PARAMETER  i i i n 0  n 0  T T Let us prove the postulation (v) of Theorem 1 on energy inequalities. Let us turn to inequalities (40)-(42),  2  (t)i i | i |2 dxdt  lim  (t)i,i  2 n n | i,n |2 dxdt. which hold on solutions of Galerkin equations. It is easy 0  0  to see that, by properties (62) and (97), the following Thus, inequality (40) for i,n = i, , ,n , relation is valid: ˆ ( , u (i))  ˆ ( , u (i)) в D(I ).  (i)  (i) implies the inequality (13) for the limit i,n n i T u n = u  , ,n Since the functional u    | u |2 dxdt is convex and (as n   ) functions i =  i, , , u = u  (i)  (i)  , . The 0  continuous in 0 L2 (I ,W 1,2 ()) for any nonnegative inequality (15) follows from (41) in a manner similar to the previous one. function   C 0(I ) , we can assert by virtue of property (59) that PROOF OF THE A PRIORI ESTIMATES (VI) OF THEOREM 2.1 T T n  (t) | u (i)|2 dxdt  lim  (t) | u (i)|2 dxdt, i = 1,2. From inequalities (49), (50) and formulas (59), due to the weak semicontinuity from below, we obtain 0  n 0  From the properties of (67), (104) it follows (passing, if necessary, to subsequences) that estimates (15)-(16). From (61), (64), the estimate (17) follows. The estimate (18) follows from (60). The estimate (19) is a corollary of (78), (79).

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