Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary
The problem of distortion of viscous incompressible uid with a great number of solid particles with given velocities is considered. The diameters of particles and the distance between them tend to zero, and the number of particles tends to infinity. The asymptotic behavior of the solutions of the l...
Gespeichert in:
| Datum: | 2007 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/7612 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Homogenization of a linear nonstationary Navier - Stokes equations system with a time-variant domain with a fine-grained boundary / N.K. Radyakin // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 342-364. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860089883381465088 |
|---|---|
| author | Radyakin, N.K. |
| author_facet | Radyakin, N.K. |
| citation_txt | Homogenization of a linear nonstationary Navier - Stokes equations system with a time-variant domain with a fine-grained boundary / N.K. Radyakin // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 342-364. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| description | The problem of distortion of viscous incompressible uid with a great number of solid particles with given velocities is considered. The diameters of particles and the distance between them tend to zero, and the number of particles tends to infinity. The asymptotic behavior of the solutions of the linear system of Navier-Stokes equations is considered. In a homogenized model there appears an additional term containing the strength tensor of a single particle.
Розглянуто задачу про збурювання в'язкої нестислої рідини потоком великої кількості твердих часток, що рухаються з заданими швидкостями. Досліджено асимптотичну поведінку розв'язків лінійної системи рівнянь Нав'є - Стокса, які описують цю задачу, коли розмір часток і відстань між ними зменшуються, а кількість часток необмежено зростає. Одержано усереднені рівняння, що описують рух суспензії.
|
| first_indexed | 2025-12-07T17:22:26Z |
| format | Article |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 3, pp. 342�364
Homogenization of a Linear Nonstationary
Navier�Stokes Equations System with a Time-Variant
Domain with a Fine-Grained Boundary
N.K. Radyakin
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:radyakin@ilt.kharkov.ua
Received May 5, 2006
The problem of distortion of viscous incompressible �uid with a great
number of solid particles with given velocities is considered. The diameters
of particles and the distance between them tend to zero, and the number of
particles tends to in�nity. The asymptotic behavior of the solutions of the
linear system of Navier�Stokes equations is considered. In a homogenized
model there appears an additional term containing the strength tensor of a
single particle.
Key words: Navier�Stokes equations, solid body dynamics, homogeniza-
tion, suspension.
Mathematics Subject Classi�cation 2000: 35B27, 35Q30, 74Q10, 76M30.
1. Introduction
This problem appeared in relation to the construction of a homogenized model
of suspensions of small solid particles in a viscous incompressible �uid. Originally
it is a rather complex one; it is described by the Navier�Stokes equations and
the equations of solid body dynamics for particles. It is necessary to study the
asymptotic behavior of the solutions of this system when the radii of particles and
the distances between them tend to zero. A direct analysis of the problem faces
some di�culties, as the domain occupied by the �uid is not known beforehand.
It is natural to divide the problem into two parts, the �rst of which is to study
the asymptotic behavior of the carrier �uid, disturbed by small particles, the
trajectories of which are known. This problem for the linear NS system is solved in
this paper. Our asymptotic analysis allowed to determine a homogenized system
c
N.K. Radyakin, 2007
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
of equations for the carrier �uid. In fact, this system happens to be incomplete
and for its completion it is necessary to study the in�uence of the carrier �uid on
the particles (to �nd the Stocks-forces). A similar result is announced in [4].
2. Problem Statement and Formulation of the Main Result
Consider a large number of small solid particles Qi
" that move in a �uid �lling
the volume
� R
3 . The mass centers of the particles move according to given
trajectories ~xi"(t) while the particles themselves rotate around ~xi"(t) with given
angular velocities ~�i"(t). Assume that the functions ~xi"(t),
~�i"(t), as well as the
surfaces @Qi
" of the particles and the border @
belong to the class C2. Assume
also that the particles while moving do not collide with each other and with the
boundary @
(they remain at positive distances from each other).
Let us introduce the notation: Qi
"(t) is a location of the particle with number
i at the moment t (the domain occupied by the particle), QiT
" is a trace of the
particle i in R4 = R
3�[0;1) after moving for time T; @QiT
" is a side surface of the
trace;
T =
�(0; T ];
T
" =
T n
N"[
i=1
QiT
" ;
"(t) =
n
N"[
i=1
Qi
"(t); Q"(t) =
N"[
i=1
Qi
"(t);
"(0) �
".
Consider the following initial boundary valued problem in domain
T
" :
~u"t � ��~u" = �rp" + ~g(x; t); div~u" = 0; (x; t) 2
T
" ; (2.1)
~u"(x; t) = ~vi"(t) +
~�i"(t)� (~x� ~xi"); (x; t) 2 @QiT
" ; (2.2)
~u"(x; t) = 0; (x; t) 2 @
� [0; T ]; (2.3)
~u"(x; 0) = ~U"(x); x 2
"; (2.4)
where � > 0 is the viscosity of �uid, ~u"(x; t) and p"(x; t) are the �elds of velocities
and pressures in the �uid, ~xi"(t) is a center of mass of the particle i; ~vi"(t) =
_~xi"(t)
is a velocity of the center of mass, ~�i"(t) is an instantaneous angular velocity of
particle i, ~U"(x) 2W
3=2
2 (
") is a divergent free vector function (the initial velocity
�eld of the �uid), which satis�es the following conditions: ~U"(x) = ~vi"(0)+
~�i"(0)�
(~x � ~xi"(0)) for x 2 @Qi
"(0);
~U"(x) = 0 for x 2 @
, ~g(x; t) 2 L2(
T ) is a volume
force, which act on the �uid.
This problem describes a linear approximation of evolution in the time of the
velocity �eld ~u" and pressures p" of the viscous incompressible �uid, disturbed
by the solid particles moving in it by given trajectories. The boundary condi-
tions (2.2) and (2.3) correspond to the condition of adhesion on the moving solid
particles Qi
"(t) and nonmoving boundary @
.
As it is known from [1, 2], there exists a unique solution f~u"(x; t), p"(x; t)g
of the problem (2.1)�(2.4) such that ~u" 2 W
2;1
2 (
T
" ), p" 2 L2(
T
" ), and the time
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 343
N.K. Radyakin
interval T does not depend on the number of particles N" and on their sizes. Let
us extend the vector of the velocity of the �uid ~u" onto the sets Q
iT
" in accordance
with the equalities (2.2) and consider the asymptotic behavior of the extended ~u"
as " ! 0, i.e., when the radii of particles tend to zero as O("3) and the number
of particles N" increases as O("�3).
We assume that the instantaneous angular velocities of the particles ~�i"(t)
and their derivatives are uniformly bounded with respect to ". Assume also
that the centers of masses of particles move according to the trajectories ~xi"(t) =
~�(~�i"; t), i = 1; 2; : : : ; N", where ~�(x; t) is a twice di�erentiable vector function
on R3 � [0; T ]. This function is a one-one map of
into
for any t 2 [0; T ], in
the following way ~�(x; 0) = ~x. Let ~�i" be the location of the center of mass of
particle i in the moment t = 0. Suppose that the points ~�i" are initially located
in the domain
0 �
, such that for each t 2 [0; T ], Æ-neighborhood �Æ(
0; t)
(Æ = max
1�i�N"
(di")
2=3 ) of the domain �(
0; t) belongs to
. Here and further by di"
we denote the external diameter of the set Qi
".
In order to describe the asymptotic behavior of the solution ~u"(x) as " ! 0,
let us de�ne the stress tensor of the sets Qi
", which characterizes their mass and
orientation in space.
Let Q be a bounded closed set in R
3 with a smooth boundary @Q. Now let
us consider the following boundary problem in R3 nQ:
�~v(x) = rp(x); div~v = 0; x 2 R3
nQ;
~v(x) = ~ek; x 2 @Q; ~v(x) = O(1=jxj); jxj ! 1: (2.5)
From the results described in [1] it follows that there is a unique solution ~vk
of this problem with the �nite energy kr~vkkL2(R3
nQ) <1. Suppose
Ckl(Q) =
Z
R3
nQ
(r~vk;r~vl)dx =
Z
R3
nQ
3X
i;j=1
@vki
@xj
@vli
@xj
dx; k; l = 1; 2; 3: (2.6)
It is obvious that the matrix fCkl(Q)g3k;l=1 does not depend on the shifts of
Q, and because of the linearity of the problem (2.5), when rotated, Q transforms
as a second rank tensor:
Ckl(�Q) =
3X
i;j=1
Cij�ik�jl; (2.7)
where f�ikg
3
i;k=1 is the rotation matrix. It is easy to see that under homothetic
contraction of Q the components of this tensor decrease proportionately to the
diameter of Q.
344 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
We de�ne fCkl(Q)g3k;l=1 as a stress tensor of the set Q as it is similarly to
newtonian capacity (see [3]) characterizes both the massiveness of the set Q and
its orientation in space.
We denote by C(Qi
") = fCkl(Q
i
")g
3
k;l=1 a stress tensor of the set Q
i
";
X
(G)
C(Qi
")
is the sum over those values of the index i, for which Qi
" is strictly inside the
domain G �
; T (Qi
") is the minimal ball containing Qi
"; r
i
" is the distance from
T (Qi
") to
N"[
j 6=i
T (Qj
")
[
@
; di" = sup
x0;x002Qi
"
jx0 � x00j is the diameter of the set Qi
".
The main result of the paper is the following theorem:
Theorem 1. Let the following conditions hold as "! 0:
1) lim
"!0
max
1�i�N"
di" = 0, d" < C1"
3 (C1 > 0).
2) For each arbitrary G �
and each t 2 [0; T ]
lim
"!0
X
(G)
Ckl(Q
i
"(t)) =
Z
G
Ckl(x; t)dx;
where C(x; t) = fCkl(x; t)g
3
k;l=1 is a continuous tensor in
T (the sum
X
(G)
is over
those values of index i, for which Qi
"(t) 2 G).
3) For each t: t 2 [0; T ] di" < C(ri"(t))
3, where C does not depend on i, " and t.
4) The sequence of the initial vector functions f~U"(x); "! 0g, extended to the
set
N"[
i=1
Qi
"(0) by using the equation (2.2), weakly converges in W 1
2 (
) to a vector
function ~U(x) 2W 1
2 (
).
Then the sequence f~u"(x; t); " ! 0g of solutions of the problem (2.1)�(2.4)
converges in L2(
T ) to the solution ~u(x; t) of the following problem:
~ut � ��~u+ �C(x; t)[~u� ~W (x; t)] = �rp+ ~g(x; t); (2.8)
div~u(x; t) = 0; (x; t) 2
T ;
~u(x; t) = 0; (x; t) 2 @
� [0; T ]; (2.9)
~u(x; 0) = ~U(x); x 2
; (2.10)
where ~W (x; t) = ~�t(�; t)j�=��1(x;t) and ��1 is an inverse mapping onto ~�.
R e m a r k 1. Let the angular velocities ~�i"(t) change smoothly with transition
from one particle to another, namely the following equality is true: ~�i"(t) =
~�(~xi"; t),
where ~�(x; t) is a vector function continuous on x and di�erentiable with respect
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 345
N.K. Radyakin
to t, and ~xi" is the position of the center of mass of particle i. Then tensor C(x; t)
from condition 2 of Th. 1 can be computed according to the following formula:
Ckl(x; t) =
3X
i;j=1
Cij(�
�1(x; t); 0)�ik(�
�1(x; t); t) ��jl(�
�1(x; t); t)
�
�����det@
~�(x; t)
@x
�����
�1
�
(�
�1(x; t));
where �
(x) is a characteristic function of the domain
, and �(x; t) is a matrix,
the columns ~�k(x; t) of which are the solutions of the Cauchy
_~�k(x; t) = ~�(x; t)�
~�k(x; t), �jk(x; 0) = Æjk. One can see it easily by using (2.7) and taking into
account that the rotation operator of a solid particle (matrix �i(t)) satis�es the
following equation _� i(t) = ~�i(t)��i(t), where ~�i(t) is the instantaneous angular
velocity of particle i, and the vector product is applied to column vectors of matrix
�i(t). Thus it is su�cient to compute the limit in condition 2 only for t = 0.
3. Additional Statements
In this section we establish some additional statements (Lems. 1, 2, and 3)
and derive apriori estimates for r~u"(x; t) and ~u"t(x; t) in L2(
T
" ), which we will
use later in the proof of Th. 1. Before formulating Lem. 1, we introduce the
following notation. Let f~vk;i(x; t); pk;i(x; t)g be the solution of the problem (2.5),
when Q = Qi
"(t), and the vector of velocity ~vk;i(x; t) is extended to Qi
"(t) by the
equality ~vk;i(x; t) = ~ek (~ek is an ort of the axis xk). Let us introduce the vector
functions ~~vk;i(x; t) to satisfy the equalities:
rot~~vk;i(x; t) = ~vk;i(x; t); (3.1)
as jx� xi(t)j � ri"(t) + di", and de�ne the vector functions
~w"(x; t) =
N"X
i=1
(
rot
3X
k=1
~~vk;i(x; t)vik(t)'
i
"(x� xi(t); t)
+
1
2
rot
3X
k=1
�ik(t)~e
k
X
j 6=k
(xj � xij(t))
2
2
'̂i"(x� xi(t); t)
9=
; ; (3.2)
~W"(x; t) = �~w"(x; t)
�r
N"X
i=1
3X
k=1
pk;i(x; t)vik(t)'
i
"(x� xi(t); t); (x; t) 2
T
" ; (3.3)
346 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
~W"(x; t) = 0; (x; t) 2
N"[
i=1
QiT
" ;
where vik(t) and �ik(t) are the components of the vectors of velocities of centers
of the mass ~vi"(t) and the angular velocity ~�i"(t) of particle i; x
i
j(t) is one of the
components of the vector ~xi(t) = ~xi"(t) of the center of mass Qi
"(t) of particle i,
'i(x; t) and '̂i"(t) are the patch functions:
'i"(x; t) = '
�
jxj � di"
ri"(t)
�
; '̂i"(x) = '
�
jxj � di"
di"
�
; (3.4)
and '(t) is a twice continuously di�erentiable function, that equals 1 for t � 0
and 0 for t > 1=2.
It follows from conditions 1 and 3 of Th. 1 that for su�ciently small " ri"(t)�
di", t 2 [0; T ], so, taking into account (3.2) and (3.4), we establish that the vector
function ~w"(x; t) satis�es the boundary conditions (2.2), (2.3) and is a divergent
free function.
Lemma 1. Let conditions 1 and 3 of Th. 1 hold. Then ~w"(x; t) converges to
zero in L2(
T ) and for any �xed t 2 [0; T ] in L2(
); the derivatives ~w"t(x; t) and
r~w"(x; t) converge weakly to zero in L2(
T ) and are bounded in L2(
) for any
�xed t. If, further, condition 2 of Th. 1 holds, then ~W"(x; t) converges weakly in
L2(
T ) to a vector function C(x; t) ~W (x; t), where ~W (x; t) = ~�t(�; t)j�=��1(x;t),
and matrix C(x; t) is de�ned in condition 2.
P r o o f. Because of the properties of functions 'i"(x; t), '̂
i
"(x), and equality
(3.1) and representation (3.2), it follows that
k~w"k
2
L2(
T )
� C
TZ
0
N"X
i=1
�
(di")
2ri"(t) +
(di")
4
ri"(t)
+ (di")
5
�
dt;
where the constant C depends only on max
t;i
j~�i(t)j, max
x;t
j~�(x; t)j.
Taking into account that di" = o(ri"), r
i
" < C and using the Cauchy�Schwartz
inequality, we obtain
k~w"k
2
L2(
r)
� CT max
i
di"max
t
(
N"X
i=1
(di")
2
(ri"(t))
3
)1=2
max
t
(
N"X
i=1
(ri"(t))
3
2
)1=2
� CTd"AB; (3.5)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 347
N.K. Radyakin
where we denote
A = sup
"
max
t
(
N"X
i=1
(di")
2
(ri"(t))
3
)1=2
; B = sup
"
max
t
(
N"X
i=1
(ri"(t))
3
2
)1=2
; (3.6)
d" = max
i
di"; " < 1:
According to the de�nition of ri"(t) and condition 3 of Th. 1, A and B do not
depend on " and t. As the balls with radii ri"(t)=2 and centers in points xi(t) do
not intersect, it follows from (3.5) according to conditions 1 and 3 of Th. 1 that
lim
"!0
k~w"kL2(
T ) = 0: (3.7)
Similarly, we obtain the following estimate for r~w"(x; t)
max
t
kr~w"k
2
L2(
)
� C
N"X
i=1
�
di" +
(di")
2
ri"(t)
�
� CAB <1: (3.8)
It follows from condition 3 of Th. 1 and (3.8) that r~w"(x; t) is uniformly bounded
on " in L2(
T ) and for each �xed t in L2(
).
The de�nition of the operator of rotation of the i-th particle �i(t) and the
linearity of problem (2.5) imply the following equalities:
~vk;i(x; t) =
3X
j=1
�i(t)~vj;i([�i(t)]�1(x� xi(t); 0)(~ek ;�i(t)~ej);
~~vk;i(x; t) =
3X
j=1
�i(t)~~vj;i([�i(t)]�1(x� xi(t); 0)(~ek ;�i(t)~ej): (3.9)
Now, using equalities (3.1), (3.2) and (3.9), the properties of the functions 'i"(x; t),
'̂i"(x) and taking into account that vector functions ~�i(t), ~vi(t) = _~x i are bounded
in C1([0; T ]) uniformly by epsilon, we obtain the following inequality:
max
t
k ~w"t(x; t)k
2
L2(
)
� Cmax
t
N"X
i=1
�
di" +
(di")
2
ri"(t)
( _ri"(t))
2
�
:
It is obvious that j _ri"(t)j � max
j
j~vj(t)j+max
j
k�j(t)kdj" , so the right-hand side
of this inequality can be estimated similarly to (3.8). As a result, the derivatives
~w" are bounded in L2(
) with respect to t and x and in L2(
T ) uniformly by ".
It follows from the uniform boundness of ~w"t andr~w" in L2(
T ) and equalities
(3.6) that ~w" weakly converges to zero in W 1
2 (
T ) and in accordance with the
348 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
imbedding theorem strongly converges to zero in L2(
) for each t 2 [0; T ]. Thus,
the �rst part of Lem. 1 is proved.
Now consider a vector function ~W"(x; t), de�ned by the equalities (3.2) and
(3.3). Using the equality (3.1) and taking into account that ~vk;i(x; t) is the solution
of the problem (2.5) with Q = Qi
"(t), and due to the divergent free of ~~vk;i(x; t):
�~~vk;i(x; t) = �rotrot~~vk;i(x; t) = �rot~vk;i(x; t), it is not di�cult to obtain the
following inequality:
j ~W"(x; t)j �C
N"X
i=1
(
3X
k=1
jpk;i(x; t)jjr'i"(x� xi(t); t)j
+
3X
k=1
1X
l=0
jDl~~vk;i(x; t)jjD3�l'i"(x� xi(t); t)j
+ 2
3X
k=1
1X
l=0
jDl~vk;i(x; t)jjD2�l'i"(x� xi(t); tj
+6
2X
l=0
jx� xi(t)jljDl+1'̂i"(x� xi(t))j
)
; (3.10)
where C = max
i
max
t
n
j~vi(t)j; ~�i(t)j
o
, ð jDl~uj =
X
j�j=l
3X
i=1
jD�uij
2
!1=2
.
Using this inequality, as well as the estimates (3.5), (3.7), (3.8), Lem. 1 and
taking into account that di" = o(ri"(t)), r
i
"(t) < C, we obtain
k ~W"(x; t)kL2(
T ) � C
8<
:
TZ
0
N"X
i=1
(di")
2
(ri"(t))
3
dt+ T
N"X
i=1
di"
9=
; � CT [A2 +AB];
where C does not depend on ", and A and B were de�ned in (2.6). This inequality
and the conditions 3 of Th. 1 imply that ~W"(x; t) is bounded in L2(
T ) uniformly
with respect to ".
Let ~ (x; t) be the arbitrary vector function from C2(
T ) and
J" =
TZ
0
Z
( ~W"(x; t); ~ (x; t))dxdt: (3.11)
Considering (3.2)�(3.3) we represent J" in the form J" = J1
" + J2
" , where
J1
" = �
TZ
0
Z
"(t)
N"X
i=1
3X
k=1
(�[~vk;i(x; t)'i"(x� xi(t); t)]
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 349
N.K. Radyakin
�r[pk;i(x; t)'i"(x� xi(t); t); ~ (xi(t); t))vik(t)dxdt;
and the following inequality is obtained for the integral J2
"
jJ2
" j � C
TZ
0
N"X
i=1
�
(di")
4 + di"r
i
"(t) + di"[r
i
"]
2
dt
� CT
h
d1=3" A2=3B4=3 + d2=3" A2=6B10=6 + d3"AB
i
;
this, together with conditions 1 and 3 of Th. 1, implies that
lim
"!0
J2
" = 0: (3.12)
Furthermore, taking into account that f~vk;i(x; t); pk;i(x; t)g is the solution of
the problem (2.5) with Q = Qi
"(t), the integral J
1
" can be represented as follows
J1
" = �
TZ
0
N"X
i=1
8<
:
3X
k;l=1
Z
R3
(r~vk;i(x; t);r~vl;i(x; t))dx � vik(t)
~ (xi(t); t)
9=
; dt
= �
TZ
0
N"X
i=1
3X
k;l=1
Ckl(Q
i
"(t))v
i
k(t)
~ (xi(t); t)dt:
Hence by virtue of condition 2 of Th. 1, the equalities ~vi(t) =
_~xi(t) =
~v(x; t)jx=xi(t), ~v(x; t) = ~�t(�; t)j�=~��1(x;t)
and the smoothness of functions ~ and
~v, it follows that
lim
"!0
J1
" = �
ZZ
T
�
C(x; t)~v(x; t); ~ (x; t)
�
dxdt:
Because the vector functions ~W"(x; t) are bounded in L2(
T ) uniformly with
respect to ", it follows from the equalities (3.11) and (3.12) that ~W"(x; t) converge
weakly in L2(
T ) to �C(x; t)~v(x; t) as "! 0. Lemma 1 is proved.
To prove Th. 1 we have to use the estimates in the space L2(
) of the partial
derivatives with respect to x and t of the vector function ~v"(x; t) = ~u"(x; t) �
~w"(x; t), where ~u" is the solution of problem (2.1)�(2.4) and ~w" is de�ned in (3.2).
To obtain these estimates consider the boundary volume problem for the function
~v"(x; t)
~v"t � ��~v" = �rq" + ~g"(x; t); div~v" = 0; (x; t) 2
T
" ; (3.13)
350 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
~v"(x; t) = 0; (x; t) 2
N"[
i=1
@Qi
"
[
@
!
� [0; T ]; (3.14)
~v"(x; 0) = ~V"(x); x 2
"; (3.15)
where
~g" = ~g(x; t)� ~w"t + �(�~w" �r
N"X
i=1
3X
k=1
pk;i(x; t)vik(t)'
i
"(x� xi(t); t);
q" = p"(x; t) + �
N"X
i=1
3X
k=1
pk;i(x; t)vik(t)'
i
"(x� xi(t); t);
~V" = ~U"(x)� ~w"(x; 0):
Multiplying (3.13) by ~v"(x; t) and integrating over
T
" , we get
TZ
0
Z
"(t)
[(~v"t; ~v")� �(�~v"; ~v")] dxdt
=
TZ
0
Z
"(t)
(rq"; ~v")dxdt+
TZ
0
Z
"(t)
(~g"; ~v")dxdt: (3.16)
Since div~v" = 0, then from (2.16) using boundary condition (3.14) and the ini-
tial condition (3.15), we have
TZ
0
Z
"(t)
(rq"; ~v")dxdt = 0;
TZ
0
Z
"(t)
(�~v"; ~v")dxdt = �
TZ
0
Z
"(t)
jr~v"j
2dxdt;
TZ
0
Z
"(t)
(~v"t; ~v")dxdt =
1
2
TZ
0
d
dt
Z
"(t)
j~v"j
2dxdt
=
1
2
Z
"(t)
j~v"(x; T )j
2dx�
1
2
Z
"(0)
j~V"(x)j
2dx: (3.17)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 351
N.K. Radyakin
Let us extend ~v" by zero on the set
N"[
i=1
QiT
" . Then, for any t 2 [0; T ]: ~v"(x; t) 2
Æ
W 1
2 (
). Now Friedreich's inequality implies k~v"k
2
L2(
T )
� Ckr~v"k
2
L2(
T
"
)
. There-
fore,
TZ
0
Z
"(t)
(~g"; ~v")dxdt � Ækr~v"k
2
L2(
T
"
) +
C2
4Æ
k~g"k
2
L2(
T
"
); (3.18)
where C is a constant, which does not depend on ", and Æ > 0 is an arbitrary
positive number.
Now from (3.16)�(3.18), we have
1
2
Z
"(t)
j~v"(x; T )j
2dx+ (1� Æ)kr~v"k
2
L2(
T
"
) �
C2
4Æ
k~g"k
2
L2(
T
"
) +
1
2
k~V"kL2(
"(0)):
We recall that ~g" = ~g � ~w"t + � ~W", ~V" = ~U" � ~w". Now the last inequality along
with Lem. 1 and condition 4 of Th. 1 imply that the partial derivatives of the
vector function ~v" with respect to x are uniformly bounded in the space L2(
T
" )
(with respect to "), i.e.,
kr~v"kL2(
T
"
) � C: (3.19)
To obtain a similar estimate for ~v"t, we make use of the following lemmas.
Lemma 2. Let f"(x) = f"(x; t) for any t 2 [0; T ] be a function from
Æ
W 1
2 (
)
and f"(x; t) = fi" = fi"(t) in Qi
" = Qi
"(t) (here t is considered as a parameter).
Let the conditions 1 and 3 of Th. 1 be satis�ed.
Then, the following inequalities hold
N"X
i=1
jfi"j
2di" < C
Z
jrf"j
2dx, where C is
a positive constant that does not depend on " and t.
P r o o f. Denote by vi"(x) = vi"(x; t) the solution of the Robin problem in
the sets Qi
" = Qi
"(t):
�vi"(x) = 0; x 2 R3
nQi
";
vi"(x) = 1; x 2 Qi
"; vi"(x)! 0; jxj ! 1: (3.20)
The Dirichlet integral of the solution of this problem is called the newtonian
capacity of the set Qi
" and is denoted by Ci
". Moreover,
2�di1" � Ci
" � 2�di"; (3.21)
352 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
where d0i" and d
i
" are the interior and exterior diameters ofQi
", respectively. In what
follows we make use of the inequalities (see [3]):
jD�vi"j � A
di"
�1+j�j
; j�j = 0; 1; 2; (3.22)
Z
T
b
jD�vi"j
2dx � A
n
(di")
2b1�2j�j + (di")
3�2j�j
o
(j�j = 0; 1); (3.23)
where � = �(x) is the distance from the point x to the minimal ball T (Qi
")
containing Qi
", Tb is the ball of radius b, which is concentric to T (Qi
") and b > di".
For any t 2 [0; T ], consider a function u"(x) = u"(x; t) (here t is a parameter),
which is the solution of the following Dirichlet problem:
�u"(x) = 0; x 2
" =
n
N"[
i=1
Qi
";
u"(x) = f"(x); x 2 @Qi
"; i = 1; : : : ; N"; u"(x) = 0; x 2 @
: (3.24)
We extend u"(x) in Qi
" by setting u"(x) = f i" as x 2 Qi
", i = 1; : : : ; N". It is
well known that u" minimizes the Dirichlet integral and, therefore,
kru"kL2(
) � krf"kL2(
): (3.25)
Since u" 2
Æ
W 1
2 (
), then it satis�es Friedreich's inequality
ku"kL2(
) � Ckru"kL2(
) (3.26)
and the multiplicative inequality
ku"kLr(
) � (48)�=6kru"k
�
L2(
)
ku"k
1��
L2(
)
; (3.27)
where r 2 [2; 6], � = 3=2 � 3=r, and C is a constant depending on the domain
only.
It follows from (3.25)�(3.27) that
ku"kL4(
) � Ckrf"kL2(
); (3.28)
where C is a constant that does not depend on " and t.
We set u"(x) = û"(x) + w"(x), where û" =
N"X
i=1
f i"v
i
"(x)'
i
"(x), v
i
"(x) is the
solution of the problem (3.20), 'i"(x) is a patch function de�ned earlier in (2.4).
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 353
N.K. Radyakin
It is clear that w" = 0 for x 2
N"[
i=1
Qi
"
[
@
and, therefore due to (3.24) w" is
orthogonal to u" with respect to the Dirichlet scalar product. Then, it follows
from the development of u", krw"k
2
L2(
)
= �(rû";rw")L2(
). In the right-hand
side of this equality we apply the integration by parts and H�older's inequality.
We have
krw"k
2
L2(
)
=
Z
N"X
i=1
f i"�(vi" � '
i
") � w"dx
�
8<
:
Z
�����
N"X
i=1
f i"�(vi" � '
i
")
�����
4=3
dx
9=
;
3=48<
:
Z
jw"j
4dx
9=
;
1=4
:
We estimate the �rst term in the right-hand side of this inequality using (3.22).
Taking into account the properties of 'i", (3.20) and H�older's inequality, we get
krw"k
2
L2(
)
� C
(
N"X
i=1
(di")
2
(ri")
3
)1=2( N"X
i=1
jf i"j
2di"
)1=2
kw"kL4(
): (3.29)
The representation of u" and (3.28) imply that kw"kL4(
) � krf"kL2(
) +
kû"kL4(
). On the other hand, from the representation of û" we deduce that
û"(x) 2W
0;1
2 (
).
To estimate the second term in the right-hand side we apply (3.27). Then
we estimate each term using (3.22), (3.23), and the inequality di" < ri" < C. We
obtain
kw"kL4(
) � Ckrf"kL2(
) + Cmax
i
di"
(
N"X
i=1
jf i"j
2di"
)1=2
:
Plugging this inequality in (3.29) and taking into account condition 3 of Th. 1,
we �nally get
krw"k
2
L2(
)
� (Æ + Cmax
i
di")
N"X
i=1
jf i"j
2di" + C=Ækrf"k
2
L2(
)
; (3.30)
where C is a constant which does not depend on ", t, and Æ is an arbitrary positive
number.
Let us estimate the Dirichlet norm of the function û" from below. We have
krû"k
2
L2(
)
=
N"X
i=1
jf i"j
2
krvi"k
2
L2(
)
+�"; (3.31)
354 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
where
�" =
N"X
i=1
jf i"j
2
Z
jrvi"j
2
�
('i")
2
� 1
�
dx
+
N"X
i=1
jf i"j
2
kvi"r'
i
"k
2
L2(
)
+ 2
N"X
i=1
jf i"j
2
Z
vi"'
i
"(rv
i
";r'
i
")dx:
The de�nition of the newtonian capacity Ci
", inequality (3.21) and condition 1 of
Th. 1 imply
N"X
i=1
jf i"j
2
krvi"k
2
L2(
)
=
N"X
i=1
jf i"j
2Ci
" � A
N"X
i=1
jf i"j
2di";
where A is a positive constant that does not depend on " and t.
It follows from condition 3 of Th. 1 that ri" � B(di")
2=3 (B > 0). Then, using
(3.22), we have
j�"j � C1
N"X
i=1
jf i"j
(di")
2
ri"
� Cmax
i
(di")
1=3
N"X
i=1
jf i"j
2di";
where C = C1B
�1 does not depend on " and t. Thus, according to (3.31)
krû"k
2
L2(
)
�
�
A� Cmax
i
(di")
1=3
� N"X
i=1
jf i"j
2di": (3.32)
Now (3.25), (3.30) and (3.32) imply the inequality
�
A� Cmax
i
(di")
1=3
� Cmax
i
di" � Æ
� N"X
i=1
jf i"j
2di" � C(1 + 1=Æ)krf"k
2
L2(
)
;
where " > 0 is an arbitrary positive number and A and C are positive constants
that do not depend on ", t and Æ.
Now from this inequality along with condition 1 of Th. 1 we immediately
obtain the statement of Lem. 2.
Lemma 3. Let conditions 1, 3 of Th. 1 be ful�lled for the sets Qi
" = Qi
"(t),
i = 1; : : : ; N".
Then for any "(x) = "(x; t) 2 L2(
), such that:
1)
Z
"(x)dx = 0;
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 355
N.K. Radyakin
2) "(x) = 0 for x 2
N"[
i=1
Qi
";
there exists a vector function ~z"(x) = ~z"(x; t) 2 W
0;1
2 (
) such that for any t 2
[0; T ]
div~z"(x) = "(x); x 2
; (3.33)
~z"(x) = 0; x 2
N"[
i=1
Qi
"(t)
[
@
; (3.34)
k~z"(x)kW 1
2
(
) � Ck "kL2(
); (3.35)
where C is a constant that does not depend on " and t.
P r o o f. For any t 2 [0; T ], we construct a vector function ~f"(x) = ~f"(x; t) 2
Æ
W 1
2 (
), such that div~f"(x) = "(x), x 2
; ~f"(x) = ~f i", x 2 Qi
" (i = 1; : : : ; N");
k~f"kW 1
2
(
) � Ck "kL2(
).
We set
~z"(x; t) = ~f"(x; t)�
N"X
i=1
rot
"
3X
k=1
~~vki(x; t)f ik"(t)'
i
"(x� xi(t); t)
#
; (3.36)
where f ik" are the components of the constant vectors ~f i", the vector function
~~vki
and the patch function 'i"(x; t) are the same as in (3.2). Taking into account
the properties of ~f", ~~v
ki and 'i", it is easy to see, that ~z" satis�es (3.33), (3.34).
It remains to show that the estimate (3.35) holds true. From (3.36) we have
k~z"k
2
W 1
2
(
)
� 2
(
k~f"kW 1
2
(
) + C
N"X
i=1
j~f i"j
2
�
di" +
(di")
2
ri"
�)
;
where C is a constant that does not depend on " and t.
It is clear that k~f"kW 1
2
(
) � Ck "kL2(
). The last two inequalities along with
conditions 2, 3, of Th. 1 and Lem. 2 imply (2.35). Lemma 3 is proved.
Let us now estimate the partial derivative ~v"t of the solution ~v" of the problem
(3.13)�(3.15).
Let ~l"(x; t) = flk" (x; t); k = 0; 1; 2; 3g be a vector �eld in
T , which is tangent
to the lateral surface of
T
" and such that
l0"(x; t) � 1; jDk
x
~l"(x; t)j < C; (k = 0; 1); (3.37)
where C is a constant not depending on ".
356 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
It is easy to see that there exists a vector �eld satisfying these properties.
In fact, consider a vector function ~�"(x; t), such that for any t 2 [0; T ]
~�"(~�; t) = ~�(~�; t) +
N"X
i=1
h
~�(~�i"; t)�
~�(~�; t) + �i
"(
~� � ~�i")
i
'
j~� � ~�i"j
di"
!
; (3.38)
where ~�i" denotes the center of mass of the i-th particle at t = 0, ~�(x; t) is
a vector function giving the motion to the centers of masses, �i
"(t) are the relation
operators generated by the rotation of the particles around their centers of mass
~�(~�i"; t), '(y) is a twice di�erentiable function, such that '(y) = 1 for y � 1,
'(y) = 0 for y > 3=2.
According to the properties of the function ~�(x; t) and conditions 1, 3 of Th. 1,
for any t 2 [0; T ] and " su�ciently small, ~x = ~�"(~�; t) is a one-to-one map into
.
Moreover, there exists a continuously di�erentiable inverse map ~� = [�"]
�1(x; t).
We set
lk" (x; t) =
@�k
"(
~�; t)
@t
j�=[�"]�1(x;t); l0"(x; t) � 1; k = 0; 1; 2; 3: (3.39)
The map ~x = ~�(~�i"; t) + �i
"(t)(
~� � ~�i") maps Qi
"(0) in Qi
"(t), and the functions
'i"(x; t) = '
j[�"]
�1(x; t)� ~�i"j
di"
!
equal 1 in QiT
" and 0 in Q
jT
" (j 6= i). Then the
vector �eld ~l"(x; t) is tangent to
N"[
i=1
@QiT
" . Since ~�(~�; t) for any t 2 [0; T ] maps
@
on @
and 'i"(x; t) = 0 on @
� [0; T ], then ~l"(x; T ) is tangent to @
� [0; T ].
The rotation operator �i
"(t) satis�es the equation _�i
"(t) = ~�i"(t) � �i
"(t) and
~�(x; t), ~�i(t), i = 1; : : : ; N", and '(y) are su�ciently smooth. Then we apply
(3.38), (3.39) and �nally obtain (3.37).
Let us denote by
d
dl"
the derivative with respect to the vector �eld ~l"; : : : ,
d
dl"
=
@
@t
+ l1"(x; t)
@
@x1
+ l2"(x; t)
@
@x2
+ l3"(x; t)
@
@x3
and set "(x; t) = div
@~v"
@l"
, where ~v"(x; t) is the solution of (3.13)�(3.15), extended
by zero in
N"[
i=1
QiT
"
[
@
.
Since the �eld ~l"(x; t) is tangent to
N"[
i=1
@QiT
"
[
@
, thus it follows from the
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 357
N.K. Radyakin
properties of ~v" that
@~v"
@l"
= 0 in
N"[
i=1
@QiT
"
[
@
, ð
Z
(t)
"(x; t)dx = 0 8t 2 [0; T ],
"(x; t) = 0 for (x; t) 2
N"[
i=1
QiT
" and
"(x; t) =
3X
i;k=1
@li"
@xk
@vk"
@xi
: (3.40)
Thus " satis�es the conditions of Lem. 2 and, therefore, there is a vector function
~z(x; t) 2
Æ
W 1
2 (
) such that 8t 2 [0; T ]: div~z" = "(x; t), x 2
; ~z"(x; t) = 0;
x 2
N"[
i=1
Qi
"(t)
[
@
k~z"kW 1
2
(
) � Ck "kL2(
):
This inequality, (3.37) and (3.40) imply
TZ
0
k~z"(x; t)k
2
W 1
2
(
)
dt < Ckr~v"k
2
L2(
T )
; (3.41)
where C is a constant that does not depend on ".
We multiply the equation (3.14) by
@~v"
@l"
� ~z" and integrate over
T
" . We have
Z Z
T
"
�
~v"t;
@~v"
@l"
� ~z"
�
dxdt = �
TZ
0
Z
"(t)
�
�~v";
@~v"
@l"
� ~z"
�
dxdt
�
TZ
0
Z
"(t)
�
rq";
@~v"
@l"
� ~z"
�
dxdt+
Z Z
T
"
�
~g";
@~v"
@l"
� ~z"
�
dxdt: (3.42)
Using (3.37) and (3.41) we estimate the left-hand side of (3.42) as followsZ Z
T
"
�
~v"t;
@~v"
@l"
� ~z"
�
dxdt =
Z Z
T
"
j~v"tj
2dxdt
+
3X
k=1
Z Z
T
"
�
~v"t; l
k
"
@~v"
@xk
�
dxdt�
Z Z
T
"
(~v"t; ~z")dxdt
358 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
� k~v"tk
2
L2(
T
"
) �C1k~v"tkL2(
T
"
)kr~v"kL2(
T
"
)
�k~v"tkL2(
T
"
)k~z"kL2(
T
"
) � (1� Æ)k~v"tk
2
L2(
T
"
) � C=Ækr~v"k
2
L2(
T
"
); (3.43)
where Æ is an arbitrary positive number and C is a constant that is independent
of ".
In a similar way for the last term in the right-hand side in (3.42), we have�������
Z Z
T
"
�
~g";
@~v"
@l"
� ~z"
�
dxdt
������� � Æk~v"tk
2
L2(
T
"
)
+
1
4Æ
k~g"k
2
L2(
T
"
) + Ck~g"kL2(
T
"
)kr~v"kL2(
T
"
): (3.44)
Since div
�
@~v"
@l"
� ~z"
�
= 0 and
@~v"
@l"
�~z" = 0 on @
"(t), then integrating by parts,
we get
TZ
0
Z
"(t)
�
r~q";
@~v"
@l"
� ~z"
�
dxdt = 0 (3.45)
and
TZ
0
Z
"(t)
�
�~v";
@~v"
@l"
� ~z"
�
dxdt
= �
TZ
0
Z
"(t)
�
r~v";r
@~v"
@l"
�
dxdt+
TZ
0
Z
"(t)
(r~v";r~z")dxdt: (3.46)
We estimate the second term in the right-hand side with the help of (3.41). Thus
TZ
0
Z
"(t)
(r~v";r~z")dxdt � Ckr~v"k
2
L2(
T
"
); (3.47)
and the �rst term can be represented in the form
TZ
0
Z
"(t)
�
r~v";r
@~v"
@l"
�
dxdt =
1
2
Z
T
"
3X
k=1
@
@xk
(lk" jr~v"j
2)dxdt
�
1
2
Z
T
"
jr~v"j
2
3X
k=1
@lk"
@xk
dxdt+
Z
T
"
3X
i;k=1
@li"
@xk
�
@~v"
@xi
;
@~v"
@xk
�
dxdt = J1+J2+J3: (3.48)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 359
N.K. Radyakin
Due to (3.37), we have
jJ2j+ jJ3j � Ckr~v"k
2
L2(
T
"
):) (3.49)
The vector �eld ~l"(x; t)jr~v"j
2 is tangent to the lateral surface of
T
" and l0" � 1.
Then applying to J1 the theorem of Gauss�Ostrogradski, we get
J1 �
1
2
Z
T
"
3X
k=0
@
@xk
(lk" jr~v"j
2)dxdt
=
1
2
Z
"(T )
j~v"(x; T )j
2dx�
1
2
Z
"(0)
jr~v"(x; 0)j
2dx: (3.50)
Now from (3.42), (3.45), (3.46), (3.48), (3.50) and the estimates (3.43), (3.44),
(3.47), (3.49) we obtain
(1� 2Æ)k~v"tk
2
L2(
T
"
) +
�
2
Z
"(T )
jr~v"(x; T )j
2dx
�
�
1
4Æ
+ C
�
k~g"k
2
L2(
T
"
) +C
�
1
Æ
+ 1
�
kr~v"k
2
L2(
T
"
) +
�
2
Z
"(0)
jr~V"j
2dx;
where Æ > 0 is an arbitrary positive number and C does not depend on " and Æ.
We set Æ = 1=4. Recall that ~g" = ~g � ~w" + � ~W", ~V" = ~U" � ~w", where ~w" and
~W" are de�ned in (3.2), (3.3). Now using Lem. 1, condition 4 of Th. 1, and (3.19)
we conclude that the derivatives ~v"t are bounded in L2(
T ) uniformly in ", i.e.,
k~v"tkL2(
T
"
) < C: (3.51)
4. Proof of Theorem 1
For any t 2 [0; T ] ~v"(x; t) 2
Æ
W 1
2 (
"). We extend ~v" by zero in
N"[
i=1
QiT
" . Due
to (3.19), (3.51) we see that the sequence f~v"; " ! 0g is bounded in
Æ
W 1
2 (
T )
and, therefore, it is weakly compact in this space. Then there is a subsequence
f~v"
k
; "k ! 0g which converges weakly in
Æ
W 1
2 (
T ) to a vector function ~v(x; t) 2
Æ
W 1
2 (
T ). Due to the imbedding theorem this subsequence converges strongly in
L2(
T ) to the vector function ~v(x; t). Moreover, it converges strongly in L2(
T )
360 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
(uniformly with respect to t 2 [0; T ]). Let us show that ~v is a solution of the
following initial boundary value problem:
~vt � ��~v + �C(x; t)~v = �rp+ ~F (x; t); div~v = 0; (x; t) 2
T ; (4.1)
~v(x; t) = 0; (x; t) 2 @
� [0; T ]; ~v(x; 0) = ~U(x); x 2
; (4.2)
where ~F = g(x; t) + �C(x; t) ~W (x; t) ( ~W (x; t) = ~�t(�; t)j�=��1(x;t)), the matrix
C(x; t) and the vector function ~U are de�ned in conditions 2 and 4 of Th. 1,
respectively.
It is clear that problem (4.1)�(4.2) has a unique solution. Therefore, the whole
sequence of extended functions fv"(x; t); "! 0g converges weakly in
Æ
W 1
2 (
T ) and
strongly in L2(
T ) ø L2(
) (8t 2 [0; T ]) to v(x; t).
Let us introduce the linear resolving operator ~�", de�ned by ~�"~u" = ~ "(x),
where ~u"(x) is a solution of the following problem:
�~u"(x)�rp"(x) = ~ "(x); div~u"(x) = 0; x 2
"(t); (4.3)
~u"(x) = 0; x 2 @
"(t): (4.4)
The energy space is denoted by
Æ
J(
). It is de�ned as a closure in L2(
") of the
divergent free vector functions with a compact support in
".
We recall that L2(
) =
Æ
J(
")�G(
"), where the subspace G(
") consists of
the gradients of the single valued functions from W 1
2 (
"). The domain D( ~�") of
~�" is the set of all the solutions of (4.3) corresponding to various ~ " 2
Æ
J(
"). It
is shown by O.A. Ladyzhenskaya that the operator �" determines the one-to-one
correspondence between D( ~�") and
Æ
J(
"). It is adjoint and negatively de�nite
on D( ~�") (see [1]). The inverse operator ( ~�")
�1 � ~R" is a compact self adjoint
operator. We extend this operator by the linearity to the whole L2(
") by setting
~R"~g = 0 for g 2 G(
") and in the space L2(
) de�ne the operator ~R = I"R"P",
where P" is the restriction operator from L2(
) to L2(
"), i.e., 8~f 2 L2(
),
P" ~f [x] = ~f(x) for x 2
"; I" is the imbedding operator from L2(
") to L2(
),
i.e.,
8~f" 2 L2(
"); I" ~f"[x] =
(
~f"(x); for x 2
";
0 for x 2
n
":
It is easy to see that R" is a compact selfadjoint operator in L2(
). In a similar
way we introduce the operator ~�C that determines the one-to-one correspondence
between ~u(x) of the solution of the problem
�~u� C(x)~u�rp = ~ (x); div~u = 0; x 2
; (4.5)
~u(x) = 0; x 2 @
; (4.6)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 361
N.K. Radyakin
and the right-hand sides of ~ 2
Æ
J(
). As in [1], one can show that the operator
~�C determines the one-to-one correspondence between its domain D( ~�C) = f~u 2
Æ
J(
) : ~�
Cu 2
Æ
J(
)g and
Æ
J(
). It is selfadjoint and negatively de�nite. Its inverse
operator R = ( ~�C)�1 is compact and selfadjoint in
Æ
J(
). We extend R by the
linearity on the whole space L2(
) =
Æ
J(
)�G(
) by setting R~g = 0 for ~g 2 G(
).
The following theorem holds:
Theorem 2. Let the conditions of Th. 1 be ful�lled. Then, for any ~f 2 L2(
)
the sequence fR"
~f; "! 0g converges in L2(
) to R~f , i.e., kR"
~f �R~fkL2(
) ! 0
as "! 0.
The proof of Th. 2 follows from [4].
Denote by Rt
" and Rt the resolving operators of problem (4.3), (4.4) (for
" =
"(t)) and (4.5), (4.6) (for C(x) = C(x; t)), respectively. These operators
are compact and selfadjoint in L2(
). Let ~v"(x; t) be a solution of (3.13)�(3.15),
extended by zero in the set QT
" =
N"[
i=1
QiT
" ). For any t 2 [0; T ] ~v"(x; t) can be
represented as ~v"(x; t) = Rt
"
~f"t[x] , where
~f"t �
~f"(x; t) = ~v"t(x; t) � ~g"(x; t) =
~v"t � ~g + ~w"t � � ~W". Let ~'(x; t) be an arbitrary vector function in
T . Denote
by (�; �)
the scalar product in L2(
). Taking into account that for any t the
operator Rt
" is selfadjoint in L2(
), and ~'(x; t) = ~'t(x) 2 L2(
), we have
TZ
0
Z
(~v"(x; t); ~'(x; t))dxdt =
TZ
0
(Rt
"
~f"t; ~'t)
dt
=
TZ
0
(~f"t;R
t
"~'t)
dt =
TZ
0
(~f"t;Rt~'t)
dt+
TZ
0
(~f"t;R
t
"~'t �Rt~'t)
dt: (4.7)
According to Lem. 1, ~g"(x; t) converges weakly in L2(
T ) to the vector function
~g(x; t)+ �C(x; t) ~W (x; t) and the subsequence f~v"
k
(x; t); "! 0g converges weakly
inW 1
2 (
T ) to ~v(x; t) as "! 0. The function ~f"(x; t) = ~v"(x; t)�~g"(x; t) converges
weakly in L2(
T ) to ~vt(x; t) � ~g(x; t) � �C(x; t) ~W (x; t) = ~f(x; t) = ~ft as " =
"k ! 0. Therefore,
lim
"="
k
!0
TZ
0
(~f"t;Rt~'t)
dt = lim
"="
k
!0
TZ
0
Z
(~f"(x; t)Rt ~'(x; t))dxdt
=
TZ
0
Z
(~f(x; t)Rt ~'(x; t))dxdt =
TZ
0
(~ft;Rt~'(x; t))
dt =
TZ
0
(Rt
~ft; ~'t)
dt
362 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
Homogenization of a Linear Nonstationary Navier�Stokes Equations System...
=
TZ
0
Z
(Rt
~ft[x]; ~'(x; t))dxdt: (4.8)
Here we make use of the fact that Rt is selfadjoint in L2(
). Then using for
any t 2 [0; T ] Th. 2 and taking into account the uniform boundness of kRt
"'tkL2(
)
on ", t we get ������
TZ
0
(f"t;R
t
"'t �Rt't)
dt
������
� kf"kL2(
T )
8<
:
TZ
0
kR
t
"'t �Rt'tk
2
L2(
)
dt
9=
;
1=2
! 0; (4.9)
as "! 0.
Thus due to (4.7)�(4.9)
lim
"="
k
!0
TZ
0
Z
(~v"(x; t); ~'(x; t))dxdt =
TZ
0
Z
(Rt
~ft[x]; ~'(x; t))dxdt:
Since ~v" converges in L2(
T ) to ~v, ð ~' as " = "k ! 0, and ~' is an arbi-
trary continuous vector function, then ~v(x; t) = Rt
~ft[x] = Rt(~vt(x; t) � ~g(x; t) �
�C(x; t) ~W (x; t). By the de�nition of Rt this means that ~v satis�es (4.1) and
boundary condition (4.2). The vector function ~v" converges weakly in
Æ
W 1
2 (
T )
to ~v as " = "k ! 0 and, therefore, in L2(
) uniformly with respect to t. Then it
follows from condition 4 of Th. 1 and Lem. 1 that v(x; 0) = U(x). Therefore, ~v is
the solution of problem (4.1)�(4.2).
Consider now ~u" of the solution of (2.1)�(2.4). Since ~u"(x; t) = ~v"(x; t) +
~w"(x; t), then taking into account ~v"(x; t) *
W 1
2
(
T )
~v(x; t) and Lem. 1, we conclude
that ~u" converges in L2(
T ) to the vector function ~u(x; t) = ~v(x; t) as
"! 0. According to (4.1)�(4.2) this vector function is the solution of (2.8)�(2.10).
Theorem 1 is proved.
References
[1] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow.
New York, London, Paris, Gordon and Breach Sci. Publ. XVIII, (1969).
[2] O.A. Ladyzhenskaya, Initial-Boundary Problem for Navier�Stokes Equations in Do-
mains with Time-Varying Boundaries. � Zap. Nauchn. Sem. Leningr. Otd. Mat.
Inst. Steklov, Leningrad 11 (1968), 97�128. (Russian)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 363
N.K. Radyakin
[3] V.A. Marchenko and E.Ya. Khruslov, Boundary Value Problems in Domains with
a Fine-Grained Boundary. Naukova Dumka, Kiev, 1974. (Russian)
[4] V.A. Lvov, The Convergence of the Solutions of a System of Initial Boundary Value
Problems in Domains with Moving Fine-Grained Boundary. � Dokl. AN USSR
(1987), No. 7. 21�24. (Russian)
[5] Eh.B. Bykhovskij and N.V. Smirnov, On Orthogonal Decomposition of the Space of
Vector Functions Quadratically Summable in the Given Domain and the Operators
of Vector Analysis. � Inst. Math. AN USSR 59 (1960), 5�36. (Russian)
[6] E.Ya. Khruslov and Z.F. Nazyrov, Perturbation of the Thermal Field of Moving
Small Particles. Studies in the Theory of Operators and their Applications. Naukova
Dumka, Kiev, 1979. (Russian)
364 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
|
| id | nasplib_isofts_kiev_ua-123456789-7612 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T17:22:26Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Radyakin, N.K. 2010-04-06T09:08:24Z 2010-04-06T09:08:24Z 2007 Homogenization of a linear nonstationary Navier - Stokes equations system with a time-variant domain with a fine-grained boundary / N.K. Radyakin // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 342-364. — Бібліогр.: 6 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/7612 The problem of distortion of viscous incompressible uid with a great number of solid particles with given velocities is considered. The diameters of particles and the distance between them tend to zero, and the number of particles tends to infinity. The asymptotic behavior of the solutions of the linear system of Navier-Stokes equations is considered. In a homogenized model there appears an additional term containing the strength tensor of a single particle. Розглянуто задачу про збурювання в'язкої нестислої рідини потоком великої кількості твердих часток, що рухаються з заданими швидкостями. Досліджено асимптотичну поведінку розв'язків лінійної системи рівнянь Нав'є - Стокса, які описують цю задачу, коли розмір часток і відстань між ними зменшуються, а кількість часток необмежено зростає. Одержано усереднені рівняння, що описують рух суспензії. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary Усереднення лінійної нестаціонарної системи рівнянь Нав'є - Стокса у області з дрібнозернистою границею, що змінюється за часом Article published earlier |
| spellingShingle | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary Radyakin, N.K. |
| title | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary |
| title_alt | Усереднення лінійної нестаціонарної системи рівнянь Нав'є - Стокса у області з дрібнозернистою границею, що змінюється за часом |
| title_full | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary |
| title_fullStr | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary |
| title_full_unstemmed | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary |
| title_short | Homogenization of a Linear Nonstationary Navier—Stokes Equations System with a Time-Variant Domain with a Fine-Grained Boundary |
| title_sort | homogenization of a linear nonstationary navier—stokes equations system with a time-variant domain with a fine-grained boundary |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7612 |
| work_keys_str_mv | AT radyakinnk homogenizationofalinearnonstationarynavierstokesequationssystemwithatimevariantdomainwithafinegrainedboundary AT radyakinnk userednennâlíníinoínestacíonarnoísistemirívnânʹnavêstoksauoblastízdríbnozernistoûgraniceûŝozmínûêtʹsâzačasom |