p

L (0, T ; Y )
H
H gV g

AB

V

C([0, T ]; Y )


A B
g

C

g

g

g

g



k · kV ′ , k · k∗

V

′

′

′

V Vg

Vg

′

Vg

h·, ·i h·, ·ig
p

| · |p
A, B

L (O)

1≤p≤∞
g

Ag, Bg, Cg


O×R ,

∇·

+

O× R ,

u(x, t) = 0

+

∂O × R ,

u(x, 0) = u0(x)

O,
u = u(x, t) = (u1, u2, u3) p = p(x, t)
ν>0

α

u0

α
α =0
ν =0

α

ν>0
u0
g

g

Og = O × (0, g)
Og
g


g
du = [ν u − (u · ∇)u − ∇p + f + F (u(t − ρ(t)))]dt
+ G(u(t ρ (t)))dW (t),

x

−

∇·

(gu) = 0,

x

u(x, t) = 0,

∈O
∈O



τ

g


•
g

•
◦

◦

g

g


◦

g

•

•

•

•

g
•

g


n

O

R (n = 2, 3)

∂O

1≤p≤∞

m

p

L (O)
p

dx = dx1 . . . dxn

p

L (O) 1 ≤ p ≤ ∞
kukLp =




m,p

W
W

m,p

p

(O)

γ

p

0 ≤ |γ| ≤ m},

(O) = {u ∈ L (O) : D u ∈ L (O)
1/p

γ

kukW m,p =

kD ukL
|γX

p

γ

(D u, D v).
|≤m

m
H0 (O)

∞

C0 (O)
p

L (0, T ; Y )

m

H (O)

C([0, T ]; Y )

Y

|| · ||
p

L (0, T ; Y ) 1 ≤ p ≤ ∞
φ : [0, T ] → Y
i)kφkLp (0,T ;Y ) :=
ii)kφkL∞ (0,T ;Y ) :=


L (0, T ; Y ) L (0, T ; Y )1/p + 1/q = 1
C([0, T ]; Y )
[0, T ] → Y
kφk
:=
C([0,T ];Y )

0≤t≤T

||φ(t)|| < ∞.

max

φ:


C([0, T ]; Y )
2

φ(s) s ∈ R

L loc(R; Y )
Y
Z

t2

2



(u, v) :=
j=1

O

3

Z
X

1

j=1

O

2

2

|u| := (u, u), kuk := ((u, u)).
∞

V = u ∈ (C0 (O))
H
1

(H0 (O))


3

∇uj · ∇vj dx, u = (u1, u2, u3), v = (v1, v2, v3) ∈ (H0 (O)) ,

((u, v)) :=

′

h·, ·i

′

V
2

2

2

2

kuk α : = |u| + α kuk , α > 0,
k·k
λ1
2

1 + α λ1

2



∂O

2

2

1

L (O, g) = (L (O)) H0 (O, g) =

2
2

Z
X

2

(u, v)g :=

uj vj g dx, u = (u1, u2), v = (v1, v2) ∈ L (O, g),
j=1

O

2

Z
X


2
∞

Vg =

2

u ∈ (C0 (O2

Hg

Vg

1

L (O,
′

′

V g ⊂ Hg ≡ H g ⊂ V g

H0 (O, g)
Vg

′

k · k∗



3

P

D(A) = (H (O)) ∩ VAu = −P u ∀u ∈ D(A)
2

3

H

(L (Ω))
B:V×V→V

′

(B(u, v), w) = b(u, v, w),

u, v, w ∈ V,

3

∂vj

b(u, v, w) = i,j=1

Z ui

X

kwk

3/4

∀u, v, w ∈ V,

,

kukkvkkwk,

ckukkvk

1/2

|Av|

1/2

∀u, v, w ∈ V,

|w|,

∀u ∈ V, v ∈ D(A), w ∈
H,

c
Ag Bg

Cg


bg (u, v, w) =

∂vj
i,j=1

X

Z

O ui

wj gdx.

∂x i

u, v, w ∈ Vg
bg(u, v, w) = −bg (u, w, v), bg(u, v, v) = 0.
Cg : V g → H g
(C u, v)
g

= (( ∇g · ∇ )u, v) = b ( ∇g , u, v), ∀ v ∈ V .
g

g
− u − ( ∇g · ∇)u,

g
1 ( ∇ · g ∇ )u =



= A u, v

g

g ·∇

ig

+(( ∇g

)u, v) , u, v

g

2

2

kuk g ≥ η1|u| g, ∀u ∈ Vg,
2

2

|Ag u|g ≥ η1kukg , ∀u ∈ D(Ag ),
η1 > 0

g

∀


u g1/2 u g1/2 v g1/2 A

b (u, v, w)
|

g

|≤

c3 u
||

1/2
g

kk
1/2
g

Au
g

|

| 1/2

v
kk



Ag w g

g

| | ∀ ∈

∀ ∈

∈

), v V , w H
g

g

∈

,
g

∈

, u Hg, v Vg, w D(Ag ),

|

g

∈

kC u(t)
g

t ∈ (0, T ),

· ku(t) kg ,
k∗ ≤ |∇g|∞
m0η1

(Ω, F , P)

t ∈ (0, T ).

1/2

Ω σ
Ω

F
P


Wt


(Ω, F , P)

Wt
{Wt}
i) W0 = 0

B(K)
K
K
X : Ω → K,
K

A
−1

X (A) = {ω ∈ Ω : X(ω) ∈ A} ∈ F .
X
Z
E(X) =

Ω

X(ω)dP(ω).
K
X:Ω→K

K

a∈K
hX, ai