Tính ổn định và ổn định hóa đối với một số phương trình tiến hóa trong cơ học chất lỏng
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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
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
cλ
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
