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p

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

AB

V

C([0, T ]; Y )


A B
g

C


g

g

g

g



H, V

Hg , V g
g




V , Vg

V Vg

(·, ·), | · |

H

((·, ·)), k · k

V

(·, ·)g, | · |g

Hg

((·, ·))g, k · kg

Vg
V

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
D(A), D(Ag)

A, Ag



u

t

(A, B)
(Ω, F , P)
h.c.c

ut(·)

ut(s) = u(t + s)
A, B



ut − α

2

u=0

ut − ν u + (u · ∇)u + ∇p = f

∇·

u(x, t) = 0

+

O

×R ,

O

× R ,

+

+

∂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


g
∂u
∂t

− ν u + (u · ∇)u + ∇p = f

(gu) = 0

∇·

u(x, t) = 0

+

O

×R ,

O

× R ,

+

+

u(x, 0) = u0(x),

∂O × R ,
O,

u = u(x, t) = (u1, u2) p = p(x, t)
ν>0
u0
g

Og = O × (0, g)

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

, t > 0,

, t > 0,

x ∈ ∂O, t > 0,

u(x, t) = ϕ(x, t),

x

, t [ τ, 0],
∈O

∈−

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

f = f (x)

u0
F (·)
W (t)

G(u(t − ρ(t)))dW (t)

ϕ

ρ : [0, +∞) → [0, τ ]
t ∈ [−τ, 0]

τ

g



g






g

g




g











g




g

g
g=







>0








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 =

1/p

Z

p

O

kukL∞
p=2

|u| dx

, 1 ≤ p < ∞,

O|u(x)|.

=

2

L (O)
Z

(u, v) = u.vdx,
O

k · kL2

2

kukL 2(O) = (u, u)


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

p

.

|≤m

W

m,p

(O)

p=2
m

H (O) = W

m,2

(O)

X



γ

((u, v))Hm =

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 ) :=

Z

0

T

p

kφ(s)k ds
0≤t≤T

1/p

<∞

1 ≤ p < ∞,

||φ(t)|| < ∞.

p

1
L (0, T ; Y )
p

q



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

kφ(s)k ds < ∞,

[t1, t2] ⊂ R.

t1

V

H
O

R

(u, v) :=

Z

((u, v)) :=

3

2

3

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

O

X

O

X

∂O

V

H
Z

3

j=1

3

1

j=1

2

2

|u| := (u, u), kuk := ((u, u)).


V = u ∈ (C0 (O))
H
1

(H0 (O))

3

:∇·u=0.

2

V

(L (O))


3

V⊂H≡H ⊂V

3

V

V

V



V

k · kV ′
V

3

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



h·, ·i



V
2

2

2

2

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

1 + α λ1

2

V
2

−2

2

kukα ≤ kuk ≤ α kukα ,


O

λ1 > 0
A
Hg

Vg

g
R

O

2

1

(H0 (O))

2

∂O

2

2

1

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

2
Z

(u, v)g :=

Z

((u, v))g :=

O

X

2

2

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

O

X

j=1

2

1

∇uj · ∇vj gdx, u = (u1, u2), v = (v1, v2) ∈ H 0(O, g),
j=1

2

g

2

|u|g = (u, u)g ||u||g = ((u, u))g
| · |g

2

k · kg

1

(H0 (O))

(L (O))

2

2


Vg =

2

u ∈ (C0 (O2

Hg

L (O,

Vg

1

V g ⊂ Hg ≡

H0 (O, g)
Vg




Hg



k · k∗

Vg

AB

(gu) = 0

)) :


Vg

∇·

.

Vg

g)
Vg

Vg



h·, ·ig




A:V→V
hAu, vi = ((u, v)),
2

u, v ∈ V.

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

∂xi

wj dx.

O

u, v, w ∈ V
b(u, v, w) = −b(u, w, v).

b(u, v, v) = 0, ∀ u, v ∈ V.

c|u|

|b(u, v, w)| ≤

1/4

−1/4



kuk

3/4

kvk|w|

1/4

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

g
Ag : Vg → Vg



hAg u, vig = ((u, v))g, ∀u, v ∈ Vg .


2

Ag = −Pg
2

L (O, g)

D(Ag ) = H (O, g) ∩ Vg
Hg

Pg

η1

Ag


Bg : Vg × Vg → Vg

hBg(u, v), wig = bg(u, v, w), ∀u, v, w ∈ Vg,
2

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 =

g

g

g

−g
g



g

h

g

g

g ·∇

(

u, v) = ((u, v)) +((

g

)u, v)

∇g

= 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



A

g



g

V .


1/2

1/2

c1|u|g
c2

kukg

||

kk

kvkg|w|g

1/2

vg

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

c4 u g v g w g

||kk| |

|

g

|
g

1/2

1/2

, ∀u, v, w ∈ Vg,
w , u V , v D(A ), w H ,

kwkg

g

g

|

w g , u D(A
| |
∀ ∈
1 /2

Ag w g

g

| | ∀ ∈

∀ ∈



), v V , w H
g

g



,
g



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

|

g






ci, i = 1, . . . , 4,
2

u ∈ L (0, T ;

C gu

Vg )
(C u(t), v) = (( ∇g · ∇ )u, v) = b ( ∇g , u, v), ∀v ∈ V ,
g
g
g
g
g
g
g
2
2

L (0, T ; Hg)
L (0, T ; Vg )
| C u(t) |g ≤ |∇g|∞ · k u(t)kg ,
m0

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

,

ii) W

,

iii) W

,

iv) Wt − Ws ∼ N (0, t − s), 0 ≤ s < t < ∞.

(K) σ

K

σ

kukK =

p

K

hu, ui

B
K

B(K)
K
K
: Ω → K,

X

K

A
−1

X (A) = {ω ∈ Ω : X(ω) ∈ A} ∈ F .
X

E(X) =

Z

X(ω)dP(ω).
Ω

K
X:Ω→K
K

a∈K
hX, ai


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