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Proceedings |
"WASCOM 2 0 0 1 " 1 f th Conference on
Waves and Stability in Continuous Media
VV Editors
Roberto M...

Author:
Roberto Monaco

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i

Proceedings |

"WASCOM 2 0 0 1 " 1 f th Conference on

Waves and Stability in Continuous Media

VV Editors

Roberto Monaco Miriam Pandolfi Bianchi Salvatore Rionero

World Scientific

Proceedings

"WASCOM 2001" f t th Conference on

Waves and Stability in Continuous Media

This page is intentionally left blank

Proceedings

"WASCOM 2001" 11 th Conference on

Waves and Stability in Continuous Media Porto Ercole (Grosseto), Italy 3-9 June 2001

Editors

Roberto Monaco Politecnico di Torino, Italy

Miriam Pandolfi Bianchi Politecnico di Torino, Italy

Salvatore Rionero Universitd di Napoli, Italy

V f e World Scientific ™b

Sinaapore*NewJersey Singapore • New Jersey »L • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

WAVES AND STABILITY IN CONTINUOUS MEDIA — WASCOM 2001 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-23-8017-5

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

This volume contains the invited lectures and contributed papers presented at the XI International Conference on Waves and Stability in Continuous Media (WASCOM 2001) held June 3-9, 2001, in Porto Ercole (GR), Italy. Ever since its initial edition organized in Catania 1981, the Conference aimed to bring together foreign and Italian researchers and scientists to discuss problems, promote collaborations and shape future directions for research in the field of stability and wave propagation in continuous media. This cycle of conferences became a fixed meeting every two years: the further conferences have been held in Cosenza ('83), Bari ('85), Taormina ('87), Sorrento ('89), Acireale ('91), Bologna ('93), Palermo ('95), Monopoli ('97) and Vulcano ('99). Every time the proceedings have been published, documenting the research work and progress in the area of waves and stability. From a scientific point of view the success of this experience is confirmed by the fact that a remarkable group of italian researchers, from many different universities, has proposed several national projects in the field. The last project, entitled "Non Linear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media", co-ordinated by Prof. T. Ruggeri (Bologna), is the main proposer of the present conference. The eleventh edition, the first of the third millennium, registered over 110 participants coming from more than 11 different countries. The topics covered by 29 main lectures and 52 short communications, within 10 sessions, were • Discontinuity and shock waves • Stability in Fluid Dynamics • Small parameter problems • Kinetic theories towards continuum models • Non equilibrium thermodynamics • Numerical applications

v

VI

The Editors of the proceedings would like to thank the Scientific Committee who carefully suggested the invited lectures and selected the contributed papers, as well as the members of the Organizing Committee, coming from the Departments of Mathematics of the Universities of Napoli, Messina and Politecnico of Torino. A special thank is addressed to all the participants to whom ultimately the success of the conference has to be ascribed. Finally, the Editors are especially indebted to the institutions which have provided the financial support for publishing this book: • Fondazione della Cassa di Risparmio di Torino (CRT) • Provincia di Grosseto Torino and Napoli, March 2002

THE EDITORS Roberto Monaco Miriam Pandolfi Bianchi Salvatore Rionero

C O N F E R E N C E DATA

WASCOM 2001 11th International Conference on Waves and Stability in Continuous Media Porto Ercole (GR), Italy, June 3-9, 2001 Scientific Committee Chairmen: R. Monaco (Torino) and S. Rionero (Napoli) G. Boillat (Clermont Ferrand), C. Dafermos (Providence) D. Fusco (Messina), A. Greco (Palermo), G. Mulone (Catania) I. Miiller (Berlin), C. Rogers (Sydney), T. Ruggeri (Bologna) B. Straughan (Durham), C. Tebaldi (Lecce) Organizing Committee Chairman: M. Pandolfi Bianchi (Torino) F. Conforto (Messina), M. Gentile (Napoli), A. Rossani (Torino) F. Salvarani (Torino) Supported by • PRIN 2000 "Non Linear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media" • Gruppo Nazionale per la Fisica Matematica - INDAM • Dipartimento di Matematica - Politecnico di Torino • Regione Toscana • Comune di Monte Argentario • A.P.T. Grosseto • Camera di Commercio Industria Artigianato e Agricoltura di Grosseto • AIPE and Pro Loco Porto Ercole

VII

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TABLE OF C O N T E N T S

Preface

v

Conference Data

vii

B. Albers Linear Stability Analysis of a ID Flow in a Poroelastic Material under Disturbances with Adsorption

1

G. Ali Compressible Flows in Time-dependent Domains: Zero Mach Number Limit

8

S. Avdonin, S. Lenhart, V. Protopopescu Recovering the Potential in the Schrodinger Equation from the N-D Map

14

F. Bagarello Multi-resolution Analysis in the Fractional Quantum Hall Effect

28

J. Banasiak, G Frosali, F. Mugelli Space Homogeneous Solutions of the Linear Boltzmann Equation for Semiconductors: A Semigroup Approach

34

E. Barbera Stationary Heat Conduction in a Rotating Frame

41

M. Bisi, M. Groppi, G. Spiga Grad's Closure in the Kinetic Theory of a Chemically Reacting Gas

48

A. M. Blokhin, R. S. Bushmanov, V. Romano Electron Flow Stability in Bulk Silicon in the Limit of Small Electric Field

55

F. Bofill, M. Dalmau, R. Quintanilla End Effects of Saint-Venant's Type in Mixtures of Thermoelastic Solids

62

IX

X

G. Boillat, A. Muracchini Characteristic Shocks in Exceptional Directions

68

G. Borgioli, M. Camprini Schrodinger-like Model for Interband Tunnelling in Heterogeneous Semiconductor Devices: A Current Estimate

75

A. S. Bormann Instabilities in Compressible Fluids

82

G. Bozis, F. Borghero An Inverse Problem in Fluid Dynamics

89

Y. Brenier Continuum Mechanics and Dynamical Permutations

95

F. Brini, T. Ruggeri Entropy Principle and Non Controllable Data for the Moment Systems

103

F. Capone, S. Rionero On the Onset of Convection in Presence of a Bounded Non Uniform Temperature Gradient

118

F. Cardin Global Geometrical Solutions for Hamilton-Jacobi Equations of Evolution Type

124

F. Cardin, M. Favretti Helmholtz-Boltzmann Thermodynamics on Configurations Space

130

S. Carillo Variational Setting of Nonlinear Equilibrium Problem: Wedge Discontinuity Lines

136

G. Cattani Wavelet Solutions in Elastic Nonlinear Oscillations

144

X. Chen, E. A. Spiegel Continuum Equations for Rarefied Gases

150

XI

F. Conforto, R. Monaco, F. Schiirrer, I. Ziegler Detonation Wave Structure Arising from the Kinetic Theory of Reacting Gases

161

H. Cornille Large Size Planar DVMs and Two Continuous Relations

169

S. De Martino, S. De Siena, F. Illuminati, G. Lauro A Constitutive Equation for the Pressure Tensor Leading to the Hydrodynamic Form of a Nonlinear Schrodinger Equation

176

P. Fergola, M. Cerasuolo Some Applications of Stability Switch Criteria in Delay Differential Systems

179

J. N. Flavin Decay and Other Properties of Cross-Sectional Measures in Elasticity

191

D. Fusco, N. Manganaro A Reduction Method for Multiple Wave Solutions to Quasilinear Dissipative Systems

204

M. Gentile, S. Rionero Stability Results for Penetrative Convection in Porous Media for Fluids with Cubic Density

214

H. Gouin Dynamics of Lines in the Spreading of Liquids on Solid Surfaces

220

A. M. Greco, G. Gambino On the Boussinesq Hierarchy

232

G. Grioli Basic Parameters in Continuum Mechanics

243

G. Guerriero Perturbative Method in the Study of Nonlinear Evolution Problems in the Diffusion of the Particles of a Mixture

253

R. Kaiser, A. Tilgner On the Generalized Energy Method for Channel Flows

259

XII

M. Lisi, S. Totaro Quasi-static Approximations of Photon Transport in an Interstellar Cloud

271

S. Lombardo, G. Mulone Double-diffusive Convection in Porous Media: The Darcy and Brinkman Models

277

M. C. Lombardo, M. Sammartino Existence and Uniqueness for Prandtl Equations and Zero Viscosity Limit of the Navier-Stokes Equations

290

F. Mainardi, G. Pagnini Space-time Fractional Diffusion: Exact Solutions and Probability Interpretation

296

L. Margheriti, C. Tebaldi Bifurcation Analysis of Equilibria with a Magnetic Island in Two-Dimensional MHD

302

S. Martin, R. Quintanilla Existence of Unbounded Solutions in Thermoelasticity

316

G. Mascali Compton Cooling of a Radiating Fluid

322

M. S. Mongiovi, R. A. Peruzza Fast Relaxation Phenomena in Extended Thermodynamic of Superfluids

328

A. Montanaro On the Reaction Stress in Bodies with Linear Internal Constraints

334

J. Miiller Integration and Segregation in a Population — A Short Account of Socio-Thermodynamics

340

A. Muracchini, T. Ruggeri, L. Seccia Second Sound Propagation in Superfluid Helium via Extended Thermodynamics

347

XIII

O. Muscato BGK Model for Simulating Electron Transport in Semiconductor Devices

360

F. Oliveri Nonlinear Waves in Continua with Scalar Microstructure

366

M. Pandolfi Bianchi, A. J. Soares A Navier-Stokes Model for Chemically Reacting Gases

373

F. Paparella Slow Eigenmodes of the Shallow-Water Equations

379

S. Pennisi A Comparison Between Relativistic Extended Thermodynamics with 14 Fields and that with 30 Fields

385

F. Pistella, V. Valente Some Numerical Results on the Development of Singularities in the Dynamics of Harmonic Maps

391

M. Pitteri On Bifurcations in Multilattices

397

J. Polewczak An i7-Theorem in a Simple Model of Chemically Reactive Dense Gases

409

M. Primicerio, B. Zaltzman Free Boundary in Radial Symmetric Chemotaxis

416

K. R. Rajagopal Modeling of Dissipative processes

428

R. Riganti, F. Salvarani On a Transport Problem in a Time-dependent Domain

438

S. Rionero On the Long Time Behaviour of the Solutions of Non-linear Parabolic Equations in Unbounded Domains

447

C. Rogers On the Geometry of Spatial Hydrodynamic Motions. Solitonic Connections

458

V. Roma, R. Lancellotta, G. J. Rix Rayleigh Waves in Horizontally Stratified Media: Relevance of Resonant Frequencies

471

V. Romano, A. Valenti Symmetry Classification for a Class of Energy-Transport Models

477

E. I. Romenski Thermodynamics and Balance Laws for Processes of Inelastic Deformations

484

A. Rossani Euler Equations Arising from Extended Kinetic Theory: Sound Wave Propagation

496

G. Russo Central Schemes for Balance Laws

503

M. Sammartino, V. Sciacca Long Time Behavior of a Shallow Water Model for a Basin with Varying Bottom Topography

515

M. Sammartino, L. Seta A Model for the Chemiotherapy of the HIV Infection with Antigenic Variation

521

F. Schiirrer, W. Roller, F. Hanser Kinetic and Fluidynamic Approaches to Four-Wave-Mixing and Thermal Acoustic Phenomena in Quantum Optics

527

M. Senthilvelan, M. Torrisi Linearization and Solutions of a Simplified Model for Reacting Mixtures

539

M. P. Speciale, M. Brocato Elastic Waves in Materials with Thin Layers

548

XV

B. Straughan Unconditional Nonlinear Stability Via the Energy Method

554

J. Torcicollo, M. Vitiello On the Nonlinear Diffusion in the Exterior of a Sphere

563

G. Toscani Entropy Methods for the Asymptotic Behaviour of Fourth-order Nonlinear Diffusion Equations

569

R. Tracind On the Symmetry Classification for a Heat Conduction Model

579

M. Trovato Hydrodynamic Analysis for Hot-carriers Transport in Semiconductors

585

D. Vivona Small Oscillations of a Spherical Liquid Bridge between Two Equal Disks Under Gravity Zero

591

LINEAR STABILITY ANALYSIS OF A I D FLOW IN A POROELASTIC MATERIAL U N D E R D I S T U R B A N C E S W I T H ADSORPTION BETTINA ALBERS Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany E-mail: [email protected] In this paper we investigate the linear stability behavior of a I D flow in a poroelastic material with respect to longitudinal as "well as transversal disturbances with mass exchange. Fields are assumed to be a superposition of a stationary (nonuniform) ID solution and of infinitesimal disturbances in the form of a one- or twodimensional linear wave ansatz. We solve numerically the eigenvalue problem for the first step field equations using a finite-difference-scheme.

1

Introduction

In general flow instabilities arise due to at least two competing mechanisms. Conceivable are problems like a kinematic nonlinearity working against viscosity or gravity competing with a temperature gradient. In the case of our multicomponent model for porous systems, where a fluid flows through the channels of a skeleton, a kinematic nonlinearity is the counterpart to the permeability of the medium. Adsorption processes contribute in a nonlinear way to field equations and influence stability properties. 2

Base flow

We investigate the stability behaviour of a ID flow in a channel of length I and width 26 under longitudinal and transversal perturbations with adsorption. The base flow (without adsorption) satisfies the following mass and momentum balances of the fluid F dpF , dpFvF ,pdvF\ dpF F (dv F Here, p is the mass density of the fluid component, vF is the fluid velocity in x-direction and IT is the bulk permeability coefficient. It describes the effective resistance of the skeleton to the flow of the fluid as well as the true viscosity of the fluid. The partial pressure in the fluid p is given by the following constitutive relation PF=PF

+ K(PF-PF),

1

(2)

2 where pF and pF are initial values of t h e pressure and the mass density in the fluid phase, K denotes the compressibility. It is assumed t h a t a deformed skeleton does not contribute to a dynamical disturbance. T h e stationary form of (1) yields pFvF = C\ = const, and it remains t o determine the unknown constants in the formal solution of (1) 2

ln

^ + ^i

(3)

= 7^-z),

where Ci is the second integration constant. Boundary conditions for the base flow have the form: " PFV?\X=Q

= TT as (CTf,g)rT:=(uf(;T),u°(;T))H.

(15)

9

Here v,f and u are solutions of (l)-(3) corresponding to the boundary controls / and g. Since u^(-,T) = UTf, we can write CT in operator form:

cT = (uTyuT. T

The operator C

TT to n.

T

is bounded in T

(16) T

since the operator U is bounded from

To obtain an expression for CT in terms of RT, we use the equations (l)-(4) for f,g e C$°[0,T] and write: 0 = /

\(iu{(x,t)+ulx(x,t)-q(x)uf(x,tUus(x,t)]dxdt

/ \uf(x,t)

=

(iu9t(x,t)+u9xx(x,t)

\uf(x,t)u9(x,t)]

+ i + / Jo

\ufx{x,t)u9(x,t) L

-q(x)u9(x,t))]

dxdt

dx - uf{x,t)u9x(x,t)\

dt. J

(17)

*=o

Finally, we have i {uf(;nu9(-,T))n

= {f,RTg)j:T

- {RTf,g)fT

•

(18)

Comparing (15) and (18) and using a standard density argument, we get: CT = i[RT-

(RTy] .

(19)

20

This formula shows that CT is selfadjoint, which is consistent with the representation (16). We implement now for the Schrodinger equation the idea proposed in Belishev10 for the heat equation (see also Avdonin 4 ), namely we recover the spectral inverse data ( { A „ , K „ } ) from the dynamical inverse data (operator RT) using the connecting operator CT and the spectral controllability of the system (l)-(3). Prom (15) and (1), we have for f,g G H^(0,T) = Tj the following equalities: (CT(if),9)^

= (u^'(->T),ii»(.,r))w =

(iu{(;T),u'(;T))n

= (-uU-,T)+qUt(-,T),u°(-,T))n

=

(£uf(-,T),u(-,T))H

= (£0«/(-,r)>u»(-,T))w. f

In deriving (20), we used that u {-,T) tion by parts in (10) gives

(20) 7

G T)(C0) for / G J ^. Indeed, integra-

oo

J2\an(T)Xn\2

nT,

WTf

=

w((-,T),

is bounded. We can prove a controllability result which shows that the operator W T is boundedly invertible: Proposition 4 When the potential q € % is known and T G (0,£] ; then for any function z S HT, there exists a unique control f £ TT such that = z(x) inUT.

w{(x,T)

(29)

Proof. According to (28), condition (29) is equivalent to the equation z(x) = -f(T

- x) + f

k(x,s)f(T-s)ds

IG(0,T).

(30)

Jx

This is the Volterra equation of the second kind with respect to / . Its solvability proves Proposition 4. Next, we recover the potential q(x) from the known spectral data, {A„,K n },n e IN. Introduce another connecting operator CT : TT —»• J1"7, but now for the wave equation (23)-(25):

(CTf,g)rT =

U(;T),w?(;T))

The operator CT is bounded and boundedly invertible, since CT = (W T ) W T . Note that our definition of the connecting operator differs from the one used for the wave equation by Avdonin 2 ' 3 and Belishev11, since it is defined by the inner product of Wt instead of w. This definition is more convenient in problems with Neumann spectral data and dynamical problems involving ND maps. For the wave equation with dissipation, a bilinear form involving wt has been used in Kurylev 17 . The solution w^(x,t) admits the (Fourier series) representation oo Wf{x,t)

Y^bn(t) 0} solving the Cauchy problem (2.3) is an extension of the operator A + B and such a result is usually insufficient for applications. The reason for this is that the semigroup G solving eq. (2.3) should be a transition (stochastic) semigroup, i.e. one should have l|G(t)/|| = Il/H,

Vi>0, V / > 0 ;

this condition expresses the fact that the total number of particles in conserved through time. A sufficient condition for G to be stochastic is T = A + B. Three situations are possible: i) T = A + B or ii) T = A + B,T ^ A + B, whereby G is stochastic, and iii) T is a proper extension of A + B, in which case G may be not stochastic. The total number of particles is preserved only if G is stochastic, so only in the first case we can claim that the obtained semigroup has physical relevance 4 . The following is an extension of the Kato-Voigt perturbation theorem that generalizes theorem by L. Arlotti x,i. Theorem 2.1 Let A, B operators in X = L 1 (fi,/u). Suppose i) (A,T>(A)) generates substochastic semigroup {GA(t),

t > 0};

ii) V{B) D V(A) and Bf > 0, V/ e V{B), f > 0; iii) f(Af

+ Bf)dn0.

(2.4)

Then, there exists a smallest substochastic semigroup {G(t),t > 0} generated by an extension T of A + B and satisfying the integral equation G(t)f = GA(t)f

+ [ G(t- s)BGA(s)f Jo

ds,

V/ G V(A), Vt > 0 .

It can also obtained by the Philips-Dyson expansion G(t)f = X^«=o Sn(t)f, feX where S0(t)f = GA(t)f, Sn(t)f = fc 5„_i(t - s)BGA(s)fds . Note that theorem 2.1 does not give any characterization of the domain of the generator T. 3

Existence of Solutions by Semigroup Theory

Let X = L 1 (K 3 ). The multiplication operator by — i/(k) plays the role of A in the abstract problem (2.3), with domain V(A) = {/ e X : v(k)f 6 X}, under the assumption that i/(k) is non-negative and belonging to L ; 1 oc (E 3 ). The role of B is played by the positive gain integral operator with symmetric

37

kernel 5(k,k') > 0, with domain V(B) = {/ G X : Kf e X}. The kernel is such that j R 3 5(k,k')) dk' £ L 1 ^ 3 ) but is only L,1^ Proposition 3.1 I|B/IILHR3)

< ||A/|| L 1 ( R 3 )

,

V/eP(A).

(3.5)

3

Proof: Let / e L ^ E ) f l P ( i ) . Evaluating the l.h.s of (3.5) and applying triangle inequality, the result follows immediately using the symmetry and the positiveness of the kernel S. • From Proposition 3.1 it follows that V(B) D 12(A). A generates the substochastic semigroup G^(t) = exp {—v(k)t}. Then it is easy to prove that

/ R (Af + Bf)dk 3

= 0,

V/eD(i),

/>0,

(3.6)

•/R

thus we can apply Theorem 2.1, obtaining the existence of the semigroup of the evolution operator of our process. This semigroup, obtained by the Phillips-Dyson expansion, is substochastic and its generator is an extension of A + B = —vI + K, where / is the identity operator. In the present context we use the method introduced by Arlotti l and adapted by Banasiak 3 ' 5 , which consists in extending suitably the domain of the collision operator. With this understanding, we have Theorem 3.1 If for any f 6 X,f > 0, such that the expression Kf(k) finite almost everywhere and such that —vf + Kf £ X, we have

f (-i/(k)/(k)+Jf/(k))dk>0,

is

(3.7)

JR3

then T = -vl + K. Before giving the characterization of T, we prove the following technical lemma L e m m a 3.1 Let Bn = {(e, u) : 0 < e < nfru>, u € S2}. Then, f (-i/(k)/(k) + ( K / ) ( k ) ) d k = , /

D{e)D{e~hu)

Jnhto

+ (nq + l) / Jnhu

I

(3.8) 0{e,e -

fo^,u,u')/(£

- /kj,u')du' dude

JS2xS2

D(e)D(e - hu) /

Q(e,e -

HUJ,U',

u ) / ( e , u') du' dude .

JS2xS2

Proof. Function e(k) is invertible on E + if considered as function of k = |k|: k = \/2a(e + as2), where a = m*/h2. From now on, we will express

38

quantities in terms of the energy e instead of k, having kdk = 2a(l + 2oe) de and /

k2dk

dk=

JR3

JO

du=

/

JS?

D(e)de

du JS2

JO

where k = fcu, |u| = 1 and D{e) = (2a) 3 / 2 (1 + 2ae)Ve + ae2. Rewriting the operators in terms of e, a straight calculation gives: /

( - K k ) / ( k ) + (/ir/)(k))dk = pnhu

= -(nq + l)

p

D(e-hu) Jhu>

g{e,e-hLO,u,u')D{e)f(e,u)dudvL'de Js^xS?

— nq I D(e + huj) / Q(e,e + Hui,u,u')D(e)f(e,u) Jo Js2xS2 -

dudu' de

/ D2(e) / G0(e,e,u,u')f(e,u)dudu'de Jo Js2xS2

+ {nq + l)

D(e) Jo

/•nhuj

+ nq

g(e,e + huj,u,u')D(£ Js2xS2

+ huj)f(e +

hD,u')dudu'de

r

D(e) / 2 G{e,e - hui,\i,u')D(e Jhui Js xs2

rnhbJ

(3.9)

- hu>)f(e —

Hoj,u')dudu'de

r

+ / D2(e) / G0(e,e,u,u')f(e,u')dvidu'de. (3.10) Jo Js2xS2 We get immediately that (3.9) + (3.10) = 0, and, shifting e variable in the other terms in a way to have f(s, u') inside the integrals and using the symmetry of the kernel Q, we obtain the r.h.s. of (3.8). Hence the lemma is completely proved. • Now we are able to characterize the generator T in the following way. Theorem 3.2 Assume that there exists no G N such that Q, as a function of e, is a strictly positive function for e > no and that there exists q < 1 such that for all n > no sup B„+1\B„

G(e,e -

HUJ,U,

n + 1 u') < q— nq

inf

G(e,e - tuu,u',u),

(3.11)

B„\B„_I

then T = -vl + K. Proof. Following Lemma 3.1 we denote by bn the right-hand side of (3.8). The proof of the characterization of the generator T relies on proving (3.7) of Theorem 3.1, that is, that the limit of the sequence bn, that exists by the

39

assumption —vf + Kf 6 X, is non-negative. So, suppose on the contrary that limn_>oo bn < 0. Then there exist no and b > 0 such that bn < —b and the assumption (3.11) holds for all n> n0. Let Gn = sup Q{e,e - fou,\i, u'), cn+1 = 4n

/

r

i>(e)/(e,u')

Proceedings |

"WASCOM 2 0 0 1 " 1 f th Conference on

Waves and Stability in Continuous Media

VV Editors

Roberto Monaco Miriam Pandolfi Bianchi Salvatore Rionero

World Scientific

Proceedings

"WASCOM 2001" f t th Conference on

Waves and Stability in Continuous Media

This page is intentionally left blank

Proceedings

"WASCOM 2001" 11 th Conference on

Waves and Stability in Continuous Media Porto Ercole (Grosseto), Italy 3-9 June 2001

Editors

Roberto Monaco Politecnico di Torino, Italy

Miriam Pandolfi Bianchi Politecnico di Torino, Italy

Salvatore Rionero Universitd di Napoli, Italy

V f e World Scientific ™b

Sinaapore*NewJersey Singapore • New Jersey »L • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

WAVES AND STABILITY IN CONTINUOUS MEDIA — WASCOM 2001 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

This volume contains the invited lectures and contributed papers presented at the XI International Conference on Waves and Stability in Continuous Media (WASCOM 2001) held June 3-9, 2001, in Porto Ercole (GR), Italy. Ever since its initial edition organized in Catania 1981, the Conference aimed to bring together foreign and Italian researchers and scientists to discuss problems, promote collaborations and shape future directions for research in the field of stability and wave propagation in continuous media. This cycle of conferences became a fixed meeting every two years: the further conferences have been held in Cosenza ('83), Bari ('85), Taormina ('87), Sorrento ('89), Acireale ('91), Bologna ('93), Palermo ('95), Monopoli ('97) and Vulcano ('99). Every time the proceedings have been published, documenting the research work and progress in the area of waves and stability. From a scientific point of view the success of this experience is confirmed by the fact that a remarkable group of italian researchers, from many different universities, has proposed several national projects in the field. The last project, entitled "Non Linear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media", co-ordinated by Prof. T. Ruggeri (Bologna), is the main proposer of the present conference. The eleventh edition, the first of the third millennium, registered over 110 participants coming from more than 11 different countries. The topics covered by 29 main lectures and 52 short communications, within 10 sessions, were • Discontinuity and shock waves • Stability in Fluid Dynamics • Small parameter problems • Kinetic theories towards continuum models • Non equilibrium thermodynamics • Numerical applications

v

VI

The Editors of the proceedings would like to thank the Scientific Committee who carefully suggested the invited lectures and selected the contributed papers, as well as the members of the Organizing Committee, coming from the Departments of Mathematics of the Universities of Napoli, Messina and Politecnico of Torino. A special thank is addressed to all the participants to whom ultimately the success of the conference has to be ascribed. Finally, the Editors are especially indebted to the institutions which have provided the financial support for publishing this book: • Fondazione della Cassa di Risparmio di Torino (CRT) • Provincia di Grosseto Torino and Napoli, March 2002

THE EDITORS Roberto Monaco Miriam Pandolfi Bianchi Salvatore Rionero

C O N F E R E N C E DATA

WASCOM 2001 11th International Conference on Waves and Stability in Continuous Media Porto Ercole (GR), Italy, June 3-9, 2001 Scientific Committee Chairmen: R. Monaco (Torino) and S. Rionero (Napoli) G. Boillat (Clermont Ferrand), C. Dafermos (Providence) D. Fusco (Messina), A. Greco (Palermo), G. Mulone (Catania) I. Miiller (Berlin), C. Rogers (Sydney), T. Ruggeri (Bologna) B. Straughan (Durham), C. Tebaldi (Lecce) Organizing Committee Chairman: M. Pandolfi Bianchi (Torino) F. Conforto (Messina), M. Gentile (Napoli), A. Rossani (Torino) F. Salvarani (Torino) Supported by • PRIN 2000 "Non Linear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media" • Gruppo Nazionale per la Fisica Matematica - INDAM • Dipartimento di Matematica - Politecnico di Torino • Regione Toscana • Comune di Monte Argentario • A.P.T. Grosseto • Camera di Commercio Industria Artigianato e Agricoltura di Grosseto • AIPE and Pro Loco Porto Ercole

VII

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TABLE OF C O N T E N T S

Preface

v

Conference Data

vii

B. Albers Linear Stability Analysis of a ID Flow in a Poroelastic Material under Disturbances with Adsorption

1

G. Ali Compressible Flows in Time-dependent Domains: Zero Mach Number Limit

8

S. Avdonin, S. Lenhart, V. Protopopescu Recovering the Potential in the Schrodinger Equation from the N-D Map

14

F. Bagarello Multi-resolution Analysis in the Fractional Quantum Hall Effect

28

J. Banasiak, G Frosali, F. Mugelli Space Homogeneous Solutions of the Linear Boltzmann Equation for Semiconductors: A Semigroup Approach

34

E. Barbera Stationary Heat Conduction in a Rotating Frame

41

M. Bisi, M. Groppi, G. Spiga Grad's Closure in the Kinetic Theory of a Chemically Reacting Gas

48

A. M. Blokhin, R. S. Bushmanov, V. Romano Electron Flow Stability in Bulk Silicon in the Limit of Small Electric Field

55

F. Bofill, M. Dalmau, R. Quintanilla End Effects of Saint-Venant's Type in Mixtures of Thermoelastic Solids

62

IX

X

G. Boillat, A. Muracchini Characteristic Shocks in Exceptional Directions

68

G. Borgioli, M. Camprini Schrodinger-like Model for Interband Tunnelling in Heterogeneous Semiconductor Devices: A Current Estimate

75

A. S. Bormann Instabilities in Compressible Fluids

82

G. Bozis, F. Borghero An Inverse Problem in Fluid Dynamics

89

Y. Brenier Continuum Mechanics and Dynamical Permutations

95

F. Brini, T. Ruggeri Entropy Principle and Non Controllable Data for the Moment Systems

103

F. Capone, S. Rionero On the Onset of Convection in Presence of a Bounded Non Uniform Temperature Gradient

118

F. Cardin Global Geometrical Solutions for Hamilton-Jacobi Equations of Evolution Type

124

F. Cardin, M. Favretti Helmholtz-Boltzmann Thermodynamics on Configurations Space

130

S. Carillo Variational Setting of Nonlinear Equilibrium Problem: Wedge Discontinuity Lines

136

G. Cattani Wavelet Solutions in Elastic Nonlinear Oscillations

144

X. Chen, E. A. Spiegel Continuum Equations for Rarefied Gases

150

XI

F. Conforto, R. Monaco, F. Schiirrer, I. Ziegler Detonation Wave Structure Arising from the Kinetic Theory of Reacting Gases

161

H. Cornille Large Size Planar DVMs and Two Continuous Relations

169

S. De Martino, S. De Siena, F. Illuminati, G. Lauro A Constitutive Equation for the Pressure Tensor Leading to the Hydrodynamic Form of a Nonlinear Schrodinger Equation

176

P. Fergola, M. Cerasuolo Some Applications of Stability Switch Criteria in Delay Differential Systems

179

J. N. Flavin Decay and Other Properties of Cross-Sectional Measures in Elasticity

191

D. Fusco, N. Manganaro A Reduction Method for Multiple Wave Solutions to Quasilinear Dissipative Systems

204

M. Gentile, S. Rionero Stability Results for Penetrative Convection in Porous Media for Fluids with Cubic Density

214

H. Gouin Dynamics of Lines in the Spreading of Liquids on Solid Surfaces

220

A. M. Greco, G. Gambino On the Boussinesq Hierarchy

232

G. Grioli Basic Parameters in Continuum Mechanics

243

G. Guerriero Perturbative Method in the Study of Nonlinear Evolution Problems in the Diffusion of the Particles of a Mixture

253

R. Kaiser, A. Tilgner On the Generalized Energy Method for Channel Flows

259

XII

M. Lisi, S. Totaro Quasi-static Approximations of Photon Transport in an Interstellar Cloud

271

S. Lombardo, G. Mulone Double-diffusive Convection in Porous Media: The Darcy and Brinkman Models

277

M. C. Lombardo, M. Sammartino Existence and Uniqueness for Prandtl Equations and Zero Viscosity Limit of the Navier-Stokes Equations

290

F. Mainardi, G. Pagnini Space-time Fractional Diffusion: Exact Solutions and Probability Interpretation

296

L. Margheriti, C. Tebaldi Bifurcation Analysis of Equilibria with a Magnetic Island in Two-Dimensional MHD

302

S. Martin, R. Quintanilla Existence of Unbounded Solutions in Thermoelasticity

316

G. Mascali Compton Cooling of a Radiating Fluid

322

M. S. Mongiovi, R. A. Peruzza Fast Relaxation Phenomena in Extended Thermodynamic of Superfluids

328

A. Montanaro On the Reaction Stress in Bodies with Linear Internal Constraints

334

J. Miiller Integration and Segregation in a Population — A Short Account of Socio-Thermodynamics

340

A. Muracchini, T. Ruggeri, L. Seccia Second Sound Propagation in Superfluid Helium via Extended Thermodynamics

347

XIII

O. Muscato BGK Model for Simulating Electron Transport in Semiconductor Devices

360

F. Oliveri Nonlinear Waves in Continua with Scalar Microstructure

366

M. Pandolfi Bianchi, A. J. Soares A Navier-Stokes Model for Chemically Reacting Gases

373

F. Paparella Slow Eigenmodes of the Shallow-Water Equations

379

S. Pennisi A Comparison Between Relativistic Extended Thermodynamics with 14 Fields and that with 30 Fields

385

F. Pistella, V. Valente Some Numerical Results on the Development of Singularities in the Dynamics of Harmonic Maps

391

M. Pitteri On Bifurcations in Multilattices

397

J. Polewczak An i7-Theorem in a Simple Model of Chemically Reactive Dense Gases

409

M. Primicerio, B. Zaltzman Free Boundary in Radial Symmetric Chemotaxis

416

K. R. Rajagopal Modeling of Dissipative processes

428

R. Riganti, F. Salvarani On a Transport Problem in a Time-dependent Domain

438

S. Rionero On the Long Time Behaviour of the Solutions of Non-linear Parabolic Equations in Unbounded Domains

447

C. Rogers On the Geometry of Spatial Hydrodynamic Motions. Solitonic Connections

458

V. Roma, R. Lancellotta, G. J. Rix Rayleigh Waves in Horizontally Stratified Media: Relevance of Resonant Frequencies

471

V. Romano, A. Valenti Symmetry Classification for a Class of Energy-Transport Models

477

E. I. Romenski Thermodynamics and Balance Laws for Processes of Inelastic Deformations

484

A. Rossani Euler Equations Arising from Extended Kinetic Theory: Sound Wave Propagation

496

G. Russo Central Schemes for Balance Laws

503

M. Sammartino, V. Sciacca Long Time Behavior of a Shallow Water Model for a Basin with Varying Bottom Topography

515

M. Sammartino, L. Seta A Model for the Chemiotherapy of the HIV Infection with Antigenic Variation

521

F. Schiirrer, W. Roller, F. Hanser Kinetic and Fluidynamic Approaches to Four-Wave-Mixing and Thermal Acoustic Phenomena in Quantum Optics

527

M. Senthilvelan, M. Torrisi Linearization and Solutions of a Simplified Model for Reacting Mixtures

539

M. P. Speciale, M. Brocato Elastic Waves in Materials with Thin Layers

548

XV

B. Straughan Unconditional Nonlinear Stability Via the Energy Method

554

J. Torcicollo, M. Vitiello On the Nonlinear Diffusion in the Exterior of a Sphere

563

G. Toscani Entropy Methods for the Asymptotic Behaviour of Fourth-order Nonlinear Diffusion Equations

569

R. Tracind On the Symmetry Classification for a Heat Conduction Model

579

M. Trovato Hydrodynamic Analysis for Hot-carriers Transport in Semiconductors

585

D. Vivona Small Oscillations of a Spherical Liquid Bridge between Two Equal Disks Under Gravity Zero

591

LINEAR STABILITY ANALYSIS OF A I D FLOW IN A POROELASTIC MATERIAL U N D E R D I S T U R B A N C E S W I T H ADSORPTION BETTINA ALBERS Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany E-mail: [email protected] In this paper we investigate the linear stability behavior of a I D flow in a poroelastic material with respect to longitudinal as "well as transversal disturbances with mass exchange. Fields are assumed to be a superposition of a stationary (nonuniform) ID solution and of infinitesimal disturbances in the form of a one- or twodimensional linear wave ansatz. We solve numerically the eigenvalue problem for the first step field equations using a finite-difference-scheme.

1

Introduction

In general flow instabilities arise due to at least two competing mechanisms. Conceivable are problems like a kinematic nonlinearity working against viscosity or gravity competing with a temperature gradient. In the case of our multicomponent model for porous systems, where a fluid flows through the channels of a skeleton, a kinematic nonlinearity is the counterpart to the permeability of the medium. Adsorption processes contribute in a nonlinear way to field equations and influence stability properties. 2

Base flow

We investigate the stability behaviour of a ID flow in a channel of length I and width 26 under longitudinal and transversal perturbations with adsorption. The base flow (without adsorption) satisfies the following mass and momentum balances of the fluid F dpF , dpFvF ,pdvF\ dpF F (dv F Here, p is the mass density of the fluid component, vF is the fluid velocity in x-direction and IT is the bulk permeability coefficient. It describes the effective resistance of the skeleton to the flow of the fluid as well as the true viscosity of the fluid. The partial pressure in the fluid p is given by the following constitutive relation PF=PF

+ K(PF-PF),

1

(2)

2 where pF and pF are initial values of t h e pressure and the mass density in the fluid phase, K denotes the compressibility. It is assumed t h a t a deformed skeleton does not contribute to a dynamical disturbance. T h e stationary form of (1) yields pFvF = C\ = const, and it remains t o determine the unknown constants in the formal solution of (1) 2

ln

^ + ^i

(3)

= 7^-z),

where Ci is the second integration constant. Boundary conditions for the base flow have the form: " PFV?\X=Q

= TT as (CTf,g)rT:=(uf(;T),u°(;T))H.

(15)

9

Here v,f and u are solutions of (l)-(3) corresponding to the boundary controls / and g. Since u^(-,T) = UTf, we can write CT in operator form:

cT = (uTyuT. T

The operator C

TT to n.

T

is bounded in T

(16) T

since the operator U is bounded from

To obtain an expression for CT in terms of RT, we use the equations (l)-(4) for f,g e C$°[0,T] and write: 0 = /

\(iu{(x,t)+ulx(x,t)-q(x)uf(x,tUus(x,t)]dxdt

/ \uf(x,t)

=

(iu9t(x,t)+u9xx(x,t)

\uf(x,t)u9(x,t)]

+ i + / Jo

\ufx{x,t)u9(x,t) L

-q(x)u9(x,t))]

dxdt

dx - uf{x,t)u9x(x,t)\

dt. J

(17)

*=o

Finally, we have i {uf(;nu9(-,T))n

= {f,RTg)j:T

- {RTf,g)fT

•

(18)

Comparing (15) and (18) and using a standard density argument, we get: CT = i[RT-

(RTy] .

(19)

20

This formula shows that CT is selfadjoint, which is consistent with the representation (16). We implement now for the Schrodinger equation the idea proposed in Belishev10 for the heat equation (see also Avdonin 4 ), namely we recover the spectral inverse data ( { A „ , K „ } ) from the dynamical inverse data (operator RT) using the connecting operator CT and the spectral controllability of the system (l)-(3). Prom (15) and (1), we have for f,g G H^(0,T) = Tj the following equalities: (CT(if),9)^

= (u^'(->T),ii»(.,r))w =

(iu{(;T),u'(;T))n

= (-uU-,T)+qUt(-,T),u°(-,T))n

=

(£uf(-,T),u(-,T))H

= (£0«/(-,r)>u»(-,T))w. f

In deriving (20), we used that u {-,T) tion by parts in (10) gives

(20) 7

G T)(C0) for / G J ^. Indeed, integra-

oo

J2\an(T)Xn\2

nT,

WTf

=

w((-,T),

is bounded. We can prove a controllability result which shows that the operator W T is boundedly invertible: Proposition 4 When the potential q € % is known and T G (0,£] ; then for any function z S HT, there exists a unique control f £ TT such that = z(x) inUT.

w{(x,T)

(29)

Proof. According to (28), condition (29) is equivalent to the equation z(x) = -f(T

- x) + f

k(x,s)f(T-s)ds

IG(0,T).

(30)

Jx

This is the Volterra equation of the second kind with respect to / . Its solvability proves Proposition 4. Next, we recover the potential q(x) from the known spectral data, {A„,K n },n e IN. Introduce another connecting operator CT : TT —»• J1"7, but now for the wave equation (23)-(25):

(CTf,g)rT =

U(;T),w?(;T))

The operator CT is bounded and boundedly invertible, since CT = (W T ) W T . Note that our definition of the connecting operator differs from the one used for the wave equation by Avdonin 2 ' 3 and Belishev11, since it is defined by the inner product of Wt instead of w. This definition is more convenient in problems with Neumann spectral data and dynamical problems involving ND maps. For the wave equation with dissipation, a bilinear form involving wt has been used in Kurylev 17 . The solution w^(x,t) admits the (Fourier series) representation oo Wf{x,t)

Y^bn(t) 0} solving the Cauchy problem (2.3) is an extension of the operator A + B and such a result is usually insufficient for applications. The reason for this is that the semigroup G solving eq. (2.3) should be a transition (stochastic) semigroup, i.e. one should have l|G(t)/|| = Il/H,

Vi>0, V / > 0 ;

this condition expresses the fact that the total number of particles in conserved through time. A sufficient condition for G to be stochastic is T = A + B. Three situations are possible: i) T = A + B or ii) T = A + B,T ^ A + B, whereby G is stochastic, and iii) T is a proper extension of A + B, in which case G may be not stochastic. The total number of particles is preserved only if G is stochastic, so only in the first case we can claim that the obtained semigroup has physical relevance 4 . The following is an extension of the Kato-Voigt perturbation theorem that generalizes theorem by L. Arlotti x,i. Theorem 2.1 Let A, B operators in X = L 1 (fi,/u). Suppose i) (A,T>(A)) generates substochastic semigroup {GA(t),

t > 0};

ii) V{B) D V(A) and Bf > 0, V/ e V{B), f > 0; iii) f(Af

+ Bf)dn0.

(2.4)

Then, there exists a smallest substochastic semigroup {G(t),t > 0} generated by an extension T of A + B and satisfying the integral equation G(t)f = GA(t)f

+ [ G(t- s)BGA(s)f Jo

ds,

V/ G V(A), Vt > 0 .

It can also obtained by the Philips-Dyson expansion G(t)f = X^«=o Sn(t)f, feX where S0(t)f = GA(t)f, Sn(t)f = fc 5„_i(t - s)BGA(s)fds . Note that theorem 2.1 does not give any characterization of the domain of the generator T. 3

Existence of Solutions by Semigroup Theory

Let X = L 1 (K 3 ). The multiplication operator by — i/(k) plays the role of A in the abstract problem (2.3), with domain V(A) = {/ e X : v(k)f 6 X}, under the assumption that i/(k) is non-negative and belonging to L ; 1 oc (E 3 ). The role of B is played by the positive gain integral operator with symmetric

37

kernel 5(k,k') > 0, with domain V(B) = {/ G X : Kf e X}. The kernel is such that j R 3 5(k,k')) dk' £ L 1 ^ 3 ) but is only L,1^ Proposition 3.1 I|B/IILHR3)

< ||A/|| L 1 ( R 3 )

,

V/eP(A).

(3.5)

3

Proof: Let / e L ^ E ) f l P ( i ) . Evaluating the l.h.s of (3.5) and applying triangle inequality, the result follows immediately using the symmetry and the positiveness of the kernel S. • From Proposition 3.1 it follows that V(B) D 12(A). A generates the substochastic semigroup G^(t) = exp {—v(k)t}. Then it is easy to prove that

/ R (Af + Bf)dk 3

= 0,

V/eD(i),

/>0,

(3.6)

•/R

thus we can apply Theorem 2.1, obtaining the existence of the semigroup of the evolution operator of our process. This semigroup, obtained by the Phillips-Dyson expansion, is substochastic and its generator is an extension of A + B = —vI + K, where / is the identity operator. In the present context we use the method introduced by Arlotti l and adapted by Banasiak 3 ' 5 , which consists in extending suitably the domain of the collision operator. With this understanding, we have Theorem 3.1 If for any f 6 X,f > 0, such that the expression Kf(k) finite almost everywhere and such that —vf + Kf £ X, we have

f (-i/(k)/(k)+Jf/(k))dk>0,

is

(3.7)

JR3

then T = -vl + K. Before giving the characterization of T, we prove the following technical lemma L e m m a 3.1 Let Bn = {(e, u) : 0 < e < nfru>, u € S2}. Then, f (-i/(k)/(k) + ( K / ) ( k ) ) d k = , /

D{e)D{e~hu)

Jnhto

+ (nq + l) / Jnhu

I

(3.8) 0{e,e -

fo^,u,u')/(£

- /kj,u')du' dude

JS2xS2

D(e)D(e - hu) /

Q(e,e -

HUJ,U',

u ) / ( e , u') du' dude .

JS2xS2

Proof. Function e(k) is invertible on E + if considered as function of k = |k|: k = \/2a(e + as2), where a = m*/h2. From now on, we will express

38

quantities in terms of the energy e instead of k, having kdk = 2a(l + 2oe) de and /

k2dk

dk=

JR3

JO

du=

/

JS?

D(e)de

du JS2

JO

where k = fcu, |u| = 1 and D{e) = (2a) 3 / 2 (1 + 2ae)Ve + ae2. Rewriting the operators in terms of e, a straight calculation gives: /

( - K k ) / ( k ) + (/ir/)(k))dk = pnhu

= -(nq + l)

p

D(e-hu) Jhu>

g{e,e-hLO,u,u')D{e)f(e,u)dudvL'de Js^xS?

— nq I D(e + huj) / Q(e,e + Hui,u,u')D(e)f(e,u) Jo Js2xS2 -

dudu' de

/ D2(e) / G0(e,e,u,u')f(e,u)dudu'de Jo Js2xS2

+ {nq + l)

D(e) Jo

/•nhuj

+ nq

g(e,e + huj,u,u')D(£ Js2xS2

+ huj)f(e +

hD,u')dudu'de

r

D(e) / 2 G{e,e - hui,\i,u')D(e Jhui Js xs2

rnhbJ

(3.9)

- hu>)f(e —

Hoj,u')dudu'de

r

+ / D2(e) / G0(e,e,u,u')f(e,u')dvidu'de. (3.10) Jo Js2xS2 We get immediately that (3.9) + (3.10) = 0, and, shifting e variable in the other terms in a way to have f(s, u') inside the integrals and using the symmetry of the kernel Q, we obtain the r.h.s. of (3.8). Hence the lemma is completely proved. • Now we are able to characterize the generator T in the following way. Theorem 3.2 Assume that there exists no G N such that Q, as a function of e, is a strictly positive function for e > no and that there exists q < 1 such that for all n > no sup B„+1\B„

G(e,e -

HUJ,U,

n + 1 u') < q— nq

inf

G(e,e - tuu,u',u),

(3.11)

B„\B„_I

then T = -vl + K. Proof. Following Lemma 3.1 we denote by bn the right-hand side of (3.8). The proof of the characterization of the generator T relies on proving (3.7) of Theorem 3.1, that is, that the limit of the sequence bn, that exists by the

39

assumption —vf + Kf 6 X, is non-negative. So, suppose on the contrary that limn_>oo bn < 0. Then there exist no and b > 0 such that bn < —b and the assumption (3.11) holds for all n> n0. Let Gn = sup Q{e,e - fou,\i, u'), cn+1 = 4n

/

r

i>(e)/(e,u')

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