This research has been supported by Facultad de Ciencias
Uniandes, Programa de Investigaciones 2020-2022.
ET us consider a bounded domain
3
R
with the
boundary
of the class
C
piecewise, and the
following system of fluid dynamics
1
1
2
2
3
3
2
1
2
3
2
0
0
0
0
div 0 , 0.
up
tx
up
tx
up
tx
t
p
t
u
u
Nu
u x t
( 1 )
In this system
1 2 3
,,x x x x
is the space variable,
1 2 3
, , , , , ,u x t u x t u x t u x t
is the velocity field,
,p x t
is the scalar field of the dynamic pressure and
,xt
is the dynamic density. We suppose that the
stationary distribution of density is described by the function
3
Nx
e
, where
N
is a positive constant. For the
compressibility coefficient
, we assume
0.
In the
model (1) the stratified fluid is rotating over the vertical axis
with the constant angular velocity
0,0, .
For non-rotational case, the equations (1) are deduced, for
example, in [1], [2].
Despite an extensive study of stratified flows from the physical
point of view (see, for example, [3-6]) we would like to
observe that there have been relatively few works considering
the mathematical aspect of the problem. We associate the
system (1) to the first boundary value (Dirichlet) condition
0.p
The following separation of variables allows us to consider the
problem of normal vibrations
4
1
5
,
,
, , .
t
t
t
u x t v x e
x t Nv x e
p x t v x e C
( 2 )
We denote
45
,,v v v v
and write the system (1) in the
matrix form
0Lv
( 3 )
where
L
On the Structure of the Spectrum of Internal Vibrations for Stratified
Rotating Compressible Flows in General Domains, in Rectangular
Domains, in General Cylinders and in Spherical Volumes
A. GINIATOULLINE
Los Andes University, Cr. 1 No 18A-12 Uniandes Dept. Mathematics, 111711
Bogota, COLOMBIA
Abstract:—For exponentially stratified rotating compressible fluid, we investigate the localization and the
structure of the spectrum of inner waves caused by the gravitational force and the Coriolis force. We find the
essential spectrum for the first boundary value problem in general domains. Our main result is the explicit
examples of the eigenvalues and the corresponding orthogonal eigenfunctions for parallelepipeds, for general
cylinders and for spherical volumes.
Keywords:—Compressible flows, computational fluid dynamics, essential spectrum, internal waves, turbulence
and multiphase flows.
Received: May 25, 2022. Revised: October 26, 2022. Accepted: December 3, 2022. Published: December 31, 2022.
1. Introduction
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L M I
and
1
2
3
1 2 3
1
1
1
1 1 1
0 0 0
0 0 0
.
0 0 0
0 0 0 0
00
x
x
x
x x x
MN
N
We define the domain of the operator
M
as follows.
3
22
0
1
2
0
11
4 2 5 2
:
,
, , ,
,
v L f L
DM v f W
v W v W
where
0
1
2
W
is a closure of the functional space
0
C
in the norm
1
2
22.f f f dx
In this paper, we investigate the structure and the localization
of the spectrum of the operator
M
for general three-
dimensional domains, and also we construct the explicit
examples of the eigenvalues and eigenfunctions for
rectangular, cylindrical and spherical domains.
From the point of view of applications, the separation of
variables (2) may serve as a tool to represent every non-
stationary motion described by (1) as a linear sum of the
stationary modes. The knowledge of the spectrum of normal
oscillations may be very useful for studying the stability of the
flows. Besides, the spectrum of operator
M
plays an
important role in the investigation of weakly non-linear flows,
since the bifurcation points where the small non-linear
solutions arise, belong to the spectrum of linear normal
vibrations, i.e., to the spectrum of operator
M
.
It can be easily seen that the operator
M
is a closed
operator, and its domain is dense in
5
2
L
.
Let us denote by
ess M
the essential spectrum of operator
M
. We recall that the essential spectrum
: is not of Fredholm type ,
ess M C M I
is composed of the points belonging to the continuous
spectrum, limit points of the point spectrum and the
eigenvalues of infinite multiplicity ([7-9]).
To find the essential spectrum of the operator
M
, we will
use the following property ([10]):
,
ess M Q S
where
: is not elliptic in sense of
Douglis-Nirenberg
C M I
Q
and
\ : the boundary conditions for the operator .
do not satisfy Lopatinski conditions
CQ
SMI
We recall the following two definitions.
Definition 1. Let us consider a differential matrix operator
11 1
1
1
...
... ... ... , , ,..., ,
... ij
N
ij ij n
n
N NN
ll
L l a D
ll
1
1
11
1
... , , ... .
...
n
n
n j n
jn
D D D D
x x x
Let
1,
N
ii
s
1
N
jj
t
be two sets of integer numbers such that,
if
0,
ij
l
then
deg
ij ij i j
n l s t
. In case
0
ij
l
, we do
not require any condition for the sum
ij
st
. Now, we
construct the main symbol of
LD
as follows.
11 1
1
...
... ... ... ,
...
N
N NN
l D l D
LD
l D l D
0 if 0 or deg
.
if deg
ij
ij ij i j
ij ij ij i j
st
l D l D s t
la D l D s t
If there exist the sets
s
and
t
which satisfy the above
conditions and, additionally, if the following condition holds,
det 0 for all \ 0 ,
n
LR
then the operator
LD
is called elliptic in sense of Douglis-
Nirenberg (see[11]).
Definition 2. Let us consider
1 2 3
, , ,
12
,,
L
the matrix of the algebraic complements of the main
symbol matrix
,L
G
is the main symbol of the
matrix
GD
which defines the boundary conditions,
,,
j
M
j
are the roots of the
equation
det , 0L
with positive imaginary part. If the
rows of the matrix
,,GL
are linearly independent
with respect to the module
,M
for
0
, then we
will say that the conditions of Lopatinski are satisfied (see
[10]).
Now we establish the following two theorems.
Remark 1. We note that, for the boundary condition
0un
, the results analogous to the Theorems 1 and 2,
were proved in [12]. The operator
M
with boundary
condition
0p
has not been considered previously.
Theorem 1. The operator
M
is skew-selfadjoint.
Proof. We represent
M
as
0 ,
N
M M B B
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where
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
,
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
N
BB
N
N
are anti-symmetric bounded operators. From [7] we have that
it is sufficient to prove the skew-selfadjointness for the
operator
0
M
with the domain
0.D M D M
We note
first that, integrating by parts, for
0
,u v D M
we obtain
22
00
, , .
LL
M u v u M v
Now, let
0.v D M
It means that
2
vL
and there
exists
2
fL
such that
22
00
, , for all .
LL
M u v u f u D M
In particular, for
5
0,0,0,0,uu
0
1
52
uW
we have
5 5 5
1, , .u v u f
Now, for
,0,0uu
we obtain
5
1div , , .u v u f
Keeping in mind that
5
vf
, we integrate by parts the
last identity and thus have that for any
22
divu L u L
, the relation holds:
55
11
, , .u n v ds u v u f
Therefore,
5 2 2
0 div ,u n v ds u L u L
from which it follows that
50,v
which implies
0
1
52 .vW
Since
0
M
is not acting on the fourth component of the vector
,u
we may consider
4 4 4 0.u v f
Summing up the obtained results, we have verified that
00
.D M D M
The reciprocal inclusion can be proved analogously and thus
the theorem is proved.
Remark 2. Since
M
is skew-selfadjoint, then its spectrum
belongs to the imaginary axis. Indeed,
v D M
, , , , 0,Mv v Mv v Mv v v Mv
from which it follows that
,Mv v
is imaginary. If
is an
eigenvalue of
M
with the corresponding eigenfunction
v
,
then
also is imaginary since
2
,M v v
v
. If we remove
from the spectrum of
M
all the isolated points which are
eigenvalues of finite multiplicity, the remaining set will form
the essential spectrum of the operator
M
.
Theorem 2. Let
min ,aN
,
max , .AN
Then,
the essential spectrum of
M
is the following symmetrical set
of the imaginary axis:
0 , ,
ess M iA ia ia iA
( 5 )
Moreover, the points
0 , ,ia iA
are eigenvalues of
infinite multiplicity.
Proof. We observe that, according to [9], [11], we can choose
the numbers
i
s
0
j
t
for
, 1,2,3,4ij
and
55
1st
.
In this way, the main symbol of the operator
L M I
is:
1
1
1
2
1
3
1 1 1
1 2 3
00
00
0 0 .
0 0 0
00
LN
N
We calculate the determinant of the last matrix
2 2 2 2 2 2 2
1 2 3
2
det .LN
( 6 )
From (6) we can see that, if
does not belong to the set (5),
then
L
is elliptic in sense of Douglis-Nirenberg. It is easy to
prove that the boundary condition
0p
satisfies
Lopatinski conditions. Indeed, if we write it as
0Gu
,
we obtain immedeately that
0,0,0,0,1G
which is a vector
row. Since
,L
is a 5x5-matrix, then
GL
is a non-zero
row with fife components and the Lopatinski condition is
satisfied.
Now, let us consider the system (3) for
0
:
5
1
5
2
5
3
1
2
1
1
1
4
3
1
0
0
.
0
0
div 0
v
x
v
x
v
x
v
v
Nv
Nv
v
Evidently, every vector-function of the form
0
2 1 3
1 1 1
, ,0, , , ,vC
x x N x
satisfies the last system and thus
0
is an eigenvalue of
infinite multiplicity. The cases
,ia iA
, are
analogous, for example, for
i
the system (3) has an
infinite set of solutions
, ,0,0,0 ,vi
2. Stratified Compressible Rotating
Fluid in General Domains
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where
0
C
and has the form
1 2 3
,x x ix x
.
In this way, the theorem is proved.
Remark 3. There exists an alternative criterion of the
essential spectrum which is attributed to Weyl [8]: a necessary
and sufficient condition that a real finite value
be a point of
the essential spectrum of a selfadjoint operator M is that there
exists a sequence of elements
n
v D M
such that
1 , tends weakly to zero, and 0
n n n
v v M I v
.
In the proof of the analogous result for the operator
M
with
boundary condition
0un
in [12], there was constructed
the following explicit Weyl sequence for the essential
spectrum of the operator
M
:
3
3
3
,
3
,
44
,
53
1 1 2 2 3 3
, 1,2,3
, , 1,2,...
ik x
kk
j j k
jj
ik x
kk
ik x
kk
i
v x e j
kx
v x e
i
v x e
k
x x x x k
3
2
00
2
0 0 0 0
1
, 1,2,...
, , 1,
k
x
x k k x x k
x C x dx
where the components
k
are defined as follows
0 1 2 0 2 1
12
2 2 2 2
00
0 3 3
3 4 5
22 22
00
0
, ,
, , 1 .
, \ 0 .
N
NN
ia iA
We observe that the above Weyl sequence is also valid for the
boundary condition
0p
, which was proved in [13].
Remark 4. The statement of Theorem 2 corresponds clearly to
the previously studied particular cases of
0
[9], [14],
where it was proved that
,
ess M iN iN
, as well as the
particular case of
0N
[9], where it was proved that
,
ess M i i
.
Remark 5. The case of essential spectrum of stratified (non-
rotational) viscous fluid was considered in [15].
We consider the boundary value problem (3) in its
component representation, where, without loss of generality,
we put
1
:
5
1
5
2
5
3
3
12
1 2 3
12
12
34
34
5
0
0
0.
0
0
0
v
x
v
x
v
x
v
vv
x x x
vv
vv
v Nv
Nv v
v
( 7 )
Theorem 3. Let
be a rectangular parallelepiped in
3
R
:
0, 0, 0, .a b c
Then, the eigenfunctions
5
v
of the
problem (7) have the form
3
12
5 , ,
22
sin sin sin ,
, , 1,2,3,...
k j n nx
kx jx
vx a b c
abc
k j n
( 8 )
and the corresponding eigenvalues are
kjn
, where
1
2
2
22
2 2 2
2
22
2 2 2
2
2 2 2
2 2 2
2 2 2
2
2 2 2
2 2 2 2
.
2
4
j
kn
a b c
j
kn
kjn a b c
j
nk
c a b
N
iN
N
( 9 )
The set
kjn
forms a discrete spectrum outside the set (5),
while the set
kjn
is dense in (5)
\0
. Every point of the
set (5)
\0
is a limit point of the eigenvalues (9).
Remark 6. After substituing
5 , ,k j n
v
in (7), the rest of the
coordinates of the eigenfunctions
1 2 3 4
, , ,v v v v
can be easily
found from the resulting algebraic system.
Proof. By consecutive differentiation and substitution, we can
exclude the unknown functions
1 2 3 4
, , ,v v v v
from the
system (7)and thus obtain the following scalar equation for
5
v
2
2 2 2 2 2 2 2 2
5
2 5 5
2
3
0
v
N v N v
x
( 10 )
with the boundary condition
50v
, where
22
22
12
2.
xx
We put
i
and solve the problem (1)0 by using the
separation of variables
53
,v x w x z x
( 11 )
where
12
,.x x x
Therefore, we obtain
2
2
3
22
22
2
22 .
dz
dx
NwN
wz
( 12 )
From (12), we solve first the problem for the function
3
zx
2
2
3
0.
00
dz
dx z
z z c
( 13 )
The solutions of the problem (13) are
3. Spectrum of Internal Waves
in Rectangular Domains
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3
2
2
3
.
sin , 1, 2,3,...
n
nc
nx
ncc
z x n
For the function
wx
, we obtain the problem
2
0, ,
0,
0
a o b
ww
w
( 14 )
where
22 2
22
22 .
nN
c
N
It is easy to see that the solutions of the problem (14) are
2
2
22
12
2
2
12
.
, sin sin , , 1, 2,3,...
j
k
kj ab
kx jx
kj ab
ab
w x x k j
Thus we conclude that the eigenvalues of the problem (10) and
(7) are found from the equation
222
2 2 2 2 2 2 2
22
,
n k j
NN
c a b
( 15 )
which can be written as
2 2 2
4 2 2 2 2
2 2 2
2 2 2 2
2 2 2 2
2 2 2 0.
k j n N
a b c
n k j
N
c a b
The roots of the last equation are
2 2 2
2 2 2 2
2 2 2
2k j n
Na b c
( 16 )
2
2 2 2
2 2 2
2 2 2
2 2 2 2
2 2 2 2
2 2 2
.
4
k j n
Na b c
n k j
N
c a b
Keeping in mind that
i
, we obtain the eigenvalues (9).
The eigenfunctions
5 , ,k j n
vx
, according to the separation of
variables (11), are represented by
3
12
5 , ,
22
sin sin sin ,
, , 1,2,3,...
k j n nx
kx jx
vx a b c
abc
k j n
We note that the sign "
" before
in
kjn
means that
the spectrum is symmetrical with respect to zero.
Evidently, the subset
kjn
forms a discrete spectrum on
the imaginary axis outside the set (5).
Let us show that the subset
kjn
is dense in (5)
\0
and
thus every point of (5)
\0
is a limit point of the eigenvalues
(9). For that, we will consider a positive function
22
2
2 2 2 2 2
1.
24
L
N L Q
fQ N L Q L N Q
( 17 )
We observe that for
L
22
2
n
c
and
2
2
22
2j
k
ab
Q
we have
that
2
L kjn
fQ
, where
2
kjn
are defined in (16).
We also represent the function
fLQ
as
2 2 2
22
2
2 2 2 2 2
2
.
4
L
L N Q
fQ N L Q
N L Q L N Q
( 18 )
Using (18), for
L
and
Q
sufficiently large, we can estimate
L
fQ
as
,
L L L
F Q f Q G Q
( 19 )
where
2 2 2 2
22 ,
LL N Q N
FQ N L Q
2 2 2 2
2 2 2 2
2.
L
L N Q N
GQ N L Q N L Q
For fixed
L
, we have
2
lim .
L
Qf Q N
( 20 )
On the other hand, for fixed
Q
, we have
2
lim .
L
LfQ
( 21 )
Evidently, the properties (20), (21) are also valid for the
functions
L
FQ
and
L
GQ
.
Now, if we denote
min ,aN
,
max ,AN
, then
we can easily see that, for sufficiently large
L
and
Q
, the
values of the functions
L
FQ
and
L
GQ
(and thus the
values of the function
L
fQ
) will belong to the interval
22
,aA
. Additionally, it can be easily seen that every point
of the interval
22
,aA
can be represented as a limit point of
the functions
L
FQ
and
L
GQ
(and thus as a limit point
of the function
L
fQ
) for appropriate election of
,.LQ
Indeed, for example, let
N
,
22
,pN
,
2 2 2
pN
,
01
. We will show, for
example, that for suitable election of
,LQ
for arbitrary
small
0
, the estimate will hold:
0.
L
p F Q
( 22 )
Indeed,
2 2 2 2
22
22
1
LL N Q N
p F Q N N L Q
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2 2 4 2
22
11
.
N L Q N
N L Q
( 23 )
In (23) we choose
1
LQ
and thus we have
42
22
1.
LQ
N
p F Q N
( 24 )
From (24) we obtain that for every arbitrarily small
0
there exists sufficiently large
0Q
such that (22) will hold,
where
1
LQ
. Therefore, every point of the interval
22
,N
is a limit point of the function
L
FQ
for
,LQ
. The same property for the function
L
GQ
can
be verified analogously and thus the Theorem is proved.
Theorem 4. Let
be a circular cylinder in
3
R
:
2 2 2
1 2 3 1 2 3
, , : , 0 .x x x x x R x c
Then, the eigenfunctions
5
v
of the problem (7) have the form
3
5 , ,
22 2
12
1
2sin exp ,
, , 1,2,3,... , arctan , ( 25 )
are positive roots of the Bessel function
k
j
k j n k
k
kj
k
jk
rnx
v x J ik
Rc
R J c
x
k j n r x x x
J
and the corresponding eigenvalues are
kjn
, where
1
22
22
22
2
22
22
2
22
22
22
2
22
2 2 2
.
2
4
k
j
k
j
k
j
n
cR
n
cR
kjn
n
cR
N
iN
N
( 26 )
The set
kjn
forms a discrete spectrum outside the set (5),
while the set
kjn
is dense in (5)
\0
. Every point of the
set (5)
\0
is a limit point of the eigenvalues (26).
Proof. As in Theorem 3, we solve the problem (10) using the
separation of variables (11). For the function
3
zx
we have
3
3
2sin , 1, 2,3,...
nnx
z x n
cc
( 27 )
For
wx
we obtain the boundary value problem
20,
0
xR
ww
w
( 28 )
where
22 2
22
22 .
nN
c
N
Solving the problem (28) in polar coordinates, we have that for
2
2 , , 1,2,3,...
k
j
kj kj
R
the solutions have the form
,
exp
,,
k
jr
R
k
kj k
kj
J ik
wr RJ
( 29 )
where
k
j
are positive roots of the Bessel function
k
J
.
In this case, the eigenvalues of the problem (10) are found
from the equation
2
2
2 2 2 2 2 2
2.
k
j
nNN
cR
Therefore, the eigenvalues of the problem (10) (and also of the
problem (7)) are:
1
22
22
22
2
22
22
2
22
22
22
2
22
2 2 2
.
2
4
k
j
k
j
k
j
n
cR
n
cR
kjn
n
cR
N
iN
N
From (11), (27) and (29) we have that the corresponding
eigenfunctions have the form:
3
5 , ,
2sin exp .
k
j
k j n k
k
kj
rnx
v x J ik
Rc
R J c
To prove the properties that the set
kjn
is dense in
(5)
\0
and that every point of the set (5)
\0
is a limit
point of the eigenvalues (26), we can follow exactly the
reasoning of the proof of Theorem 3, using the same functions
L
FQ
,
L
fQ
and
L
GQ
with
22
2
n
c
L
and
2
2
k
j
R
Q
. We only need the fact that
k
j
be infinite,
countable, do not have finite limit points and posess the
property
lim .
k
j
j
k
( 30 )
For the Bessel functions
,
k
jr
kR
J
these properties, including
(30), as well as the properties of orthogonality and
completeness in
2
L
, are established, for example, in [16].
Additionally, not only for the circle but also for more general
domains, it is well known that for the problem
0
0
ww
w
the same properties are valid (see, for example, [17]). In
particular, the eigenvalues are positive, tend to infinity, have
finite multiplicity and do not have finite limit points. The
4. Spectrum of Internal Waves
in Circular Cylinders and in
General Cylinders
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eigenfunctions form a complete system in
2
L
and can be
chosen orthogonal.
In this way, the Theorem is proved.
Remark 7. The reasoning we used in the proofs of the
Theorems 3 and 4 may be easily extended to the cases of more
general domains of arbitrary cylinders
32
1 2 3
: , , 0C x R x x R x c
.
Remark 8. For arbitrary domains
2
GR
with Lipshchitz
boundary
,
the property of positiveness of the eigenvalues
and orthogonality of the eigenfunctions for the problem
22
22 0 , , ,
0
vv
xy
G
v x y G
v
( 31 )
can be deduced directly from the following simple
calculations. Indeed, let
v
be a solution of (31). Integrating
the identity
2
2
v v v v
v v v v
x y x x y y
in the domain
G
, we have
2
2
2,
G
G G G
vv
dxdy
xy
v
v ds v vdxdy v dxdy
n
from which we obtain the positiveness of
. Now, if
km
are eigenvalues of (31) with the corresponding
eigenfunctions
k
v
and
m
v
, then, after integrating the
equation
m k m k
k m k m
k m m k
v v v v
v v v v
x x x y y y
v v v v
in the domain
G
, we obtain
0.
k m m k k m k m
GG
v v v v dxdy v v dxdy
Theorem 5. Let
be a spherical volume in
3
R
:
2 2 2 2
1 2 3 1 2 3
, , : .x x x x x x R
Then, the eigenfunctions
5
v
of the problem (7) have the form
1
2
5 , , 1
2
1
2
1
2
cos
=0,1,... =1,2,... =0, 1,..., ,
are positive roots of the Bessel function
1
are Legendre polynomials 2!
n
njm j
k j n n
n
n
jn
n
nn
n
cr
v x J P
R
r
n j m n
J
d
PP
n
21,
n
n
d
( 32 )
and the constants
njm
c
are chosen such that the normalization
condition holds:
1
2
22
2
1
2
0 0 0
1
2
1
2
1cos
!
1
21 !
Rn
jn
n
njm
nnm
j
n
r
J P rdrd d
cR
lm
RJ llm
and the corresponding eigenvalues are
nj
, where
1
22
1
2
22
2
1
2
22
2
1
2
22
1
2
2
1
2
1
2 2 2
2
2.
2
4
n
j
n
j
n
j
nn
RR
nn
RR
nj
nn
RR
N
N
i
N
( 33 )
The set
nj
forms a discrete spectrum outside the set (5),
while the set
nj
is dense in (5)
\0
. Every point of the
set (5)
\0
is a limit point of the eigenvalues (32).
Proof. We use the spherical coordinates
1
2
3
sin cos
sin sin
cos .
xr
xr
xr
Assuming that
55
,v x v r
does not depend on
, we
solve the problem (10) using the separation of
variables
5,v r z r Y
. In this way, for the function
Y
we have the problem
1sin 0
sin
d dY Y
dd
. ( 34 )
After the substitution
cos , arccosyY
we
obtain that (34) has a bounded solution only if
1
nnn
, and that the solutions of (34) are Legendre
polynomials
21
1
2!
n
n
nnn
d
Pnd
.
For
()zr
we obtain the boundary value problem
2 2 2 0
1 , 0 , 0
n
n
r z A r z
n n z z R
,
which has the solutions expressed in terms of the Bessel
functions ([17]):
1
2
1
nn
z R J Ar
r
. Proceeding
analogously as in the proof of Theorem 4, we obtain that the
eigenvalues are found from the
equation
5. Spectrum of Internal Waves
in Spherical Volumes
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2
1
2
2 2 2 2 2 2
22
1
n
j
nn NN
RR
.
Keeping in mind that
i
, we obtain finally that the
eigenvalues have the form (33). From the orthogonality and
completeness of Bessel functions in
20,LR
and Legendre
polynomials in
21
LS
, we obtain the useful property that the
found set of the eigenfunctions (32) is complete and
orthonormal in
2
L
. The rest of the proof is totally
analogous to the proofs of the Theorems 3 and 4.
For the considered particular cases of parallelepipeds,
cylinders and spheres, the explicitly calculated spectrum
clearly corresponds to the essential spectrum for general
domains.
The constructed systems of eigenfunctions (8), (25) and (32)
are complete and orthonormal in
2
L
, which can be used
for solving more general problems in various applications
modelling rotating stratified comressible fluid.
We would like to thank Professor P. P. Kumar for
suggesting the topic of this research.
[1] B. Cushman, J. Beckers, Introduction to Geophysical
Fluid Dynamics. New York: Acad. Press, 2011, ch. 2.
[2] B. Maurer, D. Bolster, P. Linden, “Intrusive gravity
currents between two stably stratified”, J. Fluid Mech.,
vol. 647, 2010, pp. 53–69.
[3] V. Birman, E. Meiburg, “On gravity currents in stratified
anbients”, Phys. Fluids, vol. 19, 2007, pp. 602-612.
[4] J. Flo, M. Ungari, J. Bush, “Spin-up from rest in a
stratified fluid: boundary flows”, J. Fluid Mech., vol. 472,
2002, pp. 51-82.
[5] P. Kundu, Fluid Mechanics. New York: Acad. Press,
1990, ch. 3.
[6] N. Makarenko, J. Maltseva, A. Kazakov, “Conjugate
flows and amplitude bounds for internal solitary waves”,
Nonlin. Processes Geophys., vol. 16, 2009, pp. 169-178.
[7] T. Kato, Perturbation Theory for Linear Operators. Berlin:
Springer-Verlag, 1966, ch. 2.
[8] F. Riesz, B. Sz.-Nag, Functional Analysis, Budapest: Fr.
Ungar, 1972, ch.5.
[9] A. Giniatoulline, An Introduction to Spectral Theory,
Philadelphia: Edwards Publishers, 2005, ch.8.
[10] G. Grubb, G. Geymonat, “The essential spectrum of
elliptic systems of mixed order”, Math. Ann., vol. 227,
1977, pp.247-276.
[11] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the
boundary for solutions of elliptic differential equations”,
Comm. Pure and Appl. Mathematics, vol. 17, 1964, pp.
35-92.
[12] A. Giniatoulline, “On the spectrum of internal oscillations
of rotating stratified fluid”, Recent Advances in
Mathematical Methods and Computer techniques in
Modern Science, Morioka: WSEAS Press, 2013, pp.113-
118.
[13] A. Giniatoulline, T. Castro, “Spectral properties of
internal waves in rotating stratified fluid for various
boundary value problems”, International J. of Math.
Models and Methods in Applied Sciences, vol. 7, issue
11, 2013, pp.915-926.
[14] A. Giniatoulline, C. Hernandez, “Spectral properties of
compressible stratified flows”, Revista Colombiana Mat.,
vol. 41, issue 2, 2007, pp.333-344.
[15] A. Giniatoulline, T. Castro, “On the spectrum of operators
of inner waves in a viscous compressible stratified fluid”,
Journal Math. Sci. Univ. Tokyo, vol. 19, 2012, pp.313-
323.
[16] G. Watson, Treatise on the Theory of Bessel Functions,
Cambridge: Cambridge University Press, 1995, ch.3.
[17] V. Vladimirov, Equations of Mathematical Physics, New
York: Marcel Dekker, 1971, ch. 5.
6. Conclusion
Acknowledgment
References
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