ON WEAK SOLUTIONS OF THE EQUATIONS OF MOTION OF A VISCOELASTIC MEDIUM WITH VARIABLE BOUNDARY V. G. pptx

ON WEAK SOLUTIONS OF THE EQUATIONS OF MOTION OF
A VISCOELASTIC MEDIUM WITH VARIABLE BOUNDARY
V. G. ZVYAGIN AND V. P. ORLOV
Received 2 September 2005
The regularized system of equations for one model of a viscoelastic medium with memory
along trajectories of the field of velocities is under consideration. The case of a changing
domain is studied. We investigate the weak solvability of an initial boundary value prob-
lem for this system.
1. Introduction
The purpose of the present paper is an extension of the result of [21] on the case of
a changing domain. Let Ω
t
∈ R
n
,2≤ n ≤ 4 be a family of the bounded domains with
boundary Γ
t
, Q ={(t,x):t ∈ [0,T], x ∈ Ω
t
}, Γ ={(t,x):t ∈ [0,T], x ∈ Γ
t
}.Thefollow-
ing initial boundary value problem is under consideration:
ρ

v
t
+ v
i
∂v/∂x
i


− µ
1
Div

t
0
exp

−
t − s
λ

Ᏹ(v)

s,z(s;t,x)

ds − µ
0
DivᏱ(v)
=−grad p + ρϕ,divv = 0, (t,x) ∈ Q;

Ω
t
pdx= 0, t ∈ [0,T];
v(0,x) = v
0
(x), x ∈ Ω
0
, v(t,x) = v

1
(t,x), (t,x) ∈ Γ.
(1.1)
Here v(t,x)
= ( v
1
, ,v
n
)isavelocityofthemediumatlocationx at time t, p(t,x)isa
pressure, ρ, µ
0
, µ
1
, λ are positive constants, Div means a divergence of a matrix, the matrix
Ᏹ(v)hascoefficients Ᏹ
ij
(v)(t,x) = (1/2)(∂v
i
(t,x)/∂x
j
+ ∂v
j
(t,x)/∂x
i
). In (1.1) and in the
sequel repeating indexes in products assume their summation. The function z(τ;t,x)is
defined as a solution to the Cauchy problem (in the integral form)
z(τ;t,x) = x +

τ

t
v

s,z(s;t,x)

ds, τ ∈ [0,T], (t,x) ∈ Q. (1.2)
The substantiation of model (1.1)isgivenin[21]. One can find the details in [12,Chapter
4]. We assume that a domain Q ⊂ R
n+1
is defined as an evolution Ω
t
, t ≥ 0ofthevolume
Ω
0
along the field of velocities of some sufficiently smooth solenoidal vector field
˜
v(t,x)
Copyright © 2006 Hindawi Publishing Corporation
Boundary Value Problems 2005:3 (2005) 215–245
DOI: 10.1155/BVP.2005.215
216 On weak solutions of the equations of motion
which is defined in some cylindrical domain

Q
0
={(t,x):t ∈ [0,T], x ∈

Ω
0
},sothat

Ω
t
⊂

Ω
0
. This means that Ω
t
=
˜
z(t;0,Ω
0
), where
˜
z(τ;t,x) is a solution to the Cauchy
problem
˜
z(τ;t,x) = x +

τ
t
˜
v

s,
˜
z(s; t,x)

ds, τ ∈ [0,T], (t,x) ∈


Q. (1.3)
Thus, it is clear that the lateral surface Γ ofadomainQ and the trace of the function
˜
v(t,x)onΓ will be smooth enough, if
˜
v(t,x) is smooth enough. We will assume sufficient
smoothness of
˜
v(t,x), providing validity of embedding theorems for domains Ω
t
used
below with the common for all t constant.
Let us mention some works which concern the study of the Navier-Stokes equations
((1.1)forµ
1
= 0) in a time-dependent domain (see [2, 5, 8, 13] etc.), by this, different
methods are used and various results on existence and uniqueness of both strong and
weak solutions are obtained. In the present work, the existence of weak solutions to a reg-
ularized initial boundary value problem (1.1) in a domain with a time-dependent bound-
ary Γ
t
is established. The approximation-topological methods suggested and advanced in
[3, 4] are used in the paper. It assumes replacement of the problem under consideration
by an operator equation, approximation of the equation in a weak sense and application
of the topological theory of a degree that allows to establish the existence of solutions on
the basis of a priori estimates and statements about passage to the limit. Note that in the
case of a not cylindrical domain (with respect to t) the necessary spaces of differentiable
functions cannot be regarded as spaces of functions of t with values in some fixed func-
tional space. Consequently, the direct application of the method of [21] is not possible.
The history of the motion equation from (1.1) is given in details in [21]. On the basis of

the rheological relation of Jeffreys-Oldroyd type the existence theorem for weak solutions
in a domain with a constant boundary was proved. The purpose of the present paper is
to prove a similar result for a domain with changing boundary.
The article is organized as follows. We need a number of auxiliary results about func-
tional spaces for the formulation of the basic results. They are presented in Section 2.
We also need some results about the linear problem in a non-cylindrical domain which
are given in Section 3. By this the proofs of the part of the results (which require the
rather long proofs) are given in Section 8.InSection 4, the main results are formulated,
in Sections 5–7 the proofs of the main results are carried out. We will denote constants in
inequalities and chains of inequalities by the same M if their values are not important.
2. Auxiliary results
2.1. Functional spaces. Let us introduce necessary functional spaces. Denote norms in
L
2
(Ω
t
)andW
k
2
(Ω
t
)by|·|
0,t
and |·|
k,t
accordingly. Denote by ·
0
anorminL
2
(Q)

or in L
2
(Q
0
)(Q
0
= [0,T] × Ω
0
), depending on a context. We will denote by D
0,t
the set
of functions, which are smooth, solenoidal and finite on domain Ω
t
. We will designate
through H
t
and V
t
acompletionofD
0,t
in the norms L
2
(Ω
t
)andW
1
2
(Ω
t
)accordingly.

We denote by V
∗
t
the conjugate space to V
t
and by |·|
−1,t
the norm in V
∗
t
. We denote
by v,h
t
an action of the functional v ∈ V
∗
t
upon an element h ∈ V
t
.Thus,thescalar
product (·,·)
t
in H
t
generates (see, e.g., [7, Chapter 1, page 29]) the dense continuous
V. G. Zvyagin and V. P. Orlov 217
embeddings V
t
⊂ H
t
⊂ V

∗
t
at every t ∈ [0,T]. It is clear that


u,v
t


≤|u|
1,t
|v|
−1,t
, u ∈ V
t
, v ∈ V
∗
t
. (2.1)
Let D be the set of smooth vector functions on Q, solenoidal and finite on a domain
Ω
t
for every t. It is easy to show that scalar functions ϕ(t) =|v(t,x)|
1,t
, ψ(t) =|v(t,x)|
−1,t
,
g(t) =|v
t
(t,x)|

−1,t
,wherev
t
(t,x)isaderivativewithrespecttot of function v(t,x), are
determined and continuous on [0,T]foreveryv ∈ D.
We denote by E, E
∗
, E
∗
1
, W, W
1
, CH, EC, L
2,σ
(Q) the completion of D accordingly in
norms
v
E
=


T
0


v(t,x)


2
1,t

dt

1/2
, v
E
∗
=


T
0


v(t,x)


2
−1,t
dt

1/2
,
v
E
∗
1
=

T
0



v(t,x)


−1,t
dt,
v
W
=v
E
+


v
t


E
∗
, v
W
1
=v
E
+


v
t



E
∗
1
, v
CH
= max
t∈[0,T]


v(t,x)


0,t
,
v
EC
=v
E
+ v
CH
, v
0
=


T
0



v(t,x)


2
0,t
dt

1/2
.
(2.2)
Let a sequence v
n
∈ D, n = 1,2, be fundamental on Q in the norm ·
0
:

T
0
|v
n
(t,x) −
v
m
(t,x)|
2
0,t
dt → 0, n,m → +∞. Then (see [19, page 224]) there exists a subsequence
v
n

k
(t,x) which is fundamental at a.e. t in the norm |·|
0,t
.Letv(t,x) ∈ L
2
(Ω
t
) be the limit
of v
n
k
(t,x). Solenoidality of functions from D implies v(t,x) ∈ H
t
at a.e. t. It implies the
possibility to get the completion of D in the norm ·
0
as a subspace of usual functions
from L
2
(Q).
It is similarly shown that an element v ∈ E is a function v(t,x)ata.e.t, v(t,x) ∈ V
t
,
v
2
E
=

T
0

|v(t,x)|
2
1,t
dt,andv ∈ E
∗
is a function v(t,x) ∈ V
∗
t
at a.e. t, v
2
E
∗
=

T
0
|v(t,
x)|
2
−1,t
dt.Forv ∈ E
∗
1
we have at a.e. tv(t,x) ∈ V
∗
t
and v
E
∗
1

=

T
0
|v(t,x)|
−1,t
dt.
Lemma 2.1. Let v ∈ E, h ∈ E
∗
. The scalar funct ion v(t,x),h(t,x)
t
= ψ(t) is summable
and

T
0

v(t,x), h(t,x)

t
dt =v,h. (2.3)
Proof. Choose such a sequences v
n
,h
n
⊂ D that v
n
− v
E
→ 0, h

n
− h
E
∗
→ 0and|v
n
(t,
x) − v(t,x) |
1,t
→ 0, |h
n
(t,x) − h(t,x)|
−1,t
→ 0ata.e.t. Then the sequence of continuous
functions ψ
n
(t) = (v
n
(t,x), h
n
(t,x))
t
converges to ψ(t)ata.e.t and, hence, ψ(t)ismea-
surable. As

T
0


ψ

n
(t)


dt 


v
n


E


h
n


E
∗
 M, (2.4)
where M does not depend on n, then the Fatou’s theorem implies summability of ψ(t).
From convergence v
n
→ v in E, h
n
→ h in E
∗
and the equality ψ(t) =v,h (2.3) easily
follows. The lemma is proved. 

218 On weak solutions of the equations of motion
The scalar product (v,h) =

T
0
(v(t,x), h(t,x))
t
dt in L
2,σ
(Q) generates the continuous
embeddings E
⊂ L
2,σ
(Q) ⊂ (E)
∗
.Here(E)
∗
is adjoint to E. Denote by v,h an action of
the functional v ∈ (E)
∗
upon h. It turns out that (E)
∗
= E
∗
.Really,E
∗
is a subspace in
(E)
∗
.IfE

∗
does not coincide with (E)
∗
, we can find an element v
0
= 0inE for which
h,v
0
=0forallh ∈ E
∗
. Choosing elements h from the set D dense in E
∗
,wegetthat
h,v
0
=(h,v
0
) = 0forallh ∈ D. This implies v
0
= 0. Therefore, E
∗
= (E)
∗
and the scalar
product (v,h)inL
2,σ
(Q) generate the continuous embeddings E ⊂ L
2,σ
(Q) ⊂ E
∗

.Note
that |v,h|  v
E
h
E
∗
.
In the Banach spaces introduced above it is convenient for us to define equivalent
norms by the rule v
k,F
=
¯
v
F
,
¯
v = exp(−kt)v, k>0. Here F is any Banach space of
functions defined on Q.
The space E
∗
is continuously embedded in E
∗
1
.Belowv,u denotes an action of
a functional v
∈ E
∗
upon a function u ∈ E. Besides, we need the set CG of functions
z(τ;t,x)definedon[0,T] × Q which are continuous with respect to all variables and con-
tinuously differentiable with respect to x. Moreover, these functions are diffeomorphisms

of Ω
t
on Ω
τ
with the determinants equal to 1. We will consider CG as a metric space with
the metrics ρ(z
1
,z
2
) =z
1
− z
2

CG
where z
CG
= max
τ
max
t
z(τ;t,x)
C(
¯
Ω
t
)
.
We denote by W
l,m

2
(Q) the usual Sobolev spaces of functions f (t,x)onQ,having
generalized derivatives up to order l with respect to t and up to order m with respect to x
which are square summable. ·
l,m
stands for their norms.
2.2. Regularization operator. Problem (1.1) involves the integral which is calculated
along the trajectory z(τ;t,x)ofaparticlex in the field of velocities v(t, x)wherebyz(τ;t,x)
is a solution to the Cauchy problem (1.2). However, even strong solutions v(t,x)ofprob-
lem (1.2), having a derivative w ith respect to t and the second derivatives with respect to
x, square summable on Q, do not provide uniquely solvability of problem (1.1). As an exit
from this situation in [21](following[9]) the regularization of the field of velocity with
the help of introduction of a linear bounded operator S
δ,t
: H
t
→ C
1
(
¯
Ω
t
) ∩ V
t
for δ>0
such that S
δ,t
(v) → v in H
t
at δ → 0andfixedt was offered. As far as the boundar y Γ

t
of a
domain Ω
t
is concerned, it was assumed to be sufficiently smooth. In the construction of
this operator a smooth decomposition of the unit for Ω
t
, some homothety transforma-
tions in R
n
and the operator P
t
of orthogonal projection in L
2
(Ω
t
)onH
t
were used. Let
v(t,x) ∈ L
2,σ
(Q). We define on L
2,σ
(Q)theoperator

S
δ
(v) =

v where v(t,x) = S

δ,t
(v(t,x))
at t ≥ 0.
As the operator P
t
is constructed by means of solutions of the Dirichlet and Neumann
problems for Ω
t
(see [18, page 20]) and Ω
t
in our case smoothly depends on t,fromthe
structure of the operator S
δ,t
at fixed t it follows that the operator

S
δ
is a linear bounded
operator from L
2,σ
(Q)inE ∩ L
2
(0,T;C
1
), and

S
δ
(v) → v in E at δ → 0. Here L
2

(0,T;C
1
)
is a Banach space which is a completion of the set of smooth functions on Q in the norm
v
L
2
(0,T;C
1
)
= (

T
0
v(t,x)
2
C
1
(
¯
Ω
t
)
dt)
1/2
.Letnowu(t,x) ∈ L
2,σ
(Q), v(t,x) = u(t,x)+
˜
v(t,x).

Let us define the regularization operator S
δ
: L
2,σ
(Q) → L
2
(0,T;C
1
) ∩ E by the formula
S
δ
(v) =

S
δ
(v −
˜
v)+
˜
v =

S
δ
(u)+
˜
v. It is clear that S
δ
(v) → v in L
2,σ
(Q)atδ → 0.

Consider problem (1.2)forv(t,x) ⊂ L
2
(0,T;C
1
). The solvability of problem (1.2)for
the case of a cylindrical domain Q was established in [10]forv ∈ L
2
(0,T;C
1
) vanishing
V. G. Zvyagin and V. P. Orlov 219
on Γ. In the same place, the estimate was obtained:


z
1
(τ;t,x) − z
2
(τ;t,x)


C(
¯
Ω
t
)
≤ M






τ
t


v
1
(s,x) − v
2
(s,x)


C
1
(
¯
Ω
s
)
ds




, (2.5)
where v
1
,v
2

∈ L
2
(0,T;C
1
), t,τ ∈ [0,T]. The same f acts are fair and for the case of a non
cylindrical Q and for functions coinciding with
˜
v on Γ. The proofs are similar to ones
resulted in [10] with minor alterations. Inequality (2.5)isrequiredtousinwhatfollows
below. We replace (1.2)for(1.1) by the equation
z(τ;t,x)
= x +

τ
t
S
δ
v

s,z(s;t,x)

ds, τ,t ∈ [0,T], x ∈ Ω
t
. (2.6)
For every v(t, x) ∈ E the function S
δ
(v) ∈ L
2
(0,T;C
1

), and, hence, problem (2.6)is
uniquely solvable. we designate by
˜
Z
δ
(v) the solution to problem (2.6). Note that S
δ
(v)
coincides with
˜
v on Γ. In particular it means that all trajectories z(τ;t,x)ofproblems(2.6)
lay in Q.
3. Linear parabolic operator on noncylindrical domain
Consider a linear operator L : E
∗
→ E
∗
defined on the set D(L) of smooth solenoidal
functions v(t,x) vanishing on Γ and at t = 0bytheformula
Lv,h=

T
0

v
t
− ∆v,h

t
dt. (3.1)

Here h(t,x) ∈ E.Obviously,D(L) is dense in E
∗
.
Let us show that the operator L admits a closure and study its properties. First we
establish auxiliary results. The following result is known (see [11, page 8]).
Lemma 3.1. Let F(t,x) be a smooth scalar function. Then
d
dt

Ω
t
F(t,x)dx =

Ω
t
F
t
(t,x)dx +

Γ
t
F(t,x)
˜
v
n
(t,x) dx. (3.2)
Here
˜
v
n

(t,x)istheprojectionof
˜
v(t,x) on the direction of the external normal n(x)at
apointx ∈ Γ
t
.
Corollary 3.2. If F(t,x) = 0 for (t,x) ∈ Γ then (d/dt)

Ω
t
F(t,x)dx =

Ω
t
F
t
(t,x)dx.
Lemma 3.3. Afunctionv ∈ D(L) satisfies the inequality:


v
t


E
∗
+sup
0≤t≤T



v(t,x)


0,t
+ v
E
≤ MLv
E
∗
. (3.3)
220 On weak solutions of the equations of motion
Proof. Integration by parts and use of Corollary 3.2 yields

t
0
(Lv,v)
s
ds =

t
0


v
t
(s,x),v(s,x)

s
−


v(s,x), v(s,x)

s

ds
=

t
0

1
2
d
ds


v(s,x)


2
0,s
+

∇
v(s,x), ∇v(s,x)

s

ds
=


t
0

1
2
d
ds


v(s,x)


2
0,s
+


v(s,x)


2
1,s

ds
=
1
2



v(t,x)


2
0,t
+

t
0


v(s,x)


2
1,s
ds.
(3.4)
From this it follows that Lv,v=1/2|v(T, x)|
2
0,T
+ v
2
E
.As|Lv,v|≤Lv
E
∗
v
E
we get from (3.4) the inequality sup

t
|v(t,x)|
0,t
+ v
E
≤ MLv
E
∗
. On the other hand,


v
t


E
∗
= sup
h∈E,h
E
=1





T
0

v

t
,h

t
dt




≤ sup
h∈E,h
E
=1





T
0

v
t
−v,h

t
dt





+sup
h∈E,h
E
=1





T
0
(v,h)
t
dt




≤Lv
E
∗
+ v
E
.
(3.5)
Thus, v
t

E

∗
≤ MLv
E
∗
. The last inequalities imply (3.3). The lemma is proved. 
Lemma 3.4. Let v ∈ W. Then |v(t,x)|
0,t
is absolutely continuous in t on [0,T],differentiable
at a.e. t ∈ [0,T] and
1
2
d
dt


v(t,x)


2
0,t
=

v
t
(t,x), v(t, x)

t
, (3.6)



v(t,x)


0,t
≤ ε


v
t


E
∗
+ M(ε)v
E
, ε>0, t ∈ [0,T]. (3.7)
Proof. Let v(t,x) be smooth. Then (3.6)followsfromCorollary 3.2.Letusprove(3.7). It
follows from (3.6)and(2.1)thatfor0≤ τ ≤ T


v(t,x)


2
0,t
≤


v(τ,x)



2
0,τ
+2

t
τ


v
s
(s,x)


−1,s


v(s,x)


0,s
ds
≤


v(τ,x)


2
0,τ

+2


v
t


E
∗
v
E
(3.8)
is valid. Supposing t ≥ T/4 and integrating over τ on [0, T/4], we have


v(t,x)


2
0,t
≤ 4/T

T/4
0


v(s,x)


2

0,s
ds+2


v
t


E
∗
v
E
≤ M

v
2
L
2
(Q)
+


v
t


E
∗
v
E


.
(3.9)
The same inequality for t
≤ 3T/4 is established by means of integ ration over τ on
[3T/4,T]. Using inequality v
L
2
(Q)
≤v
E
and standard arguments we obtain from this
inequality (3.7).
The lemma is proved. 
V. G. Zvyagin and V. P. Orlov 221
In the cylindrical case this fact is proved in [18, Lemma 1.2, page 209].
Lemma 3.5. The space W is embedded in EC and v
EC
≤ Mv
W
.
The proof of the lemma follows from Lemma 3.4.
Let W
0
={v : v ∈ W, v(0,x) = 0}. It is not hard to show that W
0
can be easily obtained
by means of the closure in the W-norm of the set of smooth on Q functions which are
solenoidal on Ω
t

at every t and vanish on ∂Ω
t
and Ω
0
.Infact,letv ∈ W
0
. By the defini-
tion of W there exists a sequence of functions v
n
smooth on Q and solenoidal at every t
such that v − v
n

W
→ 0byn →∞.Letϕ
n
(t) be a smooth nondecreasing on [0,T] func-
tion such that ϕ
n
(t) ≡ 0whent ∈ [0,T/n]andϕ
n
(t) ≡ 1whent ∈ [2T/n]. The function
u
n
(t,x) = ϕ
n
(t)v
n
(t,x) vanishes at t = 0andon∂Ω
t

at every t.Obviously,u
n
converges to
v by n →∞in the W-norm.
Theorem 3.6. The operator L admits a closure
¯
L : E
∗
→ E
∗
with D(
¯
L) = W
0
,itsrangeR(
¯
L)
is closed and
¯
L is invertible on R(
¯
L).
Proof. Fro m (3.3) it follows that L admits a closure
¯
L.ItsdomainD(
¯
L) consists of those
v ∈ E
∗
for which there exists such a sequence v

n
∈ D(L)thatv
n
→ v and Lv
n
→ u in E
∗
.
Then by definition
¯
Lv = u. Let us show that D(
¯
L) ⊆ W
0
.Letv ∈ D(
¯
L). Then there exists
such a sequence v
n
∈ D(L)thatv
n
→ v in E and Lv
n
→ u in E
∗
. Then by means of passing
to the limit we have from (3.3)forv
n
that v ∈ W
0

and the inequality holds:


v
t


E
∗
+sup
t


v(t,x)


0,t
+ v
E
≤ M
¯
Lv
E
∗
. (3.10)
Thus, D(
¯
L) ⊆ W
0
.

Let us show that D(
¯
L) ⊇ W
0
.Letv ∈ W
0
and v
n
→ v, v
n
∈ D,intheW-norm that v ∈
W
0
.From(3.1) it follows that for h ∈ E 
¯
Lv
n
,h=Lv,h=

T
0
(∇v
n
(t,x), ∇h(t,x))
t
dt +
v
n
t
,h takes place. The passage to the limit gives the validity of 

¯
Lv,h=v
t
,h +

T
0
(∇v(t,x), ∇h(t,x))
t
dt for v ∈ D(
¯
L). From the obtained above it follows that the right-
hand side part defines an element u ∈ E
∗
for any v ∈ W
0
. By this v ∈ D(
¯
L)and
¯
Lv = u.
Thus, W
0
⊆ D(
¯
L) and consequently W
0
= D(
¯
L).

From (3.10) it follows that R(
¯
L)isclosedand
¯
L is invertible on R(
¯
L). The theorem is
proved. 
Remark 3.7. From (3.4) established for smooth v by means of the passage to the limit and
the differentiation with respect to t it is easy to show that the scalar function (
¯
Lv,v)
t
for
v ∈ D(L) and a.e. t satisfies the relation
(
¯
Lv,v)
t
=
1
2
d
dt


v(t,x)


2

0,t
+


∇v(t,x)


2
0,t
. (3.11)
Theorem 3.8. The range R(
¯
L) of the operator
¯
L is dense in E
∗
.
We give the proof of this theorem in Section 8.
From Theorems 3.6 and 3.8 the next result follows.
222 On weak solutions of the equations of motion
Theorem 3.9. For every f ∈ E
∗
the equation
¯
Lv = f has a unique solution v and the esti-
mate holds:


v
t



E
∗
+sup
t


v(t,x)


0,t
+ v
E
≤ M f 
E
∗
. (3.12)
Let k>0. Ever ywhere below we set
¯
v = exp(−kt)v. It is easy to show that exp(−kt)
¯
L(v)
=
¯
L(
¯
v)+k
¯
v. From here it follows that if L(v) = f then

¯
L(
¯
v)+k
¯
v =
¯
f .
Corollary 3.10. For the solution v of the equation
¯
Lv = f by any k>0 the estimate holds:


v
t


E
∗
,k
+ v
EC,k
≤ M f 
E
∗
,k
. (3.13)
To prove it is enough to make the change
¯
v = exp(−kt)v and take advantage of

Theorem 3.6.FromCorollary 3.10 and from Theorem 3.9 there follows the following the-
orem.
Theorem 3.11. For every
¯
f ∈ E
∗
the equation
¯
L(
¯
v)+k
¯
v =
¯
f has a unique solution
¯
v and
theestimateholds:


¯
v
t


E
∗
+sup
t



¯
v(t,x)


0,t
+ 
¯
v
E
+ k
¯
v
0
≤ M
¯
f 
E
∗
. (3.14)
4. Formulation of the main results
We are interested in the solvability of the regularized problem (1.1). By this we suppose
without loss of generality µ
0
= µ
1
= ρ = 1, replace z(s;t,x)byZ
δ
(v) and restrict ourselves
with the case v

0
(x) =
˜
v(0,x), x ∈ Ω
0
, v
1
(t,x) =
˜
v(t,x), (t, x) ∈ Γ. Thus, we get the prob-
lem
v
t
+ v
k
∂v/∂x
k
− DivᏱ(v) − Div

t
0
exp(s − t)Ᏹ(v)

s,
˜
Z
δ
(v)

(s;t,x)ds =−∇p + Φ,

divv(t,x) = 0, (t,x) ∈ Q;

Ω
t
p(t,x)dx = 0, t ∈ [0,T];
v(0,x) =
˜
v(0,x), x ∈ Ω
0
, v(t,x) =
˜
v(t,x), (t,x) ∈ Γ.
(4.1)
One should mark that the case of smooth and satisfying accordance conditions func-
tions v
0
(x)andv
1
(t,x)in(1.1) can be reduced to the conditions in (4.1). Let Φ(t,x) ∈ V
∗
t
at a.e. t.Let w(t, y) = w(t,z(t;0, y)) for an arbitrary function w(t,x)definedonQ.
Definit ion 4.1. A function v(t,x) =
˜
v(t,x)+w(t,x),
w(t,x)
∈ E, w ∈ L
2

0,T;W

1
2

Ω
0


W
1
1

0,T;W
−1
2

Ω
0

(4.2)
V. G. Zvyagin and V. P. Orlov 223
is called a weak solution of problem (4.1)ifforanyh(t,x) ∈ D, h(T,x) = 0 the identity
holds:
−

T
0

v(t,x), h
t
(t,x)


t
dt +

T
0

v
i
(t,x)v
j
(t,x), ∂h
i
(t,x)/∂x
j

t
dt
+

T
0

Ᏹ
ij

v(t,x)

,Ᏹ
ij


h(t,x)

t
dt
+

T
0


t
0
exp(s − t)Ᏹ
ij
(v)

s,

Z
δ
(v)(s;t,x)

ds,Ᏹ
ij

h(t,x)


t

dt
=

Φ(t,x),h(t, x)

−

˜
v(0,x), h(0,x)

0
.
(4.3)
The following main result takes place.
Theorem 4.2. Let Φ = f
1
+ f
2
, f
1
∈ E
∗
1
, f
2
∈ E
∗
. Then the problem (4.1) has at least one
weak solution.
The proof of the theorem is organized as follows. Following [21], we need to consider

a family of approximating operator e quations with a more weak nonlinearity. Alongside
with the operator
¯
L : W
0
→ E
∗
introduced above we will consider the operators
K
i
t
: V
t
−→ V
∗
t
, i = 1,2,3,

K
3
t
(w),h

t
=

w
i
w
j

,∂h
i
/∂x
j

t
, w,h ∈ V
t
;

K
1
t
(w),h

t
=

w
i
˜
v
j
,∂h
i
/∂x
j

t
, w,h ∈ V

t
;

K
2
t
(w),h

t
=

˜
v
i
w
j
,∂h
i
/∂x
j

t
, w,h ∈ V
t
;
(4.4)
the functional
˜
g ∈ E
∗

: 
˜
g, h=

T
0

˜
v
i
˜
v
j
,∂h/∂x
j

t
dt, h ∈ E; (4.5)
the functional
˜
V ∈ E
∗
: 
˜
V, h=

T
0

˜

v
t
−
˜
v,h

t
dt, h ∈ E; (4.6)
the operator A
t
: E → E
∗
: A
t
(v),h
t
= (Ᏹ
ij
(v),Ᏹ
ij
(h))
t
, v,h ∈ V
t
;theoperatorC
t
: E ×
CG → V
∗
t


C
t
(v,z), h

t
=


t
0
exp(s − t)Ᏹ
ij
(v +
˜
v)

s,z(s;t,x)ds,Ᏹ
ij
(h)


t
, v ∈ E, z ∈ CG.
(4.7)
224 On weak solutions of the equations of motion
The operators A
t
and C
t

naturally generate the operators A : E → E
∗
,andC : E × CG →
E
∗
:

A(w),h

=

T
0

A
t
(w),h

t
dt h ∈ E
∗
, w ∈ E;

C(v,z),h

=

T
0


C
t
(v,z), h

t
dt, v ∈ E, z ∈ CG, h ∈ E.
(4.8)
Alongside with the operators K
i
t
we will consider for ε>0theoperatorsK
i
t,ε
: V
t
→ V
∗
t
and the operators K
i
ε
: E → E
∗
generated by them:

K
1
t,ε
(u),h


t
=

u
i
˜
v
j
1+ε|u|
2
,
∂h
i
∂x
j

t
,

K
2
t,ε
(u),h

t
=

˜
v
i

u
j
1+ε|u|
2
,
∂h
i
∂x
j

t
,

K
3
t,ε
(u),h

t
=

u
i
u
j
1+ε|u|
2
,
∂h
i

∂x
j

t
, u,h ∈ V
t
;

K
i
ε
(u),h

=

T
0

K
i
t,ε
(u),h(t,x)

t
dt, u,h ∈ E.
(4.9)
Let K
ε
(v) =


3
i=1
K
i
ε
(v). Let Z
δ
(w) =
˜
Z
δ
(
˜
v + w)forw ∈ E. Consider for ε>0theoperator
equation
¯
Lw − K
ε
(w)+C

w,Z
δ
(w)

=
f
ε
. (4.10)
Theorem 4.3. For any f
ε

∈ E
∗
(4.10) has at least one solution w
ε
∈ W
0
.
Let us approximate a function f
1
∈ E
∗
1
by f
1,ε
→ f
1
at ε → 0, f
1,ε
∈ L
2,σ
(Q). Let v
ε
=
w
ε
+
˜
v,wherew
ε
is a solution to (4.10). Using the passage to the limit at ε → 0 we establish

that functions v
ε
converge to the function v which is a weak solution to problem (4.1),
that is, we get the assertion of Theorem 4.2.
The proof of Theorem 4.3 is carried out in Section 6.Theoperatortermsinvolvedin
(4.10) are investigated in Section 5.
5. Investigation of propert ies of operators
To investigate the operator terms of (4.10) we need the additional properties of functional
spaces.
Let
˜
z be a solution to the Cauchy problem (1.3)andu(t, y)
=
˜
z(t;0,y), y ∈ Ω
0
.Itis
clear that u maps Q
0
in Q, Q
0
= [0, T] × Ω
0
and u(t,Ω
0
) = Ω
t
.LetU(t,x) =
˜
z(0; t,x). The

maps u(t, y)andU(t,x)atfixedt are mutually inverse and
U

t,u(t, y)

= y, y ∈ Ω
0
, u

t,U(t,x)

= x, x ∈ Ω
t
. (5.1)
V. G. Zvyagin and V. P. Orlov 225
Moreover, solenoidality of
˜
v implies


U
x
(t,x)


=


u
y

(t, y)


= 1. (5.2)
Here U
x
, u
y
are the Jacobs matrixes and |U
x
|, |u
y
| are their determinants.
Let v(t, x) be a function smooth on Q. Define the operator Υ as v = Υv, v(t, y) =
v(t,u(t, y)). It is clear that v(t,x) = v(t,U(t,x)). Using differentiation, we have
v
t
(t, y) = v
t

t,u(t, y)

+ v
x

t,u(t, y)

u
t
(t, y) = v

t

t,u(t, y)

+ v
x

t,u(t, y)

˜
v

t,u(t, y)

;
(5.3)
v
y
(t, y) = v
x

t,u(t, y)

u
y
(t, y); (5.4)
v
t
(t,x) =


v
t

t,U(t,x)

+ v
y

t,U(t,x)

U
x
(t,x); v
x
(t,x) =

v
y

t,U(t,x)

U
x
(t,x).
(5.5)
From the smoothness of
˜
v(t,x) the smoothness of u(t, y), U(t,x) and their derivatives
follow.
Denote by W

1,−1
k,2
(Q), k = 1,2 a closure of the set of functions smooth both with respect
to t and with respect to x and finite on Ω
t
for all t in the norm
v
W
1,−1
k,2
(Q)
=


T
0


v(t,x)


k
W
1
2
(Ω
t
)
dt


1/k
+


T
0


v
t
(t,x)


k
W
−1
2
(Ω
t
)
dt

1/k
. (5.6)
Spaces W
1,−1
k,2
(Q
0
), k = 1,2 are defined on analogy. Let W

0,m
k,2
(Q), k = 1,2, m = 1,−1bea
closure of the set of the same functions in the norm
v
W
0,m
k,2
(Q)
=


T
0


v(t,x)


k
W
m
2
(Ω
t
)
dt

1/k
. (5.7)

Spaces W
0,m
k,2
(Q
0
) are defined similarly. Let

F = ΥF where F is some space of functions
on Q.
Lemma 5.1. Let v(t,x) be a smooth solenoidal on Q vector function,
v(t, y) = v(t,u(t, y)).
Then
v
L
2
(Q
0
)
=v
L
2
(Q)
, (5.8)


v
t


W

0,−1
2,2
(Q
0
)
≤ M



v
t


E
∗
+ v
W
0,1
2,2
(Q)

,


v
t


E
∗

≤ M



v
t


W
0,−1
2,2
(Q
0
)
+ v
W
0,1
2,2
(Q
0
)

,
(5.9)


v
t



W
0,−1
1,2
(Q
0
)
≤ M



v
t


E
∗
1
+ v
W
0,1
2,2
(Q)

,


v
t



E
∗
1
≤ M



v
t


W
0,−1
2,2
(Q
0
)
+ v
W
0,1
2,2
(Q
0
)

,
(5.10)
226 On weak solutions of the equations of motion
Proof. Using the change of the variable x = u(t, y)and(5.5), we have



v
t
(t,x)


−1,t
= sup
|h|
1,t
=1




v
t
(t,x), h(x)

t



= sup
|h|
1,t
=1






Ω
t
v
t
(t,x)h( x)dx




= sup
|h|
1,t
=1





Ω
0
v
t

t,u(t, y)

h

u(t, y)


dy




≤ sup
|h|
1,t
=1





Ω
0
v
t
(t, y)h

u(t, y)

dy




+sup
|h|

1,t
=1





Ω
0
v
y
(t, y)u
−1
y
(t, y)
˜
v

t,u(t, y)

h

u(t, y)

dy




=

J
1
+ J
2
.
(5.11)
Further, using the smoothness of u(t, y), we have
J
1
= sup
|h|
1,t
=1






Ω
0
v
t
(t, y)
h

u(t, y)




h

u(t, y)



1,0
dy







h

u(t, y)



1,0
≤ M sup
|

h|
1,0
=1






Ω
0
v
t
(t, y)

h(y)dy




sup
|h|
1,t
=1


h(u(t, y)



1,0
≤ M


v
t



−1,0
,
J
2
≤ sup
|h|
1,t
=1







Ω
0
v
y
(t, y)u
−1
y
(t, y)
˜
v

t,u(t, y)


h

u(t, y)



h

u(t, y)



1,0
dy







h

u(t, y)



1,0

≤ M sup

|

h|
1,0
=1





Ω
0
v
y
(t, y)u
−1
y
(t, y)
˜
v

t,u(t, y)


h(y)dy




sup

|h|
1,0
=1


h

u(t, y)



1,0
≤ M


v
y
(t, y)u
−1
y
(t, y)
˜
v

t,u(t, y)



−1,0
.

(5.12)
By similar reasonings from the smoothness of u(t, y)and
˜
v(t,x) it follows that


v
y
(t, y)u
−1
y
(t, y)
˜
v

t,u(t, y)



−1,0
≤ M


v(t, y)


W
1
2
(Ω

0
)
. (5.13)
Thus, we get


v
t
(t,x)


−1,t
≤ M



v
t
(t, y)


−1,0
+


v(t, y)


W
1

2
(Ω
0
)

. (5.14)
From the last three inequalities the second inequality (5.9) follows. Other inequalities
(5.8)–(5.10) are proved in the same way more easily. Lemma 5.1 is proved. 
Let Υ
t
be the restriction of the operator Υ on the space of functions on Ω
t
.Fromthe
density of the set of smooth functions in the spaces mentioned below and the proof of
Lemma 5.1 the next lemma follows.
V. G. Zvyagin and V. P. Orlov 227
Lemma 5.2. The linear operator Υ
t
is bounded and boundedly invertible as an operator
Υ
t
: L
2

Ω
t

−→ L
2


Ω
0

, Υ
t
: W
1
2

Ω
t

−→ W
1
2

Ω
0

, Υ
t
: W
−1
2

Ω
t

−→ W
−1

2

Ω
0

.
(5.15)
The operator Υ is bounded and boundedly invertible as an operator
Υ : W
0,m
k,2
(Q) −→ W
0,m
k,2

Q
0

, k = 1,2, m = 1,−1;
Υ : W
1,−1
k,2
(Q) −→ W
1,−1
k,2

Q
0

, k = 1,2.

(5.16)
In the cylindrical case the continuous embedding takes place W ⊂ C(0,T;H
0
)([6,
Theorem 1.17, page 177]). From this and Lemmas 5.1-5.2 the next fact follows.
Lemma 5.3. For any v ∈ W
1
the function v(t, y) is weakly continuous as a function with
values in L
2
(Ω
0
).
Proof of Lemma 5.3. From the boundedness of the orthogonal projection operator P
t
:
W
1
2
(Ω
t
) → V
t
(see [18, page 24]) which is uniform with respect to t thanks to the smooth-
ness of
˜
v(t,x) it follows that


P

t
h


W
1
2
(Ω
t
)
≤h
W
1
2
(Ω
t
)
, h ∈ W
1
2

Ω
t

. (5.17)
Let v ∈ W
1
and v
n
− v

W
1
→ 0, v
n
∈ D at n →∞. Using (5.17) and the obvious rela-
tion P
t
v
n
t
= v
n
t
for v
n
∈ D it is not difficult to show that


v
n
t


W
−1
2
(Ω
t
)
≤ M



v
n
t


V
∗
t
. (5.18)
By means of the change of the variable x = u(t, y)from(5.18)andLemma 5.2 it fol-
lows that the sequence v
n
t
is fundamental in W
0,−1
1,2
(Q
0
) or that is the same in L
1
(0,T;
W
−1
2
(Ω
0
)). Since v
n

is fundamental in E then v
n
is fundamental in L
2
(0,T;W
1
2
(Ω
0
)) and
moreover in L
2
(0,T;W
−1
2
(Ω
0
)). Thus,
v ∈ W
1
1

0,T;W
−1
2

Ω
0

∩ L

2

0,T;W
1
2

Ω
0

. (5.19)
From this and [18, Lemmas III.1.1 and III.1.4], the assertion of Lemma 5.3 follows.
Lemma 5.3 is proved. 
Remark 5.4. By means of the change of the variable y = U(t,x)fromLemma 5.3 it follows
that for v(t,x) ∈ W
1
there exists a trace belonging to H
t
at all t ∈ [0, T] and the inclusion
|v(t,x)|
0,t
∈ L
∞
[0,T].
Examine the operator terms of (4.10).
Lemma 5.5. Let n
≤ 4. Then the operators K
i
ε
, ε>0, i = 1,2,3, are bounded and continuous
as operators K

i
ε
: E → E
∗
, the operators K
i
0
= K
i
are bounded and continuous as operators
K
i
: E → E
∗
1
, and the inequalities hold:


K
i
ε
(v)


E
∗
≤ M
1
/ε, ε>0;



K
i
ε
(v)


E
∗
1
≤ M
1
v
2
E
, ε ≥ 0. (5.20)
228 On weak solutions of the equations of motion
Here M
1
depends on 
˜
v
C(Q)
. Besides, the operators K
i
ε
: W → E
∗
, ε>0, i = 1,2,3 are com-
pletely continuous.

The proof of Lemma 5.5 for i = 3 is similar to the proof for the cylindrical case ([3,
Lemma 2.1 and Theorem 2.2]). For i = 1,2 the proof is easier because of the smoothness
of
˜
v.
Lemma 5.6. For any v ∈ E, z ∈ CG the inclusion C(v,z) ∈ E
∗
is valid and the map C :
E × CG → E
∗
is continuous and bounded.
The proof repeats the proof of Lemma 2.2 in [21] for the cylindrical case which is fit
for the non cylindrical case as well.
Lemma 5.7. The map Z
δ
: W
1
→ CG is continuous and for every weakly converging sequence
{v
l
}, v
l
∈ W
1
, v
l
→ v
0
, there exists a subsequence {v
l

k
} such that Z
δ
(v
l
k
) → Z
δ
(v
0
) in the
space CG.
For the proof of Lemma 5.6 it is enough to repeat the proof of Lemma 3.2 in [21]and
take advantage of Lemma 5.6 and (2.5).
Lemma 5.8. For any z ∈ CG and u,v ∈ E the estimates hold:


C(v,z) − C(u,z)


k,E
∗
≤

M/ku − v
k,E
;


C(v,z)



k,E
∗
≤

M/k

v
k,E
+ 
˜
v
k,E

.
(5.21)
Proof of estimates (5.21) repeats the proof of Lemma 2.4 of [21] with the change of Ω
0
by Ω
t
in the calculations.
Let γ
k
be a Kuratovsky’s noncompactness measure (see [1]) in E
∗
with the norm ·

k,E
∗

.LetG(v) = C(v,Z
δ
(v)).
Theorem 5.9. For sufficiently large k the map G : W
0
→ E
∗
is
¯
L-condensing with respect
to γ
k
.
The definition of the
¯
L-condensing map is given in [3].
The proof of theorem repeats the proof of Theorem 2.2 in [21] on the strength of
Lemmas 5.6–5.8 and the inequality
u − v
k,E
≤ M
¯
Lu −
¯
Lv
k,E
∗
, v,u ∈ W
0
following

from (3.13).
6. Proof of Theorem 4.3
Introduce an auxiliary family of operator equations
¯
Lv
− λ
3

i=1
K
i
ε
(v) − λC

v,Z
δ
(v)

=
f
ε
, f
ε
∈ E
∗
, λ ∈ [0,1]. (6.1)
At λ = 1(6.1) coincides with (4.10). Following [21], we will obtain a priori estimates of
solutions to this family.
V. G. Zvyagin and V. P. Orlov 229
Theorem 6.1. For any solution v ∈ W

0
to problem (4.10) the estimates
v
EC
≤ M

1+


f
ε


E
∗

,


v
t


E
∗
≤ M

1+



f
ε


E
∗

(6.2)
hold. Here M does not depend on λ but depends on ε and on 
˜
v
C(Q)
.
Proof. Let v ∈ W
0
be a solution to (6.1). From (6.1)and(3.13) it follows that
v
k,EC
≤ M

3

i=1


K
i
ε
(v)



k,E
∗
+


C

v,Z
δ
(v)



k,E
∗
+


f
ε


k,E
∗

. (6.3)
From Lemma 5.8 it follows that



C

v,Z
δ
(v)



K,E
∗
≤

M/k

v
k,E
+ 
˜
v
k,E

. (6.4)
Tak ing in to accoun t (5.20) and the last inequalities, we get
v
k,EC
≤ M

ε
−1
+


M/k

v
k,E
+ 
˜
v
k,E

+


f
ε


k,E
∗

, (6.5)
where M depends on 
˜
v
C(Q)
.
As v
k,E
≤ Mv
k,EC

, we get from this
v
k,E
≤ M

ε
−1
+


f
ε


k,E
∗
+ 
˜
v
k,E

(6.6)
for sufficiently large k. Now, t aking into account the equivalence of the n orms ·
k,EC
and ·
EC
, ·
k,E
∗
and ·

E
∗
,wecometothefirstestimate(6.2). The second estimate
(6.2) follows from the first estimate (6.2) and the boundedness in E of the maps A, K
ε
,
and C. Theorem 6.1 is proved. 
LetusgoovertotheproofofTheorem 4.3. On the strength of Theorem 5.9 and
Lemma 5.1 maps λ(

3
i=1
K
i
ε
− G)are
¯
L-condensing with respect to the Kuratovsky’s non-
compactness measure γ
k
as maps from W × [0, T]inE
∗
. Moreover, from a priori es-
timates (6.2) it follows that every equation from (6.1)atλ ∈ [0, 1] has no solutions
on the boundary of the ball
¯
B
R
⊂ W
0

of a sufficiently large radius R with the center
inzero.Hence,foreveryλ ∈ [0,1] the degree of a map deg
2
(
¯
L − (

3
i=1
K
i
ε
− G),
¯
B
R
, f
ε
)
(see [20]) is defined. As a degree of a map does not change by the change of λ,then
deg
2
(
¯
L − λ

3
i=1
K
i

ε
+ G,
¯
B
R
, f
ε
) = deg
2
(
¯
L,
¯
B
R
, f
ε
). The map
¯
L is invertible, therefore the
equation
¯
Lv = f
ε
has a unique solution v
0
∈ W
0
and v
0

satisfies estimates (6.2). Then
v
0
∈ B
R
and deg
2
(
¯
L,
¯
B
R
, f
ε
) = 1. Therefore, deg
2
(
¯
L −

3
i=1
K
i
ε
+ G,
¯
B
R

, f
ε
) = 1. The differ-
ence from zero of a degree of a map implies the existence of solutions of the operator
equation (6.1) and consequently the existence of a solution of (4.10)((6.1)atλ = 1).
Theorem 4.3 is proved.
7. Proof of Theorem 4.2
Let us establish some auxiliary facts. Approximate f
1
∈ E
∗
1
by means of f
1,ε
→ f
1
as ε → 0,
f
1,ε
∈ L
2,σ
(Q). Let f
2
∈ E
∗
, f
ε
= f
1,ε
+ f

2
. It is clear that f
1,ε
∈ E
∗
and consequently f
ε
∈
E
∗
.
230 On weak solutions of the equations of motion
Theorem 7.1. Any solution of problem (4.10)forε>0 satisfies the estimate
w
EL
≤ M

1+


f
1,ε


E
∗
1
+



f
2


E

, (7.1)


w
t


E
∗
1
≤ M

1+


f
1,ε


E
∗
1
+



f
2


E
∗

, (7.2)
where M does not depend on ε but depends on
˜
v.
Here
w
EL
= sup
t
|w(t,x)|
0,t
+ w
E
.
Proof. Let w be a solution to problem (4.10)forsomeε>0. Then
¯
L(w) − K
ε
(w)+
C(w,Z
δ
(w)) = f

1,ε
+ f
2
. Multiplying the equation by e
−kt
,weget
¯
L(w)+kw − K
ε
(w)+C

w,Z
δ

e
kt
w(t)

= f
1,ε
+ f
2
, (7.3)
where
w(t)=e
kt
w(t), K
ε
(w)(t)=e
−kt

K
ε

w(t)

, C

w,Z
δ

e
kt
w(t)

(t)=e
−kt
C

w,Z
δ
(w)

(t),
f
1,ε
(t) = e
−kt
f
1,ε
(t), f

2
(t) = e
−kt
f
2
(t).
(7.4)
Operator k
w is defined by the equality kw,h=k(w,h)forh ∈ E.
Consider an action of the functionals in the left-hand side part of (7.3) upon the func-
tion w:
1
2
d
dt


w(t)


2
0,t
+ k


w(t)


2
0,1

+


∇w(t)


2
0,t
−

K
ε

w(t)

,w(t)

t
=−

C

w,Z
δ

e
kt
w(t)

(t),w(t)


t
+

f
1,ε
(t),w(t)

t
+

f
2
(t),w(t)

t
.
(7.5)
It is known that (K
3
ε
(w(t)),w(t))
t
= 0forallt ∈ [0, T] (see [3]). Therefore, the integration
of both parts of the identity on [0,t]yields
(1/2)


w(t)



2
H
+ kw
2
0
+ w
2
E
=−

t
0

C

w,Z
δ

e
kτ
w(τ)

(τ),w(τ)

τ
dτ +

t
0


f
1,ε
(τ),v(τ)

t
dτ
+

t
0

f
2
(τ),w(τ)

τ
dτ +
2

i=1

t
0

K
i
ε

w(t)


,w(τ)

τ
dτ.
(7.6)
Taking into account the smoothness of
˜
w,wehave





2

i=1

t
0

K
i
ε

w(t)

,w(τ)

τ

dτ





≤
2

i=1

t
0



K
i
ε

w(t)

,w(τ)

τ


dτ
≤ M
˜

w
C(Q)

t
0


w(τ,x)


L
2
(Ω
τ
)


w(τ,x)


W
1
2
(Ω
τ
)
dτ
≤ M
˜
v

C(Q)


w(τ,x)


L
2
(Q)


w(τ,x)


E
.
(7.7)
V. G. Zvyagin and V. P. Orlov 231
Then using the Cauchy inequality and estimate (5.21), we get the inequality
(1/2)


w(t)


2
H
+ kw
2
L

2
(Q)
+ w
2
E
≤

M/kw
2
E
+ M
˜
v
C(Q)


w(τ,x)


L
2
(Q)


w(τ,x)


E
+


M/k
˜
v
E


w(τ,x)


L
2
(Q)
+


f
1,ε


E
∗
1
w
CH
+


f
2



E
∗
w
E
+

M/k
˜
v
E
w
E
.
(7.8)
It easy to see that


w(τ,x)


L
2
(Q)


w(τ,x)


E

≤ ε
1


w(τ,x)


2
E
+ C

ε
1



w(τ,x)


2
L
2
(Q)
, ε
1
> 0,

˜
v
E



w(τ,x)


L
2
(Q)
≤ ε
2
+ C

ε
2



w(τ,x)


2
L
2
(Q)
, ε
2
> 0.
(7.9)
Tak ing k sufficiently large, ε
1

and ε
2
sufficiently smal l and using the last inequalities,
we obtain the estimate
w
2
CH
+ kw
2
L
2
(Q)
+ w
2
E
≤ M

1+


f
1,ε


2
E
∗
1
+



f
2


2
E
∗

, (7.10)
from which the required estimate (7.1)follows.
Let us prov e estimate (7.2). Repeat the arguments of the proof of estimate (6.2). Let w
be a solution to problem (4.10)forsomeε>0. From (4.10) it follows that w
t
=−A(w) −
K
ε
(w) − C(v,Z
δ
(w)) + f
ε
. Consequently


w
t


E
∗

1
≤ M



A(w)


E
∗
+


K
ε
(v)


E
∗
1
+


C

v,Z
δ
(w)




E
∗
+


f
1,ε


E
∗
1
+


f
2


E
∗

. (7.11)
Establish the estimate K
ε
(w)
E
∗

1
in the similar to [3]way.Forn ≤ 4 the inclusion V
t
⊂
L
4
(Ω
t
) is uniformly continuous with respect to t. Then we have by definition of K
3
ε
for
w ∈ V
t
the inequality


K
3
ε
(w)


−1,t
≤ max
i, j


w
i

w
j
/

1+ε|w|
2



L
2
(Ω
t
)
≤ max
i, j


w
i
w
j


L
2
(Ω
t
)
≤ Mw

2
L
4
(Ω
t
)
.
(7.12)
The expressions K
i
ε
, i = 1,2 are estimated in the same manner. Thus, K
ε
(w)
E
∗
1
≤
Mw
2
L
2
(0,T;L
4
)
. Here we denote the completion of the set of smooth on Q functions in
the norm w
L
2
(0,T;L

4
)
= (

T
0
w(t,x)
2
L
4
dt)
1/2
by L
2
(0,T;L
4
). In virtue of the mentioned
continuity of the inclusion V
t
⊂ L
4
the inclusion E ⊂ L
2
(0,T;L
4
)iscontinuousaswell.
Then w
L
2
(0,T;L

4
)
≤ Mv
E
and consequently K
ε
(w)
E
∗
1
≤ Cw
2
E
. This estimate easily
proved inequality |(w)|
−1,t
≤ M|w|
1,t
, the boundedness of the map C on E and estimate
(7.1) admit to get the required estimate (7.2)from(7.11). The theorem is proved. 
232 On weak solutions of the equations of motion
LetusgoovertotheproofofTheorem 4.2.Letv = w +
˜
v. It is easy to show that v is a
weak solution to problem (4.1)ifandonlyifw satisfies the integral identity
−

T
0


w(t,x), h
t
(t,x)

t
dt +

T
0

w
i
(t,x)w
j
(t,x), ∂h
i
(t,x) /∂x
j

t
dt
+

T
0

w
i
(t,x)
˜

v
j
(t,x), ∂h
i
(t,x)/∂x
j

t
dt +

T
0

˜
v
i
(t,x)w
j
(t,x), ∂h
i
(t,x)/∂x
j

t
dt
+

T
0


Ᏹ
ij
(w)(t,x), Ᏹ
ij
(h)(t,x)

t
dt
+

T
0


t
0
exp(s − t)Ᏹ
ij
(
˜
v + w)

s,Z
δ
(w)(s;t,x)

ds,Ᏹ
ij

h(t,x)



t
dt
=

T
0

Φ(t,x),h(t, x)

t
dt +

˜
V +
˜
g, h

−

˜
v(0,x), h(0,x)

0
(7.13)
for h satisfying the conditions of Definition 4.1.Let f
1,ε
→ f
1

in E
∗
1
as ε → 0, f
1,ε
∈ L
2,σ
(Q).
Consider (4.10)by f
ε
= f
ε,1
+
˜
f
2
where
˜
f
2
=
˜
V +
˜
g + f
2
. It is clear that
˜
f
ε

∈ E
∗
at any ε>0.
In virtue of Theorem 4.3 there exists a solution w
ε
to problem (4.10). It is not difficult to
show that w
ε
satisfies the identity
−

T
0

w
ε
(t,x), h
t
(t,x)

t
dt +

K
1
ε

w
ε


,h

+

K
2
ε

w
ε

,h

+

T
0

K
3
t,ε
(u),h

t
dt
+

T
0


Ᏹ
ij

w
ε

(t,x)ds,Ᏹ
ij
(h)(t,x)

t
dt +

C

w
ε
,Z
δ

w
ε

=

T
0

f
1,ε

,h

t
dt +

˜
f
2
,h

−

˜
v(0,x), h(0,x)

0
(7.14)
for h satisfying the conditions in Definition 4.1.
Let us show that there exists such a number sequence ε
l
, l = 1,2, ,thatw
ε
l
≡ w
l
con-
verges to w ∈ E which satisfies identity (7.13).
Choose the sequence of positive numbers {ε
l
} converging to zero. As f

1,ε
l
→ f
1
in E
∗
1
by l →∞then the sequence  f
1,ε
l

E
∗
1
is bounded. For each number ε
l
the corresponding
problem (4.10) has at least one solution w
l
∈ W
0
. In virtue of estimates (7.1) the sequence
{w
l
} is bounded in the norm ·
EL
. Then without loss of generality we will assume that
w
l
→ w

∗
weakly in E, w
l
→ w
∗∗
-weakly in E
∗
1
.Moreover,w
l
→ w
∗
strongly in L
2
(Q).
In fact, since both w
l
t

E
∗
1
and w
l

E
are bounded in virtue of (7.1)-(7.2), then as in the
proof of Lemma 5.3 we can suppose that w
l
→ w

∗
strongly in L
2
(Q).
Show that

C

w
l
,Z
δ

w
l

− C

w
∗
,Z
δ

w
∗

,h

−→ 0, l −→ ∞ . (7.15)
V. G. Zvyagin and V. P. Orlov 233

Using Lemma 5.7, without loss of generality we will assume that
Z
δ

w
l

−→ Z
δ

w
∗

in the norm of the space CG. (7.16)
Let h ∈ E be an arbitr ary function. Consider

C

w
l
,Z
δ

w
l

− C

w
∗

,Z
δ

w
∗

,h

=

C

w
l
,Z
δ

w
l

− C

w
∗
,Z
δ

w
l


,h

+

C

w
∗
,Z
δ

w
l

− C

w
∗
,Z
δ

w
∗

,h

.
(7.17)
The second term converges to zero by virtue of the assumption (7.16)andthecontinuity
of the map C with respect to z. By means of the change of the variable z

= Z
δ
(w
l
)(s;t,x)
(the inverse change looks like x = Z
δ
(w
l
)(t;s,z)) in the first term we get

C

w
l
,Z
δ

w
l

− C

w
∗
,Z
δ

w
l


,h

=

T
0

Ω
t

t
0

Ᏹ
ij

w
l

s,Z
δ

w
l

(s;t,x)

− Ᏹ
ij


v
∗

s,Z
δ

w
l

(s;t,x)

dsᏱ
ij
(h)(t,x)dxdt
=

T
0

t
0

Ω
0

Ᏹ
ij

w

l

(s,z) − Ᏹ
ij

w
∗

(s, y)

Ᏹ
ij
(h)

t,Z
δ

w
l

(t;s, y)

dydsdt.
(7.18)
UndertheassumptiongivenabovethefirstbracketconvergestozeroweaklyinL
2
(Q).
The map (w, y) → Ᏹ(w)(s,z(s; t, y)) from the space E × CG in the space L
2
([0,T] × [0,T],

L
2
(Ω
0
)) is continuous. From this and (7.16)wegetthestrongL
2
([0,T] × [0,T], L
2
(Ω
0
))
convergence of the second factor in the last integ ral. Then the whole expression converges
to zero as l →∞as well. T his yields the proof of convergence (7.15).
The possibility of the passage to the limit for the terms with the exception of that ones
containing K
i
ε
follows from the facts given above. In [3], the convergence

T
0
(

K
3
ε
l
(w
l
),


h)
t
dt
to

T
0
(

K
3
(w
∗
),

h)
t
dt was shown. The similar fact for

K
i
ε
l
, i = 1,2 is proved easier. Using the
inverse to x = u(t, x) change of variable, we get the convergence for the terms in (7.14)
containing K
i
ε
.

To finish the proof of the theorem we pass to the limit as l →∞in (7.14). Taking into
account the convergences mentioned above and passing to the limit in (7.14)wegetthat
w
∗
satisfies the identity (7.13).
Besides, from estimate (7.1), (7.2), and Lemma 5.1 it follows that  w
l

L
2
(0,T;W
1
2
(Ω
0
))
and  w
l

W
1
1
(0,T;W
−1
2
(Ω
0
))
are bounded. Without loss of generality we can believe that w
l

converges
∗
-weakly to w in L
2
(0,T;W
1
2
(Ω
0
))

W
1
1
(0,T;W
−1
2
(Ω
0
)).
The theorem is proved.
8. Proof of Theorem 3.8
The basic moment here is the proof of the solvability of a corresponding linear Stokes
problem in a non cylindrical domain. For this purpose using the change of the variable
x
= u(t, y) =
˜
z(τ;t,x), we reduce this problem to the corresponding problem in a cylin-
drical domain. By this the methods of [17] are essentially used.
234 On weak solutions of the equations of motion

Let us show now that the range R(
¯
L)oftheoperator
¯
L is dense in E
∗
. For this purpose
we establish the solvability in the class W
1,2
2
(Q)oftheproblem
v
t
−v + ∇p = g, ∇·v = 0, (t,x) ∈ Q,

Ω
t
p(t,x)dx = 0, 0 ≤ t ≤ T; (8.1)
v(0,x) = 0, x ∈ Ω
0
; v(t,x) = 0, (t, x) ∈ Γ. (8.2)
Here ∇·v = divv. T he pair of functions v ∈ W
1,2
2
(Q), p ∈ W
0,1
2
(Q)iscalledasolutionto
problem (8.1)-(8.2) if it satisfies a.e. (8.1) and conditions (8.2).
Using the change of the variable

x = u(t, y), u(t, y) = y +

t
0
˜
v(s, y)dy, x =
˜
z(t,0,y) (8.3)
and (5.3)–(5.5)wereducetheproblem(8.1)-(8.2)totheproblem
v
t
+ D(v) −
ˆ
v +

∇

p = g,

∇·v = 0, (t, y) ∈ Q
0
;

Ω
0

p(t, y)dy = 0;
v(0, y) = 0, y ∈ Ω
0
; v(t, y) = 0, (t, y) ∈ Γ

0
.
(8.4)
Here D(v) =

v
y
C, C(t, y)isavectorfunctionwithsmoothcoefficients which are ex-
pressed by means of the coefficients of the Jacobs matrix
˜
v
y
(t, y)and

∆v= Υ(∆v),

∇

p=Υ(∇p),

∇·v= Υ(∇·v), Υv(t,x)= v(t, y) = v

t,u(t, y)

.
(8.5)
Rewrite (8.4)intheform
v
t
−v + ∇


p = f (t, y), ∇·v = ρ(t, y);

Ω
0

p(t, y)dy = 0;
v(0, y) = 0; v(t, y) = 0, (t, y) ∈ Γ
0
.
(8.6)
Here
f
=

g − D( v)+(

−)v +(∇−

∇)

p, ρ = (∇−

∇) · v. (8.7)
Note that

Ω
0
ρ(t, y)dy = 0. (8.8)
In fact, using the Ostrogradsky formula and the inverse to (8.3) change of the variable

y
= U(t,x) =
˜
z(0, t,x), we have

Ω
0
ρ(t, y)dy =

Ω
0
∇·vdy−

Ω
0

∇·vdy=−

Γ
0
˜
v
n
(t, y)dy+

Ω
t
∇·v(t,x)dx = 0.
(8.9)
Let us establish the solvability of problem (8.6). At first consider an auxiliary problem

with an arbitrary f ∈ L
2
(Q
0
)andρ(t, y) ∈ L
2
(Q
0
), satisfying some additional conditions
V. G. Zvyagin and V. P. Orlov 235
which we will impose below:
u
t
−u + ∇p = f (t, y), ∇·u(t, y) = ρ(t, y), (t, y) ∈ Q
0
;
u(0, y) = 0, y ∈ Ω
0
, u(t, y) |
Γ
= 0,

Ω
t
p(t,x)dy = 0.
(8.10)
Let us reduce the question of the solvability of problem (8.10) to the solvability of a
similar problem with ρ
≡ 0. By this we assume the condition (8.8) is fulfilled. For this
purpose we will consider an auxiliary problem for t ∈ [0,T]

−Φ = ρ(t, y), y ∈ Ω
0
, Φ |
Γ
0
= 0 . (8.11)
If ρ(t, y) ∈ L
2
(Ω
0
)ata.e.t,thenproblem(8.11)isuniquelysolvable(see[15]) and its
solution has the form Φ = R
−1
ρ and


Φ(t, y)


2,0
 M


ρ(t, y)


0,0
(8.12)
is valid. From this it follows that Φ
0,2

 Mρ
0
.HereR is a selfadjoint operator with
the domain W
2
2,0
(Ω
0
), positively defined in L
2
(Ω
0
), R
−1
is an oper ator inverse to R.Itis
clear that


R
1/2
u


2
0,0
= ( Ru, u)
0
=−(u,u)
0
= ( ∇·u, ∇·u)

0
=|u|
2
1,0
, u ∈ W
2
2,0

Ω
0

. (8.13)
In virtue of this we get the boundedness in L
2
(Ω
0
) of the following operators and the
estimates




∂
∂y
j
R
−1/2
u





0,0
 M|u|
0,0
,




R
−1/2
∂
∂y
i
u




0,0
 M|u|
0,0
,




∂
∂y

i
R
−1
∂
∂y
j
u




0,0
 M|u|
0,0
.
(8.14)
Here the bar above means the closure of an operator.
Let u
1
=−∇Φ,whereΦ is a solution to problem (8.11). Then u
1
∈ W
0,1
2
(Q
0
)and
∇u
1
= ρ.LetΨ(t, y)beasolutiontoproblem

Ψ = 0, ∇
n
Ψ |
Γ
0
=∇
n
Φ |
Γ
0
. (8.15)
Here ∇
n
is a normal derivativ e on Γ
0
.AsΦ(t, y) ∈ W
3/2
2
(Γ
0
)ata.e.t and in virtue of
(8.8)

Γ
0
∇
n
Φ(t, y)dy =−

Ω

0
Φ(t, y)dy =−

Ω
0
ρ(t, y)dy = 0, (8.16)
is valid then problem (8.15)isuniquelysolvableinW
2
2
(Ω) (see [15]).
Its solution h as the form Ψ = N
−1
γ
n
∇Φ.HereN
−1
is a linear bounded operator acting
from W
1/2
2
(Γ
0
)intoW
2
2
(Ω
0
) (see [17]), and γ
n
is the operator of taking of the normal

component of a trace on the boundary of a function defined on Ω
0
.Theoperatorγ
n
236 On weak solutions of the equations of motion
is bounded as an oper ator from W
1
2
(Ω
0
)inW
1/2
2
(Γ
0
). From here it follows that if Φ ∈
W
0,2
2
(Q
0
), then Ψ ∈ W
0,2
2
(Q
0
), and Ψ
0,2
 MΦ
0,2

.
Let now u
2
=∇Ψ and u = u
1
+ u
2
+ v. It is clear that ∇·v = 0, v |
Γ
0
= (u
1
+ u
2
) |
Γ
0
,
γ
n
(u
1
+ u
2
) = 0. The function v is a solution to the problem
v
t
−v + ∇p = w, ∇·v = 0,
v(0, y) = 0, v(t, y) |
Γ

=

u
1
+ u
2

|
Γ
0
, γ
n
(v) = 0.
(8.17)
Here w = f − u
1
t
− u
2
t
+ u
1
+ u
2
.If
u
1
∈ W
1,2
2


Q
0

, (8.18)
then from the relation u
2
=∇N
−1
γ
n
u
1
by virtue of the boundedness of operators N
−1
and γ
n
it follows that u
2
∈ W
1,2
2
(Q
0
), u
2

1,2
 Mu
1


1,2
and, hence, w ∈ L
2
(Q
0
). From
the continuity of the embedding W
1,2
2
(Q
0
) ⊂ W
3/4,3/2
2
(S
0
), S
0
= [0,T] × Γ
0
it follows that
(u
1
+ u
2
)|
Γ
∈ W
3/4,3/2

2
(S
0
). Therefore (see [15]), the following theorem takes place.
Theorem 8.1. Let u
1
∈ W
1,2
2
(Q
0
). Then problem (8.17)hasauniquesolutionv, p and the
estimate holds:
v
1,2
+ p
0,1
≤ M

w
0
+


u
1
+ u
2



W
3/4,3/2
2
(S
0
)

≤ M


u
1


1,2
. (8.19)
Let us find out the conditions under which on ρ condition (8.18) is fulfilled.
Theorem 8.2. Let
R
−1/2
ρ ∈ W
1
2

0,T;L
2

Ω
0


, ρ ∈ L
2

0,T;W
1
2

Ω
0

. (8.20)
Then u
1
=∇R
−1
ρ ∈ W
1,2
2
(Q
0
) and the estimate holds:


u
1


1,2
 M






d
dt
R
−1/2
ρ




0
+ ρ
0,1

. (8.21)
Proof of Theorem 8.2. Supposing u
1
=∇R
−1/2
R
−1/2
ρ and using the boundedness of the
operator ∇R
−1/2
in L
2
(Ω

0
)and(8.20), we get
d
dt
u
1
=
d
dt

∇R
−1/2
R
−1/2
ρ

=∇
R
−1/2
d
dt
R
−1/2
ρ ∈ L
2

Q
0

,





d
dt
u
1
(t, y)




0,0
 M




d
dt
R
−1/2
ρ




0,0
.

(8.22)
Similarly,
u
1
=∇R
−1
ρ =∇R
−3/2

R
1/2
ρ

. (8.23)
V. G. Zvyagin and V. P. Orlov 237
By virtue of the boundedness of the operator ∇∇R
−3/2
in L
2
(Ω
0
) and the conditions of
Theorem 8.2 we have the estimate



u
1



 M


R
1/2
ρ


0
= M


T
0


R
1/2
ρ(t,x)


2
0,0
dt

1/2
≤ M


T

0


ρ(t,x)


2
1,0
dt

1/2
= Mρ
0,1
.
(8.24)
From estimates (8.22)and(8.24) the statement of Theorem 8.2 follows. Theorem 8.2 is
proved. 
Theorem 8.3. Let a function ρ(t, y) ∈ W
0,1
2
(Q
0
) satisfy the integral identity

ρ(t, y),φ(y)

0
=

t

0

b(s, y),∇φ(y)

0
ds+

t
0

r(s, y),φ(y)

0
dy, (8.25)
where r,b ∈ L
2
(Q
0
) for an arbitrary function φ smooth and finite in Ω
0
. Then u
1
∈ W
1,2
2
(Q
0
)
and the inequality holds:



u
1


1,2
+ p
0,1
 M

r
0
+ b
0
+ ρ
0,1

. (8.26)
Proof of Theorem 8.3. Let us show that under the conditions of Theorem 8.3 inclusions
(8.20)takeplace.Asφ in (8.25)belongstoD(R)thenfrom(8.25) it follows that

R
−1/2
ρ,R
1/2
φ

0
=


t
0

R
−1/2
∇b,R
1/2
φ

0
ds+

t
0

R
−1/2
r,R
1/2
φ

0
ds. (8.27)
As the set of functions R
−1/2
φ is dense in L
2
(Ω
0
)thenfrom(8.27) it follows that R

−1/2
ρ ∈
W
1
2
(0,T;L
2
(Ω
0
)) and
d
dt
R
−1/2
ρ = R
−1/2
∇b + R
−1/2
r. (8.28)
The first inclusion (8.20) is established. The second one follows from the inclusion ρ ∈
W
0,1
2
(Q
0
). Thus, the conditions of Theorem 8.2 are fulfilled. Therefore, u
1
∈ W
1,2
2

(Q).
Estimate (8.26)followsfrom(8.28)andρ ∈ W
0,1
2
(Q
0
). Theorem 8.3 is proved. 
From Theorems 8.1 and 8.3, there follows the following theorem.
Theorem 8.4. Le t ρ satisfy the conditions of Theorem 8.3.Thenproblem(8.10)hasaunique
solution and the inequality holds:
v
1,2
+ p
0,1
≤ M

 f 
0
+ b
0
+ r
0
+ ρ
0,1

. (8.29)
Establish the solvability of problem (8.6)-(8.7). We will construct the approximations
v
n
,


p
n
, using the auxiliary iterative process
v
n+1
t
−v
n+1
+ ∇

p
n+1
= w
n
(t, y),

Ω
0

p
n+1
(t, y)dy = 0,
∇v
n+1
= ρ
n
(t, y), y ∈ Ω
0
; v

n+1
|
Γ
0
= 0.
(8.30)
238 On weak solutions of the equations of motion
Here n = 0,1, , v
0
= 0,

p
0
= 0, and
w
n
(t, y) = g − D

v
n

− (−
ˆ
)v
n
+(∇−

∇)

p

n
,
ρ
n
(t, y) = (∇−

∇)v
n
.
(8.31)
Show that for any n problem (8.30) is solvable. If v
n
∈ W
1,2
2
(Q
0
),

p
n
∈ W
0,1
2
(Q
0
)then
w
n
∈ L

2
(Q
0
), ρ
n
∈ W
0,1
2
(Q
0
). To take advantage of Theorem 8.3 it is enough to check the
fulfillment of condition (8.25).
Let
a
ij
(t, y) =
∂
˜
z
i
∂y
j
(t;0, y), β
ij
(t,x) =
∂
˜
z
i
∂x

j
(t;0,x), b
ij
(t, y) = β
ij

t,z(t;0,y)

.
(8.32)
As
˜
z(τ;t, y) = y +

τ
t
˜
v

s,z(s;t, y)

ds, (8.33)
then
a
ij
(t, y) = δ
ij
+

t

0
∂
∂x
k
˜
v
i

s,
˜
z( s;0, y)
∂
˜
z
k
∂y
j
(s;0, y)

ds = δ
ij
+
˜
a
ij
(t, y),
b
ij
(t, y) = δ
ij

−

t
0
∂
˜
v
i
∂x
k

s,
˜
z

s;t,
˜
z( t;0, y)

∂
˜
z
k
∂x
i

s;t,
˜
z( t,0,y)


ds
= δ
ij
−

t
0
∂
˜
v
i
∂x
k

s,
˜
z(0; s, y)

∂
˜
z
k
∂x
i
(0;s, y)ds = δ
ij
+
˜
b
ij

(t, y).
(8.34)
It is clear that at i = jb
ij
=−
˜
b
ij
. Using (8.5), we get

∇v(t, y) = b
ji
(t, y)
∂v
∂y
j
(t, y),

∇

p(t, y) =

b
ji
(t, y)
∂

p
∂y
j

(t, y)

n
i=1
,
ˆ
v(t, y) =

b
ji
∂
∂y
j

b
li
∂v
k
∂y
l

n
k=1
=

b
ji
b
li
∂

2
v
k
∂y
i
∂y
l
+ b
ji
∂
∂y
j
b
li
∂v
k
∂y
l

n
k=1
.
(8.35)
From (8.32)and(8.34) it follows that
(∇−

∇)v =
˜
b
ij

(t, y)
∂v
i
∂y
j
, (8.36)
(−
ˆ
)v =
˜
b
ji
˜
b
li
∂
2
v
∂y
j
∂y
l
+

˜
b
jl
+ b
lj


∂
2
v
∂y
j
∂y
l
+ b
ji
∂
∂y
j
˜
b
li
∂v
∂y
l
, (8.37)
(∇−

∇)

p =

˜
b
ji
∂


p
∂y
j

n
i=1
. (8.38)
V. G. Zvyagin and V. P. Orlov 239
From (8.36) it follows that
ρ
n
= (∇−

∇)v
n
=
˜
b
ij
∂v
n
i
∂y
j
,
(ρ,φ)
n
0
=


˜
b
ij
∂v
n
i
∂y
j
,φ

0
=−

v
n
i
,
∂
∂y
j

˜
b
ij
φ


0
.
(8.39)

Denote
˜
b
ij,k
= ( ∂/∂y
k
)
˜
b
ij
and (u)
y
k
= ∂u/∂y
k
.Differentiating with respect to t,wehave
d
dt

ρ
n
,φ

=−

d
dt
v
n
i

,

˜
b
ij
φ

y
j

0
−

v
n
i
,

d
dt
˜
b
ij
φ

y
j

0
=−


g
i
,

˜
b
ij
φ

y
j

0
+


D

v
n

i
,

˜
b
ij
φ


y
j

0
+


−
ˆ


v
n
i
,

˜
b
ij
φ

y
j

0
−


∇−


∇


p
n

i
,

˜
b
ij
φ

y
j

0
−

v
n
i
,

d
dt
˜
b
ij

φ

y
j

=
5

i=1
Z
i
.
(8.40)
It is easy to see that

˜
b
ij
φ

y
j
=
˜
b
ij,j
φ +
˜
b
ij

φ
y
j
,
d
dt
˜
b
ij
=−
˜
v
x
k

t,
˜
z(0; t, y)

˜
z
k
y
i
(0;t, y).
(8.41)
From formulas (4.7), (8.41) it follows that
d
dt


ρ
n
,φ

0
=

q
n
,∇φ

0
+

r
n
,φ

0
. (8.42)
Here the vector functions q
n
and r
n
are products of smooth functions. By this they
are expressed by means of
˜
v multiplied by the derivatives up to the second order of
v
n

and the first order of

p
n
with respect to the spatial variables and items which are
products of smooth functions and components of g. Therefore, q
n
,r
n
∈ L
2
(Q
0
). Besides,

Ω
0
ρ
n
(t, y)dy = 0. This equality can be checked in the way as in (8.9), using the
Ostrogradsky formula and taking into account that v
n
(t, y) = 0onΓ
0
.
Thus, from Theorem 8.3 it follows that the consecutive approximations v
n
,

p

n
are
defined for all n.
Establish the convergence of v
n
,

p
n
.Letφ
n+1
=

v
n+1
− v
n
, ψ
n+1
=

p
n+1
−

p
n
.Then
φ
n+1

y
−φ
n+1
+ ∇p
n+1
=−D

φ
n

− (−
ˆ
)φ
n
+(∇−

∇)ψ
n
,
∇φ
n+1
= ( ∇−

∇)φ
n
, φ(0, y) = 0, φ(t, y)|
Γ
0
, φ
n

|
Γ
0
= 0.
(8.43)
It is easy to see that
D

φ
n

= φ
n+1
y
j
c
j
, c
j
(t, y) =
d
dt
z
j
(0,t;x), x = z(t,0, y). (8.44)