一类Navier-Stokes方程组的弱解

1
#
]L^P ?ME Navier-Stokes _,h6* A Stokes $* 4Q$E $* .] Hbqdb6. kHbqdbE 8 A K%$*WITTT EhWEhP ( _h 7 &H) 6!x> E K 1D oJ21 Navier-Stokes _,hE M 3SP KUhP oovSKK!%U ME Navier-Stokes _,h * Ω 6 RN (N ≥ 2) YE57J 8 9o 1995-11-22, l = v< M 8da u
Qt T 0
µ1 r+2
|∇ϕ|r+2 dxdt
QT
−
f , ϕ − v dxdt, ∀ϕ ∈ W,
C* v 6mU (1.1)–(1.4) E d5 , 1.1 O* 1.3 YW a = 0 E" U6hc< _ =Y !H3YE v.
@ a = 0 / ooU ( u = v − a
736
B
|∇v | =
ooLj (1.1)–(1.2) E u 8 Dirichlet SZ( 02Z(
v = 0, ∀(x, t) ∈ Γ, ∀x ∈ Ω. v |t=0 = a(x),
∂vi 2 ( ∂x ) j i,j =1
N
1 2
k
1
2
N
, (v · ∇)v =
N j =1
N
vj ∂j v1 ,
j =1
vj ∂j v2 , · · · ,
N j =1
vj ∂j vN .
Γ = ∂ Ω × (0, T ),
(1.3) (1.4)
_,h (1.1)–(1.4) EO(hPmUh
− ∇ · [(µ0 + µ1 |∇v |r )∇]v + (v · ∇)v = f − ∇p, ∇ · v = 0, v = 0, ∀x ∈ Ω, ∀x ∈ ∂ Ω. ∀x ∈ Ω, (1.5) (1.6) (1.7)
Abstract In this article we discuss solvability of the solutions for Navier-Stokes system. We obtained more general existence, uniqueness and regularity results. We also apply these results into description of motion of viscous incompressible fluids. Keywords Navier-Stokes equation, Weak solution 1991 MR Subject Classification 35Q30, 76D05, 76D07 Chinese Library Classification O175.29
L∞ [(0, T ); J (Ω)], vt ∈
T 0
1 L2 [(0, T ); (J3 (Ω))∗ ]
1 1 = 0, v ∈ L2 [(0, T ); J2 (Ω)] ∩ Lr+2 [(0, T ); Jr +2 (Ω)] ∩
mg d!H3
[ϕt · (ϕ − v ) + µ0 ∇v · (∇ϕ − ∇v ) − (v · ∇)ϕ · v ]dxdt + µ1 r+2 |∇v |r+2 dxdt ≥
1.1[1] ξ −→ P (ξ ) Rm (P (ξ ), ξ ) ≥ 0, ∀ξ ∈ {ξ | |ξ | = ρ},
1 2
40
G
0 f+EK!%p_ 0J 1DW67@E ρ > 0, 5 Y ξ = (ξ1, ξ2 · · · , ξm ), η = (η1 , η2 , · · · , ηm ) ∈ Rm , m ξi ηi , |ξ | = (ξ, ξ ) . CA |ξ | < ρ v N(IBK!E ξ , mg P (ξ ) = 0. (ξ, η ) = i=1 " 1.2[1] * p 6 % [2, +∞) $ErOLA C ∃c1 > 0, 1D Ω(|∇u|p−2∇u 6 [t1 , t2 ] $Ebi_ A CW %bi_ A ϕ(t) : ϕ(t) ≤ c0 + K (s)ϕ(s)ds, ∀t ∈ [t1 , t2 ], 5 ϕ(t) ≤ c0 e K(s)ds . k !lS ,r LRlEmUR6A Ω ⊂ R3 YE f W Ω ⊂ R2 #+ kE `"OW6 x 1.1 * r ≥ −1, Ω ⊂ R3 6F5b XE57J 8 rO0S a(x) ∈ J21(Ω)∩Jr1+2(Ω), d5 f ∈ L2 [(0, T ); J (Ω)], C_,h (1.1)–(1.4) V&:A%p 3 1 x 1.2 * r ≥ 5 , Ω ⊂ R 6F5b XE57J 8 rO0S a(x) ∈ J (Ω), f ∈ 1 ∗ L [(0, T ); (Jr +2 (Ω)) ], C _, h (1.1)–(1.4) V & : A% p * 5 3 XE57J 8 rO0S a(x) ∈ J (Ω), f ∈ x 1.3 * r ≥ 1 2 , Ω ⊂ R 6 F 5b 1 ∗ 2 a(x) ∈ W4 (Ω) ∩ J (Ω), L [(0, T ); (Jr+2 (Ω)) ], C_,h (1.1)–(1.4) E *56g%E mg vt ∈ f ∈ L∞ [(0, T ); J (Ω)], ft ∈ L2 [(0, T ); J (Ω)], C _, h (1.1)–(1.4) E * 5 v 1 1 2 L∞ [(0, T ); J (Ω)] ∩ Lr+2 [(0, T ); Jr+2 (Ω)], v ∈ L∞ [(0, T ); Jr+2 (Ω) ∩ W2 (Ω )], Ω ⊂⊂ Ω; 7/ 1 2 ¯ ∂ Ω ∈ C 2 , oo N(DB v ∈ L∞ [(0, T ); Jr +2 (Ω) ∩ W2 (Ω)]. x 1.4 * r ≥ 1, Ω ⊂ R3 6F5b XE57J 8 rO0S a(x) ∈ J21(Ω) ∩ Jr1+2(Ω), d55 u) X i) v(x, t) 0,2+r −→ f ∈ L2 [(0, +∞); J (Ω)]. C_,h (1.1)–(1.4) E +∞ α τ f (τ 0,2 dτ < +∞, (α > 0 (A), C@ 0 < α ≤ 1 / v (t) 0,2 = 0, t → +∞; ii) 0 − − v (t) 0,2 = O(t ). O(t ); @ α > 1 / 1 x 1.5 * r ≥ 5 , Ω ⊂ R3 6F5b XE57J 8 rO0S a (x), a (x) ∈ J (Ω), f (x, t), f (x, t) ∈ L2 [(0, T ); J (Ω)], C_,h (1.1)–(1.4) }.6$@02Z(E *5 v , v + t 2 CT a −a 2 ; ii) u!H3 i) v (x, t) − v (x, t) 2 0,2 ≤ 0,2 + 0 f (τ ) − f (τ ) 0,2 dτ e t C 2 CT a −a 2 . $@q3Y ∇v (x, t) − ∇v (x, t) 2 0,r+2 ≤ µ 0,2 + 0 f (τ ) − f (τ ) 0,2 dτ e C 6U'T6 Ω 02Z(EM(A 3 XE57J 8 rO0S a(x) ∈ J (Ω), f ∈ x 1.6 * r ≥ 1 5 , Ω ⊂ R 6 F 5b i) @ µ1 → 0 / vµ , L2 [(0, T ); J (Ω)], C W 6 _, h (1.1)–(1.4) E * 5 v µ , o o 5 A L2 [(0, T ); J21 (Ω)] Y 8`6 v; vµ A L∞[(0, T ); J (Ω)] Y ∗ 8`6 v, v mg dH3
(1.1) (1.2)
KY f (x, t) = (f1 (x, t), f2 (x, t), · · · , fN (x, t)) 6rOEc] µ0 > 0, µ1 > 0, r ≥ −1 6{O E(A T 6/% t ∈ [0, T ], x ∈ Ω, QT = Ω × (0, T ). jPb v(x, t) = (v1(x, t), v2 (x, t), · · · , ∂ ∂ ∂ ∂ ∂k = ∂x , TTJe ∇ = ( ∂x , ∂x , · · · , ∂x ), vN (x, t)) 6hWEhI p = p(x, t) 6
t 0
r+2 r+1
[(0, T );
vτ , ϕ dτ +
Qt t
[(µ0 + µ1 |∇v |r )∇v · ∇ϕ − (v · ∇)ϕ · v ]dxdτ
1 ∀ϕ ∈ Lr+2 [(0, T ); Jr +2 (Ω)],
=
0
f , ϕ dτ,
∀t ∈ (0, T ),
C* v 6mU (1.1)–(1.4) E *5 x 1.3 * r ≥ −1, 0S a(x)
∂p ∂vi ∂τij ∂vi − + vj = fi − , ∂t ∂xj ∂xj ∂xj j =1 j =1
N j =1 N N
∀(x, t) ∈ QT ,
(1.8)
∂vj = 0, ∂xj
∀(x, t) ∈ QT .
(1.9)
KY (τij )N ×N 6 .]Hb τij = τji, i, j = 1, 2, · · · , N . τij 'T6 skl , Y (skl )N ×N 6 ∂v ∂v .]Hb (τij ) 7. kHb (sij ) Q%E|r6b| . kHb skl = 1 2 ∂x + ∂x .