Making use of all these limiting processes and taking the limit in the integral identity (7) with respect to the selected subsequence gV~0 we obtain that the limit function ~(x,~) will satisfy the integral identity (3), i.e., it will be one of the strong solutions of the problem (i), (2). Finally, from inequality (8), in which the constant G~ does not depend on >0 , by a limiting process as ~[~ ~0 we obtain for ~• the solution of the problem (i),

(2) --the inequality (4). Theorem i is completely proved.

LITERATURE CITED

i. A. P. Oskolkov, "On the uniqueness and the solvability in the large of boundary-value problems for the equations of motion of aqueous solutions of polymers," J. Soy. Math., 8, No. 4 (1977).

2. A. P. Oskolkov, "On certain model nonstationary systems in the theory of non-Newtonian fluids," Tr. Mat. Inst. Akad. Nauk SSSR, 127, 32-57 (1975).

3. A. P. OskOlkov, "On the theory of nonstationary flows of nonlinear viscoelastic fluids," J. Soy. Math., 34, No. 5 (1986).

4. A. P. Oskolkov, "On nonstationary flows of viscoelastic fluids," Trudy Mat. Inst. Akad. Nauk SSSR, 159, 103-131 (1983).

5. A. P. Oskolkov, "Functionalmethods in the theory of nonstationary flows of linear visco- elastic fluids," Preprint LOMI R-2-83, Leningrad (1983).

6. A. P. Oskolkov, "Initial-boundary-value problems for the equations of motion of visco- elastic fluids," Author's Abstract of Candidate's Doctoral Dissertation, Leningrad (1983).

7. O. A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach (1969).

8. O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Amer. Math. Soc. (1977).

SPECTRAL PROPERTIES OF A LINEARIZED MAGNETOHYDRODYNAMICS MODEL

G. D. Raikov UDC 517,43:533.95

One considers the linear oscillations of a perfectly conducting nonviscous plasma in an external rectilinear magnetic field. One studies the frequency spectrum cor- responding to a fixed harmonic mode. One has investigated the asymptotic behavior of the discrete spectrum at infinity and around the boundary of the essential spec- trum.

i. One considers a magnetohydrodynamic model of the linear oscillations of a perfectly conducting nonviscous plasma of nonzero pressure p [i]. The plasma fills the cylinder

~=~ ~ [0,%~] , where ~C R ~ is a bounded domain. The boundary ~ of the domain ~consists of a finite number of segments of class G ~ intersecting under nonzero angles. We shall assume that the angular points of ~ are points of convexity. The displacement vector 7 of the plasma satisfies the equation

Here cO is the frequency of the oscillations. The magnetic f ield-~is directed along the

axis of the cylinder and is described by one function 5=5(x,~)~0, (X,9~ ~_~ . On P*[0,%r

the normal component of~ is equal to zero and on the bases of the cylinder ~ satisfies the

periodicity conditions. By virtue of the equilibrium equation we have ~p=~- ~z. Fur- ther, ~ =~(X,~) is the density and ~>~ is the ratio of the heat capacities, A similar

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 120-123, 1985.

866 0090-4104/87/3701-0866512.50 �9 1987 Plenum Publishing Corporation

model in the case when ~-~ is a parallelepiped while $ and ~ depend only on N has been con- sidered in [2], Sec. 5, where one has investigated the essential spectrum.

Everywhere,~4we shall assume that for some ~>~ the functions ~ and~ belong to the

Sobolev WPai: ro~dW~' (c~ and, consequently, are continuous in ~ . In addition, ~>0 , p >0 i n . t u . A-- quantities V %-~-< ~ , I/~-~f~p iV A is the alveolar and V S is

the sound velocity) and V~'=. V ~" + V~, V~=V"zV~ VA"

We denote by [~ e$] and [~.~e+] the closed intervals covering the values of the func-

tions V~ and VA on ~. Then

0 ~ @< < @_ (1)

for B~ 0 on~ . If, however, ~ vanishes in ~ , then ~:~_ : 0.

We note that in the case (i) we may have ~ < ~_.

Expanding ~ in Fourier series, we obtain for the coefficients ~=~(~) , ~ , the equa-

tion ~(~0~--oo~ in ~, where

A +re m , / ]~x=- L~I~ x ,D~=- b~ ID~ ~ and the boundary condition ~ @4+ ~%9~IV = 0 holds, where ~F =

(~4 ~r is unit vector of exterior normal to ~'. Under this condition, ~(~) is formally selfadjoint in the Hilbert space L~(~) . To it there corresponds the quadratic form

which is considered first on ~0~-~%(~)'~j~WL(~)~ ~=I,~ ~+~%1~ =O}and then one

closes it in ~(~@) . To the closed form (2) there corresponds a selfadjoint operato r

A (4~) in L~(~)~) . Results regarding its spectral properties constitute the content of _his paper. By E(A )A) we denote the spectral projection of A ~ ) , corresponding to the

set z~ C ~ ~( A#)~ A): ~b~E(A(~),~)~'~,~,~). By ~(A (~)) we shall denote the spectrum of

A (&) , while ~e(A (~}) and N(A(~)) are its essential and discrete spectra.

2. THEOREM i. For any $ ~ to the left of the point $~e~ the spectrum of the oper- ator A (~) is empty.

THEOREM 2. The essential spectrum of the operator A (/~) is not empty:

THEOREM 3. For every ~o>~+ for ~--~o we have the asymptotics

the closure of the quotient set ~(~)/~e~$+ in the metric defined by the form S i . We

shall say that V~ assumes tamely the value @• if the form

is compact i n ~ ! .

We give the simplest sufficient conditions for the compactness of the form p+ in ~• We denote by A+ the subset of ~ , on which V A takes the value e~. We assume that one of the following two conditions holds:

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a) A+ is a one-point set {~o} 'Z0~ and near Zo one has the estimates

e+-v, czl•

b) A+ is a simple arc of class C 4 , lying in ~ . Let ~(~) be the distance from the point Z ~ to A+ and let

>

Then the form p+ is compact in ~+ . Similar conditions ensure the compactness of p_ in~_.

Other variants of the sufficient conditions are also possible. By virtue of the local char- acter of the question, these sufficient conditions can be combined.

The following two theorems exclude the accumulation of the eigenvalues of the operator

A (~) to the points ~e+ and . ' ~ e - .

THEOREM 4. For some ~0> 0 we have

(A {~ r3 Co, Xo) =

If V A assumes tamely the value e,: , then for some ~0=~o(~)>e+we have

THEOR~ 5. Let 0-r and e~<e_. I f V~ assumes tamely the value e_ then for some ~n=~o(~.)~Ce~,e_ 3 we have

(A c~) n ( ~ o , , M e _ ) = ~, It remains to consider the discrete spectrum to the right of the point ~ (we assume

~ 0 and @~<e_). Here, in general, the spectrum aacumulates to the right of the point

~ This is so, for example, if V~EG~(~) and attains the value e~ at least at one

interior point in ~'~ . In particular, the last condition is sufficient for the existence of eigenvalues inside the gap (~e~e_). Under certain additional assumptions one can write out the first term of the asymptotics of the corresponding distribution function of the spectrum. If the maximum is attained at an isolated interior point and it is not degenerate, then this asymptotics is logarithmic. In the case of degeneracy (for example, if V# at-

tains the value ~ on a smooth simple contour inside ~-~ ), then the asymptotics has a power- like character.

The author expresses his deep gratitude to his scientific adviser Professor M. Sh. Birman for his help and for numerous discussions and to A. E. Lifshits for consultations and for his interest in this paper.

LITERATURE CITED

i. C. Mercier and H. Luc, "The MHD approach to the problem of plasma confinement in closed magnetic configurations," in: EUR 5127e, Commission of the European Communities, Luxem- bourg (1974), pp. 5-157.

2. A. L. Krylov, A. E. Lifshits, and E. N. Fedorov, "On the resonance properties of magneto- spheres," Fiz. Zemli, No. 6, 49-58 (1981).

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