On one equation of internal waves

Equation of internal waves in a homogeneous incompressible non-rotating fluid is described by the Poincare equation

^ u# + N 2 ( u + U yy ) = 0,                                 (1)

where N ² is buoyancy frequency. Earlier, the equation was considered in the works of S.A. Gabov and P.A. Krutitsky [1] when working with the excitation of nonstationary internal waves in a two-layer model of a stratified fluid, the behavior of the solution at a large time was studied. Yu.D. Pletner [2] studied some initial-boundary value problems and the fundamental solution to the equation of internal waves in the case of an incompressible exponentially stratified fluid. L.V. Perova and A.G. Sveshnikov [3] generalized the works of the authors devoted to the propagation of small perturbations in stratified and rotating fluids. The equation of internal waves can be written in another form, for example, given in [4]. Interest in the study of problems with the equation of internal waves is still topical. This problem is quite interesting from a mathematical point of view, since the solution to the Cauchy-Dirichlet problem is obtained within the framework of the theory of polynomially bounded operator pencils.

Let D be a bounded domain from R3 with a smooth boundary dD. Consider the Dirichlet condition u ( x, y, z, t) = 0, ( x, y, z, t )e6D x R                                 (2)

and the Cauchy condition

u ( x,y,z ,0 ) = u₀,u _t ( x,y , z ,0 ) = u 1 .                                    (3)

The solution to problem (1)–(3) is found in the framework of the theory of Sobolev-type equations. In this paper, we use methods based on the theory of semigroups (groups) of operators [5, 6]. The monographs [6, 7] present the results of studying Sobolev-type equations and equations that are not solved with respect to the high-order derivative. In the paper [7] different classes of Sobolev-type equations are introduced and, by structure, equation (1) is considered to be a simple Sobolev-type equation.

• 2.    (A, p)-bounded Operators

Let U,F be Banach spaces and the operators A,B0,B1,^,Bn-1 eL(F;U) . Denote by B a pencil formed          by          the          operators          B0, B1,..., Bn-1.          The          sets pA (B ) = {^e C: (^A - ^-1 Bn _, -... - ^Bx - B0)- e L (F; U)} and a A (B) = C \ pA (B) are called an A-resolvent set and an A-spectrum of the pencil B , respectively. The operator-function of a complex variable RA (B) = (Ц A - Ц 1 Bn- -... - цР1 - B0 )  with the domain pA (B) is called an A-resolvent of the pencil B .

Definition 1. [9] The operator pencil B is called polynomially bounded with respect to an operator A (or polynomially A -bounded) if Ba e R + Vц e C(|Ц > a) ^ (R^ (B) e L(F;U)) .

Remark 1. [9] If there exists an operator A ^-1 e L ( F ; U ) ) then the pencil B is polynomially A -bounded.

To decompose the spaces U,F into a direct sum of subspaces, it is necessary to construct projectors. Condition (4) is necessary for the existence of projectors [8].

J ^R A ( B ) = O, k = 0,1,..., n -2,                               (4)

Y where the circuit у = {цe C: | ц| = r > a}.

Lemma 1. [9] Let the operator pencil B be polynomially A -bounded and condition (4) be fulfilled. Then the operators

^P = ¹ J R A ( ^{B ц -1} Ad ц Q = < J ^{ц -1} AR A ( ^B d " П                      Lnt

У                             у are projectors in the spaces U and F respectively.

Denote     U ⁰ = ker P, F ⁰ = ker Q, U ¹ = im P , F ¹ = im Q .    According    to    Lemma    1,

U = U ⁰ ® U 1 , F = F ⁰ Ф F ¹. Denote by A^k ( B l ) a restriction of the operator A ( B l ) on U^k ( F^k ) , k = 0,1; l = 0,1,..., n - 1.

Theorem 1. [9] Let the operator pencil B be polynomially A -bounded and condition (4) be fulfilled. Then

• ( i )    A^k e L ( u^k , F^k ) , k = 0,1;

• ( ii )    B l e L ( u^k , F^k ) , k = 0,1, l = 0,1,..., n - 1;

• ( iii )    the operator ( A ¹ )   e L ( u ¹, F ¹ )

  exists;

  
  

(iv) the operator ( B 0 ) e L ( u ⁰, F ⁰ ) exists.

Using     Theorem     1,     construct     the

operators:

H 0 = ( B 0 ⁰ ) ^-1 A ⁰ e L ( U ⁰ ) ,

Ц _ / D ⁰ A   D⁰    T (T T ⁰ A      IT _ / r>0 A   D⁰      T It J ⁰ A                                                      Onrl

H 1 = ( ^B 0 ) ^B 1 ^{e L} ( ^U ) ,.'., H n - 1 = ( ^B 0 ) ^Bn - 1 ^{e L} ( ^U )                                             ^and

C _ ( J ¹ A   D¹ T (т Г¹ A C _ ( 1 A   D¹ T (т Г¹ A C     _ 1 J¹ A   D¹   _c T (f T ¹ A

^ 0 = ( A )   ^B 0 ^{e L} ( ^U ) , ^ 1 = ( A )   ^B 1 ^{e L} ( ^U ) ,—, S n - 1 = ( A )   B n - 1 ^{e L} ( ^U ) •

Definition 2.    [9] Define the family of operators   { K^v_q, K ^,..., K ^n }   as follows:

• K 0 = O, s ^ n , K у = I, ¹          t-^2                                    nⁿ

^K1 = H 0 , ^K1 = ^{- H} 1 ,..., ^K1 = - H s - 1 ,..., ^K1 = H n - 1 ,

• ¹    j^ ⁿ       J^²   j^¹      nⁿ                 s - ^{1 -1}    nⁿ             s^^s   j^ ^{n -1}    nⁿ

^Kq  ^Kq - 1 H 0 , ^Kq  ^Kq - 1   ^Kq - 1 ^H 1 ^, ^ ^, K q  ^Kq - 1   ^Kq - 1 H s - 1 , ^ , ^Kq  ^Kq - 1   ^Kq - 1 H n - 1 , q  ^1,2,...

The A -resolvent of the pencil B can be represented by the Laurent series

- 1     -^              - 1

q = 0

q

+ 2 Ц ^q ( M ^{- 1} S n - 1 + ... + M S + S 0 ) L^XQ.

q = 1

Definition 3. [9] The point ^ is called

• (i)    a removable singularity of an A -resolvent of the pencil B , if K = O, s = 1,2,..., n ;

• (ii)    a pole of the order p e N of an A -resolvent of the pencil B , if 3 p : K^s _p^ O, s = 1,2,..., n, but K p + 1 = O, s = 1,2,..., n ;

• (iii)    an essential singularity of an A -resolvent of the pencil B , if K qn^ О qJq e N .

• 3.    Abstract Problem

Further, for brevity, a removable singularity of an A -resolvent of the pencil B is called a pole of the order 0. If the operator pencil B is polynomially A -bounded and the point ^ is a pole of the order p e {0} u N of an A -resolvent of the pencil B , then the operator pencil B is called polynomially ( A , p )-bounded.

Consider the Cauchy problem u (0) = u0, ut (0) = u1

for the second-order Sobolev-type equation

Autt = B1ut + B0u.(6)

Operator-functions U _k ^t , k = 0, 1, of the form

U0 = ^ (RM? (B)(mA - B1)e^dM,(7)

2 n i

Y

U =    J RM (B) AeMtdM(8)

Y are propagators. Here the circuit y e C bounds a domain containing the A-spectrum of the operator pencil B . The solution to problem (5), (6) in terms of the theory of degenerate groups was obtained in [8], under the condition that the operator pencil B is polynomially bounded with respect to the operator A.

Theorem 2. [10] Let the operator pencil B be polynomially A-bounded, condition (4) be fulfilled, and ^ be a pole of the order p e {0} u N of the A -resolvent of B. There exists a unique solution u e C” (IR ,U) to problem (5), (6) of the form u (t) = U1 u1 + U0u0 ,                                      (9)

where u _k e im U 1 ⁰ = im U 0 , k = 0,1,im U 1 ⁰ and im U 0 is a subspace in U .

• 4.    Internal Wave Equation

Consider the cases when the domain D is a parallelepiped or a cylinder. Let the domain D = [0, a ] x [0, b ] x [0, c ] c R ³ be the parallelepiped. Problem (1)-(3) can be reduced to Cauchy problem

• (5 ) for equation (6).

Introduce the spaces U = W ^ ^{+ 2} ( D ) , F = W l ( D ) and define the operators in the given spaces

Л2 Л2    Л2

a = ^ + ^ + ^ , B = O, B o =- N ²

d x² d y ²    d z ^{2   1       0}

' d ² d ² ^ _vd x ²    d y ² _v

For      any       l e {0} u N ,      the      operators

A , B 1 ,B ₀ e L ( U , F ) .

Define

- ^kmm , n = — ( n k/a ) ² - ( n m/b ) ² - ( n n/c ) ² to be the eigenvalues of the Dirichlet problem for the Laplace operator. Denote by ^ _k , _m , _n = sin ( n kx/a ) sin ( n my/b ) sin ( n nz/c ) the orthogonal eigenfunctions that correspond to { - 2 ^ , _m , _n }in the sense of the scalar product in L ² ( D ) .

Since {Фк,m,nI c Cда(D), then да _                           _

^P A ^— ^ B 1 ^— B 0 — E ^^{— p} ^ k , m , n ⁺ N ^ k,m J < ^ф к,m , n , * >  ^ф к,m,n к , m, n — 1

where B ₀ ф _kmn — —Л_k m_г ф_{k mn}, and < •,• > is the inner product in L 2 ( D ) . Construct the equation to determine the A -spectrum:

д 2    2, ^2.2 __n  „1,2  _   N4 , m .

—.

^ к , m , n ^p  + N ^ к , m   ⁰, ^р к , m,n    ± II ~ ^{i .}

у ^Лк, m , n

The A -spectrum a A ( В ) — { p ^² } is bounded, because pOmn | <  N . Since the operator A is continuously invertible in the given spaces, then condition (4) is satisfied. As a result, the conditions of Lemma 1 hold.

Construct propagators by formulas (7), (8) as follows:

да

U0u0 — E cos к, m, n—1

N

t < ^фк, m, n , ^U 0 > ^фк, m, n *>

N^1, m U1u1— E / 2 ’ sin к, m, n—1 J Лк, m, n

N^m к, m

L 2

v^/Lк, m, n

t < ^фк,mn,

, m, n•

By Theorem 2, the solution to problem (1)–(3) has the form да

u(x,y,z,t)—Ecos к, m, n—1

, v^   N^-к, m  •

⁺^ TT^{=sin к},^m,^n—1а^к, m, n

NO, m lb 2 \/^/Lк, m, n

t < Фк.

, m, n , ^U 0 > ^фк, m, n +

N^m

, m v/Lк, m, n

, m, n•

Now consider the case when the domain D is a cylinder. Similarly, as in the case of the parallelepiped, problem (1)-(3) can be reduced to Cauchy problem (5) for equation (6). Use the operators A, B1, B0 of the form d2    1 d 1  d d2

+   + dr2   r dr  r2 дф2

•

9r²  r dr  r²дф²  dz²

Write down the equation to determine points of the A-spectrum:

p²((^_кⁿ ))²+ (nml )²) + N²(^кⁿ)) — 0, where В₀ф_к,_m,_n= v⁽ⁿ'⁾ф>_(с,_m,_n and (•,•) is the scalar product in L²(D). We get the A-spectrum of the form

P^m,n^— ±⁽N^In74(^кn^{))2 +}Wl⁾²)i•

Let us construct propagators according to formulas (7), (8) as follows:

^U0^u 0 ^—

Nv^{( n})

да

E cos к, m, n—1

t< ^фк, m, n ,^U 0 > ^фк, m, n ,

да к, m, n—1

--П-----sin

Nvi"¹

NV1 n * (V^Tinm/)⁷

t< ^фк, m, n , ^U1 > ^фк, m, n •

By Theorem 2, the solution to problem (1)–(3) has the form

« u ( x, y, z, t )= к, m, n=1

nV ”)

.             kt

V 4

< ^фк, m, n

u 0 > Фк, m, n⁺

^

+ z к, m, n=1

----------Гл-------sin

N^k”)

■             ^k              t

_V4^^Vk”^{))2 + (nm}ll⁾²>

< ^фк, m, n ’ u1 > ^фк, m, n •

As a result of the work, we obtain solutions to initial-boundary value problem (2)–(3) in a closed form on the considered domains for equation of internal waves (1).