On small motions of a “liquid-gas” system in a bounded domain
We study small motions and free oscillations of a compressible stratified liquid, the structure of the spectrum, and the basis property of a system of eigenvectors and obtain asymptotic relations for eigenvalues.
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| Date: | 2006 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3535 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509641348218880 |
|---|---|
| author | Vronskii, B. M. Вронский, Б. М. Вронский, Б. М. |
| author_facet | Vronskii, B. M. Вронский, Б. М. Вронский, Б. М. |
| author_sort | Vronskii, B. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:10Z |
| description | We study small motions and free oscillations of a compressible stratified liquid, the structure of the spectrum, and the basis property of a system of eigenvectors and obtain asymptotic relations for eigenvalues. |
| first_indexed | 2026-03-24T02:44:20Z |
| format | Article |
| fulltext |
UDK 517.9:532
B. M. Vronskyj (Tavryç. nac. un-t, Symferopol\)
O MALÁX DVYÛENYQX SYSTEMÁ „ÛYDKOST|-HAZ”
V OHRANYÇENNOJ OBLASTY
Small motions and proper oscillations of a compressible stratified fluid are studied. The structure of a
spectrum and the base property of a system of eigenvectors are investigated, asymptotic formulas for
eigenvalues are obtained.
Vyvçeno mali ruxy i vlasni kolyvannq styslyvo] stratyfikovano] ridyny, doslidΩeno strukturu
spektra, bazysnist\ systemy vlasnyx vektoriv, oderΩano asymptotyçni formuly dlq vlasnyx
znaçen\.
1. Postanovka zadaçy y pryvedenye ee k operatornoj forme. 1.1. Posta-
novka naçal\no-kraevoj zadaçy. Pust\ nepodvyΩn¥j sosud celykom zapolnen
ydeal\noj sΩymaemoj Ωydkost\g. Ûydkost\ predpolahaetsq stratyfycyro-
vannoj, t. e. ee plotnost\ v sostoqnyy pokoq yzmenqetsq vdol\ vertykal\noj
osy Oz po zakonu ρ ρ0 0= ( )z . Oblast\, zanqtug Ωydkost\g, oboznaçym çerez
Ω, a ee hranycu (tverdug stenku) — çerez S. Sçytaem, çto systema naxodytsq
pod dejstvyem syl¥ tqΩesty s uskorenyem
� �
g gk= − , hde
�
k — ort osy Oz.
Budem rassmatryvat\ sluçaj ustojçyvoj stratyfykacyy, kotoraq ymeet
mesto pry v¥polnenyy uslovyq (sm. [1, 2])
0 < N−
2 ≤ N z2( ) ≤ N+
2 ≤ ∞ ,
(1)
N z2( ) ≡ N z
g
c0
2
2
( ) −
, N z0
2( ) ≡ – g z(ln ( ))ρ0 ′ ,
hde c — skorost\ zvuka v Ωydkosty. Velyçynu N z2( ) prynqto naz¥vat\ çasto-
toj plavuçesty yly çastotoj Vqjsqlq – Brenta.
Mal¥e dvyΩenyq system¥ opys¥vagtsq uravnenyqmy (sm. [1, 2])
∂
∂
2
2
�
w
t
= –
1 1
0 0ρ ρ
ρ∇ −p g k
�
( v Ω ) , (2)
ρ ρ ρ+ ′ +w wz 0 0 div
�
= 0 ( v Ω ) , (3)
ρ ρ+ ′wz 0 = c p gwz
− −2
0( )ρ ( v Ω ) , (4)
kraev¥m uslovyem
� �
w n⋅ = 0 ( na S ) , (5)
v¥raΩagwym uslovye neprotekanyq na tverdoj stenke, y naçal\n¥my uslovyq-
my
� �
w x( , )0 =
� �
w x0( ),
∂
∂
� �
w x
t
( , )0 =
� �
w x1( ). (6)
Zdes\
� � �
w w x t= ( , ) — pole smewenyq çastyc Ωydkosty ot sostoqnyq ravnove-
syq, p p x t= ( , )
�
— otklonenye polq davlenyq ot ravnovesnoho, ρ ρ= ( , )
�
x t —
otklonenye polq plotnosty ot ravnovesnoho,
�
n — vneßnqq normal\ k S, x =
= ( , , )x x z1 2
— toçka v R
3.
1.2. Metod ortohonal\noho proektyrovanyq. Naçal\no-kraevug zadaçu
(2) – (6) pryvedem k dyfferencyal\nomu uravnenyg v nekotorom hyl\bertovom
prostranstve.
Vvedem v rassmotrenye prostranstvo vektor-funkcyj
→
L2( )Ω so skalqrn¥m
proyzvedenyem
© B. M. VRONSKYJ, 2006
1326 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O MALÁX DVYÛENYQX SYSTEMÁ “ÛYDKOST|-HAZ” … 1327
( , )
� �
v u L2
=
ρ0( )z u d
� �
v ⋅ ∗∫ Ω
Ω
. (7)
Oboznaçym çerez
→
J0 0( , )Ω ρ podprostranstvo
→
L2( )Ω , poluçagweesq zam¥kany-
em po norme
→
L2( )Ω mnoΩestva hladkyx funkcyj
→
J ( )Ω =
� � � �
u C u u n S∈ = ⋅ ={ }1 0 0( ) : ,Ω Ωdiv v na . (8)
V kaçestve druhoho voz\mem podprostranstvo
→
G( )Ω =
→
∈ = ∇
� �
v vL( , ) :Ω Φρ
ρ0
0
1 . (9)
Skalqrn¥e funkcyy Φ Φ= ( , )
�
x t , poroΩdagwye podprostranstvo
→
G( )Ω ,
obrazugt prostranstvo, kotoroe budem oboznaçat\ W2
1
0
1( ),Ω ρ− . Skalqrnoe pro-
yzvedenye y norma v nem zadagtsq po formulam
( ), ,Φ Ψ Ω1 = ρ0
1− ∗∇ ⋅∇∫ Φ Ψ Ω
Ω
d , Φ Ω1
2
, = ρ0
1 2− ∇∫ Φ Ω
Ω
d . (10)
Krome πtoho na funkcyy Φ Ω∈ −W z2
1
0
1( ), ( )ρ nalahaetsq normyrugwee uslovye
Φ Ω
Ω
d∫ = 0.
Lemma01. Prostranstvo
→
L2 0( , )Ω ρ dopuskaet ortohonal\noe razloΩenye
→
L2 0( , )Ω ρ =
→
J0 0( , )Ω ρ � G( )Ω . (11)
Yz (11) sleduet, çto lgboj vektor
�
w ∈
→
L2 0( , )Ω ρ moΩno predstavyt\ v vyde
�
w =
�
u z+ ∇−ρ0
1( ) Φ ,
�
u ∈
→
J0 0( , )Ω ρ , ρ0
1− ∇( )z Φ ∈
→
G( )Ω . (12)
V dal\nejßem funkcyy
� �
w x t( , ), p x t( , )
�
pry lgbom t > 0 budem sçytat\ πle-
mentamy prostranstva
→
L2 0( , )Ω ρ . Funkcyg
�
w budem yskat\ v vyde (12), a
ρ0
1− ∇p — sçytat\ πlementom
→
G( )Ω . Uslovye
� �
w n⋅ = 0 na S pozvolqet za-
klgçyt\, çto
∂
∂
Φ
n
= 0 na S.
Sproektyruem uravnenye (2) na
→
J0 0( , )Ω ρ y
→
G( )Ω . V rezul\tate poluçym
d u
dt
P N z u k P
N z
z z
k P g z kz
2
2 0 0
2
0
0
2
0
0 0
1
� � � �
+ + ∂
∂
+ − ∇−( ) ( ( ) )( )
( )
( )
( )
ρ
ρΦ Φdiv = 0,
d
dt
z P N z u k
N z
z z
k g z kG z
2
2 0
1
0
2 0
2
0
0
1( ) ( )( ) ( )
( )
( )
( )ρ
ρ
ρ− −∇ + + ∂
∂
− ∇
Φ Φ Φ
� � �
div +
+ ρ ρ ρ0
1
0
2
0
1− −∇ + ∂
∂
− ∇
g u g
z
c zz
Φ Φdiv( )( ) = 0,
hde P0 y PG — proektor¥ na podprostranstva
→
J0 0( , )Ω ρ y
→
G( )Ω sootvetst-
venno. Vvedem v poslednem uravnenyy funkcyy Ψi , i = 1, 2, 3, takye, çto ΨiN∈
∈ W z2
1
0
1( ), ( )Ω ρ− y
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1328 B. M. VRONSKYJ
ρ0
1
1
− ∇Φ = P g z kG( ( ) )( )− ∇−div ρ0
1 Φ
�
, ρ0
1
2
− ∇Φ =
P
N z
z z
kG
0
2
0
( )
( )ρ
∂
∂
Φ �
,
ρ0
1
3
− ∇Φ = P N z u kG z( )( )0
2
�
.
Zatem proyntehryruem eho po prostranstvenn¥m peremenn¥m. V rezul\tate po-
luçym systemu
∂
∂
+
2
2 0 0
2
� �u
t
P N z u kz( )( ) +
+
P
N z
z z
k P g z k0
0
2
0
0 0
1( )
( )
( )( ( ) )
ρ
ρ∂
∂
+ − ∇−Φ Φ
� �
div = 0, (13)
∂
∂
+
2
2 0
Φ
t
g uzρ +
+ g
z
c z
∂
∂
− ∇ + + +−Φ Φ Ψ Ψ Ψ2
0
1
1 2 3div( )( )ρ = 0. (14)
1.3. Operatornoe uravnenye zadaçy. Dlq perexoda ot system¥ (13), (14) k
dyfferencyal\nomu uravnenyg v hyl\bertovom prostranstve vvedem operator¥
Aij y Bij , i, j = 1, 2, sledugwym obrazom:
A u11
�
= P N z u kz0 0
2( )( )
�
, A12Φ =
P
N z
z z
k0
0
2
0
( )
( )ρ
∂
∂
Φ �
,
A u21
�
= Ψ3 , A22Φ = Ψ2 ,
B12Φ = P g z k0 0
1( ( ) )( )− ∇−div ρ Φ
�
, B u21
�
= g uzρ0 ,
B22Φ = g
z
∂
∂
+Φ Ψ1, B u11
�
= 0,
B0Φ = – c z2
0
1div( )( )ρ− ∇Φ .
Teper\ systemu (13), (14) moΩno zapysat\ v vyde
d U
dt
AU BU
2
2 + + = 0, U( )0 = U0
, ′U ( )0 = U1
,
(15)
U ≡ ( ),
�
u TΦ ∈ H ≡
→
J0 0( , )Ω ρ � W z2
1
0
1( ), ( )Ω ρ− .
Skalqrnoe proyzvedenye v prostranstve H zadaetsq po formule
( ),U U H1 2 = ( ) ( ) ( ), , ( ) ( ),
� � � �
u u z u u z dL1 2 1 2 1 0 1 2 0
1
1 22
+ ⋅ + ∇ ⋅∇∗ − ∗∫Φ Φ Φ Φ ΩΩ
Ω
ρ ρ .
Operator¥ A y B yz (15) ymegt vyd
A =
A A
A A
11 12
21 22
, B =
0 12
21 22 0
B
B B B+
.
1.4. Svojstva operatorov.
Lemma02. Operator A : H → H qvlqetsq samosoprqΩenn¥m y neotryca-
tel\n¥m, pryçem A N z N= ≡max ( )0
2
0
2
.
Dokazatel\stvo sostoyt v sostavlenyy bylynejnoj form¥ ( ),AU U H1 2 ,
hde U1 , U2 ∈ H, y prymenenyy opredelenyj operatorov Aij , i, j = 1, 2, vekto-
rov Ui y sootvetstvugwyx skalqrn¥x proyzvedenyj.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O MALÁX DVYÛENYQX SYSTEMÁ “ÛYDKOST|-HAZ” … 1329
Krome toho, moΩno pokazat\, çto A N≤ 0
2 . Yspol\zovav to, çto σ ( )A11 =
= [ ],0 0
2N [4], poluçym A N= 0
2 .
Lemma03. Operator B0 qvlqetsq neohranyçenn¥m, samosoprqΩenn¥m y po-
loΩytel\no opredelenn¥m v prostranstve W z2
1
0
1( ), ( )Ω ρ− .
Dokazatel\stvo. Sostavym bylynejnug formu ( ), ,B0 1 2 1Φ Φ Ω , hde Φ1 , Φ2N∈
∈ D ( B0 ) . Ymeem
( ), ,B0 1 2 1Φ Φ Ω = c z z z d2
0 0
1
1 0
1
2ρ ρ ρ( ) ( ) ( )( ) ( )div div− − ∗∇ ∇∫ Φ Φ Ω
Ω
= ( ), ,Φ Φ Ω1 0 2 1B .
V formule dlq ( ), ,B0 1 2 1Φ Φ Ω poloΩym Φ1 = Φ2 = ΦN∈ W z2
1
0
1( ), ( )Ω ρ− . Tohda
( ), ,B0 1Φ Φ Ω = c z z d2
0 0
1 2
ρ ρ( ) ( )( )div − ∇∫ Φ Ω
Ω
.
Netrudno ubedyt\sq v tom, çto operator B0 πllyptyçeskyj. Dlq takyx ope-
ratorov v¥polnqetsq neravenstvo
c
W1
2
2
2Φ
Ω( )
≤ B L0
2
2
Φ Ω( ) ≤ c
W2
2
2
2Φ
Ω( )
, c1 , c2 > 0.
V¥raΩenye dlq B L0
2
2
Φ Ω( ) ymeet vyd
B L0
2
2
Φ Ω( ) = c z z d4
0 0
1 2
ρ ρ( ) ( )( )div − ∇∫ Φ Ω
Ω
.
Sravnyvaq v¥raΩenyq dlq ( ), ,B0 1Φ Φ Ω y B L0
2
2
Φ Ω( ), a takΩe yspol\zuq
pryvedennoe neravenstvo, pryxodym k v¥vodu, çto norma v πnerhetyçeskom
prostranstve operatora B0 πkvyvalentna odnoj yz norm prostranstva W2
2( )Ω .
Otsgda y yz teorem vloΩenyq sleduet, çto, vo-perv¥x, ( ), ,B0 1Φ Φ Ω = Φ B0
2 ≥
≥ Φ Ω1
2
, , t. e. operator B0 — poloΩytel\no opredelen, y, vo-vtor¥x, lgboe
mnoΩestvo, ohranyçennoe v norme πnerhetyçeskoho prostranstva operatora B0 ,
kompaktno v norme prostranstva W z2
1
0
1( ), ( )Ω ρ− .
Lemma04. Operator B0
1−
prynadleΩyt klassu � p pry p > 3 2/ .
Dokazatel\stvo. Kompaktnost\ sleduet yz pred¥duwej lemm¥. Krome
toho, yzvestno, çto sobstvenn¥e znaçenyq operatora B0
1−
ymegt asymptotyçes-
koe povedenye:
λn B( )0
1− = c n o
B0
1
2 3 1 1−
− +/ ( ( )) pry n → ∞ , (16)
t. e. operator B0
1−
prynadleΩyt klassu � p pry p > 3 2/ .
Lemma05. Operator D A B≡ + neotrycatelen.
2. Sobstvenn¥e kolebanyq. Perejdem k yssledovanyg sobstvenn¥x kole-
banyj system¥, t. e. k yzuçenyg svojstv reßenyj zadaçy (15), zavysqwyx ot vre-
meny po zakonu exp( )i tω . V rezul\tate poluçym spektral\nug zadaçu
λU AU BU= + , λ ω= 2 . (17)
Otmetym, çto tak kak operator D A B= + neotrycatelen, spektr zadaçy
(17) vewestvenn¥j y raspoloΩen na luçe [ 0, + ∞ ) .
2.1. Akustyçeskye kolebanyq. Perepyßem uravnenye (17) v matryçnoj
forme y budem sçytat\, çto λ > N0
2 . Ot uravnenyq (17) perejdem k systeme
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1330 B. M. VRONSKYJ
( )I A u− −λ 1
11
�
= λ−1
12D Φ ,
(18)
λ Φ = D u D B21 22 0
�
+ +( )Φ , i, j = 1, 2,
yz kotoroj, ysklgçyv
�
u , poluçym zadaçu na sobstvenn¥e znaçenyq
λ Φ = λ λ− + +1
21 12 22 0D R D D B( ) Φ Φ Φ ,
hde R I A( ) ( )λ λ≡ − − −1
11
1
— analytyçeskaq operator-funkcyq, prynymagwaq
znaçenyq na mnoΩestve ohranyçenn¥x samosoprqΩenn¥x operatorov ( )λ > N0
2 .
V¥polnym v poslednem ravenstve zamenu Φ = −B0
1 2/ ζ y prymenym k obeym
çastqm operator B0
1 2− / . V rezul\tate poluçym zadaçu
L( )λ ζ ≡ ( )( )I S B F+ − +− −λ λ λ ζ0
1 1 = 0, (19)
hde
S ≡ B D B0
1 2
22 0
1 2− −/ / , F( )λ ≡ B D R D B0
1 2
21 12 0
1 2− −/ /( )λ .
Lemma06. Operator-funkcyq F( )λ y z (19) pry λ > N0
2
prynymaet zna-
çenyq na mnoΩestve lynejn¥x, ohranyçenn¥x y samosoprqΩenn¥x operatorov.
Dokazat\ nuΩno tol\ko ohranyçennost\. Dlq πtoho pokaΩem, çto operator¥
Q D B12 12 0
1 2≡ − /
y Q B D21 0
1 2
21≡ − /
ohranyçen¥. Poskol\ku ony vzaymno soprq-
Ωen¥, to dostatoçno dokazat\ ohranyçennost\ Q12 . Ymeem Q A B12 12 0
1 2= − / +
+ B B12 0
1 2− / . Pervoe slahaemoe ohranyçeno (y daΩe kompaktno) v sylu svojstv
operatorov A y B0
1 2− /
. PokaΩem, çto y B B12 0
1 2− /
takΩe ohranyçen. Dlq πtoho
v¥çyslym B B12 0
1 2 2− / ζ :
B B12 0
1 2 2− / ζ = ( )/ /,B B B B12 0
1 2
12 0
1 2− −ζ ζ = ( ),B B12 12Φ Φ =
= g z z d2
0 0
1 2
ρ ρ( ) ( )( )div − ∇∫ Φ Ω
Ω
,
ζ 2 = ( ),B0Φ Φ = c z z d2
0 0
1 2
ρ ρ( ) ( )( )div − ∇∫ Φ Ω
Ω
.
Otsgda sleduet
B B12 0
1 2 2− / = g c2 2/ .
Sledovatel\no, operator¥ Q12 y Q21 ohranyçen¥ y dlq yx norm spravedlyv¥
ocenky
Q12 = Q21 ≤ λ1
1 2
0
1
0
2/ ( ) /B N g c− + ,
hde λ1 0
1( )B− — pervoe sobstvennoe znaçenye operatora B0
1− . Takym obrazom, oh-
ranyçennost\ operator-funkcyy F( )λ dokazana.
Lemma07. Operator S qvlqetsq samosoprqΩenn¥m y kompaktn¥m.
Dokazatel\stvo. SamosoprqΩennost\ sleduet yz struktur¥ operatora S
y svojstv vxodqwyx v neho operatorov. PokaΩem eho polnug neprer¥vnost\.
Dlq πtoho predstavym S v vyde summ¥ S S SA B= + , hde S B B BA ≡
− −
0
1 2
22 0
1 2/ / ,
S B B BB ≡
− −
0
1 2
22 0
1 2/ / . Operator SA neotrycatelen y, krome, toho, SA p∈� pry
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O MALÁX DVYÛENYQX SYSTEMÁ “ÛYDKOST|-HAZ” … 1331
p > 3 2/ . Yzuçym svojstva operatora SB. Dlq πtoho sostavym v¥raΩenye dlq
( ),SBζ ζ :
( ),SBζ ζ = – 2 0
1g z
z
ddiv( )( )ρ−
∗
∇ ∂
∂
∫ Φ Φ Ω
Ω
.
Yspol\zovav neravenstvo Koßy – Bunqkovskoho, poluçym
( ),SBζ ζ ≤ 2 0
1g Bζ ζ ζ( ),− .
V¥berem posledovatel\nost\ { }ζn n=
∞
1 takug, çto ζn → 0 (slabo), n → ∞ .
Dlq nee suwestvuet c > 0 takoe, çto dlq vsex n ∈ N
ζn = c z z dn
2
0 0
1 2
1 2
ρ ρ( ) ( )( )
/
div − ∇
∫ Φ Ω
Ω
< c
y v sylu polnoj neprer¥vnosty operatora B0
1−
v¥polneno uslovye
( ),B n n0
1− ζ ζ =
Ω
Φ Ω∫ − ∇ρ0
1 2( ) )z dn → 0 pry n → ∞ .
Teper\ v sylu ocenky dlq ( ),SBζ ζ poluçym, çto ( ),SBζ ζ → 0 pry n → ∞
dlq lgboj slabo sxodqwejsq k nulg posledovatel\nosty { }ζn n=
∞
1. Otsgda
sleduet, çto SB p∈� pry p > 3. Krome toho, dokazano neravenstvo
SB ≤ 2
2 1
1 2
0
1g
c
Bλ / ( )− .
Lemma dokazana.
2.2. Faktoryzacyq operatornoho puçka. Dlq yssledovanyq operator-
funkcyy L( )λ yz (19) vospol\zuemsq teoremoj o faktoryzacyy yz [3, c. 178].
Pered tem kak prymenyt\ πtu teoremu v¥polnym v (19) zamenu λ µ= 1/ spekt-
ral\noho parametra. V rezul\tate poluçym operatorn¥j puçok
M( )µ = µ µ µ µ µI B S S FA B− + + +− −
0
1 2 1( ), µ ∈ −[ ],0 0
2N . (20)
Yspol\zovav ocenky dlq norm operatorov, sostavlqgwyx πtot puçok, y teo-
remu o faktoryzacyy yz [3], prydem k sledugwemu utverΩdenyg.
Teorema01 (dostatoçnoe uslovye faktoryzacyy). Pry v¥polnenyy uslovyq
λ1 0( )B > max ,N g
c
g
c
g
c0
2
2
4 2
2 2
44+ −
(21)
operator-funkcyq M( )µ dopuskaet faktoryzacyg vyda M( )µ = M+( )µ ×
× ( )µI Z− , hde M+( )µ holomorfna y holomorfno obratyma v nekotoroj ok-
restnosty otrezka [ ( ) ],− + −ε ε1 0
2
2
1N , a operator Z takoj, çto σ( )Z ⊂
⊂ [ ( ) ],− + −ε ε1 0
2
2
1N pry ukazannom v¥ße v¥bore çysel ε1 y ε2 .
S pomow\g metoda neopredelenn¥x koπffycyentov lehko proveryt\, çto
Z M B I T B= = +− − −
0
1
0
1
0
1( ) , hde T ∈ ∞� , t. e. Z — slabovozmuwenn¥j kompakt-
n¥j operator, pryçem Ker Z = { }0 .
2.3. O polnote system¥ mod akustyçeskyx voln.
Teorema02. Esly v¥polneno uslovye faktoryzacyy (21), to zadaça (19) pry
λ > N0
2
ymeet dyskretn¥j spektr { }λk k=
∞
1, λ µk k Z= −[ ( )] 1, sostoqwyj yz ko-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1332 B. M. VRONSKYJ
neçno kratn¥x sobstvenn¥x znaçenyj s edynstvennoj predel\noj toçkoj λ =
= + ∞ .
Sootvetstvugwye ym sobstvenn¥e vektor¥ { }ζk k=
∞
1 obrazugt polnug y
mynymal\nug systemu v prostranstve W z2
1
0
1( ), ( )Ω ρ− .
Dokazatel\stvo sleduet yz teorem¥ M.NV.NKeld¥ßa o slabovozmuwennom
operatore.
Teorema03. Esly v¥polneno uslovye faktoryzacyy (21), to reßenyqm zada-
çy (19) { }λk k=
∞
1 y { }ζk k=
∞
1 pry λ > N0
2
sootvetstvugt mod¥ kolebanyj
U uk k k
T= ( , )
�
Φ , k = 1, 2, … , ymegwye xarakter akustyçeskyx voln. A ymenno:
pry normyrovke
�
uk L k2
2
1
2+ Φ Ω, = 1, k = 1, 2, … ,
ymegt mesto asymptotyçeskye formul¥
�
uk L2
→ 0, B k
k
k0
1
1
1− −Φ Φ
Ωλ ,
→ 0, k → ∞ .
Dokazatel\stvo. PoloΩyv Φ Φ= k y λ λ= k , k = 1, 2, … , zapyßem
systemu (18) v vyde
�
uk = λ λk k kI A D− − −−1 1
11
1
12( ) Φ ,
λk kΦ = D u D Bk k21 22 0
�
+ +( )Φ .
Zatem, v¥polnyv v nej zamenu Φk kB= −
0
1 2/ ζ , posle nekotor¥x preobrazovanyj
poluçym
�
uk = λ λk k kI A D− − −−1 1
11
1
12( ) Φ ,
λ ζ ζk k kB0
1− − = Q u Sk k21
�
+ ζ ,
posle çeho perejdem k predelu pry k → ∞ . Yz pervoho yz sootnoßenyj ymeem
�
uk L2
0→ , a yz vtoroho — B k k k0
1 1
1
0− −− →ζ λ ζ
,Ω
. Perexodq ot ζk k Φk ,
poluçaem utverΩdenye teorem¥.
2.4. Dvustoronnye ocenky sobstvenn¥x znaçenyj. Po-preΩnemu, budem
sçytat\, çto v¥polneno uslovye (21). V πtom sluçae sobstvenn¥e znaçenyq
{ }λk k=
∞
1 zadaçy naxodqtsq na yntervale ( ),N0
2 +∞ , obrazuq tam dyskretn¥j
spektr s edynstvennoj predel\noj toçkoj λ = +∞ . Perejdem k puçku s ohra-
nyçenn¥my operatoramy. Dlq πtoho v¥polnym zamenu ζ = B0
1 2/ Φ . V rezul\ta-
te poluçym
λ
ζ ζ ζ
I
B
u A Q
Q S
u
I
u0
0
0 0
00
1
11 12
21
−
−
−
� � �
= 0,
yly v kompaktnoj forme (oboznaçenyq qsn¥)
L ( )λ ≡ ( )λR A J− − z = 0, z = ( ),
�
u Tζ . (22)
UmnoΩyv skalqrno (22) na z, moΩno sdelat\ v¥vod, çto sobstvenn¥e znaçe-
nyq λk naxodqtsq sredy znaçenyj funkcyonala Releq
λk z( ) =
( , ) ( , )
( , )
A J
R
z z z z
z z
+ ,
pryçem moΩno sçytat\, çto ( , )R z z = 1. Netrudno vydet\, çto ( ( ) )( ) ,L λ z z z =
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
O MALÁX DVYÛENYQX SYSTEMÁ “ÛYDKOST|-HAZ” … 1333
= 0 y ( ( ) )( ) ,′ >L λ z z z 0. Takym obrazom, para ( ( ) ), ( )L λ λ z obrazuet systemu
Releq na yntervale ( ),N0
2 +∞ .
Dlq çastot λk N∈ +∞( ),0
2
spravedlyv varyacyonn¥j pryncyp [1]
λk = sup inf ( )
M z M
k k
z
∈
λ , k = 1, 2, … , (23)
hde Mk — k-mernoe podprostranstvo.
Dalee poluçym dvustoronnye ocenky dlq sobstvenn¥x znaçenyj λk . V sylu
svojstv vxodqwyx v v¥raΩenye dlq λ( )z operatorov
( , )
( , )
J
R
z z
z z
≤ λ( )z ≤
( , ) ( , )
( , )
A J
R
z z z z
z z
+ .
Dal\nejßee preobrazovanye pryvodyt k dvojnomu neravenstvu
( , )B0Φ Φ ≤ λ( )z ≤ ( , )DU U , U = ( , )
�
u TΦ .
Vospol\zovavßys\ lemmojN5, ocenym sverxu v¥raΩenye dlq ( , )DU U :
( , )DU U ≤
Ω
Φ Ω∫ + ∇ ⋅−ρ ρ0 0
2
0
1 2
( ) ( ) ( )( )z N z u z k d
� �
+
+ c z z d2
0 0
1 2
Ω
Φ Ω∫ − ∇ρ ρ( ) ( )( )div +
+ 2 0 0
1
0
1g z u z k z d
Ω
Φ Φ Ω∫ + ∇ ⋅ ∇− −ρ ρ ρ( ) ( ) ( )( ) ( )
� �
div +
+
Ω
Φ Ω∫ + ∇ ⋅−ρ ρ0 0
2
0
1 2
( ) ( ) ( )( )z N z u z k d
� �
+
+ c z z d2
0 0
1 2
Ω
Φ Ω∫ − ∇ρ ρ( ) ( )( )div +
+
2 1
0 0
2
0
1 2
1 2
gc z N z u z k d− −∫ + ∇ ⋅
Ω
Φ Ωρ ρ( ) ( ) ( )( )
/
� �
×
× c z z d2
0 0
1 2
1 2
Ω
Φ Ω∫ − ∇
ρ ρ( ) ( )( )
/
div ≤
≤ N B gc B0
2
0
1
0
1 22+ + −( , ) ( , ) /Φ Φ Φ Φ .
Prymenyv teper\ varyacyonn¥e pryncyp¥ (23) y Kuranta, poluçym trebuem¥e
ocenky dlq sobstvenn¥x znaçenyj λk :
λk B( )0 ≤ λk ≤ λ λk kB gc B N( ) ( )/
0
1 1 2
0 0
22+ +− ,
otkuda sleduet asymptotyçeskaq formula
λk = λk B o( )( ( ))0 1 1+ , k → ∞ .
2.5. Symmetryzator spektral\noj zadaçy. Vvedem v rassmotrenye ope-
rator F po formule
F = 1
2
1
π
µ µ
µ
i
M d
t
−
=
∫ ( ) , λ1 0
1( )B− < t < N0
2− .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1334 B. M. VRONSKYJ
Tak Ωe, kak y v pp.N2.1, dokaz¥vagtsq sledugwye utverΩdenyq.
Lemma08. Operator F qvlqetsq samosoprqΩenn¥m, ohranyçenn¥m y sym-
metryzugwym sprava operator Z, t. e. ( )ZF ZF∗ = .
Lemma09. Operator F qvlqetsq poloΩytel\no opredelenn¥m y ymeet
strukturu F I F= + 1, hde F p1∈� (pry p > 3).
2.6. Bazysnost\ system¥ mod akustyçeskyx voln. Nalyçye symmetryza-
tora F operatora Z pozvolqet, pry v¥polnenyy uslovyq (21), dokazat\ sledu-
gwug teoremu.
Teorema04. Pry v¥polnenyy uslovyq (21) systema sobstvenn¥x vektorov
zadaçy (19) obrazuet p-bazys (pry p > 3) prostranstva W z2
1
0
1( ), ( )Ω ρ− .
Dokazatel\stvo povtorqet sootvetstvugwye rassuΩdenyq yz pp.N2.2.
2.7. Voln¥, poroΩdenn¥e stratyfykacyej. Budem sçytat\, çto λ N∈
∈ [ ],0 0
2N . Perepyßem yssleduemug systemu v vyde
λ
�
u = A u D11 12
�
+ Φ ,
( )λ I D B− −22 0 Φ = D u21
�
.
Na otrezke [ ],0 0
2N operator λ I D B− −22 0 obratym vsgdu, za ysklgçenyem ko-
neçnoho çysla toçek. Dejstvytel\no,
λ λI D B B I S B B− − = − − −− − −
22 0 0
1 2
0
1
0
1 2/ /( ) .
Sohlasno teoreme M. V. Keld¥ßa [6] operator-funkcyq I S B− − −λ 0
1
obratyma
vsgdu, za ysklgçenyem ne bolee çem sçetnoho mnoΩestva yzolyrovann¥x toçek
s predel\noj toçkoj λ = ∞ . Na otrezke [ ],0 0
2N πtyx toçek ne bolee
koneçnoho çysla, t. e. utverΩdenye ob obratymosty operatora λ I D B− −22 0
dokazano.
V tex Ωe toçkax, v kotor¥x operator λ I D B− −22 0 obratym, ymeem spekt-
ral\nug zadaçu
λ
�
u = ( ( ))A Q Q P u11 12 21− + λ
�
,
hde operator-funkcyq P ( λ ) prynymaet znaçenyq na mnoΩestve samosoprqΩen-
n¥x vpolne neprer¥vn¥x operatorov.
Dalee nam snova ponadobytsq rezul\tat rabot¥ [6] o predel\nom spektre
operatora A11. Opyraqs\ na neho y rassuΩdenyq, analohyçn¥e sootvetstvug-
wym rassuΩdenyqm yz pp.N2.2, moΩno dokazat\, çto spravedlyva sledugwaq te-
orema.
Teorema05. Predel\n¥j spektr zadaçy (17) sovpadaet s otrezkom [ ],0 0
2N .
1. Brexovskyx L. M., Honçarov V. V. Vvedenye v mexanyku sploßn¥x sred. – M.: Nauka, 1982. –
335 s.
2. Kopaçevskyj N. D., Krejn S. H., Nho Zuj Kan. Operatorn¥e metod¥ v lynejnoj hydrodyna-
myke: πvolgcyonn¥e y spektral\n¥e zadaçy. – M.: Nauka, 1989. – 416 s.
3. Markus A. S. Vvedenye v spektral\nug teoryg polynomyal\n¥x operatorn¥x puçkov. –
Kyßynev: Ítyynca, 1986. – 260 s.
4. Kopaçevskyj N. D., Car\kov M. G. K voprosu o spektre operatora plavuçesty // Ûurn. v¥-
çyslyt. matematyky y mat. fyzyky. – 1987. – # 3. – S. 548 – 551.
5. Suslyna T. A. Asymptotyka spektra nekotor¥x zadaç, svqzann¥x s kolebanyqmy Ωydkos-
tej. – L., 1985. – 79Ns. – Dep. v VYNYTY, #N8058-V.
6. Keld¥ß M. V. O polnote sobstvenn¥x funkcyj nekotor¥x klassov nesamosoprqΩenn¥x
lynejn¥x operatorov // Uspexy mat. nauk. – 1971. – 24, v¥p.N4 (160). – S. 15 – 41.
Poluçeno 24.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
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| id | umjimathkievua-article-3535 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:20Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/72/590d22eb69b493b55c15e267cf6e7672.pdf |
| spelling | umjimathkievua-article-35352020-03-18T19:57:10Z On small motions of a “liquid-gas” system in a bounded domain O малых движениях системы „жидкость-газ" в ограниченной области Vronskii, B. M. Вронский, Б. М. Вронский, Б. М. We study small motions and free oscillations of a compressible stratified liquid, the structure of the spectrum, and the basis property of a system of eigenvectors and obtain asymptotic relations for eigenvalues. Вивчено малі рухи i власні коливання стисливої стратифікованої рідини, досліджено структуру спектра, базисність системи власних векторів, одержано асимптотичні формули для власних значень. Institute of Mathematics, NAS of Ukraine 2006-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3535 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 10 (2006); 1326–1334 Український математичний журнал; Том 58 № 10 (2006); 1326–1334 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3535/3806 https://umj.imath.kiev.ua/index.php/umj/article/view/3535/3807 Copyright (c) 2006 Vronskii B. M. |
| spellingShingle | Vronskii, B. M. Вронский, Б. М. Вронский, Б. М. On small motions of a “liquid-gas” system in a bounded domain |
| title | On small motions of a “liquid-gas” system in a bounded domain |
| title_alt | O малых движениях системы „жидкость-газ" в ограниченной области |
| title_full | On small motions of a “liquid-gas” system in a bounded domain |
| title_fullStr | On small motions of a “liquid-gas” system in a bounded domain |
| title_full_unstemmed | On small motions of a “liquid-gas” system in a bounded domain |
| title_short | On small motions of a “liquid-gas” system in a bounded domain |
| title_sort | on small motions of a “liquid-gas” system in a bounded domain |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3535 |
| work_keys_str_mv | AT vronskiibm onsmallmotionsofaliquidgassysteminaboundeddomain AT vronskijbm onsmallmotionsofaliquidgassysteminaboundeddomain AT vronskijbm onsmallmotionsofaliquidgassysteminaboundeddomain AT vronskiibm omalyhdviženiâhsistemyžidkostʹgazquotvograničennojoblasti AT vronskijbm omalyhdviženiâhsistemyžidkostʹgazquotvograničennojoblasti AT vronskijbm omalyhdviženiâhsistemyžidkostʹgazquotvograničennojoblasti |