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

Lp(0, T ; Y ) C([0, T ]; Y )

HgVg

A B

α = 0

ν = 0

A : V → V ′hAu, vi = ((u, v)), u, v ∈ V.

D(A) = (H2(O))3 ∩ VAu = −P u ∀u ∈ D(A) P

B : V × V → V ′(B(u, v), w) = b(u, v, w), u, v, w ∈ V,

3

∂vj

b(u, v, w) = i,j=1Z ui wj dx.

∂xi X

kuk2g ≥ η1|u|2g, ∀u ∈ Vg,

|Ag u|g2 ≥ η1kukg2, ∀u ∈ D(Ag ),

g

c1|u|g 1/2kukg 1/2kvkg|w|g 1/2kwkg 1/2, ∀u, v, w ∈ Vg,

Wt

KK

Q W (t), t ≥ 0 K

∞

X p

W (t) = λ′nβn(t)en, t ≥ 0,

n=1

hW (t), e

ni ,

p

λn′

βn(t) = 0,

a b

dx ≤ ax + b,

dtx(t) ≤ (x(0) + b at − b

u0

ku(t) − u∗k ≤ Ce−δt, ∀t ≥ 0.

Oω = O\ω,

Vω = u ∈ (C0∞(Oω))3 : ∇ · u = 0

(νAu + kP (1ω u), u) ≥ (νλ1 ∗(ω) − ε)|u|2, ∀u ∈ V.

|b(u, v, w)| ≤ γ|u|kvkH β |w|, u ∈ Vω, v ∈ (Hβ (O))3 ∩ V, w ∈ Vω,

γ∗(u∗) := sup {|b(u, u∗, u)| : |u| = 1} ≤ γ ku∗kH α

d(φ + α2Aφ) + [νAφ + B(u, u) − B(u∗, u∗)]dt = σ(I + α2A)φdWt

φ(0) = u 0 − u∗.

(I + α2A)−1

dφ + (I + α2A)−1[νAφ + B(u, u) − B(u∗, u∗)]dt = σφdWt

φ(0) = u 0 − u∗.

dφ + (I + α2A)−1[νAφ + B(u, u) − B(u∗, u∗)]dt + 1

λ1 1 2 2− c0 |f |) > 0

ℓ := 1 + α2λ1(ν + σ α2 λ3/4ν

1

ku(t) − u∗kα2 ≤ ku(0) − u∗kα2e2σW t e−2ℓt.P

lim Wt = 0,t

gO

dt (w(t), v)g + ν((w(t), v))g + ν(Cg w(t), v)g

w = v∗ − u∗ u∗ = v∗

∂ω f ∈ L2(O, g)

δ > 0

µ1k = inf ν kukg2 + k Z

ω |ϕ1k|g2gdx : u ∈ Vg, |u|g = 1 .

µk1 ≤ νη1∗(ω), ∀k ∈ N∗,

εη∗ Z0 z (s) 2

ds z(0) 2

ε{zn} L∞ (0, T ; Hg) ∩ L2 (0, T ; Vg)

z

n

zn ⇀∗z

+ νAgz + νCg z + Bg(z, z) + Bg0z + kPg (1ω z) = 0 L2(0, T ; Vg′).dt

1 d |z(t)|g2 + νkz(t)kg2 + k(Pg (1ω z(t)), z(t))g

2 dt

1 |∇g| ∞ η∗(ω

)

γ∗(u∗)

ε

M0 2 2 2

|ϕ − Ih(ϕ)|g≤ m0 c0h kϕkg , ∀ϕ ∈

Vg,

z(0) = u(0) − u∗ =: z0,

M 0

ω02m02k0η12k kL ∞

(0,T per ;D(A g ))

L ∞ (0,T per ;D(A g ))|Agyper|g.

−bg(yper, yper, Agyper ) ≤ c3|yper|1g /2kyperkg |Agyper|3g /2,

≤ c3 khkL2∞ (0,T per ;D(A g ))|Agyper |g,

ω 2 η 1

0

≤ c 3 khkL ∞ (0,T ;D(A)) |yper|1/2|Agyper|3/2,

ω

0η

11

≤ c 3 khkL ∞ (0,T ;D(A)) kyperk1/2|Agyper|3/2.

ω

0η1

+ c3|yper|1g /2kyperkg|Agyper|3g /2

+ c3 khkL2∞ (0,T per ;D(A g ))|Agyper|g

+ c23/4khkL ∞ (0,T per ;D(A g))kyperkg 1/2|Agyper|g 3/2

ω0η1

+ ν khkL ∞ (0,T per ;D(A g))|Agyper|g

ω0

≤2R2 + 2Lh kF k

L2

∞ (0,T per ;D(A g )).

ω2η 1 0

ω0

√ 2Lh

z 2 0.

1

θ ∈[−τ,0]

d

dt u(t) = −νAgu(t) − νCgu(t) − Bg(u(t), u(t))

t > 0, + f + F (u(t ρ(t))),

g

∗

E|u(t) − u∗|2g ≤M0e−α0 t

, t ≥ 0,

nt0

g

•

g

g

g

g

H m

g

g

g

g

R3