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# Разрешимость краевой задачи для уравнений одномерного движения двухфазной несжимаемой жидкости.

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M P >
M
Asi Ax, tB, vi Ax, tB, pi Ax, tB, θAx, tBB
>?
Q AFBKAB #!
@
@
∂ Aρi si B
v |
D a AtB, v | D b AtB,
C ∂ Aρ s v B D E, i D F, , AFB
Ǳ
< = #
∂t
∂vi
ρi si
∂t
∂x
i x
i i i
i
C vi ∂v
∂x
−
∂
∂x
μi si
∂vi
∂x
i
D −si ∂p
C ϕi C ρi si g,
∂x
s C s D F, ϕ D K Av − v B,
ϕ D −ϕ , p − p D pc As , θB,
i
ci ρi si A
∂θ
∂t
∂θ
∂
∂θ
C vi ∂x
B D ∂x
AχAs B ∂x
B,
D
vi |t
∂θ
|x
∂x
AB
i
D vi AxB,
D θ AtB,
i x
s |t
i
D s AxB,
∂θ
|x D θ AtB,
∂x
θ |t
AB
D θ AxB.
# @ ai AtB D
bi AtB D aAtB
v |x D a AtB, v |x D
b AtB, v |x ,x D E JI # AFB # s C s D F " s v C s v D hAtBI ! M hAtBI t ∈ RE, T S <
# M
hAtB D aAtBI I K hAtB D sAE, tBa AtB M
! #
p Ax, tB =
AGB
AB
"
= Ax, tB ∈ QT D H × AE, T BI
H D AE, FB J! vi I si I pi K !I =L
i MN θ K =>
Aθ D θ D θBN g K N
ρi > EI μi > EI ci > E K !I OMM # ! i M
=LN OMM M K As BI OMM χAs B ! pc As , θB K
p Ax, tBdx D E.
ATB
∗ !" #\$% &###'(')*\$+ ,
-
". #% &"'#)'#""#.//+ &"'#)'#""#*)*\$#
UVWXUVWPV P UXYVZPV
[! M s AxBI θ AxB >? @
E < m ≤ s AxB ≤ M < F,
AB
−
E < k ≤ θ AxB ≤ k < ∞
x ∈ RE, FS M m I M I k [==? "
# AFBKATB ! M
Asi Ax, tB, vi Ax, tB, pi Ax, tB, θAx, tBBI i D F, si ∈ L∞ AE, T N W AHBB,
k− As s Bq ≤ χAs B ≤ k As s Bq ,
k− As s Bq
|
|
|
|
∂si
∈ L AQT B,
∂t
|
Avi , θB ∈ L∞ AE, T N W AHBB ∩ L AE, T N W AHBB,
pi ∈ L∞ AE, T N W AHBB,
∂vi ∂θ ∂pi
, ,
∂t ∂t ∂x
−
≤
dχAs B
≤ k As s Bq
ds
−
,
∂pc As , θB
| ≤ k As s Bq |θ|q ,
∂s
∂pc As , θB
| ≤ k As s Bq |θ|q ,
∂θ
∂ pc As , θB
| ≤ k As s Bq |θ|q ,
∂s
∂ pc As , θB
| ≤ k As s Bq |θ|q ,
∂θ
∂ pc As , θB
| ≤ k As s Bq |θ|q ,
∂s ∂θ
k D const > EI q, q , ..., q K M
? I # q ≥ EI q ≥ EI
q ≥ EI q ≥ EI q ≥ E
X ABI ?
# # t > EI t ∈ AE, T B
I # t ∈ AE, t B ? ==? " Asi , vi , pi , θB # AFBKATB
X !@
g ∈ C α,α/ AQT B, As , vi , θ B ∈ C α AHB,
∈ L AQT B,
H D AE, FB, QT D H × AE, T B,
>? AFBKAB E < s < FI E < θ < ∞ # > QT >? # #! #
M # "
# AFBKAB ! M
Avi Ax, tB, si Ax, tB, pi Ax, tB, θAx, tBBI i D F, I
=> I ? AFBKABI > I #! #
QT M
Ǳ! # AFBKATB
> >? Avi , θ B ∈ W AHB, s ∈ W AHB,
vi AEB D ai AEB,
∂θ AxB
|x
∂x
vi AFB D bi AFB,
i D F, ,
D θ AEB, ∂θ∂xAxB |x D θ AFB,
OMM K As B χAs B K MM M I Qt
? # " #I >?
si ∈ C α,α/ AQt
B, A vi , θB ∈ C α,α/ AQt
B,
g ∈ L AE, T N W AHBB,
∂pi
∈ C α,α/ AQt
B,
∂x
Aθi , ai , bi B ∈ W AE, T B
# # E < m
< M < FI
@
E < m
< M < ∞I I #
vi AEB D ai AEB, vi AFB D bi AFB, i D F, N
E < m
≤ s Ax, tB ≤ M < F,
∂θ AxB
∂θ AxB
|x D θ AEB,
|x D θ AFB.
E < m
≤ θAx, tB ≤ M < ∞, Ax, tB ∈ Qt
.
∂x
∂x
Q
RI
Ǳ! K As BI pc As , θBI χAs B K #
M I GS >? # sAx, tB ≡
s Ax, tBI uAx, tB D β v Ax, tB − β v Ax, tBI p Ax, tBI
>? >? @
θAx, tBI RAx, tB@
K As B D K As BAs s Bq ,
∂s
∂
β sh
C ∂x aAsBu C a D E,
A]B
E < k− ≤ K As B ≤ k D const,
∂t
μ
A
∂ aρ
∂t aμ u
−ν ∂∂xu
∂s
B C a AsBu ∂u
∂x C a AsBu ∂x −
−
∂s
ν aa
ss ∂u
∂x ∂x
−
G
A BC
A^B
∂s ν aa
ss u ∂x
μ
β h ∂ ρ sβ h ρ AF − sBβ h
−ρ AF − sBA
B B − ∂t
A a C
BC
aμ
aμ
μ
c
,
Cρ aρ AsBg C AF − sB ∂p
∂x
AFEB
∂θ
∂θ
C
c aAsBu C
∂t
∂x
h ∂θ
∂
∂θ
B,
Cχ AsBAβ C β B a ∂x D ∂x AχAsB ∂x
μ
χ AsB
AFFB
c
Cρ g − ∂p
C hG ≡ f As, u, RB,
∂x
>? =#
βi
aAsB D
ν
μ D μ C μ ,
D μμi ,
D ρρi ,
,
χ AsB D c ρ s C c ρ AF − sB,
c
aμ
−
∂x
β h
∂ aAsB
D aK Aβ − β B C A AF −a sμBβ δ Bu ∂x
A sa B−
−A
AF − sBβ ρ Bu ∂ A F B − A β δ B ∂ A aAsBu B−
a
∂x a
a μ ∂x a
−A
sβ ρ
sβ δ
aAsB
∂ F
∂
B
u A B−A
B
u A
BC
aμ
∂x aμ
aμ μ ∂x AF − sBaμ
μ
μ
μ
μ
μ
μ
μ
μ
μ
μ
μ
μ
μ
μ
∂
aAsB
CA βa βμ δ B ∂x
A AF −
BS.
sBa
μ
μ
W =I sAx, tBI uAx, tBI
> A]BKAFB AFB
u |x D β a AtB − β a AtB,
p Ax, tBI θAx, tBI RAx, tB
u |x D β b AtB − β b AtB,
u |t
D u AxB,
θ |t
∂θ
|x
∂x
D θ AtB,
∂θ
|x D θ AtB,
∂x
i D F, ,
s |t
ρ
μ s
G
aμ AsB D β AF − sB C β s,
aμ
Aβ − β B ∂s
μ
∂ aAsB
∂ F
A sa B C A βa ρ B ∂x
A a BC
CA βa βμ δ B ∂x
D ρ C ρ , aρ AsB D α AF −sBC α sN
AsB D α AF − sB − α s , a AsB ≡ da ,
D μ/ρ
a
sAF − sB
,
aμ
αi
∂
∂x
μ
D − Fμ a AsBaAsBuR AR − bAsBuB −
K
δ aAsB
−
u−
u·
aAsBaμ
μ aμ
∂u β AF − sB − β s
·
C
uAR − bAsBuB C
μaμ
μ
∂ aAsBu
∂ F
A a B C hRA βa ρ B ∂x
A a BC
CA aβμδ B ∂x
∂R
CU As, uB ∂R
∂t
∂x
∂x
μ
F − s·
∂
Aβ − β B ∂s B C KhAβ − β B ,
· Aμ AF − sB
∂x
aμ
∂x
ρ sAF − sBaμ
·
ρ aAsBβ hu C ρ aAsBβ hu − ρ sA β h B −
a
a
a
μ
μ
μ
μ
μ
μ
μ
∂ AF − sBu
∂ h Cβ A ah B ∂x
A a B C β ∂x
A a B S C βs h ·
∂
∂
∂
C ∂x
Aμ sβ ∂x
A a hAsB B C μ AF − sBβ ∂x
A a hAsB B−
−
μ
μ
c
μ
μ
μ
∂ h ρ
∂ h
∂ h
C β ∂x
A a B S− ρ Rβ ∂t
A a B−AF−sBuβ ∂x
A a BC
C ρ
aK
sa u − ρ
∂p∂x − g − G D E,
∂
∂
∂ F−s
D ∂x
Aμ s ∂x
A a AsB uB−μ AF−sB ∂x
A a sAsB uB−
μ
μ
aρ AsB
∂
−ρ
aAsBu B − Aρ − ρ B AaAsBuBC
aμ AsB
∂t
∂p
∂x
∂ h
∂ h
∂ −su
D ρρ Rβ ∂t
A a B−suβ ∂x
A a BCβ A ah B ∂x
Aa B
ds
R |t
D c ρ − c ρ ,
D s AxB,
D θ AxB,
p Ax, tBdx D E,
AFGB
D aAs μAxBB ∂s∂xAxB C bAs AxBBu AxBC
C μδ a AhsAEBAxBB ≡ R AxB.
μi ∂si
,
RAx, tB D R Ax, tB−R Ax, tB, Ri Ax, tB D ρi vi C
μ
si ∂x
Ǳ aρ
, δ D μ ρ − μ ρ , g ≡ Aα − α Bg, !" A]BKAFB bAsB ≡ ρ
aμ
U As, uB ≡ a AsBu −
β β h
,
aμ
∂s
∂t
B
D −a AsBuAAR − bAsBuB aAμsB − aaAsBμhδ
−
μ
UVWXUVWPV P UXYVZPV
−
aAsB aAsBhδ
β β h
AA
R − bAsBuB
−
B−
aμ
aμ
aμ μ
∂u
≡ f Au, s, RB,
∂x
∂u
∂
∂u
−ν
L As, u, R, θB ≡
C
b AsB
∂t
∂x
∂x
−aAsB
AFB
∂θ
− b AsBg − hG D E,
∂x
∂
∂θ
∂θ
−
L As, u, R, θB ≡
C
b AsB
∂t
∂x
∂x
a AsBaAsBuδ
aμ μ
C
μ
b AsBuC
aAsBδ
B−
C F b AsBa AsBu A− aaAsμBδ BCνb AsB aaAAssBB ∂u
A
∂x a μ
C
Cb AsBu AR − bAsBuB C b AsBu ∂u
∂x
Cb AsBu AR − bAsBuB C b AsB ∂u
AR − bAsBuB C
∂x
−b As, θB
C βaβ AR − bAsBuB aAμsB − aaAsBμδh
μ
Cb AsBu − b As, θBAR − bAsBuB−
D bb AAssBB
G
−νb AsB
μ
a AsB
a AsB
δ
u − AR − bAsBuB aAsB
aμ
μ
μ
C a aAsBμδ h
μ
CG AR, s, u, hB.
Ǳ E ≤ sAx, tB ≤ F OMM
aρ AsB, aμ AsB, b AsB, χ AsB > AFB E < 7.2Aα , α B ≤ aρ AsB ≤ 7*;Aα , α B < F,
E < 7.2Aβ , β B ≤ aμ AsB ≤ 7*;Aβ , β B < F,
7.2Aβ , β B ≤ b AsB ≤ 7*;Aβ , β B < ∞,
E < 7*;A
∂θ
α , α B
7.2Aα , α B
AFTB
Ab AsB C b AsBB ∂x D E,
E < 7.2Ac ρ , c ρ B ≤ χ AsB ≤ 7*;Ac ρ , c ρ B < ∞.
∂R
∂R
F
C
U As, uB
D
− a AsBaAsBuR·
∂t
∂x
μ
OMM bj AsB, j D E, .., F >
K
δ aAsB
· AR − bAsBuB −
u
−
u·
aAsBaμ
μ aμ
b AsB b AsB b AsB b AsB
C − ≤ A
,
,
,
B ≤ C,
aAsB aq− AsB aAsB χAsB
∂u β AF − sB − β s
·
C
uAR − bAsBuB C
∂x
μaμ
|b AsB|
aμ AsB
, aμ AsBb AsB,
b AsBB ≤ C,
C − ≤ A
∂pc
χ
A
s
B
a
A
s
B
aAsB
Cρ g − ∂x C hG ≡ f As, u, RB. AFB
|b AsB| C |b AsB| C |b AsB| ≤ C,
J! >? =#
@
|b As, θB| ≤ C AsAF − sBBq |θ|q ,
aμ AsB
β β aAsB
b AsB D
, b AsB D
,
aρ AsB
ρ μ aρ aμ AsB
|b As, θB| ≤ C AsAF − sBBq |θ|q ,
C K !
I ?
b AsB
b AsB D a AsB, a AsB D b AsBa AsB C aAsB
,
!
#
b AsB
uAx, tB D ω Ax, tBC ψ Ax, tB, θAx, tB D
Ǳ
aAsB
b AsB θ_Ax, tB C φAx, tB, ψ Ax, tB D AF − xBu AtB C
F
b AsB D a AsB
, a AsB D a AsBb AsBCa AsB
,
μ
b AsB xu AtB, φAx, tB D − −x θ AtB C x θ AtBI D
D
E
D
A
ω
|
ω
|
,
ω
|
ω
x
B D u AxB −
x
x
t
a AsB
aAsB
b AsB D −
A
b AsB
− b AsBB,
∂ θ
∂ θ
ψ Ax, EB, ∂x |x D ∂x |x D E J
AFBK
μ
aAsB
uAx, tB # ω Ax, tBI θAx, tB # θ_Ax, tB
AFB
Kb AsB
b AsB aAsB ∂pc As, θB
b AsB D
,
A
,
b
s,
θ
B
D
`" As, ω, θ_, RB AFBKAFB ρ aAsBaμ AsB
ρ
μ
∂s
>? #! #
b AsB aAsB ∂pc As, θB
aAsB
=
b As, θB D
, b AsB D c
,
" Asn , ωn , θ_n , Rn BI ωn Ax, tB ρ
μ
∂θ
χ AsB
θ_n Ax, tB >
# @
χAsB
χAsB aAsB
n
n
b AsB D
,
, b AsB D −c ωin AtB +.2AπixBI
θ_jn AtB )(+AπjxBI n D F, , ...
χ AsB
χ AsB μ
i
j
OMM uni AtBI i D
h
aAsBhδ
b AsB D χ AsBAβ C β B
, b AsB D
,
F, , ..., nI θjn AtBI j D E, F, , ..., n aμ AsB
aμ AsBμ
∂θ
∂θ
− b AsB AR − bAsBuB
Cb AsBu ∂x
C
∂x
I # AFBI
AFTB >
=@
L Asn , ω n , θ_n , Rn B +.2AπixBdx D E,
AF]B
i D F, , ..., n,
_n
Cb Asn BARn − bAsn BAωn C ψBB ∂ Aθ ∂xC φB C
_n
D Ab Asn B C b Asn BB ∂ Aθ ∂xC φB S )(+AπjxBdx.
c sn Ax, tBI
"
#
L Asn , ω n , θ_n , Rn B )(+AπjxBdx D E,
AF^B
∂sn
∂t
Rn Ax, tB
D s AxB. AFB
D E, F, , ..., n.
W ωin AtBI θ_jn AtB "
∂R
n ∂R
n n
n _n
n
∂t C U
∂x D f Aω , s , R , θ B ≡ f ,
# " = MM
AB
! Rn |t D R AxB,
dω
n
n
n _n
_n n n
dt D ai Aω , ..., ωn N θ , ..., θn N s , R B,
n
n
n
n
n
AF^B@
AEB n U ≡ U Ans , ωn B D na As nBω a
s δh
∂s
/∂x
D
a
A
s
BA
R
−
b
A
s
BA
ω
C
ψ
BB
/μ
−
dθ
a s μ n
n
n _n
_n n n
dt D λj bj Aω , ..., ωn N θ , ..., θn N s , R B,
Q>?
AB #
RS@
n
ωi AEB D ω AxB +.2AπixBdx,
dy t,ξ D U n AyAt, ξ B, tB, y |t D ξ,
dt
AGB
n
n
dR
n
dt D f Ay At, ξ B, t, R B, R |t D R Aξ B,
θ_n AEB D θ_ AxBdx,
n
y At, ξ B D xI R At, ξ B D Rn Ay, tB t,ξ
I ≡ ∂y∂ξ
_θjn AEB D θ_ AxB )(+AπjxBdx, j D F, , ..., n.
j
D f Aωn , sn , Rn B ≡ fn ,
n
sn |t
n
n
i
n
n
j
μ
n
n
t
J! λ D FI λj D I j D F, , ..., nI
a At B D n
i
∂
R−A ∂ψ
B
−ν
∂t
∂x
I
∂ Aω n C ψ B
b As B
∂x
n
−
n
n
.
a D a Asn B ∂ Aω ∂xC ψB C a Asn BAωn C ψB ∂s
∂x
−b Asn BAω n C ψ B ARn − bAsn BAω n C ψ BB −
I = ! Rn Ax, tB x D E # U n AE, tB > E A x D F # U n AF, tB < EB Z
> M> hAtB # " AEBKABI
hn AtB D sn AE, tBa AtB A !"
n h B
W =I = "
Aωn , sn , θ_n , Rn B # " AEBK
AB d!
"! O # M n "Ǳ = MM
! # t I # RE, t S " n O # #! n ωn , sn , Rn , θ_n #
∂ Aω n C ψ B
− b Asn BAω n C ψ B ·
−b As BAω C ψ B
∂x
∂ Aω n C ψ B
·
· ARn − bAsn BAω n C ψ BB − b Asn B
∂x
· ARn − bAsn BAω n C ψ BB − b Asn BAω n C ψ B−
n
n
−b As, θ_n C φBAR − bAsBAω n C ψ BB−
−b Asn , Aθ_n C φBB
∂ Aθ_n C φB
− b AsBg
∂x
S +.2AπixBdx,
_n
∂
b AtB D R ∂x
Ab Asn B ∂ Aθ ∂xC φB B−
n
j
−b Asn BAω n C ψ B
D &;6 a AyAτ, ξ B, τ Bdτ,
∂ Aθ_n C φB
C
∂x
UVWXUVWPV P UXYVZPV
sn , θ_n Ǳ
zn AtB D ω n AtB C Rn AtB C ωxn AtB C
Cκ
AB
n
ωxx
Aτ B dτ,
E<m≤s
n
A]B
d
n
Rn AtB ≤ Aε C ε Bωxx
At B C
dt x
CRxn AtB C θ_n AtB C θ_xn AtB C
t
Cψx AtB C ψAtB C |hAtB| C
C|hAtB| C |hAtB| B,
CC Aε , ε BAg x AtB C zn AtB C zn AtB C zn AtBC
C &;6 GC Aq C FB z AtB C ψn AtB C ψn AtB C
n
κ
≤ M < F,
x
Cψxn AtB Cψn AtB CψnAtBCψxnAtBC|hAtB| C
E < m
≤ θ_n ≤ M < ∞.
C|hAtB| C |hAtB| C |hxAtB| C
|Dx Rxn |dxB.
A^B
∂ω
ωAtB D ω Ax, tBdx, ωx D ∂ω
∂x , ωxx D ∂x ,
J!
κ ∈ AE, FB K ? aAsn B n
P > RI GS
D Asn , ω n , Rn , θ_n B D pcs Asn , θ_n B
AR − bAsn B·
μ
C
F
≤ C &;6A zn AtB C ψx AtBC
sn Ax, tBAF − sn Ax, tBB
κ
·Aω n C ψ BB C pcθ Asn , θ_n BAθ_n C φBx .
W! # = C|h| C ψAtB B.
AB
θ_n Ax, tB θ_xn Ax, tB@
J! C K !
I
?
n
F d Aθ_n AtB C θ_n AtB BC
ωn Ax, tB ωxn Ax, tB >?
x
dt
RI GS@
d
n
dt ω
At B C ν
ωxn AtB ≤ C Ag AtB
C
CznAtB C zn AtB C Dn AtB C |hAtB| C
C
n b Asn BA|θ_xn | C |θ_xx
| Bdx ≤ Czn AtBZ AtB,
ATB C|hAtB| C |hAtB| C |hAtB| B,
Z At B D
d n
n
n
ωx AtB C ν ωxx
A
t B ≤
εi ωxx
At B C
dt
i
b Asn B
F C 7*; b Asn B C
C
7*;
7*;
≤x≤ b Asn B
≤x≤ b Asn B
≤x≤ b Asn B
b Asn B
b Asn B
C 7*;
C
7*;
C
≤x≤ b Asn B
≤x≤ b Asn B
CC Ag AtB C Dn AtB C zn AtB C zn AtBC
C &;6 GC Aq C FB z AtB C ψAtB C ψAtB C
Ab As BaAs BB C φ AtB C φ AtB C
C 7*;
t
x
≤x≤
b Asn B
n
κ
C|hAtB| C |hAtB| C |hAtB| .
Cψx AtB C ψxx AtB C ψt AtB C
P M b I b I b I b C|hAtB| C |hAtB| C |hAtB| C |hAtB| B. AB >?
@
J! >? =#
@
b AsB
F ≤ C AsAF−sBB−q ,
≤ C AsAF−sBB−q ,
n
n
n _n
n
n
n
n
_
DAs , ω , R , θ B D b As , Aθ C φBBAR − bAs B·
b AsB
b AsB
·Aω n C ψ BB C b Asn , Aθ_n C φBBAθ_n C φBx .
b AsB
b AsB
≤ C AsAF−sBBq , ≤ C AsAF−sBB−q ,
b AsB
b AsB
[ Rn Ax, tB Rxn Ax, tB > @
Ab AsBaAsBB ≤ C AAsAF − sBBq C AsAF − sBBq B.
d
n
R AtB ≤ C Ag AtB C zn AtB C zn AtBC
b Asn B
dt
< AB ?> Czn AtBC D AtB C&;6 C AqκC FB zn AtBC ψAtB C k > E k > EI ? ! n
n
C I q ? n I
k &;6Ak zn AtBB W d
dt
C
# Z AtB
Aθn AtB C θxn AtB BC
n b Asn BA|θxn | C |θxx
| Bdx ≤
≤
C φx AtB C |hAtB| B,
AGB
C n P AGB n #! zn AtB AGEB t
≤ t I φt AtB
≤ Ck Az n AtBB &;6Ak zn AtBB.
d
Aωn AtB C ωxn AtB C Rn AtB C Rxn AtB BC
dt
Cν
At B C ν
n
ωxx
At B
≤
i
n
εi ωxx
At B
CD AtB C D AtB C
n
T
Ag Aτ B C
Cg x Aτ B C ψt AtB C ψx AtB C ψxx AtB C
CψAtB C φt AtB C φx AtB Bdτ }B.
Ǳ = t ABI AGEB AGFB I # n @
C
CC Ag AtB C g x AtB C zn AtBC
Czn AtB C zn AtB C &;6 GC|qκC F| zn AtBC
n
t < C − &;6A−C{zn AEB C C
Q
ATBKA^B #
ωxn
CψtAtB C ψx AtB C ψxx AtB C ψAtB C
E < m ≤ sn Ax, tB ≤ M < F, Ax, tB ∈ H × AE, t B,
7*;
ωxn AtB C 7*; θ_xn AtB C
≤t≤t
≤t≤t
|Dx Rxn |dxC
Cψt AtB C ψx AtB C ψxx AtB C
CψAtB C |hAtB| B.
AGFB
[ ! # AGFB Ǳ! b Asn , θn B ≤ C AsAF −
sBBq |θ|q b Asn , θn B ≤ C AsAF − sBBq |θ|q I ?> k > E k > EI ? ! C I qI q I q I q ? n
I # DAtB C D AtB ≤ k &;6Ak zn AtBB
I # pcss I pcsθ I pcθθ # A^B I #
C 7*;
Rn AtB C 7*; Rxn AtB C
≤t≤t
≤t≤t
C
t
n
n
Aωxx
AtB C θ_xx
AtB Bdt ≤ C
AGGB
C ? n O
"
>? M@
E < ≤x≤
7.2 θ AxB ≤ θAx, tB ≤ 7*;
θ AxB #
≤x≤
> Qt
A RSB
Ǳ#" [
AGGB
>
! !
F b Asn B|θn | dxC
|Dx Rxn |dx ≤
n
n
n _n
xx
{s , ω , R θ } ?
! Ǳ# n
> ! = ?
n
CC ARx AtB C k &;6Ak zn AtBBB,
!@
k > EI k > E ! C I q , ..., q n
ω n , Rn , θ_n , ωxn , Rxn ω, R, θ_, ωx , Rx
Q AGEB AGFB =
= L AE, t N L AHBB,
ε − ε εi D ν − κ A ν ≤ FI i
κ D ν / ε − ε = n
, ωtn ωxx , ωt = L AQt
B,
ωxx
εi D Gν /N ν > FI κ D F/I
i
Qt
D AE, t B × AE, FB.
εi = εi D ν − F/B P ωn L AE, t N L AHBBI
i
> #! # I W AQT B ωtn L AQt
B ! !! ωn I ! ?>
ω
L AE, t N L AHBB RTS P AGGBI #I dzn AtB
≤ C Ag AtB C g x AtB C &;6 Czn AtBC
!
! sn s L AQt
B. < #
dt
UVWXUVWPV P UXYVZPV
AFB
B
D
b Asn BAω n B Rn F dQt
→
⎛
Qt
⎝
b AsBω RF dQt
, n → ∞, ∀F ∈ L AQT B.
→
t
| B |≤
B
D
b Aξ BAs −sBAω
n
B
n R F dQt
→ E, n → ∞,
·⎝
⎝
t
! ⎞/
Rn F dt⎠
,
L AQt
B.
`
B
D
b AsBAω B ARn − RBF dQt
→ E, n → ∞,
Qt
| B |≤
⎛
ω n − ω dt⎠
ωn → ω
n
Qt
| b AsBAω n C ω B | ·
⎞/ ⎛
Qt
[#I B D B C B C B . Ǳ#
7*;
x,t,θ∈QT
t
7*;
x,t,θ∈QT
= L AQt
B,
b AsB, ωn − #.
Ǳ! I ? hn hnt I =#
! !
hn AtB hAtB = !> hnt AtB ht AtB L AE, T B
Ǳ! AF]BI
AF^BI AFBI AB I # !
M {sAx, tB, ωAx, tB, RAx, tB, θAx, tB} > =
=? " RE, t S
Ǳ" ==? "
# A >? " #B I = RGI FEN F]TS
| b Aξ BAω n Ax, tBB | ·
Rn R
⎞ / ⎛ t
⎞/
sn − s dt⎠ ⎝ Rn F dt⎠ ,
sn → s
V#
B
D
! L AQt
B.
b AsBAω n − ω B·
Qt
·Aω n C ω BRn F dQt
→ E,
n → ∞,
\$ " F e Q #
M K UI F^^
K UI F^]
Ǳ VVI V Pe J# d
[VI Q <VI
! ZZ d >? K Z
=# K UI
=I EE K <PZPWPI fGg
F^T
G Ǳ VV # T d
[V # M M! K hI EE^
# M K UI F^G
` hdI i ZZ Q
```
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