Weighted Energy Methods in Fluid Dynamics and Elasticity (Lecture Notes in Mathematics, 1134) 3540156453, 9783540156451

Citation preview

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1134 Giovanni R Galdi Salvatore Rionero

Weighted Energy Methods in Fluid Dynamics and Elasticity

Springer-Verlag Berlin Heidelberg New York Tokyo

Authors

Giovanni P. Galdi Salvatore Rionero Dipartimento di Matematica "R. Caccioppoli" Mezzocannone 8 80134 Naples, Italy

Mathematics Subject Classification: 73C 10, 73C 15, 76E25, 76E30 ISBN 3-540-15645-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15645-3 Springer-Verlag New York Heidelberg BerlinTokyo

Library of Congress Cataloging-in-Publication Data. Galdi, Giovanni P. (Giovanni Paolo), 1967Weighted energy methods in fluid dynamics and elasticity. (Lecture notes in mathematics; 1134) Bibliography: p. Includes index. 1. Fluid dynamics. 2. Elasticity. 3. Differential equations- Numerical solutions. I. Rionero, Salvatore. II. Title. III. Series: Lecture notes in mathematics (Springer-Verlag); 1134. QA3.L28 no. 1134510 s 85-12662 [QA911] [532'.05] ISBN 0-387-15645-3 (U. S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translating, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

PREFACE As is known, by energy methods (EM) one denotes a suitable mathematical device for deriving estimates of solutions to differential equations.

The name of the

method is due to the fact that it is usually founded upon "conservation laws" which must be obeyed by solutions.

A typical example of EM, and maybe one of the earliest,

is given by the approach introduced at the end of the nineteenth century by the Russian mathematician A.M. Liapounov for studying the stability of solutions to ordinary differential equations.

Actually, this approach is based on the existence

of a suitable functional which must be positive and non-increasing in time (the socalled "Liapounov functional").

Another no

less important example of EM is furnish-

ed by the method proposed by J. Serrin in 1959 and successively generalized and deepened by D.D. Joseph and his co-workers, for studying nonlinear stability of viscous incompressible flows in bounded domains.

Roughly speaking, the method con-

sists in forming the "kinetic energy" of perturbations to a given basic flow and in studying its behavior in time (see Chapter I).

Of course, there are several other

examples of applicability of EM, such as uniqueness, existence, etc., and for some of them the reader is referred to the papers cited throughout these Notes. Aside from the above EM, there are the so-called weighted energy methods (WEM). Unlike the former, the latter explicitly involve the use of suitable auxiliary functions (weight functions) whose task may vary from case to case, depending on the kind of problem one is dealing with.

In fluid dynamics, they were originally intro-

duced by the writers in 1975 for studying uniqueness of unsteady viscous flows in unbounded regions.

By using suitable spatially weighted norms, they were able to

prove uniqueness without prescribing the velocity field at large spatial distances. Since then, WEM have been widely applied to other branches of mathematical physics (magnetohydrodynamics, elasticity, viscoelasticity, etc.) and used to solve different kinds of questions such as existence, continuous data dependence and stability as well.

M::>reover, they turn out to be useful also in investigating well-posedness

problemS-in bounded domains.

IV

The aim of these Notes is to give an account of WEM and to show how they work in solving several problems arising in fluid dynamics (Part One) and elasticity (Part Two).

Of course, we did not consider all possible applications of the method

and, among others, we left out important questions such as uniqueness and continuous data dependence of viscuous incompressible flows in unbounded domains; however, we refer the interested reader to the recent M::mograph of B. Straughan (Ref. 5 to Chapter II).

Acknowledgments These notes mostly summarize the research developed by the authors in the last years with the support of GNFM of Italian CNR and a contract MP1 40%.

Both

Firms are, therefore, gratefully acknowledged. These Notes are also, in part, the content of a series of lectures which the first author gave at the ''Workshop on Exterior Domain Problems" at Howard Universi ty , Washington, D.C., June 20­24, 1983.

He wishes to thank the Director of the Workshop,

Professor Isom H. Herron, for his kind invitation and his unselfish help, and the Chairman of the Department of Mathematics, Professor J .A. Donaldson, for providing him with a Howard University Visiting Scientist appointment during . the tenure of which these Notes were completed.

He also thanks all his colleagues and friends at

Howard for their kind and warm hospitality. Last but not least, the authors are grateful to Mrs. 11. McCalop for her superb job of typing.

TABLE OF OJNl'ENTS

PREFACE • • • . . • . . . • • • • •

iii PART ONE

WEIGlITED ENERGY ME1HODS IN FLUID DYNAMICS Chapter I.

INTRODUCIDRY TOPICS ON STABILITY OF VISOJUS

1.1

Statemerrt of the Stability Problen

1.2 The Linear Stability 1.3 Energy

. . .

for Nonlinear Stability in Bounded Dcxnains

References to Chapter I II.

ENERGY MITHODS IN UNBOUNDED OOMAINS:

3 7

. . . . . . . . . . . . .

13

THE CASE OF A HALF SPACE

14

Preliminary Considerations .

14

2.2 Weighted Poincare Inequality

15

2.3

17

2.1

The Energy Relations in a Half Space

2.4 Universal Stability

20

2.5 Asymptotic Stability

24

2.6 Variational Formulation

29 .

32

References to Q1apter II

35

2.7 Applications

III.

FL(l.JS

ENERGY METOODS IN UNBOUNDED OOMAINS:

THE CASE OF AN EXTERIOR OOMA.IN.

36

3.1 Weighted Poincare Inequalities

36

3.2 The Energy Reiations

39

3.3

40

Universal Stability

3.4 Asymptotic Stability

41

3.5 Variational

45

3.6 On the Applicability of the Stability Criteria References to Chapter' III

. 48 . 48

VI

Chapter

N.

SCME PROPERTIES OF STEADY SOLUTIONS IN EXTERIOR IXMAINS

49

4.1

. . . . . . . . .

49

4.2 D-solutions and Their Properties

49

4.3

Preliminaries

Further Properties of D-solutions for Arbitrary Reynolds Nunbers

.

References to Chapter N . . . . . . . . . . . . . . . .

V.

54

EXCI'WG: OF STABILITIES, SYM1ElRY AND 'IRE CONNECTION BEIWEEN LINEAR AND IDNLINEAR STABILITY . . . . . • .

55

5.1

55

Symretry, Linear and Nonlinear Stability . . . .

59

5.3 Nonsymretric Operators and Stability

63

5.4 Stability of Couette Flow Between Coaxial Cylinders Within the Radius Approximation in the Narrow Gap Limit

68

5.2 Applications of Theorem 5.3

5.5

Energy Stability in Bio-convection . . . . .

5.6 N:m.linear Stability of a Diffusion Equation References to Chapter V VI.

51

. . . . . . . . . .

71 73 77

SKEW-SYM1ETRIC OPERA.10RS AND IDNLINEAR STABILIZATION: 'mE HYDROMAGNETIC BENARD PROBLEM . . . . . . .

78

6.1

Skew-symretric Operators and Stabilization

78

6.2

Statement of the Problem . . . . . . . . .

79

6.3 Generalized Energy Theory for the Linearized Problem

81

6.4 Stability of the N:m.linear Problem

87 91

References to Chapter VI PART 'lID

WEIGHTED ENERGY ME1HODS IN ELASTICITY Chapter VII.

WEIGHTED ENERGY ME'IIDDS IN ELAS1DSTATICS ON UNBOUNDED lXMUNS

92

7.1 Uniqueness in Linear Elastostatics:

92

7.2

Statanent of the Problem

Uniqueness and Continuous Dependence in Linear Elastostatics

95

7.3 A Saint-Venant Principle for the Semi-infinite Elastic Cylinder: Preliminaries. . . . . . . . . . . . . . . . . .

98

VII

Chapter 7.4.

VIII.

Asymptotic Behavior in the Nonlinear Elastic Beam .

102

References to Chapter VII . . . . . . . . . . .

107

WEIGl:ITED ENERGY METHODS IN LINEAR ELASTODYNAMICS IN

UNBOUNDED DCMAINS • . • • • • • • • • • • • • • • • .

108

8.1.

Uniqueness and Continuous Dependence in Linear Electrodynamics with Definiteness Assurptions on the Elasticities . . . . . . 108

8.2.

Continuous Data Dependence without Definiteness Conditions on the Elasticities . . . .

112

References to Chapter VIII

1 19

. . . . . . . . . . . . . . .,

PART ONE:

WEICHI'ED ENERGY METIIOD IN FLUID DYNAMICS

GlAPTER I

INTRODUCTORY TOPICS ON STABILITY OF VISCOUS FLOWS

1.1.

Statement of the Stability Problem. Assume we want to study in the time interval [O,T) (T

hOIlK)geneous and viscous fluid dimensional space IR

3•

F occurring in a fixed region

= 0)

=

(x,y,z)

and on time t

to

and boundary data (on

the generic motion of

(0,00]) the mot.Ion of a

of the physical three-

We shall suppose that the flow is driven by a body force

depending on the space variable ;lS initial (at t

to

to

£

[O,T), and by

As a mathematical model describing

F we shall adopt the Navier­Stokes equations:

(1.1.1) \l

Here y.

=

•

Y.

=

o.

represents the velocity field, p

the pressure field (divided

by the constant density), and v (> 0) is the coefficient of kinematic viscosity.

To

(1.1.1) we must, append the initial and boundary conditions which we shall write in

the form (1.1. 2) (x,t) where

to

x

[O,T)

(divergence free) and Y*(fS,t) are ascribed vector functions.

As far as

(1.1.2)2 is concerned, it means that the fluid adheres completely to the boundary 3Q and it is certainly verified in most applications whenever

is rigid.

Within the

totality of motions performable by F a particular interest is held by the steady motions.

They are defined by the condition that the velocity field

sure p do not depend on time explicitly, namely,

=

and p

and the pres-

= pUs).

As a con-

sequence of (1.1.1) we then have that a steady motion obeys the following equations:

2

(1.1. 3) =

'l

0

to which we append the boundary condition (1.1.4) Of course, a necessary condition in order to realize a steady motion is that the body force

£ does

not depend explicitly on time.

Now, let m =

be a given motion of F (i.e., a solution to (1.1.1)-(1.1.2)

or (1.1.3)-(1.1.4)) corresponding to given data and body force. basic ,flow.

Assume that at a given instant (t

amount YO = YO(x).

=

We shall call m a

0, say) Y, is varied by a certain

The fluid F will then perform a new motion m' = (y, + Y(;lS,t),

P + 'IT(;lS,t)) corresponding to the same boundary data as m, to the same force the initial condition Y,(O,;lS) + YO'

E and to

Since all (unsteady) motions of F must obey

(1.1.1), we deduce, by subtraction, that the perturbation (y,lT) satisfies the following initial-boundary value problem:

au

at 'l

+ u • 'lu + V • 'lu + u • 'lv

•

Y = 0

(1.1. 5)

The stability problem of the basic flow m is reduced to studying the behavior in time of the perturbation ()J,lT) or, equivalently, to studying problem (1.1.5),

If

it happens that for)JO "small" )J(;1S,t) is "small" for all time t, then m is stable; on the other hand, if M(;1S, t) "may grow" irrespective of the "smallness" of MO' then m is unstable.

We shall now formulate these concepts in a mathematically rigorous

way [1]. DEFINITION 1.1. The motion m is said to be (energy) stable if and only i f 'ir£ > 0,3.0(£) > 0:

3

Conversely, m is said to be tmStable i f and only i f it is not stable. DEFINITION 1. 2. A motion m is said to be (energy) asymptotically stable if and only if it is stable and, moreover, 3 If y

= 00,

y

c (0,00]:

< y -

lim t--

0.

m is unconditionally asymptotically stable.

We speak of "energy" stability because according

to the above definitions we

have that i f m is stable then the kinetic energy difference between m and m' remains "small" for all time t, provided it is initially "small".

In this respect, we re-

call that for systems having infinitely many degrees of freedom (continuous systems), the concept

of stability is intimately related to the metric we choose to "measure"

the perturbation.

Actually, it can happen that the same motion is stable with re-

spect to a given metric and unstable with respect to another one (cf. [2]).

Later

in these Notes we will have the opportunity to show how the stability conditions on a basic flow may depend on choosing the metric appropriately (see Chapters V, VI). Another question which may be raised when we consider the above definitions is that they are silent about the perturbation to the pressure field.

In fact, what can be

proved is that when one employs energy methods for studying stability (see Section 1.3) the conditions ensuring asymptotic stability imply also that J>2lvrrj2d>2 must tend to zero as t tends to infinity [3]. 1.2.

The Linear Stability Method. We have seen in Section 1.1 that the rigorous study of the stability of m is

reduced to investigating the behavior in time of solutions to (1.1.5).

However,

for quite a long time this research presented itself as a very difficult one and only qualitative results were obtained in this sense [4,5,6,7].

Therefore, in the

wake of the early pioneering work of Rayleigh and Reynolds [4,8] one was led to face the problem in a mathematically approximate way by using the so-called linearized stability method.

The aim of this section is to sketch briefly the main points of

4

this method, referring the reader for a much more complete understanding of it to the monographs [1,9,10,11]. Let mO =

(;IS)) be a steady motion of F and assume that for all ;l£,t the

perturbation (g, 11) to m is so "small" that we can neglect in (1.1. 5) 1 the nonlinear O term g

Ilg.

We thus obtain the following system of linearized equations:

V'JJ=O

(1. 2.1)

(;lS,t)

E

an

x

[O,T).

This system is linear and autonomous and therefore we may look for solutions of the following form: (1. 2.2) where

G

is a priori a complex parameter.

Substituting (1.2.2) back into (1.2.1),

we obtain:

(1. 2.3)

Mathematically, (1.2.3) is an eigenvalue problem in which and eigenvalues, respectively.

and

G

are eigenfunctions

Since the behavior in time of perturbations (1.2.2)

is governed by the sign of the real part of 0 (re (0) ), the stability problem for the linearized system (1.2.1) is reduced to studying system (1.2.3). if re(o)

a for all

G

rna is linearly stable (or stable according

we shall say that

to the linear theory); if re(o)


0, while s


Pc'

Though

this seems reasonable, it is not always true and the dependence of s on P must in principle be ascertained from case to case [11]. culating Pc is very simple.

Sometimes, the procedure for cal-

Actually, let us rewrite system (1.2.3) as (1.2.5)

where L is a (suitably defined) linear operator which depends on P and, in the case of periodic perturbations, on Cl

i.

We thus have, in general, L

=

L (P,Cl

i).

Now,

suppose exchange of stabilities holds, namely, (1.2.6) or, in other words, the first eigenvalue 01 is real at criticality (s

0).

In this

circumstance (1.2.5) thus furnishes (1.2.7) This equation is now ooderstood for each Solving this problem gives P

= P(Cli)

Cl. as an eigenvalue problem in P. 1

and, therefore, we have

6

We wish to spend a few words concerning condition (1. 2.6).

From the physical

point of view, this means that instability sets in as a steady secondary mot.ion, From the mathematical .point of view, we should notice that there are many instances where (1.2.6) can be proved and this certainly happens when the operator L is symmetric or syrnmetrizable (see Chapter V).

However, there are also other cases, such

as Couette flow between coaxial rotating cylinders, where though in some range of parameters the observed secondary motion is steady (Taylor vortices) the proof of (1.2.6) has not yet completely been achieved.

If (1.2.6) is violated, namely, if

at the onset of instability an oscillatory motion prevails, one says that one has a case of overstability. We end this section with two further mathematical considerations. we have seen that i f s zero as t -+"'.

>

0 then all perturbations of the type (1.2.2) must tend to

One then may wonder what happens to other solutions to (1. 2.1) which,

possibly, are not of the form (1. 2.2).

In this respect, we notice that it can be

proved [IS] that when n is bounded, the requirement s lowing statement:

for all t

>

First of all,

>

0 is equivalent to the fol-

there exist positive constants A, B such that

0 and for all solutions g to problem (1.2.1).

A second consideration concerns the connection between linear stability, i.e., s > 0, and nonlinear stability in the sense of Definitions 1.1 and 1. 2, i. e , , the behavior in time of solutions to the non-linear problem (1.1. 5) .

In this respect,

we quote the so-called principle of linearized stability, whose proof can be found under several different. assumptions in [12,13,14], which states that i f s > 0 then there exist constants A'B', and

y

such that

ve

0

(1.2.8)

whenever

Estimate (1.2.8) is satisfied along solutions to the nonlinear problem (1.1.5). Conversely, if s


0 for all values of the Reynolds number R and hence we should have nonlinear stability for all R, according to the above principle [20].

However, the

experimental observations show that plane Couette flow becomes unst.abl.e when R = Rc' where R is several thousand. c

The "paradox" can be explained if we conjecture that

the dependence of y on s is such that for R sufficiently close to R y is, in pracc tice, zero.

We thus mderstand how Impor-tant; it is to know explicitly the dependence

of y on the pararreters associated with the basic flow. noticed.

Another thing should be

There are several cases where the linear stability results are given under

the assunption of the validity of exchange of stabilities (1.2.6).

Though many of

these results seem to capture the essential physics of the problem (e.g., Couette flow between coaxial cylinders, stabilizing effect of magnetic field on convection, etc.), they cannot be considered, in the strict sense, as nonlinear stability results via the linearization principle.

In all the above cases it thus becomes necessary to

have a different nonlinear .analysis for the ultimate justification of the stability criteria of the linear theory (see Chapters V, VI).

We may thus state that the roost

:iIIportant value of the principle of linearized stability consists in that it ensures nonlinear instability whenever there is linear instability (s < 0). 1.3.

Energy Methods for Nonlinear Stability in Bomded Danains.

In the previous section we have seen that the linear theory reduces the stability question to that of studying a suitable eigenvalue problem (see (1.2.3». ftmdarraltal paper in 1959 [16J J. Serrin proved, among other things, that also

In a

8

nonlinear stability can be reduced to studying an eigenvalue problem that is tightly connected with (1.2.3).

The method introduced by Serrin, which holds for

Q

bounded

in at least one direction, is the so-called energy method and since then it has been further developed and generalized to include other physical phenomena by several authors.

Among others, we may wish to quote the works of D.D. Joseph, S.H. Davis

and S. Rionero (cf. [1]).

The aim of this section is just to present the method and,

in doing this, we shall follow the approach suggested more recently in [15,17]. begin to notice that dotting (1.1. 5) 1 by jd, integrating by parts over

Q

We

and taking

into account (1.1.5)2 , 4 we obtain the so-called energy equality (1. 3.1) where Q

Sym(VX)

is the deformation rate tensor associated with the basic flow m,

Assume first m steady (the unsteady case will be treated later) and set

(jd,X)

(1.3.2)

Denote further by S the class of admissible perturbations, i.e., of solenoidal vector functions vanishing on aQ.

Thanks to a theorem of S. Rionero [18], we have that in

S there exists

A

min G(jd,y). UfOS

Therefore, from (1.3.1) and (1.3.2) we deduce

and hence the following. 1HEOREM 1.1.

!i. the

minimum A of the functional G is positive, the basic flow m

tionally asymptotically stable and it holds

along solutions to the nonlinear problem (1.1. 5) .

uncondi-

9

Let us now draw some consequences of Theorem 1.1.

If the minimum exists we may

form the Euler-Lagrange equations associated with the variational problem (1.3.3) to obtain that A is the minimum eigenvalue of the following problem:

(1. 3. 4)

As far as (1.3.4) is concerned, analogously to problem (1.Z.3), it can be shown that i f problem (1. 3.4) is set up in a suitable funct ion space, there exists a denumer-

able number of real eigenvalues {)1n}ndN which can be ordered as follows: )11 .;;; )1Z .;;; ... .;;; )1n .;;; ... We thus deduce A =

and, as in the linear theory, that A depends on the basic flow 1 through a dimensionless parameter and on the wave numbers of perturbations, in case \1

they are periodic in suitable directions.

Now, it arises in quite a natural way the

problem of investigating the connection between the eigenvalue problems (1. 2. 3) and (1. 3.4).

To this end let us introduce the following "operators":

Lsi

= -V6z!

+

D (1. 3. 5)

where .2 is the "vorticity" tensor associated with m.

It can be easily checked that

L and LA are symmetric and skew-symmetric operators, respectively, with respect to 5 the L2(Q)-scalar product (notice that both problems (1.2.3) and (1.3.4) are set in

2(Q)). funct ion spaces which are certainly contained in L

In fact, we have

E.:

S.

10

Comparing (1.2.3) and (1.3.4) in the light of (1.3.5) we thus obtain a first interesting result, namely, that while linear stability is reduced to the study of the sign of the first eigenvalue of the whole of L, the energy method reduces nonlinear stability to the same study but for the symmetric part of

Is.

As a consequence,

whenever LA :: 0 the two problems are the same and we have the coincidence of linear and nonlinear stability criteria.

We shall return to this result in Chapter V in a

much more general context. The comparison of eigenvalue problems (1. 2.3) and (1. 3. 4) allows us to deduce another interesting fact, which, on physical grounds, should be expected.

Precisely,

we have (1.3.6)

A > 0 -ec- s > 0,

i.e., the nonlinear stability of m according to the energy method implies that m is also linearly stable.

A proof of (1. 3.6) can be found in [15].

Another consequence which follows from Theorem 1.1 is that it allows the formulation of universal stability criteria, i.e., stability criteria which hold whatever is the "form" of the basic flow and whatever is the "geometry" of the domain

Q,

pro-

vided only that it is bounded. To state the above criteria, we need to recall a Precisely, setting d = diamQ, the

result due to L. Payne and H. Weinberger [19]. following Poincare inequality holds for all

c S (1. 3. 7)

where a

80.

denoting by

Inequality (1.3.7) allows us to give an estimate of A.

Actually,

an eigenfunction corresponding to A, from (1.3.3), (1.3.7) we have

o•

tRI.

with K = max Q

0

0

J2 • jd dQ + v fn'iJjd

0

Therefore, indicating by R

0 2

dQ)/fQ(jd ) dQ;;;' -K +

ali

2 d

(1. 3. 8)

2 Kd / 1i a Reynolds number associated

with m, from (1. 3. 8) we deduce that R




0

a

C(T)

= sup

[sup

>

0

Ivyi (1

t)

II z=O

+ Z/L)2)
0) and the assuaptaon made

inJd • vJd

.(y - Yoo)dQ

vZ

2

2V inu (1

z -2

+ r)

v

dQ + "2

*Notice that, as can be easily shown, the numbers particular choice ofL.

IS.

0I1 V 'We

also obtain (2.4.5)

and

Rz

are independent of the

22

Substituting (2.4.5) into (E)II and using inequality (2.2.1) thus yield 2L2 2 2 4V t fnu (;lS,t)dQ:;;; fQu (;lS,tO)dn + [-v- - v] f t

(2.4.p)

o

which proves energy stability. REMARK 2.2.

The stability criteria (2.4.1) and (2.4.3) are analogous to those given in (1.3.9), (1.3.10) in the case of a bounded domain. REMARK 2.3.

The method described above presents itself as the natural generalization of the energy method to the case when

is a half space.

Q

However, one may ask if, according

to the methodology ejnployed, the energy approach can be further improved to include, for example, a wider class of unperturbed motion or to increase the numbers 1/4 and 1/2 representing the "critical Reynolds numbers" for the validity of the Universal Stability Criteria.

The answer to these questions is, in general, negative.

In fact,

on the one hand, the classes C and C cannot be enlarged to allow for Q or y to be 2 l an order of spatial decay weaker than (1 + -2 and (1 + -1, respectively. This is

r)

r)

a consequence of the invalidity of the weighted Poincare inequality, as shown in Remark 2.1.

On the other hand, at least in the class of perturbations which are

periodic, the stability criteria (2.4.1) and (2.4.4) cannot be improved.

This

follows from the fact that the number 4 in inequality (2.2.1) cannot be taken smaller, as we are going to show.

Let

n

=

1,2, ... , be a sequence of functions of the

following type:

w

"11

=

(w (z),O,O), n

where w will be specified below. n

.

2 f Q wn (l

4 - Lim j n

+ z)

I7w : I7w

Q "11

dQ

"11

"TI

Notice that V

prove that it is possible to choose

_

w (0)

=0 = 0

for any n.

We want to

such that

4

2

. f 0 wn (1 + z) - Lim dW n (af) 2dz

dQ _

00

To this end, let us make the change of variable y

=

4

1 (1 + z).

dz

(2.4.7)

We thus have

23

a Wn2 (2) (1 + 2) -2 dz J'"

(2.4.8)

(k'i

Joo(-1l)2dz adz w

n

1

- 1). Moreover, the boundary condition on lh(l) =

a.

becomes (2.4.9)

Fran Lemna. 2.1, one has 1 2 J O un(y)dy du 1 J Y dy

°

,;;; 4,

\Tn.

(2.4.10)

Y

Further, since Y E (0,1), from (2.4.10) we also have

(2.4.11)

Now, choose u (y) n

=

y-l/2 J (a yl/2n) n O,n

where I n (I;) is the Bessel function of the first kind of order n and aO,n is the first (nonzero) zero of J. It is obvious that u satisfies the following differenn n tial equation: 2 2 dUn 2 dUn aO,n 1/n-2 (2.4.12) -:-z-+--+-:-z-Y u 0. dy Y dy 4n n Multiplying both sides of (2.4.12) by y2Un, integrating by parts over [O,lJ and employing (2.4.9), we deduce yl/n l 2(dun)2 d f OYa:yy

(2.4.13)

From (2.4.11) and (2.4.13) it follows that

(2.4.14)

24

On the other hand, it is well known (see, e.g., [6, p. 516]) that 2

--z-- =

· n 1 lID

n

(2.4.15)

l.

aO,n

Therefore, (2.4.7) follows from (2.4.8), (2.4.14) and (2.4.15). 2.5.

Asymptotic Stability. The lack of validity of the Poincare inequality (1.3.7) in unbounded domains

does not allow us in this case to deduce the decay of the perturbation energy from (2.4.3) or (2.4.6) under the assumptions (2.4.1) or (2.4.4), respectively.

There-

fore, the question of whether the above energy decays in time to zero remains open. However, when D is an exterior domain, we are able to answer this question affirmatively, provided the initial data are chosen suitably

next chapter).

As far

as the half space is concerned, we do not know, at the moment, if the energy decays ul timately to zero, but we can prove nonetheless that the perturbations must fall off as t tends to infinity in a suitable sense. Precisely, we have the following result. THEOREM 2.2. Let (y,p) verify the assumptions of at least one

[v] ,.; c, I (ayat) I ,.; C(l

+

z) -1 uniformly in t.

USC and, further, providedjd(x, to) is suffi-

ciently smooth and "small", one has lim

lim

(2.5.1)

0

and, if (v,p) is steady,

I""

Ia

1

2 dn ,.; (1 C + t) ,

(2.5.2)

where C depends on the initial data and on the basic

M::lreover, for

(in case (2.5.1)) and

In(z)

u

2

C(Z) ,.; (1 + t)1/2

(in case (2.5.2))

z>

0,

(2.5.3)

25 where

PROOF.

The proof of this theorem is quite long and can be found in [1].

Here,

we shall restrict ourselves to sketch a formal proof in the case when (:!:,p) is steady and satisfies the assumption of USC I.

-

We shall use the following notation:

at UQ

II)JII

UQ

and assume , for simplicity, to = O. =

p

>

1

1/2

From (£)1 and (2.4.1) we have (with

to)) (2.5.5)

Differentiate (1.1.5)1 with respect to t , multiply by

and integrate over Q to

obtain (2.5.6) MJreover, multiply (1.1. 5) 1 by 2 l u t I2

= -

and integrate over Q to have

(u • 'lu U ) - (v •

U ) ""=t

Let us now estimate the right-hand sides of (2.5.6) and (2.5.7).

2

•

(2.5.7)

In doing this we

use here and there the Holder inequality, inequality (2.2.1) and the following one

(cL [7]):

(2.5.8) We have (we assume , for simplicity, m = 1)

26

(2.5.9)

(!!t Moreover, for all n

>

0, it is Vl 0).

Universal Stability. As in the case of the half space we can now give tmiversal. stability criteria

also for an exterior da:nain.

= L, ... ,n).

rli(i

In the case rl

CZ(rl) where

We choose as a length scale the min:imun diameter of

=

{(;t,p):

is a constant vector.

= IR 3 ,

L will be any canparison length.

V:= sup

sup rl

(r/L)

t) -


0 for which

1 d 2 2 2" dt II u] + k*lIull

ex

cIIull

This differential inequality may be rewritten as (5.3.11) where 0, Y > 0 and n > 1. l z(t) Let w(t)

Fran (5.3.11) we obtain at once

z(O) exp(-ot)

z(t) exp( ot).

+ Yl exp (e St )

f

t Os n e z ds. O

The last inequality :iIIplies that w(t)

y(t) , where y(t)

is the solution to the fo11CMing initial value problem: (5.3.12) y(O)

=

z(O)

Integrating (5.3.12) from 0 to t we deduce w(t)

n-l

z(O) / [1 -

1

(1 _ eo(l-n)t)J l-n

Therefore, provided

1 lu O"

=

z(O)
0).

(5.6.7)

75 Hence (5.6.6), (5.6.7) imply that (5.6.8) From (5.6.5\, (5.6.8) one deduces that i f A < 1 the solution (5.6.1) is nonlinearly asymptotically stable.

Actually, by an easy calculation one shows that i f

then

E(t) (see (15) for details).

E(O) exp(A - l)t The next step is to check when the condition A < 1 is

To this end we write the Euler-Lagrange equations associated with the

satisfied.

maximun problem for A and al.Low periodicity in the x-direction.

-¢"

where u = 2

I

+

(U¢)'

+ a 2¢

2

= u Csech y)¢,

¢

E

We thus find (5.6.9)

H'

denotes differentiation with respect to y, a is a wave nunber, and

(f -

1).

We wish to prove that u > 0 for all a f O. Consider equation (5.6.9)

at criticality, i.e., when u

_",,, + '"

(U"') I '"

=

0:

+ a 2", '" = O

0

,

¢

E

(5.6.10)

H'.

It is noteworthy that equation (5.6.10) coincides with the one obtained from the linear theory at criticality, namel.y, L¢

=

O. Obviously, this is due to the sym-

rretrizability of L in the weighted scalar product.

From the analysis developed in

[16] it follows that (5.6.10) admits a solution belonging to H' a O = O.

Thanks to this fact, let us shoe that (5.6.11)

thus proving coincidence of linear and nonlinear stability. (5.6.9) corresponding to a and ].l(a). parsmetera . follows that

Let ¢ be a solution of

This solution depends continuously on these

Consider two such solutions and label then with subscripts.

It

76

2 sech Y¢l

(5.6.12)

2 2 Multiply (5.6.12)1 by cosh Y 0)

and integrate by parts over rl

Multiply (7.1.1) by

R

to

obtain f

gc .. kne. ne. . drl

rlR

lv

Klv

+

fn H

(7.2.1)

o. If we assure that Fi' u

i'

e

i j,

c

i jk

£

"grow" at most as r

k

(k > 0) we may let R +

00

in (7.2.1) to deduce

(7.2.2)

The identity (7.2.2) which for g

=

1 reduces to the energy relation (7.1.4) is

called the weighted energy equality (see [6,8,9]). prove the following continuous dependence theorem.

Starting fran (7.2.2) we wish to

96

1HEOREM 7.1. Assune that for some positive constants M, E, Co

mr-;>z-E

n=3

,,;

(7.2.3)

{ mr- c

n = 2

for all symnetric

Then, for all 0 c(O,l) sup(lu(x) I, Iw(x)

for all R ,,; 0 PROOF.

-s

I,

IF(x)

(s > 0),

I)

< 0

k is

f'J

,,;

R

kOP

(p > 0)

positive constant depending only on M and n.

Fran (7.1.2) and (7.2.3)1 we deduce the following implication for all

symnacrtc tensors

Yij and for any

Il >

0

Therefore, we have

2 ,,; a 2 f Qgu2dQ

+

i

(7.2.4)

Moreover, by assumption, it is also

(7.2.5) where C,,< depends only on M and n-L -y n-L C '" 2 llM fOe y dy. 00

Substituting (7.2.4) and (7.2.5) back into (7.2.2) we thus have

97

1

Jrfcijk.Q,eijl\JI,em

i

2

-n

2" Jngu em + Coa

(7.2.6)

+ C*o.

Now, anploying (7.2.3)2 we obtain(t)

(7.2.7) where C**

l . :J 00 -y 1-£ 2n- 1TM- JOe y dy.

Fram (7.2.6), (7.2.7) we thus have (7.2.8) where A is a suitable constant depending only on M and n.

In view of (7.2.3)3'

(7.2.8) in turn inplies for all R (7.2.9) On choosing

o
0)

Integrating by parts over il and assuming

(7.3.1)

100

1·

e

-oz f

D(z)

{u 2 + (du)2}dD dZ

=

°

(7.3.2)

we easily obtain

(7.3.3) where use has been made of Schwarz's inequality.

Since (7.3.1) 2 holds,

U

satisfies

the Poincare inequality

where y is a constant depending only on the geanetry of D.

Employing (7.3.4) on

the left-hand side of (7.3.3) we thus obtain

(7.3.5) for all

c

(0,2/y).

'Iheref ' . ore,letrang ex';- 0'm (735) .. glves (7.3.6)

2 which proves, am:mg other things the finiteness of f"l\luI ett. assured only (7.3.2). for all z

°

Notice that we

Let us now prove the following trace inequality which holds

and for all

E

(0,00)

(7.3.7) where c depends only on D and z .

For any t >

u 2(x,y,z ) = u 2 (x,y,z + t )

°

and fixed (x.y ,z) we have

_ 2 fz+t u Z

and so therefore we have 2 u (x.y ,z)

z+t 2 z+t 2 2 u (x,y,z + t) + f z U dr + f z j \lui dr .

On integrating (7.3.8) over the unifonn cross section D, we obtain

(7.3.8)

101

2 f D(z) u dD

2 z+t f D(z+t) u dD + f z de

{J D(z+t)u

2 dD} 2

1 1 + f z+t z de {fD(z+t) vU dD}.

Hence, an integration of (7.3.9) fran t O t o t =

JL

(7.3.9)

with the use of (7.3.4) yields

(7.3.7) . Let us employ (7.3.7) in (7.3.6) to obtain

(7.3.10) We nCM multiply (7.3.1)1 by ug and integrate by parts over r1(z)

2 (} 2 frl(z) glvul drI - :2 frl(z)gu drI

=

=D

x

[z,oo) to obtain

dU a 2 g(z) {- fD(z) U 8Z dD + 2 fD(z)u dD}. (7.3.11)

Fram (7.3.4), (7.3.6) we deduce

We may therefore apply Lebesgue dominated convergence theorem in (7.3.11) in the limit a

0 to deduce (7.3.12)

Using Schwarz's inequality and (7.3.4) in (7.3.12) and explicating the integration over Q(z) we thus deduce

(7.3.13) Setting 00

G(z) = f z {fD(t)

2 dD}dt,

Iv u [

(7.3.13) can be rewritten as

'"

G(z) :;;; - y "G' (z) which gives

(7.3.14)

102

From (7.3.7), (7.3.10) and (7.3.14) we thus obtain fD(z)u2dD which shows the exponential decay.

c2

2dD exp(- y'"z)

It is worth remarking that to obtain (7.3.15)

we have only assuned the "growth" condition (7.3.2) with 7.4.

(7.3.15)

00 E.

(0,2/y).

Asynptotic Behavior in the Nonlinear Elastic Beam. We now suppose that a hanogeneous nonlinear elastic body in its natural state

occupies the region

$1

defined in the previous section.

The elastic body is defomed

to an equilibriun position in which the body force is zero, surface traction is distributed over the base, D(O), and the lateral sides are maintained at zero displacerrent.

The following calculations are easily adapted to include lateral sides

that are stress-free (see [18]), but then the base load must additionally be selfequilibrated.

In the deformed position, the Piola-Kirchhoff stress

G

i j is related

to the spatial gradient of the displacen:ent u. by means of the constitutive 1.

relation (*) (i,j,k,l

1,2,3)

and satisfies the equilibrium equations G •••

a.j ,j

=

0

(7.4.1)

in

subject to the displacerrent being held zero on the lateral sides, Le., u.

1.

=

0

on

3D x [0,00).

(7.4.2)

To ccmplete the description of the boundary value problem we must specify the asymptotic behavior at large (axial) distances.

To this end, we introduce the

weight ftmction g(z) given by g(z)

=

exp{-y(z

+ a)oo}

where a.v ,« are given normegative constants, and suppose that

(*)We shall assume throughout the notation

,s

- 3/ 3X ' (s s

=

1,2,3).

103

(7.4.3)

lim a

2

+ 1) 2 (a-l) S(z)dz

00

a+O

where

0

=

p)1. We shall assune further that there exists a positive constant c

l

such that the

following inequality holds: (7.4.4)

f D ( )li. . u . . dD:;;; c l f D ( )0, .u. . dD. z z

We remark that a sufficient condition for (7.4.3)2 can be proved to be (cf. [19, Section 11 J) S(z) :;;; A(Z

o :;;; u

+ a)k [log(z +

< 1, k :;;; 1.

We also notice that for problem (7.3.1) we do not need the assumption (7.4.3) l' while (7.4.4) is automatically satisfied.

However, in the linearized case (i.e.,

see Section 7.1) it can be proved that both assumption (7.4.3)1

=

and (7.4.4) can be replaced by the following (7.4.5)

for suitable c

2,c3

> 0 and for all syrrnetric tensors E;ij'

Before proceeding to the

asymptotic behavior estimates we recall the following Sobo1ev embedding inequality holding for any (sufficiently smooth) function v:

+

IR

q > 1

where c

(7.4.6)

depends only on the "gearetry" of D (cf. [20)). 4 Analogous to what we proved for the s:illlple problem (7.3.1), we now show the

following result (cf. (7.3.10».

104

Lenma 7.1. The solution u

the boundary value problem (7.4.1), (7.4.2) and subject to

(7.4.3), (7.4.4) satisfies the inequality (7.4.7)

J,.,u. " ,J. u . . d1';; A JD(O) o'30'3dD

where A is

PROOF.

positive ccmputable constant. We multiply the equilibriun equation (7.4.1) by g(z)u and after an

i

integration by parts, use (7.4.3)1 to obtain

_

J,.,g(z)o .. u . . drG - -g(O) JD(O) o'3u.dD " a, Consider the last tenn on the right.

. + ay J,.,g(z)(z + a)

o'3u.drG.

(7.4.8)

An application of Holder's inequality followed

by (7.4.8) leads to

00

.;; c,.ay J '+

lr

222 c 4a y 00 .;; ---.,,-- J . 0).

This kind of result is intuitively expected and already may be found in the analysis made in [22] for the linear nonprisrnatic cylinder. ized elasticity (i.e., (J ..

--

= C.

However, in the case of linear-

',.ne. ., see Section 7.1) by using the sane kind of KJI,

method employed here, the exponential decay order (see (19,16]) can be proved.

The

kind of procedure in this case does not differ too much frxm that adopted for problem (7.3.1) and therefore it will be emitted.

Moreover, as we already said. in the

linear case the hypotheses (7.4.3)2 and (7.4.4) can be replaced by the single assumption that the elasticities are positive definite and bounded above (see (7.4.5»).

107

References to Chapter VII 1.

M.E. Gurtin, Theotuj

0

n EfM:t sup fa u.u.(x,s)ds < By

where B depends only on N and T and

t

ql

(> 0) depends on 6(= tiT) (ql ->-

{t).Nai:Ik:lly, there exists a finite cone r such that each point x vertex of a finite cone r c congruent to r (see, e.g., [8]).

x-

[O,T)

E

E

°

as t

->-

is the

T).

118

PROOF.

Let

US

a

Est.

fix arbitrarily

= ys

(s > 0), C

= fa

00

e

Setting

-y2

y dy

we easily obtain

F(T)

N(l + T4 +

Therefore, from (8.2.1) we deduce for s < 1/3 (8.2.12) where B* and q are suitable positive constants depending only on N,T and tf «, respectively.

Therefore, denoting by h the height of the (finite) cone defining the

cone property of st, fran (8.2.12) and by assunption we deduce (8.2.13) since unifonnly bounded in st

x

l..k , 9-

is

[O,T] we may employ (8.2.11) to obtain

which, along with (8.2.13), ccmpletes the proof of the theorem.

References to Chapter VIII 1.

M.E. Gurtin, Ref. [1] to Chapter VII.

2.

G.P. Galdi & S. Rionero, In-t. J. Engng.

3.

R.J. Knops &B. Straughan, Int. J. Engng. SU., 14 (1976) 555.

4.

R.J. Knops & L.E. Payne, Un.-ique.ne.M The.o/te.m6

Su., 17 (1979) 521.

Tracts in Natural Philosophy, Vol. 19 (1971).

5.

B. Straughan, Ref. [5] to Chapter II.

..[n

U..ne.aJt Efu6:ticUy, Springer

119

6.

R.J. Knops &: L.E. Payne, Sa:Je lliiqueness and Continwus Dependence 'Iheorems for Nonlinear Elastodynamics in Exterior Donai.ns , Applicable. AYla£.y.6.{J."

to be

published. 7.

G.P. Galdi &: S. Rionero, PMC.. Roya£. Soc: EcUYlbU!1.gh, 93A (1983) 299.

8.

R. Adams, Ref. [9] to Chapter II.

9.

G.P. Galdi &: S. Rionero, Ref. [19] to Chapter VII.

10.

G.P. Galdi &: S. Rionero, Ref. [3] to Chapter III.

SUBJECT Asymptotic spatial behavior Asymptotic suction 32 Basic flow 2

INDEX

99, 102

Benard Problem 59 between spherical shells 60 in Magnetohydrodynamics 60,79,81 compressible 79 Bessel functions 23, 38 Bio-convection 71 Chandrasekhar number 62, 80 Continuos data dependence in Elastodynamics 109 without definiteness assumption on the elasticities Continuous data dependence in Elastostatics Couette flow between coaxial cylinders Counterexamples to improvements on USC half space 22 exterior domain Criticality 5

37

Decay of kinetic energy in bounded domain 8, 12 in exterior domain 43 for a model equation 75 Deformation rate tensor Diffusion equation 73 D-solutions 49 uniqueness 51 further properties Eigenvalue problem in linear stability

8

51

5. 56 in energy stability 9. 57 Ekman layer 33 Elasticities 92 Energy relations in bounded domain 8 in half space 17 in exterior domain 39 Exchange of stabilities 5 strong

63

68

95

112

121

Functionals maximization of Generalized energy Instability 4, 7

30, 46, 64 81, 87

Kinematic viscosity 1 Linearization principle 6, 7 Mean radius approximation 68 Navier-Stokes equations perturbed 2 Normal modes 5

1

Operator symmetric 6, 9, 52 symmetrizable 6, 63 - with compact resolvent 55 Oscillatory Stokes layer 6 Overs 6 Periodicity cell

15, 45

Poincare inequality 10 counterexamples 17, 37 weighted 15, 37 Prandtl number magnetic

59 61

Rayleigh number Reynolds number Shear layer model

59 10, 11, 20, 21 40, 41 73

Stability asymptotic 24, 41 asymptotic energy 3, 24, 41, 57 energy 2, 57 - linear 3, 4, 56 and skew-symmetric operators 63 and symmetrizable operators 63 Stabi

in unbounded domain 78 effect of magnetic field

Trace inequality 100 Uniqueness in Elastodynamics Uniqueness in Elastostatics in bounded domain 93 in exterior domain 95 Uniqueness of seady flows

53

109, 115

61, 85

122

Universal Stability Criteria (USC) in bounded domain

10, 11

in exterior domain 40 in half space 20, 21 Variational formulation of stability for a general evolution equation in bounded domain 9 in exterior domain 45 in half space 29 Weak compactness 30 Weighted convexity inequality Weighted energy equality 95 Weight function method in Elasticity 95 in Fluid Dynamics

18

112

64

GENERAL BIBLIOGRAPHY N.B. Numerals in square brackets denote the chapter in which the bibliografic reference is relevant. Adams, R., Sobo,(l2-v Spaev, , Academic Press (1976) Babenko ,

[;II,III,VI,Vn,VIII]

K.1., Math. USSR SbOfLi'Uk , 20 (1973) 1 [III, IV]

Barre de Saint­Venant, A.J.C., Mvn. VA.-veJU,12- SavaVl-t6

, 14 (1855) 233 IYn]

Bousstneso .ij , , Appuea-Uon des Potl2-i'lJ;{I2-U a ,('Efudl2- dl2- ,('EqtUtibfLl2- des Mouvl2-ml2-n.v, dv, Soudv, EfMtA-qUV, , Paris, Gauthier­Villars (1885) [VII]

Eusse,E., J. F,(uA.-d Ikeh. , 71 (1) (1975) 193 IYI]

Chandrasekhar , S. , HydILodynamA.-e and HydfLomagnwe Stabilily , Oxford at Clarendon Press (1961) Childress, S., Levandowsky H. & E.A. Sniege1, J. Ffi..Ud Ikeh.,63 (1975) 591 [V] Davis, S.H. & C. von Kerzceck, AlLeh. Raft

/.kch. AnaL,52 (1973) 112 [IJ

Drazin, p.n. & "l­T.H. Reid, HydtwdynamA.-e Stabilily , Cambridge Univ. Press (1980) [I,n] Dudis, J.J. & S.H. Davis, J. Fti..Ud Ml2-ch.,47 (1971) 405 Ericl:sen, J .L., On thl2- FOfLmu,(a-tCon 0& thl2- Sunt-Vl2-nant Pit., Nonlinear Analysis and Hechanics, Heriot­Hatt Symn., Vol I, Pitman­London (1977) 158 [VII] "'ichera, G., Ann. Seuola Noltm.

"'.

r irm, R.,

, (3)

Alteh. Raft Ill2-eh. Ana,(. ,

1

w _. lIll1, R., AlLeh. Raft Ikeh. Ana,(.,

"'. __ l!'U1,

(1950) 35 [VII]

(1959) 181 Q;V1 (1960) 371 [IV]

R., Acta Math. , 105 (1961) 197 LIV]

Finn, R., AJr..ch. Raft Ml2-eh. Ana,(. , 19 (1965) 363 !]:v] "'inn, R., Roeky Mount. 11ath J.

,1

(1973) 107 [III ,IV]

Foias, C. & R. Temam, Comm. PUf1..12- AppL Math. , 30 (1977) 149 [IV] Fujita, H., J. Fae.. Suo Ui'Uv. Tokyo Ga1di,

n.r..

AlLeh.

, 9 (1961) 59

Q;V]

Raft Ml2-eh. Ana,(., 59 (1975) 1 [I]

Ga1di, G.P., RA.-ceJtehe Mat. , 27 (1978) 387 1):IJ Ga1di, G.P., Ll2-ctUf1..e Notv, On HydILodynamA.-e StabA.-Wy , Surmer School in Hatherna.tical Physics, Ravello, Italy, (1980) [IJ r;a1di, G.P., Soml2- pJr..opeJtt- bt Un11ethods in the Dynamics of Fluids bound. VarnUM, Proc. Symn. and Ionized Gases", Trieste Urriv. Press (1981) 45 [III,IV] Ga1di, G.P., The Mam!1m ModufM TheOfLem &0Jr.. UnMfL ElMtiWy on Ex:teUolt VomUM, Institute of Hathernatics, University of Naples, Int. P­.ep. (1981) [vn]

124

Galdi, G.P., EneJLg1j meA:hod6 and Symme.b1.y Stab;'LLty 06 V..L6C.OlL6 F.f'0l The,oILeJM

u.neCU1. Ef.a.-6u,e.uy, Springer-Ver-

Knops, R.J. & L.E. Payne, Meh. Rail. Meeh. AncU'., 81 (1983) 1 [VII) KnODS, R.J. & L.E. Payne, Same Uniqueness and Continuous Dependence Theorems for Nonlinear Elasticity in Exterior Domains, App-f.. AnaL , to be published

J

'

KnODS, R.J. & B. Straughan, Int. J. Engng Suo , 14 (1976) 555 [VIII] KnODS, R.J. & E.H. Hilkes, Ef.M-ue Stab-Uily, Hand. Phys. , Bd VI a/3 (1973)

Ladyzhenskai.a , O.A., The McvtheJrla:ti.ecU' Theolly 06 2nd Edition, Gordon & Breach (1969) rII, III Lalas, P.D. & S. Carmi., PhYl>. Lerav, J.,

J. Mcvth. PUlLi2J.l App-f.., 12 (1933) 1 [IV

F-f.ow ,

1

ri'.i.UdJ, , 15 (1972) 212 [VI

[I J

1

J

Leray, .J., Ac.:ta Mcvth, 63 (1934) [III J (1975) 296

Levandowsky , N., Childress, S. & LA. Spiegel, J. Lin, C.C.,

Stab--LWy , Cambri.dge Univ. Press (1955) [I]

The theally 06

McF. Orr, H., Pn.oc., Roy. Iwh Acad. Haremonti, P.,

Rend.

Natroshivily, D.G.,

Scm.

[V J

(A), 27 (1907) 69

Mcvt. Un. Pad., to be published I

nu« Ullcwneniyc(

01einik, O.A., & G.A. Yosifian, Alleh. Rcvtf.

J

[I

,14 (1978) 1649 (Russian) [VII ]

Me.eh. AncU'.,

(1982) 29 [VII)

Padula, H., Nonlinear Stability of Compressible Benard Problem, BoU. U.i\l. I., to be published [VI] Padula, 11., On the Traction Value Problem in Linear IJ.Lc''''L-J.\C",LPllOC. Roy. Soc. , to be published

in Exterior Domains,

Pai., S.,

]

F-f.ow TheMy , Van Nostrand, Vol I (1956)

Payne, L.E. & H.F. Heinberger, in NonLLneCU1. Pllob-f.cm1l Hisconsin Press (1963) [I J Prom,

G.,

Rend. Scm. Mcvt. Univ. Padova

Rayleigh, Lord,

,Ed. R.E. Langer, Univ. of

,32 (1962) 374

J

Pap0'C-6 , Cambridge Uni.v, Press, 3-6 (1913) [IJ

Reynolds, 0., PW. TILan-!>. Ray. Sac. (Al, 186 (1895) 123 [IJ

126

Rionero, S., AM. Meu:. PLU1.a App!. Rionero, S.,

,78

(1968) 339 LIl

Meu:. , 17 (1968) 64' [V,V:g

Romanov, V.A.,

i: Ega P!l.ilazhaMya ,

Rudraiah, N., Publ MVL. Sattinger, D.H.']. Math.

Soc.

Japan,

(1981) 721 [YI)

Me,ch., 19 (1970) 797

Serrin, J.B., A!l.ch. Ratl Mecho An,

l

?... (1973) 62 (Russian)[IJ

IIJ

(1959B) 1 It,IV]

Shir, C.C. & D.D. Joseph, A!l.ch. Ratl Mee-h. Au ,.; 30 (1968) 38 [V] Spiegel, E.A.,

J,

l

(1965) 1068 [VI]

Straughan, B., London (1982) [II,VIII]

and

Thomas, T.Y., Pno c., Nat. Acad. Sc.i.; ,

(1943) 243

En. Method;."

Pitman

[11

Thompson, W.B., Ph;'.l. Mag. Sen, 42 (1951) 1417 [V] Toupin, R.A., Aneh. Ratl Mee-h. An., Unno , W., Kato, S.

& M. Makita, Pu.

(1965) 83 [VII)

A;.,tn. S. Japan

von Kerczeck, C. & S.H. Davis,St. App!. Math. Watson, G.N.,A (1944) [II] Wilcox, C.H.,]. E!a;.,t.

,12

(1960) 193 [VI]

,51 (1972) 239

[ri;]

on the Theony 06 Be;.,;.,el Fun., Cambidge Univ. Press

,

(1979) 221 [VII ]