New Trends in Mathematical Physics: In Honour of the Salvatore Rionero 70th Birthday Proceedings of the International Meeting Naples

NEW TRENDS IN MATHEMATICAL PHYSICS

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Proceedings of the International Meeting

NEW TRENDS IN MTHEMTICAL PHYSICS in honoutrof the salvatorerionero70thbirthday Naples, Italy

24-25January2003

Editors

Paolo Fergola Florinda Capone Maurizio Gentile Gabriele Guerriero Universita degli Studi di Napoli Federico II, Italy

wp World Scientific N E W JERSEY ' LONDON

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NEW TRENDS IN MATHEMATICAL PHYSICS

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PREFACE The International Meeting on “New Trends in Mathematical Physics” has been organized to celebrate the 70th birthday of Professor Salvatore Rionero. Born in Nola on January 1st 1933, Professor Salvatore &onero is Full Professor of Mathematical Physics at the Faculty of Sciences of the University of Naples Federico I1 and Director of the PhD School in Mathematics. He has been the Vice-president of the Board of Directors of INdAM and, for many years, the President of the Scientific Council of the Italian Group of Mat hematical Physics (G.N .F.M.) . His main research interest concerns the nonlinear stability theory of the solutions of P.D.Es., where he constantly gives fundamental contributions with his original results (see, for instance, the papers of B. Straughan and G. Mulone in the present volume). His long and continuous scientific activity has awarded to him a great national and international renown. In 2002 he got the Honorary Degree of Doctor in Science of the National University of Ireland of Galway. His efforts, devoted both to promote the research in the field of Mathematical Physics and to create a remarkable school of young researchers, have been and are completely successful due both to the Summer School of Mathematical Physics in Ravello (he directed since the foundation in 1976) and the promotion (in 1979) of the well known international conference “Waves and Stability in Continuous Media” (WASCOM). Due to the very high level of the mathematicians involved, from all over the world, to give lectures, many young qualified mathematical physicists took great benefits from both these events. Author of more than 120 original works, including papers and books, he is an active member of many prestigious academies like, for instance, the Accademia Nazionale dei Lincei. Besides 22 original papers which represent a scientific tribute to Rionero’s figure of researcher, this volume also contains the talks (in Italian) given in the opening session of the meeting. From all these contributions it comes out his very high level of scientist, his care in the scientific and teaching activities, his curiosity and enthusiasm in the scientific research as well as his rare human qualities like great equilibrium, capability of work and generosity.

The Editors P. Fergola F. Capone M. Gentile G. Guerriero July 2004 V

Professor Salvatore Rionero Universith di Napoli Federico I1

SCIENTIFIC COMMITTEE Chairman: P. Fergola (Napoli) J. N. Flavin (Galway), M. Maiellaro (Bari) G. Mulone (Catania), B. Straughan (Durham), T. Ruggeri (Bologna)

ORGANIZING COMMITTEE Chairman: P. Fergola (Napoli) B. Buonomo (Napoli), F. Capone (Napoli) M. Gentile (Napoli), G. Guerriero (Napoli), M. Maiellaro (Bari) G. Mulone (Catania), I. Torcicollo (Napoli)

SUPPORTED BY 0

Universitb degli Studi di Napoli Federico I1

0

Dipartimento di Matematica ed Applicazioni “Renato Caccioppoli”

0

0

Accademia delle Scienze Fisiche e Matematiche della Societb Nazionale di Scienze, Lettere ed Arti (Napoli) Gruppo Nazionale per la Fisica Matematica (G.N.F.M.) dell’INdAM

vii

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CONTENTS Preface

V

Conference Committee

vii

A Time Dependent Inverse Problem in Photon Transport A . Belleni- Morante Torsionless Conformal Killing Tensors and Cofactor Pair Systems S. Benenti

1

12

Sulle Distorsioni del Tip0 Volterra Applicate a un Cilindro Iperelastico Cavo Omogeneo Trasversalmente Isotropo G. Caricato

24

Heat and Mass Transport in Non-Isothermal Partially Saturated Oil-Wax Solutions A . Fasano and M. Primicerio

34

New Applications of a Versatile Liapunov Functional J. N . Flavin

45

Remarks on the Propagation of Light in the Universe D. Galletto and B. Barberis

54

Thermodynamic Limit for Spin Glasses S. Grafi

68

Rigid Motions in Celestial Mechanics. Keplerian Motions G. Grioli

77

Stability Switches on Four Types of Charateristic Equation with Discrete Delay 2. Ma and J. Li

83

On the Best Value of the Critical Stability Number in the Anisotropic Magnetohydrodynamic BQnard Problem M. Maiellaro

95

A BGK-Type Model for a Gas Mixture with Reversible Reactions R. Monaco and M. Pandolfi Bianchi ix

107

X

Stabilizing Effects in Fluid Dynamics Problems G. Mulone

121

An Alternative Kinematics for Multilattices M. Pitteri

132

On Contact Powers and Null Lagrangian Fluxes P. Podio Guidugli and G. Vergara Caffarelli

147

Models of Cellular Populations with Different States of Activity M. Primicerio and F. Talamucci

157

Flows of a Fluid with Pressure Dependant Viscosities Between Rotating Parrallel Plates K. R. Rajagopal and K. Kannan

172

Control Aspects in Gas Dynamics P. Renno

184

A Functional Framework for Applied Continuum Mechanics G. Romano and M. Diaco

193

Global Existence, Stability and Nonlinear Wave Propagation in Binary Mixtures of Euler Fluids T. Ruggeri

205

Exchange of Stabilities in Porous Media and Penetrative Convection Effects B. Straughan

215

Effects of Adaptation on Competition among Species D. Lacitignola and C. Tebaldi

219

Wasserstein Metric and Large-Time Asymptotics of Nonlinear Diffusion Equations J. A . Carrillo and G. Toscani

234

Opening Talks

245

Acknowledgements

263

A TIME DEPENDENT INVERSE PROBLEM IN PHOTON

TRANSPORT A. BELLENI-MORANTE Dipartimento di Ipzgegneria Civile, Via S.Marta 3, 50139 Firenze, ltalia E-mail: belleniOdma.unzfi.it A Salvatore Rionero, in occasione del suo settantesimo compleanno, con affetto. Assume that the boundary surface C = aV of the region V C R3 occupied by an interstellar cloud, the scattering cross section us and the total cross section D are known. Suppose also that the UV-photon density arriving a: a location P, far from the cloud, is measured at times &,, ti = t o 7,.. . , t ^ ~= t o 57. Then, we prove that it is possible to identify the time behaviour of the source q that produce UV-photons inside the cloud.

+

+

Keywords: inverse problems, photon transport, interstellar clouds.

1

Introduction

Inverse problems in photon transport are of great interest (in particular) in astrophysics. In fact, consider an interstellar cloud that occupies the (bounded but “large”) region V c R3 and assume that one is interested in evaluating some physical or geometrical quantities (such as the cross sections, the UV-photon sources, the shape of the boundary surface C = aV), which characterize the behaviour and the evolution of the cloud. Assume also that the value of the UV-photon flux is measured ar a location i far from the cloud (for instance, by terrestrial astronomers). Then, a typical inverse problem may be stated as follows: given the photon “far field”, measured at i ,it is possible to determine one of the characteristics (physical or geometrical) of the cloud? Remark 1. We recall that interstellar clouds are astronomical objects that occupy large regions of the galactic space: the diameter of an average cloud may range from 10-1 to lo1 parsec, i.e. from lo3 to lo5 times the diameter of our solar system. Clouds are composed of a low density mixture of gases and dust grains (mainly hydrogen molecules with some 1- 2% of silicon grains); typical particle densities may be of the order of 104particles/cm3, i.e. times the density of earth atmosphere at sea level, In this paper, we shall consider the following time dependent inverse problem. Assume that the boundary surface C = aV of the region V c R3 occupied by an interstellar cloud, the scattering cross section us and the total cross section (T are known. Suppose also that the UV-photon density arriving . at . .a. location ?, “far” from the cloud, is measured at times &, z$ = & 7 , .. . , t j = t o j ~. . ., , t ^ ~= & 57. Then, we prove that it is possible to identify the time behaviour of the source that produce UV-photons inside the cloud. Remark 2. The literature on time independent inverse problems in photon transport (in particle transport) is rather abundant, see the references listed in ’. On the other hand, only a few papers deal with time dependent inverse problems, see

’.

+

+

1

+

2 for instance 3, 4 , 5 , ‘, ’. Remark 3. The time step T must obviously such that Iq(t T ) - q ( t ) l / q ( t )<( 1, where q = q ( t ) represents the UV-photon source. Since Q ( t + T ) - q ( t ) ‘2 T q ( t ) , T must be chosen so that T I q ( t ) / / q ( t ) << 1, i.e. T << q ( t ) / l q ( t ) l ,T << min{q(t)/lq(t)l, t E

+

[~o,i.Il>. 2

The mathematical model

Let the interstellar cloud under consideration occupy the bounded and convex region V c R3 and assume that the boundary surface C = dV is closed and “regular” (in a sense that will be explained later on). If we further assume (for simplicity) that the cloud is homogeneous, then the scattering cross section us and the total cross section are given positive constants (with us < u) inside V and are equal to zero outside. Correspondingly, the photon transport equation, the boundary condition and the initial condition have the form

s,

d cus -N(x, u, t ) = -cu . V N - CUN(X,u,t) + at 47r N ( y , u , t )= 0 if y E C and u . v ( y ) < 0, N ( x ,u, 0) = No(x,u), x E uE

v,

s.

N(x,u‘,t)du‘ + q(x,t ) ,

(14

XEK,UES, t>0, (1b) (1c)

I n . ( l ) , N ( x , u , t ) is the number density of photons which, at time t , are at x and have velocity v = cu (where c is the speed of light). Moreover, u E S is a unit vector, S is the surface of the unit sphere (with du‘ = sinO‘dO’dqY), K is the interior of V (hence, V = U C), and ~ ( y is ) the outward directed normal to C at y. Finally, q(x,t)represents the UV-photon source and we shall assume that, at each t , q = q(x,t ) is a positive constant (independent of x) if x E VOand q = 0 if x 6 VO(where VOis a convex region, bounded by the “regular” surface Co and such that VOc V,,see Figure 1). In order to write the abstract version of system (1) in the Banach space X = L(V x S),we introduce the following operators:

Bf = -cu.Vf -cuf, D ( B ) = {f : f E X,u ’ V f E X, f satisfies the boundary condition (lb)}, R ( B ) c X, (2)

s,

ccs Kf =f(x,u‘)du’, D ( K ) = x,R ( K ) c x. (3) 47r The relevant properties of the operators K and B are listed in the following lemma.

3

Figure 1. The convex regions V = Vi U C and VO= Voi U Co, with Vo C Vi.

Lemma 1 (a)

K E B ( X ) , i e . K is a bounded operator, with IJKIJ6 cgs;

(ii) B E G(1, -cu; X ) , i e . B is the generator of the strongly continuous semigroup {exp(tB), t 0 } such that Ilexp(tI3)lJ6 exp(-cut) V t 3 0, *. Proof. (i) immediately follows from definition (3) because

(ii) Consider the resolvent equation fot the operator B : ( X I - B)f = g7

(4) where the known term g is a given element of X I X = (Y + ip is a complex number and the unknown f must belong to D ( B ) . We have from definition (2)

cu . Of(x - ru, u) where 0 6 r

+ (A + cu)f(x - ru, u) = g(x - ru, u),

< R(x,u), see Figure 1. It follows that

4

+

d {exp [ - Txr ] cu

dr

f ( x - r u , u ) } = -;exp 1

[--TI

x + cu

g(x-ru,u)

and so integration with respect to r , between 0 and R(x, u), gives

where we took into account that f must belong to D ( B ) , i.e. f must satisfy the boundary condition (lb) (thus, f ( x - R(x, u)u,u) = 0). We now observe that ( 5 ) defines an element f belonging to the space X , Vg E X . In fact, we have

where a = Rex. It follows that

where S = max{R(x,u), ( x , u ) E V x S } is the diameter of V and x v ( y ) = 1 if YE XV(Y) = 0 if Y 6 Remark 4. We msume that the boundary surface C = aV is such that R(x, u) is a continuous function of (x, u) E V x S (with R(x, u) = 0 if x E C and u is directed towards K). If we put y = x - r u , then we obtain

v,

v.

.

where V’ = { y : y = x - ru, x E V}. Hence,

1

<-a + cu 11g11, + co > 0.

provided that Q We conclude that f E X and

5 Vg E X,a: = Re X

> -cu.

Note that the element f E X , defined by ( 5 ) satisfies the boundary condition (lb) because R(x, u) = 0 if x E C and u . V(X) < 0, see Remark 4. Further, since it is not difficult to verify that such an f satisfies equation (4), see ’, we have that u.Vf = -*f $g E X.Hence, f E D ( B ) . Finally, D ( B ) is dense in X because D ( B ) 3 C r ( V x S ) . We conclude that B E Q(1,-cu; X ) , 8, with

+

Remark 5. It follows from (7) that (i) ((a1- B)-’g)(x,u)E C(V x S ) (or E L ~ ( V x s)) Vg E C(V x S) (or E Lc”(V x S ) ) ;

(ii) ((a1- B)-’g)E X + Vg E X + , where X+ = {f: f E X, f(x,u) 3 0 at a.a. (x,u)E V x S} is the closed positive cone of X ; (iii) if g E X+ and g > 0 along a finite portion of the half straight line = {y : y = x - r u , r > O } , s e e F i g u r e l , t h e n ((al-B)-lg)(x,u)>OV(x,u).

By using some standard results of perturbation theory, *, we have from Lemma 1 that ( B K ) E G(l,-c(a - u s ) ; X ) , i.e. ( B K ) is the generator of the strongly continuous semigroup {exp[t(B K ) ] ,t 2 0} such that I(exp[t(B+K)](( exp[-c(u - us)t]V t 0. Consider then the abstract version of system ( l ) , ’:

+

+

>

$N(t) = (B

+

+ K ) N ( t )+ q ( t ) ,

<

t > 0, (9)

( N ( 0 ) = No, where N ( t ) = N ( . , .,t ) and q ( t ) = q(., t ) are now maps from [0,+03) into the Banach space X, and No = NO(.,.) is a given element of X (or, to be more precise, a given element of D ( B K ) = D ( B ) ) . The unique strict solution of the initial value problem (9) can be written as follows:

+

6

+

Remark 6. Relation (10) holds provided that (a) NO E D ( B K ) = D ( B ) ; (b) q = q(t) is a (strongly) continuously differentiable map from [0,+03) into X. However, since we assume that the source term q in equation (10) is independent of x E VO(the UV-photon source is spatially homogeneous a t each t 2 0), it is enough to suppose that the real function q = q(t) (from [0,+03) into R+) has continuous and bounded (elementary) derivative. 1 We finally note that (10) is equivalent to the following (implicit) expression for N(t):

N ( t ) = exp(tB)No

+

+

exp[(t - s ) B ] { K N ( s ) q ( s ) } d s .

(11)

Time-discretization procedures

3

As anticipated in the Introduction, assume now that the values N j = N ( f ,u, ij) of the photon number density are measured at some location x far from the cloud i Voi # 0, see Figure 1, and with (far-field measurements), with u such that ~ 2 , fn i. - t ^ 0 j ~ j ,= 0 , 1 , . . . , J . Then, we also have that

+

Nj

= N ( X ,u,ij) = N ( i ,u7 t 3.),

where i is the “first” intersection of y2,fi with C and t j = ij-i with t^ = If--il/c. (In what follows, we shall assume that i,-, = i, i.e. that to = 0 and t j = (io+jr)-i= j ~ . ) Correspondingly, (10) gives

However, it is not easy to “to extract” some information on the time behaviour of the source q from (12), even of the J left hand side N j are known (e.g. from experimental measurements). In fact, it seems much more reasonable to discretize (9) as follows, lo:

- ( B + K)mj+l + q(tj),

j = O,1,. . . , J

(13) (mo = No,

where that

{

mj

mj+1

= mj(x,u) “approximates” N ( x ,u, t j ) = Nj(x,u). We have from (13)

= [I- T(B

+ K)]-lmj + T [ I - T(B + K)]yq(tj),

j = 0 , 1 , .. .,J ,

(14)

mo = No,

7 and so

because t j = j r , j = 1 , 2 , . . . , J . From (15) and (16), we obtain

By taking into account that ( B

+ K ) E G(1,

-C(O

IIexp[t(B + ~ 1 . f- ( I - tA)-’fII

+

x)C G(1,o; x)and So

- gs);

t2 < ?II(B + Kl2fII

(17)

, t 2 0, (see chapter 9 of 11), it is not difficult t o prove that llNj - mjll/r < a positive constant V j = 1 , 2 , . . . ,J , (18)

Vf E D ( ( B K)’)

+

provided that (i) NO E D ( ( B K)’) = D(B’) and (ii) the real function q = q(t) is regular enough. Remark 7. Relation (15), that gives the “approximated” value mj as a function of NO and of the source term q , is rather complicated t o be used, for instance, in connection with numerical experiments. In fact, the explicit expression of [I r ( B K)]-’ is not known because ( B K ) is an integrodifferential operator.

+

+

An alternative discretization procedure of (9) is the following:

where nj = nj(x,u) still “approximates” Nj(x,u) = N ( x , u , t j ) . We have from (19)V(x,u) E v x

s

nj+1 (x,u) = ( ( I - TB)-lnj) (x,u) +T

( ( I - rB)-l[Knj +&)I)

(x,u), j = 0 , 1 , . . . , J - 1 (20)

nab, 4 = Noh, 4, where the explzcit expression of ( I - rB)-l is known, see (7).

8

Remark 8. The two discretization procedures (13) and (19) are equivalent. In fact, we have from (13) and (19) that

1

1

IlU - TB)-1 I1 - -1K-I 7 7

- By11 6

~

1 1 +cur'

see (8)' we obtain

On the other hand, (15) gives mj+l - m j = T [ I- r ( B

+ K)]-j-'(B + K)No + T [ I- T ( B+ K)]-'q(tj) j

+ r2C[I- r ( B + K)]+l(B + K ) q ( t j - i ) . i=l

Since

we have IImj+i - mjll

6 rll(B + K)Noll + W + r[j.]Q 6 r11(B + K)Noll + rq + riQ

where 4 = max{llq(t)ll, t E [O,fl} and Q = max{\l(B By using (22), inequality (21) becomes

+ K ) q ( t ) ( (t, E [O,q}.

and so we have

where 1 = 0 , 1 , . . . , J - 1. Since AO = mo - no = 0, we finally obtain

(22)

9

llAl+lIl

<

cusr t { ( ( ( B + K ) N o ( l + q + f Q } .

(23)

Inequality ( 2 3 ) shows that llA~+~ll = Ilml+l - nl+l((4 0 as r + O+. We conclude that both mj and nj are “good” approximations of Nj if T is small enough.

4

Identification of the source

Consider now a location 2 “far” from the cloud (2 @ V, see Figure 1) # 0. Assume that the phoand a unit vector ii such that -yk,fi U V O ~ ton number density N is measured a t 2 , a t times &,il,. .. ,ij. Thus, N(k, ii,$), N(2,u,il), . . . , N ( 2 ,u, t ^ ~ ) are known quantities. Correspondingly, N(i,ii,to = N(i,U,O), N ( i , U , i l - i)= N(2,U,tl),..., N(i,U,iJ - 0 = N ( i ,u,t J ) are also known. This implies that (in some sense) the approximated nl(i,ii),. . . , n ~ ( i , i iare ) given. densities n o ( i , u ) (= No(i,u)), LF’rom (20), with x = 2, u = u and j = 0, we have

-0

(

)

r (I-rB)-1q(to) (2, ii) =

(:(-I

)

- rB)-lq(to) (2, ii)

because q ( y ,t o ) = 0 if y @ VOand q ( y ,t o ) = q ( t 0 ) = a positive constant if y E VO. Thus, we obtain

nl(i,ii)= { ( ( I - T B ) - ~ ( I + T K ) N O(i,ii)} )

( 1+7 cur 1 + c f f rli-iol)-exp I2 - Yol)] +7 1 + cur [exp (----cr

(?(to).

(25)

Since n1(i,ii) is “measured” and the first term on the right hand side is known, relation ( 2 5 ) gives the expZzcit value of q ( t 0 ) . Note that, by substituting such a value into the right hand side of (20) with j = 0, the density n1 (i,u) can be explicitly obtained a t any ( 2 , u ) E V x S. Correspondingly, ( 2 0 ) with x = f, u = u and j = 1 becomes

10

1

+ COT

cr

where n Z ( 2 , i i ) is measured and the first term on the right hand side is known. Hence, (26) gives the explicit value of q(t1). Substitution of much a value into the right hand side of (20) with j = 1 gives then n2 = (k,ii) ‘d(k,ii) E V x S. In an analogous way, by using (20) with x = 2, u = ii and j = 2 , 3 , . . . , J - 1, the values q ( t z ) , . . . ,q ( t J - 1 ) can be explicitly determined. 5

Concluding remarks

1. Once the J values q(to), . . . ,q ( t J - 1 ) have been evaluated, the (continuous) time behaviour of the source term can be obtained by using sone suitable interpolation procedure. 2. Consider for instance relation (25), that gives q ( t 0 ) ; to be precise, the left hand side n1 (2, u) is only an approximated value of the measured photon density N ( 2 ,u,t l ) . Of course, since N ( 2 ,ii,t l ) is an experimental value, it is reasonable to assume that n l ( 2 , u) coincides with N ( 2 ,h,t l ) within the experimental error. If, on the other hand, we want to distinguish between the two values, then IN(%,ii,t l )n l ( 2 , u ) I < E , see Section 3, and so (25) gives q(t0) with an error of the order E. 3. If q = q ( x ,t ) > 0 V x E VOand q ( x , t ) = 0 if x $ VO(i.e., if q is position dependent within VO),then (24) becomes n l ( i , u )= ( ( I - ~ B ) - ~ ( I + ~ K( )i ,N u )~ )

dr exp

(--.>+ cur 1

q(2 - ru, to).

(27)

Then, it is not obvious how (27) determines the function q ( x , t o ) , x E VO.In this case, a suitable family of positive functions, defined ‘dxE VO,must be chosen where to look for q ( x ,t o ) , q ( x ,t l ) ,. . . , q ( x ,t J - 1 ) , by using the fact that the photon densities n l ( 2 , ii), n z ( 2 , ii), . . . are in some sense “strictly increasing” functions of the source term. A further paper will be devoted to this problem. References 1. Dyson J.E. and Williams D.A., T h e physics of the interstellar medium, Inst. of Physics Publishing, Bristol 1977. 2 . Belleni-Morante A. and Mugelli F., Identification of the boundary surface of a n interstellar cloud from a measurement of the photon f a r field, Math. Meth. Appl. Sci. 2003, in print. 3. Prilepko A.I. and Volkov N.P., Inverse problems of finding parameters of a nonstationary transport equation f r o m integral ouerdeterminations, Differ. Equations 23, 91-101 (1987).

11

4. Sydykov G.M. and Sariev A.D., O n inverse p r o b l e m f o r a time-dependent transport equation in plane-parallel geometry, Differ. Urain. 27, 1617-1625 (1991). 5. Prilepko A.I. and Tikhonov I.V., Reconstruction of the inhomogeneus t e r m in a n abstract evolution equation, Russian Acad. Sci. Izv. Math. 44, 373-394 (1995). 6. Ying J., He S., Strom S. and Sun W., A two-dimensional inverse problem f o r the time-dependent transport equation in a stratified half-space, Math. Engrg. Indust. 5, 337-347 (1996). 7. Prilepko A.I., Orlovsky D.G. and Vasin I.A., Methods f o r solving inverse problems in mathematical physics, Marcel Dekker, New York 2000. 8. Belleni-Morante A. and McBride A.C., Applied nonlinear semigroups, J.Wiley, Chichester 1998. 9. Greenberg W., Van Der Mee C. and Protopopescu V., Boundary value problems in abstract kinetic theory, Birkhauser, Base1 1987. 10. Vitocolonna C. and Belleni-Morante A., Discretization of the t i m e variable in evolution equations, J. Inst. Maths. Applics. 14, 105-112 (1974). 11. Kato T., Perturbation theory for linear operators, Springer, Berlin 1980.

TORSIONLESS CONFORMAL KILLING TENSORS AND COFACTOR PAIR SYSTEMS S. BENENTI Department of Mathematics, University of Turin, Via Carlo Albert0 10, 1-10123 Torino, Italy E-mail: [email protected] In this lecture we propose a geometrical reinterpretation of recent interesting results on the so-called cofactor and cofactor-pair systems, showing their link with a special class of separable systems.

1

Introduction

Let L and G two contravariant symmetric 2-tensors on a manifold Qn. Let us consider the characteristic equation det(L - u G) = det [Lij - u @ j ] = 0.

(1)

We call the n roots of this algebraic equation the eigenvalues of L w.r. to G. In the following we shall examine the case in which: (a) G is a metric tensor (of any signature) i.e., det G # 0. (b) L has simple and real eigenvalues ( u i )w.r. to G. (c) L is a conformal Killing tensor w.r. to G. (d) L is torsionless w.r. to G. Let us call L-system a pair (L,G) of this kind, and L-tensor a tensor L satisfying the above conditions. The interest of considering such a system is due to the following Theorem 1.1 T h e symmetric 2-tensors K,, a = 0 , 1 , . . . ,n - 1, defined by the sequence

KO= G, K, = t r (Ka-1L) G - K,-1 L, a # 0 , (2) are independent Killing tensors in involution if and only i f L i s a L-tensor. Since all these tensors have common eigenvectors, they define a Stackel system. This means that: (i) the eigenvectors are normal i.e., orthogonally integrable or surface-forming: each one admits an orthogonal foliation of hypersurfaces (submanifolds of codimension 1). The set of these n foliations forms an orthogonal web which we call, in this case, separable orthogonal web or Stackel web. (ii) Any local parametrization of this web i.e., any coordinate system ( q i ) such that each foliation is locally described by equation q i = constant, is a separable orthogonal coordinate system: these coordinates separate the geodesic HJE. This theorem summarizes results established in for positive-definite metrics. However, under the assumption that the eigenvalues of L are real, these results can be extended to indefinite metrics. This matter has been recently revisited in '. L-systems are only a special class of Stackel (orthogonal separable) systems. However, they have the following nice property: all the Killing tensors i.e., all the quadratic first integrals of the geodesic p o w , which are related t o the separation, ''I2

'p2

12

13 can be constructed in a pure algebraic way by the sequence (2) starting from the tensor L. It is remarkable the fact that this algebraic procedure does not reuuire the knowledge of the eigenvalues of L. Hence, in this case, L plays the role of a generator of the involutive algebra of first integrals associated with the separation. Furthermore, for finding the separable coordinates we have only to examine the tensor L, which contains all information: the web orthogonal to its eigenvectors is indeed a Stackel web. Note that we do not require the functional independence of the eigenvalues ui of L; some of them may be constant. The essential requirement is that they must be pointwise distinct, ui# u j . However, Theorem 1.2 If the eigenvalues (ui) are independent functions, then they are separable coordinates. is examined in 5 . It can be shown that, The functional independence of (ui) Theorem 1.3 The eigenvalues (ui)of L are independent if and only if L is not invariant 1u.r. to a Killing vector of G . In it was shown that the definition (2) can be replaced by other equivalent definitions, which however, require the knowledge of the eigenvalues. Among them we recall the following one: a

Ka =

C(-l)k

ua-k

Lk.

(3)

k=O

Here, a,(g) denotes the elementary symmetric polynomial of order p in the variables

u = (Ui). In fact, sequences of the kind (2)-(3) appeared in the literature many years before within a completely different realm. In the Ricci calculus of Schouten 30 they are recalled from Souriau 31 and Fettis 14: they are used for computing in a fast way the eigenvectors of a matrix L (knowing the eigenvalues). In the book it is also remarked that, in our notation and terminology, the tensor

Q(z) = cof(L - z G )

(4)

is a polynomial of degree n - 1,whose coefficients are the tensors K, defined in (3). However, if we go further in the past, we find in a paper of the young LeviCivita l9 the construction of geodesic first integrals by a formula similar to (4),in connection with the problem of finding the most general metric tensor admitting a geodesic correspondent. Indeed, if two (positive-definite) metric tensors G and G are such that the eigenvalues of G w.r. to G are simple, then they have the same unparametrized geodesics if and only if G admits a L-tensor. This matter have been recently investigated by Bolsinov & Matveev and Crampin The aim of this lecture is to show how the scheme of L-system fits with and with related the schemes of cofactor and cofactor-pair systems, lotl.

29,20~21922,23

topiCs.6,7,~,17,27,28,32,34,35

The notion of special conformal Killing tensor introduced by Crampin, Sarlet and Thompson 13, associated with other special tensors, which arise in connection with the problem of the geodesic equivalence, gives to the matter presented here a general and elegant setting. An extensive paper on this topic, entitled Special

14

two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems, is in preparation. 2

Notation and basic definitions

If L = (LZ,-j)is a contravariant symmetric tensor on a manifold by PL the polynomial function on T*Q defined by .

Qn,

then we denote

.

pL = Lt.4 p a * . . p j . Thus, we can define the symmetric tensor product 0 between these tensors by setting

P L ~=KPLPK. By means of the Poisson bracket, defined by

d F d G dGdF { F ,G } = --- -api aq2

api 64%'

we define a Lie-bracket by setting P[L,K]= {pL,pK).

Two tensors are said to be in involution in [L,K] = 0. If G is a metric tensor, then L is a conformal Killing tensor (CKT) if

[ L , G ]= C O G .

If C = 0 i.e., [L, GI = 0, then L is a Killing tensor. If K , L, . . . are symmetric contravariant 2-tensors and if a metric tensor G = (@j) is present, then we define their algebraic product LK by setting (KL)ij= KihLkjghk, where [ g i j ] = [G'jjl-' are the covariant components of the metric. We say that K and L commute when K L - LK = 0. This gives the meaning of some formulas of 31. The torsion H of a (1,l)tensor T = (q!) is the (1,2)-tensor defined by

2 H&(T)

q:dlhlLt] - TLa[iT$

This definition does not depend on the choice of the coordinates and the partial derivatives Oi = d/aqa may be replaced by the covariant derivatives Vi w.r. t o any symmetric connection. When H(T) = 0, T is said to be torsionless. The torsion in order t o establish criteria has been introduced by Nijenhuis and Haantjes for the normality of the eigenvectors of a ( 1 , l ) tensor. If we start from a (symmetric) contravariant 2-tensor L, then we can define the torsion w.r. to a metric tensor [ g i j ]= [@I - ' by considering the associated (1,l)-tensor L: = Lhj gh'a ' From results of 26,16 it follows that, Theorem 2.1 If a symmetric tensor L has simple and real eigenvalues w.r. to a metric G , then H ( L ) = 0 if and only if (i) there are local coordinates (qa) in which both L and G are diagonalited, (ii) &uj = 0 for i # j , being ua the eigenvalue of the eigenvector ai. 26y16915,

15 3

Elliptic-parabolic tensors on

R"

We shall work on the manifold Qn = R", referred to the Cartesian coordinates g = (xc")centered at the origin 0 = (0,. . . ,O). We denote by r = OP the position vector of the generic point P : its Cartesian components coincides with the coordinates of

P. We shall study contravariant symmetric 2-tensors of the form

E = C +m r @ r

+ w 0r,

(1)

where C = (Cap) is a constant symmetric contravariant 2-tensor, w is a constant vector and m E R. Here, 0 denotes the symmetric tensor product (of vectors): a 0b A (a @ b b @ a ) .In the papers of S. Rauch and coworkers they are called elliptic matrices and denoted by G ( a = m, p = tw). Here we prefer to use the symbol G for a generic metric although, as we shall see, we shall interpreted a tensor E as a metric. Tensors of this kind were introduced in as planar inertia tensors of a system of points with (positive or negative) masses, and related to the separable webs of R" endowed with the standard Euclidean metric. The scalar m is the total mass (it may be zero). It is remarkable the fact that a tensor E is a torsionless CKT w.r. t o the standard Euclidean metric, so that, if it has simple eigenvalues, it is a L-tensor and it generates a L-system. In it is shown that the case m # 0 corresponds to the elliptic-hyperbolic web (i,e., to the separation in confocal elliptic-hyperbolic coordinates) as well as the case m = 0 and w # 0 corresponds t o the parabolic web (i.e., to the separation in parabolic coordinates). In these two cases we call E elliptic tensor or parabolic tensor, respectively. The trivial case m = 0 and w = 0 corresponds to the separation in Cartesian coordinates. Our aim is to show th%t Theorem 3.1 If E and E are two tensors of the kind (1) such that det(E) # 0 and E has pointwise real simple eigenvalues W.T. to E, then the tensor

4

+

L = det(@ E

(2)

i s a torsionless conformal Killing tensor 2u.r. to the metric tensor

G A det(G) E.

(3)

In other words, the pair (L, G) is a L-system on the manifold R". For brevity we shall prove this theorem only in the case of an elliptic tensor E (6# 0). Going back to the definition (l),we observe that E has a n a f i n e character (it does not depend on a metric). Collecting results of we can affirm that Theorem 3.2 (i) If m # 0, then there exists a unique point 0' such that E assume the form E = C' + m r' @r',

r' = O'P,

C' = C - & w 8 w.

(4)

(ii) If m = 0 , then there exists a unique point 0' such that Eo,(w) = 0 and w is a n ezgenvector of E at all points P of the line parallel t o w and containing 0'. In item (ii) we refer to the standard Euclidean metric of R".

16

Proof. (i) If 0' Since

# 0 is

any other fixed point, then r = v

+

+ r', with v = 00'.

+

r 8 r = r' 8 r' 2r' 0v v 8 v, w Or = w 0r'+ w 0v,

it follows that

E = C' + mr' 8 r' + w' 0r', For m

C' = C w' = w

+ v~ (mv + w), + 2mv.

# 0, by choosing v = - 2m I w7

we get w' = 0 and (2). (ii) For m = 0 see l. As a consequence of this theorem we can always find a point 0, in general different from the point 0 E R", such that a tensor E has the form

{ EE = CC+ w O r , =

+mr@r,

C(w)=O,

for m # 0, for m = 0 .

(5)

Moreover, we can choose orthogonal Cartesian coordinates (x") with origin at 0 such that

Eap = c" Sap + m x" xfl, E"D = ca 6"s + i~ 2 (xa 610 + xp P ) , c1 = 0 , w = w1, 4

for m # O , for m = 0.

(6)

Commutation relations

A first remarkable property of the elliptic-parabolic tensors is the commutation formula

As we shall see below, the interest of this formula is that the Lie-commutator of E and E' is a sum of two terms which factorize in E and E' themselves. Note that the vectors A, more precisely the vectors N = A, have been already introduced in 28. To prove this formula we observe that, in Cartesian coordinates, PE = ~

+

+

p p ~ pmp( ~ ~ p , waxPp,pp, ) ~

where Cap and wL"are constant. Then we can easily prove the following conimutation relations: { P C , Pr) = 2 p c , {PC, (Pr)2} = ~ P c P ~ , {Pc,PwPr} = 2PCPW, { P w , P r } = Pw, {Prpw, ( P r I 2 } = ~ p w ( P r ) ~ .

17

It is a surprising fact that, by introducing the vectors A we get

so that

which is equivalent to (1). 5

Elliptic metric tensors

Let us consider an elliptic tensor E, m # 0. As we have seen (formula (2) of 31) we can always find orthogonal Cartesian coordinates (x")such that

E"D = b a P c a + m x a x P

(1)

Note that the constants ca are the eigenvalues of C i.e., of E at r = 0, w.r. to the standard Euclidean metric. Let us consider a contravariant symmetric tensor of the kind

e,a = baO

- < P E P xa xP.

We have

xa x') EYoe,o = (670c' + m ZY xp) (hop E" - E = 62 CYEa + m xy xn En - (&' cy xy En 2" + m x' En xa CP&P (z"2) = 62CYE" 27 xa En [m- E ( E 7 C 7 + m CP€ P ( Z P ) " ] .

<

+

Equation EYPeap = 62 is satisfied if and only if C"&O

This proves

= 1,

<(€YC'

+ m CPED (xP)2)= m.

18 Theorem 5.1 The tensor E (with m ca # 0. The covariant metric e,p is

# 0) is a metric tensor

if and only if all

-1

e,p =

6,p - E

(2)

&oxaxp,

The metric is not defined on the hyperquadric 1 Besides

E

+ m Cp€0

= 0.

(3)

it is convenient t o consider the quantity

c 2 C , &U(X,)2 , so that

m C = L - L~, l-[C=$. (4) I+mC' The tensor e,p has a remarkable geometrical interpretation, which will be of help in the following. In Rn+' = R" x R = (xa,xnfl),with the canonical basis (c,, t), we consider the surface JHI, described by the vector-parametric equation

E=-

L

+ - + C , E " ( X ~t)=~ r + z t ,

q = xac,

with domain D+ c R" where the tangent spaces of W, is

e,

E > 0. Since

= a,q = c,

+ z-l

&z =

P xa t

z 2 t-4,

(6)

z-~Px the~natural , basis of

= c , ~ fEa xa t .

If in Rn+' we introduce the metric tensor g(u,v) = C , €'"uava- unf'vn+l,

(7)

then

{

g(ca,cp) = 6,p g(ca,t) = 0 ,

Ea,

{

g(t,t) = - 1, g(q,q)= - l / m .

(8)

It follows that the metric tensor g induces on JHI, the metric tensor e,p = g(e,, ep>= hap

-

1 P&B 22

xax~.

This metric tensor, reduced t o R",is exactly that defined in (2). The conclusion is that we can work on the covariant metric tensor E by interpreting it as the metric tensor induced on W, by the pseudo-Euclidean metric (7) Theorem 5.2 Let ( p , q ) be the number of the positive and negative constants cat respectively. Then, for m > 0 the signature of E is ( p , q ) and f o r m < 0 the signature is ( p - 1, q 1 ) . Proof. The vector q is orthogonal t o W,, g(q,e,) = 0. The signature of g is (p,q 1). From ( 8 ) we see that if m > 0, q is timelike. Since it is orthogonal t o W,, it follows that the signature of the metric induced on W, hence, the signature of E is ( p , q ) . If m < 0, q is spacelike, so that the signature is (p - 1,q 1). Theorem 5.3 The metric E has constant negative curvature.

+

+

+

19

1 ape, = - e,p t.

(9)

z

We consider the unit vector orthogonal to W,, second fundamental form of HI,, Bap = g(aaep,U) = - 1

u = lmlf q, for computing the

4 4 e,p.

The eigenvalues of B,p w.r. t o the metric e,p are the all equal t o -1mlf. This shows that W, has a constant intrinsic curvature. The sign of this curvature is a matter of convention. We note that the eigenvalues do not depend on the sign of the In the case where they are all positive, the metric g is a Minkowski metric, and W, is the hyperboloid of the unit timelike vectors oriented in the future. Its constant curvature is known t o be, by convention, negative (as well as for a unit sphere the curvature is assumed t o be positive). Theorem 5.4 The Christoffel symbols of the metric e,p are

qp= Proof. By definition,

- m xY e,p

= E(d,ep, ell) = g(d,ep, ep). Because of

(9) and ( 6 ) ,

20

6

The torsion of E w.r. to

E

Let E be a n elliptic-parabolic tensor. Assume that E is an elliptic metric tensor (fi # 0). We apply to it all the results of the preceding section. We recall the last formula (by interchanging E with E):

G n ~ =zz y

-

( m z u p - f i E u p ) + x(E 4zP ( m b : - f i E , Y ) + i ( w r z u p + . w p 6 2 ) . m

(1)

Hence,

E,"TuEz =

[

= E," zy

F

i

(m Eup - f i Eup) + Tp zp (m6: - f i E,Y)+ (wyE,,p

-

m

F = x y (mEaP - f i ~ :+~ )E" zP (mE2 - f i ( ~ ~ 1 + 2 3) ( w y E,P m This proves that

-

+ waE:).

-

E ~ V , E=~&~7ozp m (mE2 - f

-

+

i ( ~ 2 ) ~ + ) WOE;

. . . = terms symmetric in (a,p). On the other

+ .. .

(2)

hand, we can write (1) in the form

v a E p- $ 2 zp (md; - f i E;) 0-m so that

1

+ wp 62)

+ f wp 6: + . . . ,

-

-

t 8 x p ( m E 2 - f i ( E 2 ) z )+ wp E,Y + . . . . Ep' V a E P- 7 0-m The comparison with (2) shows that H(E) = 0. Thecorem 6.1 T h e elliptic-parabolic tensor E i s torsionless w.r. t o the elliptic metric E.

i

7 Elliptic-parabolic tensors as conformal Killing tensors It can be shown

+ m xa zp, E = det[Ea4] = a,(c) + m C, a P 1 ( ~ ) ( z a ) 2 .

that for EaP = ca 6"P

(1)

It follows that

aaE = 2 m uZP1xa,

EapaaE

= 2 m a,"-l za(cahap

(2)

Let us recall the commutation relation

[E,k]=2(x@E-A@k),

+ m xa z P )= 2 m E xp.

A=2mr+w,

-

A=2fir+%.

21

For any function f on R" we have

+

OE +

[fE,fE]= f 2 [E,E] f [E,f ] f [f,E]O E = 2 f 2 ( X i E - A A E ) + f [ E , f ] @ E +f[f,E:]oE. Thus, an equation of the kind [fE,f Z] = V 0 E is fulfilled iff 2fii

+ [f,El = 0,

i.e.,

[E,fl = 2 (

-+

2 ~ r W)f.

For G # 0 we can consider W = 0 without loss of generality: This equation is equivalent to

E%?,

[E,f ] = 4% f r.

f = 2Gf so.

Due t o (2), this equation is solved by f = det(E). Thus, Theorem 7.1 If the tensor G = d e t ( E ) E is a metric tensor, then L = d e t ( E ) E is a conformal Killing tensor. Theorem 7.2 The tensor L is @rsionless W.T. to the metric G . Proof. E is torsionless w.r. to E. The eigenvalues and the eigenvectors of E w.r. to E are the same of L w.r. to G . If the eigenvalues of E are simple and real, then also L is torsionless. This proves Theorem 3.1.

Acknowledgments This research is sponsored by Istituto Nazionale di Alta Matematica, Gruppo Nazionale per la Fisica Matematica.

References 1. Benenti, S., Inertia tensors and Stackel systems in the Euclidean spaces. Rend. Semin. Mat. Univ. Polit. Torino 50, 315-341. 2. Benenti, S., Orthogonal separable dynamical systems. In Proceedings of the

3.

4.

5. 6.

7.

5th International Conference on Differential Geometry and Its Applications, Silesian University at Opava, August 24-28, 1992, 0.Kowalski & D.Krupka Eds. Differential Geometry and Its Applications 1, 163-184. Web edition: ELibEMS, http://www.emis.de/proceedings. Benenti, S., An outline of the geometrical theory of the separation of variables in the Hamilton-Jacobi and Schrodinger equations. In Symmetry and Perturbation Theory - SPT 2002 (proceedings of the conference held in Cala Gonone, 19-26 May 2002), S. Abenda, G. Gaeta, S. Walcher Eds., World Scientific. Benenti, S., Hamiltonian Optics and Generating Families. Napoli Series on Physics and Astrophysics, Bibliopolis (Napoli, 2004). Benenti, S., Separability i n Riemannian manifolds. Royal Society, Phil. Trans. A. (forthcoming, 2004). Blaszak, M., Separability theory of Gel'fand-Zakhareuic systems o n Riemannian manifolds. Preprint, A. Mickiewics University, Poznari. Blaszak, M. & Badowski, L., From separable geodesic motion to bihamiltonian dispersionless chains. Preprint, A. Mickiewics University, Poznari.

22 8. Blaszak, M. & Ma, W.-X., Separable Hamiltonian equations o n R i e m a n n m a n ifolds and related integrable hydrodynamic systems. J. Geom. Phys. 47,21-42. 9. Bolsinov, A. V. & Matveev, V. S., Geometrical interpretation of Benenti systems. J. Geom. Phys. 44,489-506. 10. Crampin, M., Conformal Killing tensors with vanishing torsion and the separation of variables in the Hamilton-Jacobi equation. Diff. Geom. Appl. 18, 87-102. 11. Crampin, M., Projectively Equivalent R i e m a n n i a n Spaces as Quasi-biHamiltonian Systems. Acta Appl. Math. 77 (3), 237-248. 12. Crampin., M., Sarlet, W., A class of non-conservative Lagrangian systems o n R i e m a n n i a n manifolds. J. Math. Phys. 42 (9), 4313-4326 (2001). 13. Crampin., M., Sarlet, W., Thompson, G., Bi-differential calculi, biHamiltonian systems and conformal Killing tensors. J. Phys. A: Math. Gen. 33,8755-8770 (2000). 14. Fettis, H.E., A method for obtaining the characteristic equation of a matrix and computing the associated modal columns. Quart. Appl. Math. 8, 206-212. 15. Frolicher, A. & Nijenhuis, A., Theory of vector-valued differential forms. Proc. Kon. Ned. Ak. Wet. Amsterdam 59-A,338-359. 16. Haantjes, J., O n X,-forming sets of eigenvectors. Proc. Kon. Ned. Ak. Wet. Amsterdam A 58 (2), 158-162. 17. Ibort, A., Magri, F. & Marmo, G., Bihamiltonian structures and Stackel separability. J. Geom. Phys. 33,210-228. 18. Kalnins, E. G. & Miller Jr., W., Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations. SIAM J. Math. Anal. 11, 10111026. 19. Levi-Civita, T., Sulle trasformazioni delle equazioni dinamiche. Ann. di Matem. 24, 255-300. 20. Lundmark, H., A new class of integrable N e w t o n systems. J. Nonlin. Math. Phys. 8, Supplement, 195-199. 21. Lundmark, H., Newton Systems of Cofactor Type in Euclidean and Riemann i a n Spaces. Linkoping Studies in Sciences and Technology, 719 (2001). 22. Lundmark, H., Multiplicative structure of cofactor pair system in R i e m a n n i a n spaces. Preprint (2001), in [Lundmark, 2001bl. 23. Lundmark, H., Higher-dimensional integrable N e w t o n systems with quadratic integrals of motion. Studies in Appl. Math. 110,257-296 (2003). 24. Marciniak, K. & Blaszak, M., Separation of variables in quasi-potential systems of bi-cofactor f o r m . J. Phys. A: Math. Gen. 35,2947-2964. 25. Marshall, I. & Wojciechowski, S., W h e n is a Hamiltonian system integrable?. J. Math. Phys. 29,1338-1346. 26. Nijenhuis, A., X,-l-forming sets of eigenvectors. Nederl. Akad. Wetensch. Proc. 54A,200-212. 27. Rauch-Wojciechowski, S., From Jacobi problem of separation of variables t o theory of quasipotential N e w t o n equations. To appear in Royal Society, Phil. Trans. A. 28. Rauch-Wojciechowski, S. & Waksjo, C. Stackel separability for N e w t o n systems of cofactor type. Preprint, University of Linkoping.

23 29. Rauch-Wojciechowski S., Marciniak, K. & Lundmark, H., Quasi-Lagrangian systems of Newton equations. J. Math. Phys. 40(12), 6366-6398. 30. Schouten, J. A., Ricci Calculus. Berlin: Springer. 31. Souriau, J.M., Le calcul spinoriel et ses applications. Recherche Aronautique 14,3-8. 32. Topalov, P., Hierarchies of cofactor systems. J. Phys. A 35,L175-L179. 33. Waksjo, C., Stackel Multipliers in Euclidean Spaces. Linkoping Studies in Sciences and Technology, 833. 34. Waksjo, c.,Determination of Separation coordinates f o r Potential and Quasipotential Newton Systems. Linkoping Studies in Sciences and Technology, 845. 35. Waksjo, c. & Rauch-Wojciechowski, S., How to find separation coordinates for the Hamilton-Jacobi equation: a criterion of separability for natural Hamiltonian systems. Math. Phys. Anal. Geom. (to appear).

SULLE DISTORSIONI DEL T I P 0 VOLTERRA APPLICATE A UN CILINDRO IPERELASTICO CAVO OMOGENEO TRASVERSALMENTE ISOTROPO G. CARICATO Dipartimento di Matematica - Universita‘ ”La Sapienza”

1

- Roma

Considerazioni preliminari

Poco pili di un anno fa, il 21 novembre 2001, su invito del car0 amico Salvatore Ftionero, in un’aula di questo istituto, esposi i risultati di una mia ricerca, comunicata a1 XIV congress0 Nazionale Aimeta e pubblicata su Meccanica l. Riguardava l’estensione della teoria delle distorsioni (che Volterra aveva sviluppata in un cilindro elastico cavo, omogeneo e isotropo 6, ad un cilindro elastico cavo, omogeneo e anisotropo, ma dotato di un asse di simmetria elastica coincidente con l’asse geometric0 del cilindro, brevemente trasversalmente isotropo, second0 la definizione di Love ‘. La ricerca, pur avendo tracciato le linee fondamentali di sviluppo, non era ancora completa. E’ stata quindi ripresa da E. Laserra e M. Pecoraro ’,che l’hanno ulteriormente sviluppata, come ora preciser6. L’ipotesi che il cilindro cavo omogeneo sia trasversalmente isotropo implica che, nell’ambito dell’elasticitb linearizzata e ammettendo che il cilindro sia soggetto a un campo di spostamenti a una data temperatura a partire da uno stato naturale, ossia esente da stress e deformazione, la densitb di energia di deformazione del corpo possa essere scritta nella forma (4 p. 160):

ove T indica la temperatura costante della trasformazione, u = { u k }6 il vettore spostamento del generic0 punto P* del cilindro nel suo stato naturale C,: le funzioni

sono le componenti linearizzate del tensore di deformazione, che possono essere scritte anche nella forma di caratteristiche di deformazione vT = qTT ( T = 1 , 2 , 3 ) , e le quantitb 776 = 27112 = 779-1-2, 774 = 27723 = 779-2-3, 775 = 27713 = 779-1-3,

rappresentano i 5 moduli elastici del corpo. 24

25 2

Le leggi costitutive. Le equazioni indefinite di equilibrio.

Le leggi costitutive, che in generale hanno la forrna

dove 9 = IIX:l’II 6 il tensore di stress (di Cauchy) simrnetrico che agisce in ogni punto P* interno a C,; nel caso in oggetto diventano

o equivalentemente

E le equazioni di equilibrio di Cauchy in assenza di forze di massa d i v 9 = 0 VP* E C:

(7)

associate alle condizioni a1 contorno

essendo f 6 il vettore densit&delle forze di contatto sulla frontiera aC: ed n = nhch il versore della normale interna a aC:, in questo prohlema assumono le rispettive

26

forme

dU1 + N -danUx 221 + L-n3 +(A ax3

I

I

dU2

-

2N)-n1t 8x2

+~a.11Z ae3n 3 + F au3 z n 2 + LGn3= - f 2 Lgn'

dU1 aU2 3 + A-n3 + L-n2 + A-naau2 + 8x1 ax3 x2

i 5 moduli elastici A , C ,F , L , N essendo costanti. Ricordiamo ora che una distorsione di Volterra pu6 essere eseguita nel mod0 seguente: supponiamo di trasformare il cilindro a connessione doppia in uno semplicemente connesso mediante un taglio lungo una sezione generatrice, parallela all'asse di simmetria, che indichiamo con T ; applichiamo quindi a uno dei due lembi del taglio uno spostamento rigido trasltorio di vettore h E 7r e uno rotatorio di vettore k E T a. Lo spostamento rotatorio pu6 portare i due lembi del taglio a sovrapporsi o allontanarsi, a seconda del verso del vettore k; nel primo caso si sopprime in mod0 opportuno yn p6 di materia, nel second0 se ne aggiunge. Si portano quindi i due lembi cosi modificati a contatto e si saldano. I1 cilindro ha subito una distorsione di Volterra di vettori caratteristici h e k. E' stato dimostrato che la scelta di uno spostamento rigido, fatta da Volterra per eseguire una distorsione. 6 una condizione necessaria

a

27

+

x'

Figure 1. I1 cilindro cavo n e h stato naturale e una Sua sezione retta.

I1 procedimento matematico che permise a Volterra di realizzare una distorsione, fu d a lui eseguito nel caso isotropo. 10 ho sviluppato un procedimento analog0 nel caso in cui il cilindro ammetta 5 diversi moduli di elasticit& (cilindro trasversalmente isotropo) 3

Determinazione di un insieme di soluzioni delle equazioni di equilibrio e delle relative condizioni a1 contorno

Ho cominciato col cercare una soluzione del sistema delle equazioni di equilibrio (9) del tipo: 1 u(P*)= -(h 2T

+ k x OP') 6 + [ ( a .OP* + a4) cl+ + (b . OP' + b4) ~2 + ( l . O P *+ /4)cg] logp'

(11)

con 0 = arctan $, h = h' c,, k = k' c, vettori assegnati costanti, a = a' c,, b = bT cT,1 = l'c, vettori incogniti da determinare, e ugualmente gli scalari costanti a4, b,, l4 incogniti; infine p esprime la distanza di P' dall'asse del cilindro. Sostituite le componenti cartesiane dello spostamento (11) nelle equazioni a derivate parziali (9) ho trovato la soluzione delle stesse nella forma:

dove le costanti a1 = b2, 1 3 , 14 sono arbitrarie. Laserra e Pecoraro hanno verificato tutte le formule tramite il programma C.A.S. Mathematica e hanno riscontrato che il parametro l 3 6 a sua volta espresso

28

tramite il parametro arbitrario u1 3

1 =

Nk3-4nAUl 27T(F L )

+

'

(13)

e hanno di conseguenza corretto e dedotto esattamente tutte le formule successive, nonchk hanno riottenuto facilmente le formule orinali di Volterra (relative a1 caso isotropo) nell'ipotesi (dello stesso Volterra) che il parametro arbitrario I4 si annulli. Pertanto in virtti delle (12),(13) le caratteristiche linearizzate di deformazione divent an0 :

e lo stress soluzione delle equazioni di equilibrio (7) diventa

-(N

k3 + A)-2n +(N

k3 -(N + A ) 2n

Xi:) = -F

+(N

(T + ) 2"?: -

-

N k3 logp2 A)-A 2n

-

N k A)-? logp2 A 2n

N k3 logp2 1 - - F-A2n

3:

29

E’ interessante rilevare che la revisione effettuata ha portato a una semplificazione della componente Xg3, in quanto i miei calcoli avevano assegnato a1 coefficiente di k2 il valore

Lo stress (15) deve soddisfare le condizioni a1 contorno (8). A tale scopo consideriamo il contorno aC: come somma delle due superfici laterali, interna C * l , esterna C*2 e delle due basi a;, a; e applichiamo separatamente ad ognuna di quese parti le condizioni a1 contorno (8). Otteniamo 1) S U C * ~( n = c l c o s O + c z s i n O , x l = r c o s O , x ~ = r s i n B ,0 5 z 3 5 d )

2) su C*2 ( n = -c1 cose - cgsint?,

X I = RcosB,

x 2 = RsinB, 0 5 x 3 5 d )

= (fda.1

30 4)

S U C Y * ~( n = - c g ~ ( O , O , - l ) ,

x3=d)

Non i: difficile verificare che la sollecitazione cui B soggetto il contorno del cilindro, ora individuata, B equilibrata. I primi termini presenti nei secondi membri di (16) sono Nh2 Nh’ L -9 -~ , -2-14 7i-r xr r e gli analoghi in (17)

Nh’ L -,irr 2-14 7i-r r esprimono due sollecitazioni costanti, parallele e discordi ripartite la prima su C; , la seconda su C; , ciascuna equivalente a1 proprio risultante applicato nel centro del cilindro. Valendo le uguaglianze

_-N h 2

1

l;

F d C ; = F. N h 2 2irrd = 2Nh2d

Nh2

-

Nh2

--

irR

.2irRd

=

-2Nh’d

le due sollecitazioni (20),(21)si fanno equilibrio. Passando ai termini residui dei secondi membri di (16) e (17) B verificabile in mod0 analog0 che per ogni valore di x3 essi esprimono una coppia di braccio nullo. Percib l’intera sollecitazione agente sulla superficie laterale del cilindro B equilibrata. Infine, le sollecitazioni agenti sulle basi a; , a; sono anch’esse equilibrate come mostrano le (18),(19)non appena si ammette che il parametro l 3 risulti nullo. Questa ulteriore condizione implica che dalla formula (13) si tragga

N k3 4Air Osservo che, senza l’intervento della condizione ( 2 2 ) determinata d a Laserra e Pecoraro, io avevo potuto dimostrare l’equilibratezza della sollecitazione agente sul contorno del cilindro solo invocando il teorema di d a Silva. In definitiva il campo vettoriale di spostamenti, soluzione delle equazioni indefinite di equilibrio b espressa dalle formule a1=-.

u’= & [(h’ - k 3 z 2 +

+ k1z3+ Zk3x1) l0gp2] (23)

31 In esse B visibile che il campo di spostamenti dipende unicamnete dal rapport0 dei due moduli di elasticita N e A. A1 contrario il campo di stress soluzione delle equazioni ( 7 ) , che si ottiene d a (15) dopo aver tenuto conto di (13),(22), assume la forma

In queste ultime & visibile che lo stress soluzione dipende d a quattro dei cinque moduli elastici, oltre che dal parametro arbitrario 14. Infine, in virtu di (22) (corrispondente alla condizione i3 = 0) lo stress agente sul contorno del cilindro, gi&espresso da (16), (17), (18), (19), assume la seguente espressione definitiva:

Utilizzando le (23), Laserra e Pecoraro hanno anche verificato che, se le cinque costanti elastiche A, C, F, L, N introdotte nel caso di un cilindro trasversalmente isotropo soddifano le condizioni A=C, L=N, F=A-2N

(29)

32 e si pone A=A+2p, N = p , /4=0,

(30)

si ricade nel caso di un cilindro isotrop e le (23) danno, come caso particolare, lo spostamento utilizzato da Volterra per sviluppare la sua Teoria:

In definitiva la ricerca di soluzioni del problema a1 contorno reiativo a1 sistema di equazioni a derivate parziali del 2’ (9) con le condizioni (10) sulla frontiera ha portato a1 campo di spostamenti (23) e a1 campo di stress (24)’ entrambi dipendenti da un parametro arbitrario, l4 . 4

Genesi di una distorsione

Nel precedente paragrafo I11 abbiamo verificato che se il cilindro circolare reto preso in esame, subisce uno spostamento polidromo isotermo C: + espresso da (23) e conseguentemente uno spostamento una deformazione regolare, a partire da una configurazione iniziale C: (a temperatura T ) che sia uno stato naturale, e quindi esente da forze esterne di massa e superficiali, nella configurazione di equilibrio sono presenti uno stress dato da (24) in ogni punto interno, e sul contorno una sollecitazione espressa dalle relazioni (25)-(28) che risulta equilibrata. Supponiamo ora di considerare il cilindro in uno stato naturale a temperatura I-, C: , e di agire sul suo contorno con la sollecitazione equilibrata (25)-(28); possiamo individuare uno spostamento regolare u’ # u”, eventualmente polidromo, ma con il medesimo asse di polidromia O x 3 ,che porta il cilindro in una diversa configurazione di equilibrio C: . Cib premesso, pensiamo di eseguire sulla configurazione C: lo spostamento

cT

cT

u” ( P * )= u ( P ” )- u’ ( P * )

(32)

cui corrisponde un tensore di deformazione linearizzato q” = q - q’, regolare, non identicamente nullo, e quindi uno sterss interno XLk. I1 cilindro, inizialmente nella configurazione C:, per effetto dello spostamento (32) assume una configurazione di equilibrio C: sulla cui frontiera non agiscono forze esterne e che pertanto i:

[ “1

una configurazione di equilibrio spontaneo. Le funzioni u sono polidrome e ammettono l’asse di simmetria del cilindro come asse di polidromia. E’ lo spostamento u” ( P * ) applicato al cilindro nella configurazione C: che volterra chiamb una distorsione e suggeri di eseguire come ho indicato nel precedente paragrafo 11, ed ho verificato nella mia Nota In andogia con quanto fece, Volterra su particolari corpi isotropi ‘j, occorre determinare uno spostamento u # u e sperimentare la teoria su qualche tip0 di cristallo che ammette un asse di simmetria elastica.

’.

33 References 1. G. Caricato, O n the Volterra’s distortions theory, Meccanica, 35,pp. 411-420, 2000. 2. G. Grioli, Le distorsioni elastiche e l’opera di Vito Volterra, International Congress in memory of Vito Volterra, Roma, 8-11 October 1990, printed in Atti dei Convegni Lincei, n. 92, 1992. 3. E.Laserra and M.Pecoraro, Volterra’s theory of elastic dislocations for a transuersally isotropic homogeneous hollow cylinder, Non Linear Oscillations, v01.6, No.1, (2003), pp. 56-73. 4. A. E. H. Love, The Mathematical theory of elasticity, Cambridge University Press, Fourth edition, 1952. 5. F. Stoppelli, Sull’esistenza di soluzioni delle equazioni dell ’elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio, Ricerche di Matematica 1957, 6,pp. 241-287, 1958, 7 pp.71-101, 138-152. 6. V. Volterra, Sur l’equilibre des corps elastiques multiplement connexes, Annales Scientifiques de 1’Ecole Normale Superieure, vol. 24, 1907, pp. 401-518. 7. C.C.Wang and C.Truesdel1, Introduction to Rational Elasticity, Nordhoff International Publishing, Leyden, pp.500-503.

HEAT AND MASS TRANSPORT IN NON-ISOTHERMAL PARTIALLY SATURATED OIL-WAX SOLUTIONS* ANTONIO FASANO AND MARIO PRIMICERIO Dipartimento di Matematica “U.Dini ”, Uniuersita’ di Firenze, Viale Morgagni 6 7 / A , 501 34 Firenze, Italia

Deposition of wax a t the wall of pipelines during t h e flow of mineral oils is a phenomenon with relevant technical implications. In this paper we present some general ideas about one of t h e main mechanisms at t h e origin of wax deposition, i.e. diffusion in non-isothermal solutions. We formulate a mathematical model taking into account heat and mass transfer in the saturated and in the unsaturated regions, as well as the process of segregation (and dissolution) of solid wax and its deposition on t h e boundary.

1

Introduction

The aim of this paper is to better understand a phenomenon which is of crucial importance for instance in the pipelining of waxy crude oils (WCO’s), i.e. of mineral oils containing heavy hydrocarbons (with the generic name of wax or in more specific cases of wax). The presence of wax makes the rheology of WCO’s extremely complicated. The literature devoted to the technology of WCO’s is quite large The peculiar phenomenon inspiring and we refer to the recent survey paper this paper is the wax deposition on the pipeline wall. Although the question is somehow controversial, various authors propose that wax migration to the wall is mainly driven by two mechanisms: (1) displacement of crystals suspended in oils saturated with wax, due to presence of a shear rate, (2) molecular diffusion to the wall generated by a radial concentration gradient in the saturated oil, induced by a thermal gradient (a typical situation encountered in submarine pipelines, where heat loss to the surroundings takes place in a significant way). Although there are flow models including (1) and (2) (separately or simultaneously) a theoretical investigation of such processes is missing. While (1) has a purely mechanical origin and is connected to the flow, the latter mechanism can be studied also in static conditions and in a small scale laboratory device. The paper is a first attempt to derive wax diffusivity from experimental observation of deposition on a cold wall. Here we want t o investigate the specific problem of diffusion-driven migration in saturated solutions in a non-uniform thermal field, including the phenomenon of deposition of the segregated material on part of the boundary. In this problem has been considered precisely in the framework of WCO’s pipelining. In our paper the physical situation we want to discuss is different because we deal with static conditions in general geometry, allowing some diffusivity of the segregated phase and the onset of desaturation.

’.

*WORK PERFORMED IN THE FRAMEWORK OF THE COOPERATION BETWEEN ENITECNOLOGIE AND I2T3

34

35 Our approach will be mainly focussed on the mass transport process, in the sense that the (rather weak) coupling between this process and the evolution of the thermal field will be neglected in a first instance. However we will also give some hints on how to deal with the case in which such a coupling is taken into account. The aim of this paper is to present some general ideas, but further developments will be necessary to deal with the WCO’s flow problem, where the transport of the various components is likely to be substantially influenced by the strain rate of the mixture. In the next section we illustrate some general features and we present the classical statement of the problem. A weak solution is defined in Sect. 3. Generalizations with (i) a coupled thermal field, and (ii) segregated phase in a gel state are subsequently discussed.

2

The basic model

Consider a bounded domain R c !R3 with smooth boundary, filled by a WCO at rest in which the dissolved substance S is monocomponent or behaving as such. z E R, t 2 0 the temperature of the Denote by T ( z , t ) and by CTOT(Z,~), mixture and the total concentration of S. For T in the range in which the system is liquid a saturation concentration cs(T)is defined, so that at all points where CTOT > cs a solid segregated phase is present with concentration

where [ f ] + = m a z ( f ,0 ) . According to experience cs(T)is a given function, positive, increasing and smooth.

Remark 1 Although c,(T) represents the solute concentration in the liquid phase, when we write (2.1) we use it as the concentration in the whole system (liquid plus the segregated phase). T h i s is acceptable if the solvent i s relatively abundant, a situation which will be assumed throughout the paper. Then, we define the concentration of the dissolved substance

c ( z , t )= min[crro~(z,t),cs(T(s,t))l,

(2.2)

so that in any case

+ G(z,t).

CTOT(z,t) = C(zc,t)

For the specific case of WCO’s the densities of S (both in its dissolved and segregated phases) are essentially equal to that of the solvent and their variations with temperature are negligible in the range of interest. Therefore we assume that the density p of the mixture is constant and that sedimentation due to gravity can be neglected, at least on the time scale we are interested in.

Remark 2 Consistently with Remark 1 , we suppose that solvent i s at rest so that convection is systematically neglected. In the same spirit, even in the presence of

36 a growing layer of solid wax on the boundary, neither the thickness of the layer nor the displacement of its front affect the transport process. We will resume this question in a more general framework in a forthcoming paper adopting the point of view of mixture theory. The approximation adopted here can be of help in most practical cases. Consider the region R s c R where G > 0 (saturated region). There, the dissolved phase has concentration c s ( T ( x ,t ) ) and its mass balance equation

-acs _

DV2cs = Q ( x ,t ) ,x E R s , t > 0 (2.3) at provides the expression of the volumetric rate at which the segregated component S dissolves (Q > 0) or is produced (Q < 0) in terms of the thermal field: Q ( x , t )= c L ( T ) ( T~ D V 2 T ) - D c ~ ( T ) ( V T ) E~ ,RZs , t > 0.

(2.4)

The corresponding mass balance for the segregated material is

aG at If the region R \ R s is non-empty, there C

-- D G V ~ G = -Q(x, t ) ,x

E Rs,t

> 0.

(2.5)

= CTOT and we have pure diffusion

ac

- - DV2C=0,a: E R s , t > 0. at

(2.6)

Let us assume that the part of the boundary of aRs not lying on a R is a set r ( t )which is the union of a finite number of connected components that are smooth surfaces. There, we impose the continuity of CTOT implying

Glr = 0 , Clr

= cs(Tlr)

(2.7)

and the continuity of mass flux, i.e.

Discussing boundary conditions on 6’R is more delicate, because we have to distinguish between its ”warm” part r w (i.e. where

aT

aT > 0, n being the outan -

ward normal) and its ”cold” part T c (where - < 0). Indeed, only on the latter an deposition may take place. On l?w we impose that the total mass flux vanishes:

aG DGan

+ D-acs an

ac

-=0, an

= 0 , on

on

rws= aRs n rw,

rwu=rw\rws,

(2.10)

According t o Remark 2, we neglect the thickness of the deposit so that deposition front is assumed to lie on 8 0 .

37 Thus, we formulate the conditions on rc starting with rc f l a R s = rcs. We assume that a given fraction x E (0,1] of the incoming flux on TCSis (irreversibly) converted into a layer of solid deposit. Hence (2.11)

Of course x = 0 (no deposit and total recirculation) corresponds t o the trivial case in which mass flux vanishes on the whole of dR, whereas in case x = 1 (all incoming dissolved substance is withdrawn from the diffusion process) the no-flux condition applies to the phase G. Passing to rcu = rc\rCs, two different situations can arise: either C = cs or

ac E an

acs [-,O], an reformulate the condition on I'cu as

C < cs. In the former case

ac

ac

while in the latter - = 0. We can

an

ac

cs 5 0 , - < 0 , (c- cs)= o , x E rcu (2.12) an an i.e. a Signorini-type unilateral condition. To complete the formulation of the mass transport problem an initial condition has to be specified

c

-

Let us come to the equations for the thermal field. We have aT x x E Rs,t > 0, - - asV2T = --Q, at PYO dT - - auV2T = O , X E R\Rs,t > 0.

(2.14)

at

In (2.14) 70represents specific heat and X is the latent heat required to dissolve the unit mass of segregated phase. Moreover as and au are the thermal diffusivities in the two regions. Usual conditions of continuity of temperature and thermal flux are meant to hold on r. We also prescribe boundary and initial temperatures

T ( x , t )= T B ( x ) , xE aR,t > 0 , T(x,O)= T o ( z ) , zE R.

(2.15)

Remark 3 Of course, neglecting the thickness of Cld as well as convection, along ~ 1, is a very relevant simplification of the problem, with the assumption C T O T /<< but the model is still realistic in the practical case of WCO pipelining over a not too long time interval. Although in principle the formation of a thick deposit m a y occur, such a situation m u s t be avoided in practice. T h e model stated in the fixed domain R allows anyway the computation of the growth rate of the deposit. Denoting by Vn the normal component of the growth velocity, we have (2.16) (2.17)

38 Now we introduce non-dimensional variables

C = CfC,

G = GfC,

Es(T) = cs(f'F)/C, f i with

~ / Li ,= 1,2,3, t^ = t/t^,

C,T ,L suitably selected (fcr instance T = maxTlan, C = cs(T),L =- diainS2,

and if

=x

T=T/T,

=

cs

L2

-). D

We can immediately realize that the rescaled velocity IV,l

L

is small

- << 1, consistently with our approach. P

We denote by

e2,9 the operators acting on the new space variables and we set aT - V2T) - EgT)(QT)?

Q=

(2.18)

Equations (2.5), (2.6), (2.13), (2.14) become (2.19)

aT 1 - - -v2T ~

at^

=

ES

_ "- ezd

at^

-6Q,

= 0, in

aT -1v -2 -~= 0, at^ E l l

__

in {G > o>, t

> 0,

{G = o>, t > 0 ,

in { G = o>, t

(2.20)

(2.21)

> 0,

(2.22)

= a s / D ,;E 1 = a u / D ,

(2.23)

where

0 = Dc/D, and

xz1

@=-

(2.24) PCJYOT' The initial and boundary conditions can be written accordingly. In particular, condition (2.11) becomes

-8-

a6 = (1- x)- X S ( T ) , f E rcs, an

an

(2.25)

and (2.9) becomes (2.26) We re-write (2.8) as a jump condition

[-I-+aC i. -&-I-aG =

an

an r + '

(2.27)

39 where ?+ (resp. )?I is the side of dC i., . [XIi.-is the jump.

? facing the

set {G

> 0} (resp. {G 5 0)) and

d dn + C, G -+ G and so on) to simplify notation.

Note that in the above expressions - is also non-dimensional. From now on we eliminate hats

(c

Weak formulation

3

Using generalized derivatives in a standard way, we derive a unified formulation of the mass transfer process, still assuming that equations (2.18)-(2.22) hold classically in their respective domains and that the set {CTOT= cs}nR is a smooth connected surface r(t)for any t . In the following we assume x = 1 for simplicity and we adopt the notation of for functional spaces. We introduce the function

u ( z , t )= CTOT(z,t) - c S ( T ( z , t ) ) , i n

R,

(3.1)

so that

u(z,t ) = G(z, t ) ,where G > 0,

(34

G = 0. u ( z ,t ) = C(z, t ) - cs(T(z, t ) ) where ,

(3.3)

and

Thus u > 0 corresponds to the saturated region (i.e. to the region where segregated phase exists). Moreover, u is continuous across ,'I where it vanishes

ulr,

= ulr- = 0.

(3.4)

If we assume that equation dU

- - V . { [1+ H(u)(O- l)]Vu} = -Q, at

zE

R,t > 0

(3.5)

holds in a suitable generalized sense, we not only encompass (2.19) and (2.21) in their respective domain of validity, but we include the Rankine-Hugoniot condition

which is precisely (2.27). Writing

A(u) and defining on

dR

1

+ H(u)(O- l),

(3.7)

40

we can synthetize (2.9), (2.10), (2.11), (2.12), i.e. the boundary conditions on rwsu rwuu rcsu rcu,as foiiows acs

A(u)- = - ~ ( - , u ) - , acs z an an an aU

E dR,t

> 0.

(3.9)

Coming back to (3.5), it is clear that the function space in which u will be sought has to have enough regularity so that (3.9) is meaningful. Let r be a fixed positive constant and 4 ( z , t )be a test function belonging the space W;,'(R x ( 0 , ~ ) with ) f$(x,r) = 0 ,

x E

a.

(3.10)

Assume that T ( z ,t ) is given and that our problem has a classical solution with regular interface r. Then, passing to the variable u, we write

Then, since for any t E ( 0 , r ) and R1

CR

using (3.4) and (3,6) we have

(3.12) Taking into account (3.9) we finally have that, if the problem admits a classical solution as specified above, then the following equation holds for any r and for any choice of the test functions in the selected space

In (3.13) U O ( Z ) is obviously given by

Then we give the following definition

Definition 1 A weak solution to the mass diffusion problem in QT = R x (0,r)is a function u E HPIP/~(Q,)n W 1 i o ( Q T )for , some /3 E ( 0 ,l ) , satisfying (3.13) for any E W i l ' ( Q T )vanishing f o r t = r. Now, assuming u is known, we go back to the thermal problem. Writing 1 a(u)= EU

+ H ( u ) (ES-1

-

1 -)

EU

(3.15)

41

we combine (2.20) and (2.22) as follows

Tt - V . ( ~ ( u ) V T =)-OH(u)Q, x E R, t

> 0.

(3.16)

Equation (3.16) includes continuity of temperature and heat flux across I?, so that the problem is in the class of the so-called ”diffraction problems” (see 6).

Remark 4 We note that in the approximation 0

N

0 and

E(I

= 6s = E , implying

1

a ( u )= -, the thermal problem is completely uncoupled, so that T can be regarded E as a known function of x and t with the required regularity at the boundary

4

An alternative model

So far diffusion has been considered as the transport mechanism of the segregated phase. It makes sense, however, t o deal with the extreme case in which DG = 0. A remarkable example is the one of those WCO’s in which the segregated wax in static conditions aggregates producing a gel structure. In this case we can no longer rely on the diffusivity of G to ensure instantaneous equilibrium between the phases and we have to revise our approach drastically, admitting the possibility that for instance we have G > 0 even in presence of desaturation. In that case the dissolution of the segregated phase will not be instantaneous, but will develop with some relaxation, i.e. according to some kinetics. This point of view is not necessarily peculiar t o the case we are discussing. We may think of an intermediate situation in which DG > 0, but diffusion is not effective enough t o supply all the material that would be necessary e.g. t o prevent immediate desaturation at the dG dcs(T) D= 0, ”warm” wall. In that case, instead of the balance condition DGan an we would have a condition of the type

+

where

&

is a positive constant, if we have chosen e.g. a simple linear dissolution

kinetics. Thus, if &G

< D-

the solution will become desaturated and, dG aC instead of the former condition - = 0, we have now DG- = 0. an an an Many more changes are necessary. Here we want to deal briefly with the case DG = 0. Once this choice has been made, various scenarios are still possible. Indeed we may or may not allow a substantial degree of oversaturation. If we admit oversaturation, now denoting by C ( x ,t ) the concentration of the dissolved substance, we can describe both segregation and dissolution by means of a kinetic equation of the form

dn

ac

where

+

42

(a) f > 0 if C < cs(T) and G > 0 (dissolution), with f(C,T,O) = 0, f being continuously differentiable w.r.t. C,T and also w.r.t. G, for G > 0, while it is only required t o be Holder continuous for G = 0 (b)

8.f = 0 (segregation) f < 0 if C > cs(T),with -

aG

(c) for C = cs(T) we can take f = 0, if we suppose that phase segregation is generated only through supersaturation. However, we can choose to exclude supersaturation and keep (4.1) only for dissolution, replacing it by

aG = -Q

at

when Q , defined by (2.4), is negative, meaning that segregation takes place with no relaxation. Clearly, C satisfies in any case

If we impose the obstacle C 5 cs(T),a saturated region may still exist but of course the conditions (2.7), (2.8) on the interface r must be modified as follows

Clr = cs(T\r),

(4.4)

All other boundary conditions must be changed. First of all, only total deposition makes sense. If there is no supersaturation, the growth rate of the deposit is still related to -as in (2.16) with

dn

x = 1.

If however the solution is supersaturated the deposition mechanism can be described by a law of the type

V,

1

= --F(C

P

dT an

- cs(T))H(--),for

C > cs(T),

with F' > 0 and F ( 0 ) = 0 , expressing that the deposit growth rate is proportional to the supersaturation degree and is not zero only at those points of the boundary when heat flows out of the system. We can extend (4.6) to the whole of dR by taking F ( 6 ) = 0 for 6 5 0. Accordingly, the boundary condition for C will be

aC Dan

= F ( C - cs(T)).

(4.7)

Of course we must specify the initial value Co(z) of C. Equations (4.1) or (4.2) require the initial condition G(z, 0 ) = Go(z). Extinction of G at a place where Go(.) > 0 can take place in a finite time only if the function f in (4.1) is not Lipschitz for G = 0.

43 Passing to the thermal problem, if we do not neglect the latent heat, we must ex-

aG

press the source term as A--, irrespectively of the way we model phase transition. at Further discussion of this model is out of the scope of the present paper.

5

Analogy with fast chemical reaction problems

In order to analyze problem (3.13), (3.16), we first consider the simplified case in which heat diffusion is much faster than mass diffusion. As a matter of fact, taking ES = EU >> 1 , O << 1, temperature satisfies

V2T=0,

x

E fl,t > 0

(5.1)

with conditions

T ( z , t )= T B ( Z ) ,

zE

afl,

(5.2)

and problem (5.1), (5.2) can be solved independently of the knowledge of u. Thus we are led to considering (3.13) where Q, given by (2.18), is an assigned smooth function of x and t , while v, given by (3.8), is a prescribed graph of u, depending in a known way on x and t . In this case, we can identify (3.13) with the weak formulation of a problem modelling the transport of two chemical substances diffusing in a solvent and undergoing an immediate reaction at the reaction front (playing the role of r in our model). The concentration of the two species are G and cs(T)- C. The fast reaction problem of two diffusing species has been studied in and in '. There are some differences with respect to the scheme treated here (the most important is the presence of u in the boundary term in (3.13)), but the same technique can be used in our case to prove well-posedness. We will not deal with such details. The model including the thermal problem is obviously more difficult. We envisage the following strategy. Let h = r / n and define u(x,t ) ,T(x,t ) for t E (0,h ) to coincide with the initial data. Then, for t E (h, 2h), (i) solve (3.16) where u ( z ,t) is replaced by u(x,t - h); (ii) solve (3.13) where, in 77 and Q, T(x,t ) is replaced by

T(x,t - h).

Of course, (i) is a standard "diffraction" problem, while (ii) is a problem of the fast chemical reaction type just considered. Iterating the procedure we find a pair (Th,u h ) . Convergence can be proved on the basis of a compactness argument. Again, we postpone the analysis of the details to a forthcoming paper.

6

Conclusions

We have modelled mass transport in non-isothermal solutions in the presence of a segregated phase in the case in which all the components (including the solvent) have the same density. A relevant application is the one of waxy crude oils, where

44

such a phenomenon (molecular diffusion) is one of the main mechanisms of wax deposition on the pipe wall during transport. We consider the case in which the solvent is relatively abundant and the thickness of the deposit is negligible, leaving t o a forthcoming paper the study of a more general situation. As long as the segregated phase is present - in equilibrium with the solution the model describes the following processes: a) diffusive mass Aow within the solution towards the cold wall, induced by the thermal gradient. b) the convexflow of the segregated phase towards the warm wall c) the mass exchange between the solute and the segregated phase. Although we disregard the geometric and kinematic effects of deposition, the corresponding boundary condition for mass transport is discussed in detail. The situation is much more complicated when a region appears in which the concentration of the solute is below saturation. In this case we give a generalized formulation of the corresponding free boundary problem (the free boundary, in simple geometric cases, is the surface separating the unsaturated solution from the saturated region) including the nontrivial analysis of the boundary conditions that are formulated in terms of unilateral (or Signorini type) constraints. We also note that, in simple geometric situations, the problem can be essentially reduced to the parabolic free boundary problem modelling a fast chemical reaction.

References 1. A. FASANO, L. FUSI,S. CORRERA, Mathematical models f o r waxy crude oils. To appear W a x diflusivity: i s it a physical 2. S. CORRERA,M . ANDREI,C. CARNIANI, property or a pivotable parameter? Accepted or publication on Petroleum Science and Technology. 3. L. FUSI,O n the stationary Bow of a waxy crude oil in a loop. Nonlinear Analysis, 53 (2003) 507-526. C.D. HILL,O n the movement o f a chemical reaction interface. 4. J . R . CANNON, Indiana Math. J. 20 (1970) 429-454. A. FASANO, Boundary value multidimensional problems in fast 5. J.R. CANNON, chemical reactions. Arch. Rat. Mech. Anal. 53 (1973) 1-13. 6. O.A. LADYZENSKAYA, V.A. SOLONNIKOV, N.N. URALCEVA, Linear and quasilinear equations of parabolic type. AMS Translations of Mathematical Monographs 27, Providence R.I. (1968).

NEW A P P L I C A T I O N S OF A V E R S A T I L E L I A P U N O V FUNCTIONAL J. N. FLAVIN Department of Mathematical Physics, National University of Ireland, Galway. Ireland. E-mail: James.FlavinOnuigalway.ie. I n onore d i Salvatore Rionero, il mio fratello italiano,da cui ho imparato molto sia riguardo alla matematica che alla vita, nell’ occasione del suo settantesimo compleanno. The paper considers the nonlinear diffusion equation where the diffusivity depends on the dependent variable. Unsteady and steady states, corresponding to Dirichlet boundary conditions, independent of time, are addressed. The rate of convergence, as the time t --f m,of the unsteady to the steady state is studied by obtaining an upper estimate for a Liapunov functional governing the perturbation. A similar estimate is obtained for the i.b.v.p. for the perturbation backwards in time, and it is proved that the solution fails to exist for sufficiently large time. In all of the foregoing a diffusivity appropriate to a (particular case of a) porous medium is assumed. An analogous issue is considered for steady state diffusion in a right cylinder with Dirichlet boundary conditions on its lateral surface, independent of the axial coordinate. The rate of convergence of the solution to the corresponding two-dimensional solution - as one recedes from the plane ends - is studied, using a methodology similar to that used in the previous context.

1

Introduction

In previous papers a novel, versatile Liapunov functional was used to obtain timedecay/asymptotic stability estimates for perturbations to steady states in a variety of nonlinear thermal and thermo-mechanical contexts “11- [3]], Moreover, it was shown in [4] that the versatility of the functional extends to certain nonlinear elliptic boundary value problems in a right cylinder, the axial variable in this context replacing the time variable in the previous one.The results presented in this paper represent further developments of this ongoing work: issues addressed in [l]and [4] are reconsidered under different assumptions. In the first case, an initial boundary value problem is considered for the diffusion equation, where the diffusivity depends on the dependent variable, and it is supposed that the dependent variable is specified on the boundary as a function of position only . The rate of convergence (as the time t -+ m) of the unsteady to the steady state is considered by obtaining an upper inequality estimate for a positive definite measure (Liapunov functional) for the perturbation. This is acccomplished by means of first order differential inequality techniques. In [I] the fundamental assumption is that the diffusivity is bounded below by a (given) positive constant, and it is found that the decay rate is (at least) exponential. In this paper, the fundamental assumption is changed: it is supposed that the diffusivity is proportional to the square of the dependent variable and that the dependent variable is positive/ non-negative e.g. porous medium. In this context the estimate obtained gives a different (upper) decay rate. Various aspects of this decay estimate are discussed. Moreover, the backwards in time initial boundary value problem for the perturbation is considered: a similar methodology is used to obtain a lower bound for the 45

46

relevant Liapunov functional, and one may conclude from this that the solution fails to exist for a sufficiently large (computable) time. In the second context, a steady state diffusion problem is considered for a right cylinder. The transverse diffusivity depends on the dependent variable while the axial diffusivity is constant. It is supposed that the dependent variable is specified on the boundary of the cylinder, its values on the lateral boundary being independent of the axial coordinate. The issue is the convergence of the solution to the solution of the corresponding two-dimensional problem induced by the lateral boundary conditions. This is done by obaining an upper inequality estimate for a cross-sectional, positive definite measure (cf. Liapunov functional in the previous paragraph). This is accomplished by second order differential inequality techniques. In [4]the fundamental assumption is that the transverse diffusivity is bounded below by a given positive constant, and it is found that the cross-sectional estimate for the perturbation decays (at least) exponentially away from the two plane ends. In this paper, the fundamental assumption is changed: it is supposed that the transverse diffusivity is proportional to the square of the dependent variable, and that the dependent variable is positive/ non-negative [analogous to the assumption made in the previous section.] 2

Steady, U n s t e a d y , and Perturbation Problems.

Consider a spatial region R with smooth boundary a R . Consider T ( x ,2) satisfying (with k ( T ) denoting the diffusivity at T ,t being the time)

subject to

T ( x , t )= T(x) on 80 and subject to

T(x, 0 ) = f ( x ) in 0. This is referred to as the unsteady state problem. The corresponding steady state solution U(x) satisfies

subject to

U(x) = T ( x )on 80. The perturbation defined by

u=T-U

(3)

47 satisfies, with G

U

q u ;U ) = I d

u J’ k ( 7 + U)d7-,

0

0

the initial boundary value problem

subject to

u(x,t) = 0 o n dR

(9)

u(x, 0) = f(x) - U ( x ) .

(10)

and

In (8)and subsequently the subscript u means partial differentiation with respect to u. The issue to be addressed is the rate of convergence, as t --t 03, of the unsteady to the steady state. Let us recall a result obtained inter alia in [l]: Assume that the diffusivity k(0)satisfies

w.1 2 ko

(11)

for all values of the argument, ko being a positive constant. Defining

E ( t )=

s,

@dV,

where is defined as in (7), it may be proved, under assumption ( l l ) , that the measure defined by (12) is positive definite in u;it is thus termed a Liapunov functional. One may prove, via a first order differential inequality, that

E ( t ) 5 E ( 0 )exp(-2kooXlt)

(13)

where XI is the usual, lowest ‘fixed-membrane’ eigenvalue for the region R. In the present paper the assumption (11) is dropped and we suppose that k(r) = r2

(14)

corresponding to ( a particular case of) a porous medium. In these circumstances the measure defined by (11) is again positive definite in u ( and is, again, thus termed a Liapunov functional): that this is so is readily seen from the explicit form of (7) under the assumption (14) i.e.

q u ,U ) = (u2/12)[(.

+ 2U)2 + 2U2].

(15)

In the derivation of the asymptotic properties of (12) in the present circumstances the following assumptions are made: (i) The dependent variable is supposed to be non-negative i.e.

48 (implying, in particular, that 5?, f - arising in (2),(3) -are non-negative). (ii) Classical solutions are implicitly assumed, although the results obtained are, in fact, valid for suitably defined weak solutions. Straightforward calculation (as in [l]) gives

d E = -/(VB,)2dV dt

n

X1 being the lowest 'fixed membrane' eigenvalue of 0. To make progress, we need an inequality (treated in the appendix) of the type where K, 6 are positive constants - given by

K = 819, S = 312. One thus deduces from (17)-(19) that

< -KX1

dt -

(19)

1

G3l2d0.

Applying the Holder inequality to (20) one deduces the differential inequality for E

where V denotes the volume of the region R.Integration of (21) yields the following: Theorem 1. T h e Liapunou functional (12) for the perturbation u , defined by (12), satisfies

E ( t ) 5 E(O)[l+ 2 - 1 K X 1 V - 1 / 2 E 1 / 2 ( 0 ) t ] - - 2

(22)

where K is given by ( 1 9 ) , V is the volume of the region R, and XI i s the lowest fixed membrane eigenualue for the region fl - assuming that the diffusiuity is given by (14) and that T , U are non-negatiue. Remark 1. It is clear from (22) that

E ( t ) 3 0 as t

co,

giving convergence of the unsteady state T to the steady state U,in the measure E. Remark 2. The question arises as t o which of the upper estimates (13),(22)corresponds to the slowest decay (assuming that both are valid): It may be proved, by elementary means, that the latter gives slower decay than the former provided

E 1 / 2 ( 0 ) V - 1 / 2 < 2koo/K (i.e. provided that the initial state is sufficiently close t o equilibrium); if, however,

E 1 / 2 ( 0 ) V - 1 / 2 > 2ko/K

49

(i.e. provided that the initial state is sufficiently far from equilibrium), the former gives slower decay than the latter for sufficiently small (easily computed) times. Remark 3. It is possible to deduce from ( 2 2 ) , similar decay properties in the L1, L2, norms (making certain assumptions): (i) The L1 estimate follows from (22) together with (15):

on using Schwarz's inequality and implicitly assuming the convergence of the integral involving U . (ii) Assuming classical solutions together with

T 1 M,

M being a positive constant, it follows from the maximum principle etc. that

The Lz estimate follows from this and (22). We now consider the backwards in time problem for the perturbation u: this is formally identical to that defined by (8)-(10) etc., except that -&/at replaces dufat in (8). Again assuming non-negative solutions T , U , a similar analysis gives

using the same notation as previously. Integration yields

E-1/2(0)- 2-lKX1V-lj2t > E - l f 2( t ) .

(24)

Choose t so large that the left-hand side of (24) is negative i.e.

t

> 2E-'I2 ( 0 )V1I2/ (KX1).

(25)

In these circumstances, (24) would contradict the non-negative character of E ( t ) , whence one deduces non-existence of solution for times satisfying (25). For times for which there is existence of solution,(24) may be expressed as

E ( t ) 2 E(0)[1- 2-'KX1V/-1/2E1/2(0)t]-2

(26)

One summarizes these results in the following Theorem: Theorem 2. For the backwards in time initial boundary value problem in u , specified b y (8),(10) etc., but with -dufdt replacing duldt, and assuming nonnegative solutions T , U , one has non-existence of solution for times t satisfying (25), but for times for which the solution exists one has the estimate ( 2 6 ) .

50 Analogous Estimate for a Steady Diffusion Problem in a Right Cylinder

3

In this section we consider a matter relating to a steady state diffusion problem in a right cylinder which is somewhat analogous to the issue just addressed, and which is moreover amenable to a somewhat analogous treatment. Let x = ( x ~ , x z 2) ,3 denote rectangular Cartesian coordinates, and consider the right cylinder ~

(

o <) Z 3 < e,

2x ~

D(x3) being the cylinder cross-section (at 2 3 ) , C being a constant. We consider the steady state boundary value problem therein: T ( x ,2 3 ) satisfies

+

V1 { b ( T ) V i T } T,33 = 0

(27)

T = T ( x ) on a D ( z 3 ) x 0 < 2 3 < l

(28)

T ( x , O ) , T ( x , t= ) specified.

(29)

subject to

and

Let U (x) now denote the corresponding two-dimensional solution induced by the lateral boundary conditions: U ( x ) satisfies

V1 {b(U)V1U}= 0 in D

(30)

U = p ( x ) on d D .

(31)

subject to

Defining the perturbation u = u ( x ,2 3 ) , as in the previous section:

T ( x ,2 3 ) = U(X)

+U ( X ,231,

it is found that u satisfies the boundary value problem - on again defining (7) -

V ; a u+ ' 1 1 , ~=~ 0 in D ( x 3 )x 0 < z3< C

(32) as in

(33)

subject to

u = 0 on d D ( x 3 ) x 0

< 23
(34)

and

u ( x ,0 ) , u ( x ,l ) = specified

(35)

(in terms of the assigned values of T , and of U ) ; o : denotes the Laplacian in the plane. The issue to be addressed is the rate of convergence of T ( x ,2 3 ) to U ( x ) as one recedes from the plane ends (e.g. with 2 3 ) . We recall a result obtained in [4]:

21, 2 2

51

Assume that the transverse diffusivity k satisfies (11). Defining the cross-sectional measure of the perturbation u by

(at the cross-section 2 3 ) , it follows from assumption (11) that this measure is positive-definite in u (cf. the Liapunov functional in §2).Moreover, one may prove, via a second order differential inequality for E ( x 3 ) ,that

E(x3) 5 E(O)exp[-d%%~3]

+ E ( L )e x p [ - d G ( e

- x3)]

(37)

A1 now denoting the lowest 'fixed membrane' eigenvalue for the cross-section. This shows that the effects of the perturbations on the ends decay therefrom at least exponentially. In fact, somewhat stronger results are obtained in [4]. Let us know suppose that Ic(7.) = 7.2,

(38)

as in (14). Let us furthermore assume positive, classical solutions throughout although the final results continue t o be valid for non-negative weak solutions. One may establish [as in [4]] that the second derivative E" of E (with respect t o 2 3 )satisfies

E"(z3) 2

AilD,

QPEd-4

(39)

13)

Using (18)-( 19) and applying Holder's inequality,we obtain the second order differential inequality E"(Z3)

- K A I A - ' / ~ E ~ 2/ ~ 0, ( ~ ~ )

(40)

where K is given by (19) and A is the area of the cross-section. One may obtain an upper bound for E(23) by invoking a theorem given in [ 5 ] : in the present circumstances, one obtains

subject to

G(O) L

JV-4,G ( [ ) 2 F ( [ ) .

(43)

One possible choice for G(x3) is now made, which does not, however, exhaust all the possiblities. Try

G ( x 3 )= E(0)[1+C

Y ~ ~ ] - ~

(44)

where p , a are positive constants chosen with a view t o securing equality in (42). One readily finds that p = 4, a = J m { F ( 0 ) } ' / 2

(45)

52 where

m = KXlA-l/’/20

(46)

X1 being the lowest ‘fixed membrane’ eigenvalue for the cross-section, A being its area. Bearing (41) - (46) in mind, one finds

+

I E ( o ) [ ~~ { E ( O ) ) ” ~ Z ~ ] - ~

~ ( 3 3 )

(47)

provided that

+

E(O)[I ~ { E ( o ) ) ~ / ~ 2 cE(c). ]-~

(48)

One notices that (48) is satisfied, in particular, if E(C) = O.The convergence of the solution of the steady state problem (27)-(29) etc. to that of the corresponding two dimensional problem (30)-(31) etc., in certain circumstances, is thus conveyed by the following theorem. Theorem 3. Assuming that the transverse diffusivity is give by (38))and assuming non-negative classical solutions of (27)-(29) etc., and of (30)-(31) etc., the cross-sectional measure (36) of the perturbation u satisfies the decay estimate given by (47)) (46), provided that the condition (48) holds. 4

Closing remarks

A more general treatment of Section 2, using a somewhat different methodology, will appear in [6]. 5

Appendix

We require an inequality of the type

2 K@ where K , S are positive constants, i.e. we require an inequality

(u + U ) 3 - U 3 2 9K[(u2/12){(u + 2U)’

+ 2U2}I6

provided (in both cases) that

u + u 2 0,u 2 0. Now (50) may be expressed (provided U

# 0 pro

+

tem.)

U ~ [ ( ~ / U1)3 - 1122 9 ~ [ i 2 - ~ ( ~ / u ) ~ + { (q2 ~ /+u2}]6u46 For ‘universality’ we require

S = 3/2, comparing the powers of U on either side. Putting p =u/u+ 1

53 for convenience

(U# 0), (52) may be expressed (p3 - q22 ~ . 9 . 1 2 - ~ / -~ [ ( ~

+ q2+ 2}i3?

(54)

Putting p = 0 gives

K 5 23.3-2.

(55)

Let us take

K = 23.3-2, noting that this gives equality in (50) when U = 0.

(56)

Excluding the trivial case U = 0,(54) holds with K given by (56) provided that, for p 2 0, (p3 -

q23 3-3/2[(p - I ) ~ { ( I~ )++~ 2}13/”

This is true provided that

(p2+p+114 ~ 3 - ~ ( ~ - 1 ) 2 ( ~ 2 + 2 ~ + 3 ) 3 , which is (in turn) true provided that (p2 + p

+

2 3-3(p - 1)2(3p2+ 3p + 3)3,

which is (in turn) true provided that

p2+p+lL(p-1)2 which is (in turn)true as p 2 0. Hence the inequality (49) holds for U given by (53) and (56) respectively.

+ u 2 0, U 2 0,provided that 6, K

are

References

J.N. Flavin and S. Rionero, J. Math. Anal. Appl. 228, 119 (1998). J.N. Flavin and S. Rionero, J. Mech. Appl. Math. 52, 441 (1999). J. N. Flavin and S. Rionero, Cont. Mech. Thermodynam. 11, 173 (1999). J.N. Flavin and S. Rionero, Math. Meth. Appl. Sc. 25, 1299 (2002). M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1967). 6. J.N. Flavin and S. Rionero, J. Math. Anal. Appl. (to appear).

1. 2. 3. 4. 5.

REMARKS ON THE PROPAGATION OF LIGHT IN THE UNIVERSE*

DIONIGI GALLETTO Dipariimento di Matematica, Universiia di Torino, Via Carlo Alberto 10, 10123 Torino, lialy E-mail: dionigi.galleiio @ uniio.ii

BRUNO BARBERIS Dipariimenio di Maiemaiica, Universita di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail: [email protected] Taking into account the three geometrical cases which are possible for the Universe (characterized by positive, negative or zero curvature), a fundamental formula which expresses the property of light of being independent from its source in the point where it is measured is deduced. From this formula it follows that: the classical law of composition of velocities holds true; if the velocity of light for an observer were the same in any point of the Universe, the latter must necessarily be static; the formula relating the red-shift to the expansion; the theory of the horizons and all the various and well-known results of relativistic cosmography connected to the propagation of light. Besides, the local character of special relativity is highlighted by observing that such a theory holds true only at the level of clusters of galaxies, which do not participate in the expansion of the Universe. Finally, once the Friedmann-Robertson-Walker metric is deduced, it is observed that, in agreement with what we have now said, at the level of clusters of galaxies this metric becomes the metric of the Minkowski space-time. In the first part of the paper, for the sake of clarity and for the role that they play in the paper itself, some well-known metrics of the three-dimensional manifolds of constant curvature are deduced.

1 The Cosmological Principle Astronomical observation indicates that, when considered on a large scale, and precisely for distances of at least 100 Megaparsec (Mpc)’, the Universe is homogeneous, in the sense that, wherever one considers a region of it that may be likened to a sphere’ with a radius of at least such an order, the number of galaxies that are to be found in that region is substantially always the same. Astronomical observation indicates moreover that, always on a large scale, the Universe is isotropic, in the sense that it does not present privileged directions on a large scale when seen from our galaxy. This would seem to place our galaxy at the centre of the Universe, in a situation of singular privilege of the Ptolomaic type, a situation that is certainly not logically sustainable and which therefore leads us to consider that the Universe looks exactly the same if it is observed from any other typical galaxy. It is * Research supported in part by G.N.F.M. of the Italian C.N.R. 1 Mpc=106 parsec. One parsec (circa 3.26 light years) is the distance in correspondence to which the

’ ’

radius of the Earth’s orbit, which is assumed to be circular, is subtended by one second of arc. The meaning to attribute to this term is obvious also in the case in which the Universe does not have the geometrical characteristics of the ordinary Euclidean space.

54

55

precisely this conclusion that leads to the assumption that on a large scale the Universe is isotropic not only with respect to our galaxy, but also with respect to every other galaxy3. Leaving aside those aspects of the Universe that, on a local scale, differ from homogeneity and isotropy, and resorting therefore to a rather schematic view of the Universe, the few considerations that have now been made lead, as a conclusion, to the so-called cosmological principle, on which are founded the relativistic treatments of the cosmological models that are now almost universally accepted. This principle, in its usual formulation: 1) the Universe is homogeneous and isotropic4at any time, will be set at the basis of this paper. By leaving aside those aspects of the Universe that, on a local scale, vary from homogeneity and isotropy, and given the cosmological principle, it follows that the objects constituting the Universe (galaxies, quasars, etc.) are to be considered uniformly distributed, and that the Universe may therefore be represented by a continuous medium, the so-called cosmological fluid which is at the basis of cosmology, a fluid that will be indicated by Zt.

2 The Intrinsic Geometry of the Fluid 22 Considering, obviously, that the Universe is expanding, the assumption that the cosmological principle is valid entails (as the statement of the principle itself recites) that each particle of Zt must be considered to be associated with a clock, and that all these clocks must be synchronised among them. We may thus consider 24 to be associated with a universal time, the so-called cosmic time, which will be indicated by t and for which we will assume as its origin (i.e. t = 0) the instant in which the process of expansion began. Without the introduction of this time it would not be possible to speak, for example, of homogeneity for Zt, which entails that the density p of 7% (i.e. the density of the Universe considered on a large scale) is a function of time only: ,u = p (t). Likewise, assumption 1) entails that the manifold V which originated from the configuration assumed by Zt at any instant is a manifold which has the same Riemannian curvature K in every one of its points, i.e. that in such an instant it is a manifold with constant curvature. It follows that at every instant the manifold that originated from the configuration assumed by Zt is a manifold whose Riemannian curvature K depends only on t: K = K (t). In the case of K(t) f 0, indicating by R(t) the correspondent curvature radius, which is connected to K ( t ) by: At this point it is possible to observe that isotropy, excluding privileged directions in every point of the Universe, itself entails homogeneity. Indeed, if the Universe at any given moment were to present itself as being different in two of its different points, to an observer situated in a point of the Universe equidistant from these two points, isotropy would no longer exist. In the case in which the Universe has the geometric characteristics of the ordinary Euclidean space, the assumption of isotropy can be substituted by the very weak assumption that typical galaxies recede from our own with a radial motion (i.e. with their velocities directed constantly from the centre of mass of our galaxy towards them), without assuming that such velocities are solely functions of their distances from our galaxy. Such an assumption necessarily entails isotropy, as has been seen very briefly in [7], 9 and as will be explained in detail in a forthcoming paper.

56 1

K ( t ) = fR2(t) ' with the sign depending on whether it is K(t) > 0 or K(t) < 0, with regard to the curvature K(t) there are three possibilities for Zt: K(t) > 0, where Zr! has the same intrinsic geometry as a spherical hypersurface of radius R(t), which will be indicated by S 3; K(t) = 0 , where 2% has the same intrinsic geometry as the ordinary Euclidean space E 3; K(t) < 0, where Zt has the same intrinsic geometry as a pseudospherical hypersurface of radius R(t) embedded in the ordinary pseudo-Euclidean Minkowski space, a hypersurface which will be indicated by PS Leaving aside the many cases of isotropic (and therefore homogeneous) manifolds that can exist in correspondence with the three cases seen above, in accordance with what is generally done in the relativistic treatment of homogeneous and isotropic models of the Universe, in this paper we shall consider the cases that appear to be the most natural, i.e. the cases in which the Universe, and that is K, has the configuration of S (with radius R(t)), or of E or of PS (with radius R(t)),and precisely of one of the two sheets that compose it.

'.

',

3 The Metrics for Kin polar coordinates in Cases S and PS Leaving aside for the moment case E for which a finite radius of curvature does not exist, Zt may be imagined as embedded in a Euclidean space E or in the ordinary pseudo-Euclidean Minkowski space PE '. The cosmological principle consequently entails that, however a particle P of Zt (i.e. a point of S or of PS 3, is considered, the trajectory it describes in E or in PE ', is situated on a half-line starting in the centre C of 2,i.e. of s or PS In both cases (S e PS 3, these half-lines identify a frame of reference ;Pc (in E and in PP), into which it is possible to introduce a system of orthogonal Cartesian coordinates XI, x2,x3,x4 whose origin is in the origin C of ;Pc. In case S 3, with respect to this system of coordinates this hypersurface is represented by the parametric equations

',

'.

',

'

x1 = R(t)sinXsinficosrp,

x 2 = R(t)sinxsinz?sinq, x3 = R(t)sinXcosI?, x4 =R(t)cosX,

x

with 0 I 5 , 0 I 6 5: n , 0 2 cp 5 2n, from which it follows that the metric of S in the polar coordinates I?, cp is expressed by

x,

do2 = R2(t) ( d x 2 + sin2X ( d f i 2+ sin21? d q 2 ) ) .

(3.1)

57 In case PS

’,in the aforesaid system of coordinates the metric of P E

is

ds2 = ( d ~ ’+)(dx2)2 ~ + ( d ~- (dx4)2 ~ ) ~, while the equation of PS is (X’)2

+(X2)2 +(

x ~-)( ~x ~= )-R2(t) ~ ,

with its sheet contained in the half-space x4 > 0 represented by the parametric equations

I

x1 = R(t)sinhXsinflcosp,

x 2 = R(t)sinhxsinflsinp,

x3 = R(t)sinhXcosfl,

(3.2)

x4 = R(t)coshX,

x

with 0 I: < 03 , 0 I 6 2 n,0 I q~ I 2 n . Remembering what was said at the end of n. 2, it will be assumed to be one of the three possible is this sheet that in case PS configurations of 2, a sheet that from now onwards, unless specifically stated otherwise, will still be indicated by PS From (3.2) it follows that its metric in the polar coordinates X,6, q~ is expressed by

’

’.

d o 2 = R2(t)( dx2 + sinh2X (df12+ sin2fl dp2)),

(3.3)

a metric that is the same for the entire pseudospherical hypersurface PS and which is positive definite (which means that PS despite its being embedded in the Euclidean space P E , is a strictly Riemannian three-dimensional manifold).

’,

’

4 Case E and Hubble’s Law

’,

Always with reference to cases S and PS let 0 and P be any two particles of 2, where 0 is considered to be fixed. Their distance d(t) at the instant t (length of the geodetic arc OP ) is the so-called proper distance (or geometrical distance) in cosmology treatments. In both cases it is expressed by

once we have chosen the polar coordinates

x,6, q~ in the frame of reference ZCso that x

h

is the costant angle’ OCP .

x

For the sake of brevity, we call an angle also in case PS

’.

58 We will indicate by ti the instant in which R(ti)= 1, as soon as the unit of distance for these two cases has been chosen. That said, let us considering now case E However the particle 0 of Zt has been fixed, according to the cosmological principle in this case the trajectories described by the particles of Zt with respect to 0 are half-lines starting in 0, which give rise to a frame of reference ;Po with its origin in 0. Once the instant ti has been arbitrarily fixed and any particle P has been considered, let us make the position OP = d(t) (and therefore let us now also indicate by d(t) the proper distance at the instant t of P from 0).By letting d(ti)= x , the cosmological principle entails

’.

4 t ) = Wt) x

(4.1’)

9

an equality which, in its scalar form, as the instant ti is arbitrary, differs from (4.1) solely because the factor R(t) (scalarfactor) is now indeterminate. It is obvious that R(tJ = 1 . Introducing the Hubble parameter defined by6

.

for all three cases from (4.1) Hubble’s law immediately follows:

d ( t ) = h ( t ) d ( t ).

(4.3)

’,

At this point we can observe that: in case E equations (4.1’) [and therefore (4.1)) and (4.3) remain unchanged both if we interpret the fluid Zt (i.e. the Universe) to be we consider E to be expanding in E and i j in analogy to cases S and PS expanding, with the particles of Zi? at rest in it and therefore dragged by E in its expansion process. While the first interpretation does not place any limit on the velocity of the particles of Zt in E the second leaves the particles of 2 at rest with respect to E and transfers result (4.3) to the points of E Therefore, in this second interpretation, result (4.3) must be regarded as the expression for the veloicity of recession of the points of E in analogy with what happens with (4.3) for the points of S and PS More precisely, with reference to the Universe, both in case E and in cases S and PS , it is clusters (or even clusters of clusters) of galaxies that are to be understood as being dragged in the expansion process. Due to their intense gravitational field, space is not expanding inside them. I n this paper, in case E we will always and only resort to the second of the interpretations we have just seen.

’

’,

’,

’,

’

’

’

’.

’, ’

’

’

’.

’

’,

’,

In case E even though R(r) is indeterminate, h(r) is determinate. And indeed in all three cases we can easily obtain: p(f)R’(I) =cost., from which

follows both for case E and for cases S and PS

’

59 5 On the Metrics of Zt in Cases E

3,

’,

S and PS

In case E once we have introduced into the frame of reference ;Po and at the fixed instant ti a system of polar coordinates 19,p ,with the radial coordinate (previously introduced while writing (4.1’) in scalar form), from (4.1’) it follows that the metric for E in such coordinates is expressed by’

x,

x

d o 2 = R2(t) ( d x 2 + X2(dtY2 + sin2tY d q 2 ) ) .

(5.1)

In the above frame of reference, if instead we resort to orthogonal Cartesian coordinates, indicating by y ’ (i = 1, 2, 3) the components of x , or in other words the coordinates of P at the instant ti , from (4.1’) it follows that the metric for E in such coordinates is expressed by 3

d o 2 = R 2 ( t ) ci(dyi)z.

(5.1’)

1

The three metrics (3.1), (3.3), (5.1) can be represented by means of a single formula:

d o 2 = R2(t)( d x 2+ S 2 ( x ) (df12+sin26 dp2)),

(5.2)

where

S(x) = sinx in case S

’, S(x)

=

x

in case E 3 , S(x)= sinhx in case PS .

(5.3)

In analogy with (5.1’), it is also possible to assign to metrics (3.1) and (3.3) a single form in the orthogonal Cartesian coordinates y (i = 1, 2, 3) which also includes (5.1’). Indeed it is sufficient to introduce r = 2 t g L into (3.1) and r = 2 t g h g into (3.3) to 2 2 obtain

(5.2’)

It is sufficient to observe that (4.1’) entails that, however we consider the particle Q, at the instant r we have, with the evident significance of the symbols: PQ = R(r) P, Q, and therefore, in particular, dP = R(t) dP, , etc.

60 with E = 1 in case S and E = -1 in case PS . With E = 0 (5.2’) becomes (5.1’), which corresponds to case E When E = 1 the Cartesian coordinates y (i = 1, 2 , 3) are the coordinates that we obtain by resorting to the stereographic projection of S considered at the instant ti (the instant in which, as we have said at the beginning of n. 4, R(tJ = 1). This projection is made from point (0, 0, 0, -1) of E on the hyperplane x = 1 which is tangent at this

’.

=c 3

instant at S in point 0. They are related to r by r 2

i ( y i ) 2.

I

In the case of E = - 1 the interpretation of (5.2’) is entirely analogous. In this regard it should however be pointed out that, remembering what we observed for PS in n. 3, the metric (5.2’) with E = - 1 should be considered with the coordinates y restricted by the

C 3

limitation: r2 =

; ( y i ) 2< 4 .

*

I

However particle P is considered, both the polar coordinates x, 19,V, and the Cartesian coordinates y identify its position at the instant ti and they thus remain constant in time for it. In cosmology these types of coordinates are called co-moving coordinates. They are none other than the so-called Lagrangian coordinates that are introduced into continuum mechanics. W e have briefly outlined all these results both for reasons of clarity and completeness and above all for the role that they will play in the following part of this paper.

6 The Intervention of the Velocity of Light With the premises that have been made until now, in relation to the fluid Zt we will now consider the possible implications of the property of the velocity of light of being independent of its source at the point where its measurement is made. This property implies that the light emitted by any galaxy has its velocity (in the place where it is measured, i.e. on Earth) that is independent from the galaxy considered, or in other words, it is the same for every galaxy. In the scheme that was introduced at n. 1 to represent the Universe (the cosmological fluid), the aforesaid property has to be considered verified for the observations made by an observer situated in the particle of Zt representing our galaxy, a particle which will still be indicated by 0. It is therefore sufficient at this point to remember what the cosmological principle states to conclude that this property must subsist in correspondence to every particle 0’ of Zt, in the sense that a hypothetical observer situated in 0’ who measures the velocity of light in 0’ at any given instant will find the same value for it as the value that the observer situated in 0 will find in the same instant. This velocity will be considered to be independent from time, i.e. like in all treatises, it will be considered a universal constant which, as usual, will be indicated by c. At this point therefore, in addition to assumption 1) made at n. 1, we can now make the following further assumption: The coordinates y ‘ for which 3 > 4 represent the coordinates of the stereographic projection (always made from the same point (0, 0, 0, -1) of PE and on the same hyperplane x4 = 1) of the sheet of the pseudosphere contained in the half-space x4 < 0.

61

2) the velocity of light is independent from its source at the point where it is measured. 7 The Velocity c and the Law of Composition for Velocities Once we have made assumption 2), and again indicating by 0 any fixed particle of Zt, let us consider a photon F emitted by 0 which transits through particle P at the instant t. Remembering (4.1) and bearing in mind that, as we have seen at n. 4, it subsists in all three cases S PS e E let us indicate by A x the increment that has undergone in the passage of F from the instant t to the instant t + A t. The path travelled by F in this interval of time, measured by an observer situated in P, is expressed by R(t+At)AX . If we consider the average velocity of F in this interval of time and make At tend to 0, we obtain the following result for the velocity of F measured by an observer situated in P at the instant t when it transits through P:

’, ’ ’,

x

From (7.1) follows

Let us indicate by te the instant when F is emitted by 0 and let us indicate by l(t) its distance from 0 at the instant t. Bearing in mind (4.1) and what has just been said about it, from (7.2) follows (7.3)

From (7.3)~, and remembering (4.2), we obtain the following equation for the velocity of F with respect to 0 at the instant t when it transits through particle P whose distance from 0 is l(t): l ( t ) = h(t)l(t)+ c ,

(7.4)

or in other words, remembering (4.3): the velocity at which the photon F recedes from 0 is at any instant equal to the sum of c (velocity of F with respect to P) and the recession velocity with respect to 0 of the particle P through which F is transiting at that instant. Therefore it is possible to have l ( t ) = c ifand only ifthe recession velocity of P with respect to 0 is zero, i.e. ifthe Universe is static. However, this eventuality is physically speaking inadmissible, as for over seventy years it has been a well-known fact that the Universe is expanding.

62 I f the velocity of F is measured by an observer situated in P and at the instant when F

transits through it, in that case we have [ ( t )= 0 and (7.4) gives us l ( t )= C , in perfect agreement with assumption 2). At the same time result (7.4), as we have already seen in the comment to it, expresses that F, emitted by 0, is receding from 0 with a velocity that is in perfect agreement with the law of compositionfor velocities of classical mechanics. It is worth pointing out that, once result (7.1) has been established, it is possible to reach result (7.4) directly as soon as we observe that the velocity I(?) of F with respect to 0 is the velocity of F with respect to the position occupied by P (with respect to 0) at the considered instant t. Indeed the path followed by the photon in its passage from instant t to instant t + A t measured not from P but from the position occupied by P at the instant t is expressed by R(t+At)(x+Ax)- R ( t ) x , that is, leaving aside the infinitesimals of any order greater than one with respect to A t : R ( t ) x A t + R(t + A t ) A x . By dividing by A t , making A t tend to 0 and bearing in mind (4.1), (4.2), (4.3) and (7.1),result (7.4) immediately follows.

8 The Law of Composition for Velocities and the Hubble Sphere Let 0’ be any other particle of 2% that at a given instant, which will still be indicated by t, , emits a photon F ’ directed at 0. Once we have indicated by the angle (or, in case E the distance do. (ti))that separates 0’ from 0, from (4.1) (bearing in mind what was said about it at the beginning of n. 7) and from (7.3), it follows that the distance from 0

xo.

’,

(which will be indicated b y l ( t ) ) at which F ’ is at the instant t (considered before the eventual’ instant in which F ’ reaches 0) is expressed by

From (8.1) it follows that the velocity of F ’ with respect to 0 at this instant is given

bY

-z ( t ) = h(t)T(t)- c ,

(8.2)

which is wliolly analogous to result (7.4) and for which the same observations made about that result also hold true.

“Eventual” in the sense that, together with what we have said until now, we cannot exclude the eventuality that F ’ will never reach 0. In fact it could happen that, considering the Universe to be in infinite expansion (as indeed recent astronomical observations appear to indicate), the integral that appears in (8.1) is convergent as t tends to with a limit lower than . In other words it might happen that F ’ is outside the so-called event horizon at instant t,.

-

xu

e

63 Taking into account (8.2), the photon F ’ directed at 0 could also, initially, recede away from 0 instead of approaching towards it. This is precisely what occurs in the

-

eventual interval of time in which 1 0 ) >-

C

h(t)

-

(i.e. 1 ( t )> 0 ), or in other words in the

eventual interval of time in which 0 is outside the so-called Hubble’s sphere with respect to 0 and at tlie instant t”, that is outside the spherical surface” which is in this instant the position of the particles of Zt whose distance from 0 (Hubble’s distance) is equal to C . In such an instant this surface divides the particles of Zt whose recession velocity

h(t) with respect to 0 is greater than c from those with velocities lower than c.

9 About Cosmography and the Deduction of the Function R(t) Indicating again by 0 the particle of Zt that represents our galaxy, let us consider a particle 0’ that emits a photon F ’ at the instant t, which reaches 0 at the instant t o . Indicating by l+z the ratio between the wavelength of F ’ at its arrival in 0 and its wavelength at the instant of emission t, , through considerations which have as their point of departure (7.1) (i.e. assumption 2)) and which are not only formally but also conceptually identical to those developed in the relativistic treatment, we obtain the following very important and well-known result:

where z ( the “red shif” parameter) is a directly measurable quantity. Likewise, always starting from (7.1), it is possible to develop the famous Rindler theory of horizons’’ and, more generally, it is possible to obtain all the various and wellknown results of relativistic cosmography connected to the propagation of light13. To obtain these results in an explicit form it is necessary to know the function R(t). In the relativistic treatment the equations that enable us to obtain this function are deduced by resorting to Einstein’s gravitational equations applied to the Friedmann-RobertsonWalker metricI4 and they are considered among the most greatest acquisitions of the general theory of relativity. In reality (as we have proved in [7]) it is possible to deduce these equations in a completely determined form by resorting solely (besides the indications given by astronomical observation) to assumptions l), 2) and to the assumption that for thejluid Zt the principles of classical mechanics hold true, without making recourse to any theory of gravitation”. See IS], n. 2; [ 9 ] , p. 281. See note 2. “See [11]. I’ In this regard see, for instance, [12], 1 4 , 3 , 4 , 6 , 7 . j 4 For the deduction of this metric see n. 11. Is This subject will be developed in greater depth in forthcoming papers. lo

‘I

64

10 The Local Character of Special Relativity in Cosmology In addition to the orthogonal Cartesian coordinates y (i = 1, 2, 3) which appear in metric (5.1’) it is possible to introduce the orthogonal Cartesian coordinates x ’ (i = 1, 2, 3) defined by xz = R(t)y‘

(i = 1,2,3).

(10.1)

In case S these coordinates are the coordinates that are obtained by means of the stereographic projection of S considered in the instant t and made from point (O,O,O,- R(t)) of E on the hy erplane x4 = R(t) tangent at this instant at S in point 0. In case PS the interpretation of the xicoordinates is completely analogous. Bearing in mind the limitation to which the 1 are subject in this case, we obtain for them:

P .

I

With the xi coordinates defined by (lO.l), both in cases S and P S 3 and in case E metric (5.3’) in the point 0 becomes

’,

and this turns out to be the metric of the three-dimensional Euclidean space tangent in 0, a space that will be indicated by Eo and which is not in expansion. Bearing in mind what has already been observed at the end of n. 4 (i.e. that the clusters of galaxies are to be understood as being dragged into the expansion process, and that inside them due to the intense gravitational field they generate there is no expansion), we obtain that a cluster which is at 0 may be considered as belonging to the space Eo where there is a frame of reference yoidentified by the system of coordinates xi with its origin in 0. In addition to the frame of reference 70 it is possible to consider the class of the frames of reference in uniform translatory motion with respect to yo. Assumption 2), imposed by experience, implies that this assumption remains valid in each of these frames of reference, and therefore that to each of these frames of reference a different time must be associated. Taking into account, as it is obvious, that this class also includes yo , it follows that the transformations which permit us to pass from one of these frames of reference to any other are expressed by the Lorentz transformations which are at the basis of the special theory of relativity. These considerations we have just seen with reference to particle 0 are valid unchanged for every particle 0’ of 2. We can therefore add that in case E the class of the frames of reference which are in uniform translatory motion with respect to yo.is made up of frames of reference that move with accelerated translatory motion with

65 respect to the frames of reference in uniform translatory motion with respect to yo,since 90.moves with accelerated translatory motion with respect to yo,16 and vice versa. These few considerations, which are only valid in Euclidean tangent spaces, highlight the typically local character that the special theory of relativity presents in the study of the Universe.

11 The Deduction of the Friedmann-Robertson-Walker Metric

For a photon emitted by 0 from (7.1) it follows that in all three cases S have

’,PS ’,E ’ we

R 2 ( t )d x 2 - c2 dt2 = 0 , i.e., bearing in mind that the first addendum is none other than (5.2) written with an opportune choice of the coordinates 8,a, :

x,

d o 2 - C 2 dt 2 = 0 .

(11.1)

Therefore at this point it is possible to introduce, in correspondence with each of the aforesaid three cases, a Riemannian manifold (the space-time manifold which is analogous to the Minkowski space-time of the special theory of relativity), whose metric for photons must satisfy (11.1) and moreover must be such that the sections t = cost. have as their metric the correspondent expression of d o 2 given by (5.2) (or by (5.2’) due to the invariance of d o 2 with respect to coordinate transformations). Resorting to the polar coordinates 19,a, ,this metric is expressed by

x,

ds2 = R 2 ( t )( d x 2 + S 2 ( x )( d o 2 + sin2z9 d q 2 ) )- c2dt2 ,

(11.2)

with S ( x ) given by (5.3), while resorting to the Cartesian coordinates y it is expressed by

’, ’,

with E = 1,-1,O in correspondence to cases S PS E respectively.

In fact

i(0= 0

cannot hold true because from (4.3) follows

Bearing in mind what has been deduced in [ 6 ] ,12, condition h(t) + h 2 ( t )= 0 should imply p (t) = 0, i.e. that the Universe is empty.

66 Metrics (11.2), (11.2’) are two of the forms of the well-known FriedntannRobertson-Walker metric (FRW metric) which is at the basis of the relativistic treatment of the expanding homogeneous and isotropic models of the Universe, that is of the socalled Friedmann nzodels of the Universe. These metrics have been obtained in correspondence to the three most significant cases given by S PS E As we have already briefly mentioned at n. 2, they are three examples of isotropic (and therefore homogeneous) manifolds belonging to the three different three-dimensional classes of isotropic manifolds, each of which is characterised by the same intrinsic geometry and therefore by the same metric. Metrics (1 1.2), (1 1.2’) therefore have a general character, in the sense that they do not only hold true in correspondence to cases S PS E but also for each of these classes to which S PS E belong (classes that are characterised by the three cases that follow from the metric written in the form (5.2) or (5.2’)). These metrics, as we have seen, follow from the cosmological principle and from equation (7.1), which is an almost immediate consequence of assumption 2).

’, ’, ’.

’, ’,

’, ’, ’

12 The Deduction of (7.1) in the Relativistic Context. The Minkowski Metric With reference to the relativistic treatment (or, more precisely, to the treatment founded on the FRW metric), we can observe that starting from this metric written in the form (11.2) and considering the motion of a photon F (for which ds2 = 0), equation (7.1) immediately follows as soon as we consider F to be emitted by the particle 0 which is in the origin of the suitably chosen system of coordinates 8,p . Continuing at this point to indicate by l(t) = R(t)X(t)the distance of F from 0 at the instant t , from (7.1) result (7.4) (like result (8.2)) immediately follows. With regard to result (7.4) (and hence to result (8.2)), it is necessary to observe that until now it has been known only at a qualitative level (in the sense that it has never been expressed in mathematical formulae). One can say that it has been known since Eddington’s time because, in accordance with what we observed in n. 8, in 1933 he wrote”: “Light is like a runner on an expanding track with the winning-post receding faster than he can run”. In the absence of the cosmological constant and at the same qualitative level, it has been widely reconsidered by Harrison in his remarkable book Cosntology’*. In general it does not appear in treatises, with the exception, for example, of the book by Mould’’ who takes into examination always at the same qualitative level case (8.2), considering the physical space as Euclidean and the cosmological constant equal to zero. In this regard it is necessary to observe that there exist works regarding relativistic cosmology written in the past which contain incorrect statements and conclusions which are in contrast both with (7.4) and with (8.2). A quite concise deduction of the results obtained in nn. 7, 8 has been given in [7], 28, 29. By considering it valid, in [ 2 ] ,like in [4], 10 as well, we have made recourse to result (7.4) in order to deduce the FRW metric in case E Always with reference to case E and starting from (7.4), in [3] we obtained result (7.1) and starting from (8.2) we obtained result (8.1), etc. With reference to case E again, the way indicated above to reach (7.1) and (7.4) has been followed in [5], 4, a paper where considerations regarding not only the

x,

’.

’

”See [I], p. 73. See [9], p. 446. l9 See [lo], 14.8.

’

67 velocity but also the acceleration of photons in the Universe have also been developed both from the kinematical and the dynamical point of view2’. Returning to the FRW metric, with the introduction of the xi coordinates (i = 1,2,3) defined by (10.I), in point (O,O,O, t ) of the space-time manifold this metric written in the form (1 1.2’) becomes the metric of the pseudo-Euclidean space-time which is tangent to the space-time manifold in this point. This metric is given by

=xi(&)’ 3

ds2

-c2dt2,

I

i.e., in perfect accordance (as it is obvious) with what we have seen in n. 10, by the metric of the Minkowski space-time, which (in addition to Lorentz’s transformations, with respect to which it is invariant) is at the basis of the special theory of relativity.

References 1. Eddington A., The Expanding Universe (Cambridge University Press, Cambridge, 1933). 2. Galletto D., Sulla metrica dello spazio-tempo. I, Atti Accad. Sci. Torino 113 (1979) pp. 113-1 17. 3. Galletto D., Sulla metrica dello spazio-tempo. 11, Atti Accad. Sci. Torino 115 (1981) pp. 345-56. 4. Galletto D., Des principes de la micanique classique aux thtories de la gravitation de Newton et Einstein. In Atti del Convegno Internazionale “Aspetti matematici della teoria della relativitd”, Roma, 5-6 giugno 1980, (Accademia Nazionale dei Lincei, Roma, 1983) pp. 59-83. 5. Galletto D. and Barberis B., Modelli cosmologici e propagazione della luce, Giornale di Fisica 31 (1990) pp. 245-251. 6. Galletto D. and Barberis B., Principio cosmologico e modelli cosmologici. In Atti del Convegno “La matematica nelle scienze della vita e delle applicazioni”, Bologna, 1-2 giugno 1 9 9 9 ~(Pitagora Editrice, Bologna, 2000) pp. 59-128. 7. Galletto D. and Barberis B., Modelli d’universo e propagazione della luce. In Atti del Convegno Internazionale “Nuovi progressi nella fisica matematica dull ’eredita di Dario GrasJi”, Bologna, 24-27 maggio 2000, (Accademia Nazionale dei Lincei, Roma, 2002) pp. 97-1 5 1. 8. Harrison E., Hubble Spheres and Particle Horizons, Ap. J. 383 (1991) pp. 60-65. 9. Harrison E., Cosmology 2”ded. (Cambridge University Press, Cambridge, 2000). 10. Mould R. A., Basic Relativity (Springer-Verlag. New York, 1994). 11. Rindler W., Visual Horizons in World-Models, Mon. Not. Roy. Astronom. Soc. 116 (1956) pp. 662-677. 12. Weinberg S . , Gravitation and Cosmology: Principles and Applications of the General Theory ofRelativity (John Wiley & Sons, New York, 1972).

It is necessary to observe that in the deductions contained in [2], [3], [4], [5]we have implicitly assumed that the second interpretation here introduced for case E in n. 4 holds true.

’

THERMODYNAMIC LIMIT FOR SPIN GLASSES SANDRO GRAFFI Dipartimento d i Matematica, Universitd di Bologna, 40127 Bologna, Italy E-mail: [email protected] This contribution represents a short review of some recent advances on t h e thermodynamic limit in spin glasses, and is addressed t o readers not necessarily familiar with statistical mechanics.

To Salvatore Rionero on has 70-th birthday

1

Basic lexicon of classical statistical mechanics for spin systems

Standard basic references on mathematical aspects of statistical mechanics of spin systems are for instance the textbooks of Ruelle

and Simon

7; for

the particular

branch represented by spin glasses an equally basic reference is Mkzard-ParisiVirasoro

4.

Consider a subset A

c

Zd, d E

N;for the sake of simplicity we may take A a

hypercube. 1. The set contains N = IAl points, called sites. At each site x E A sits a dicotomic variable

gX

= hl, called

spin by its obvious physical interpretation.

2. The N-vector u = (ul,. . . ,U N ) , uk = f l , k = 1,.. . ,N is a spin configuration.

3. The phase space EA is the set of all spin configurations. Clearly

lc~l= 211’

=

2N.

4. The potential energy, or potential of a configuration u in a magnetic field hx depending in general on the site x E A is by definition

wit assigned boundary conditions, i.e. assigned values of the spins on aA. Here

Jxyis the so-called coupling matrix. It describes the interactions among the spins a t sites x and y in A. The term

C hxuxis the linear magnetic potential, xEA

corresponding to the constant field h, on each site. 68

69

5 . A macroscopic state is any probability measure on CA. Equivalently: a macroscopic state is an assignment of the probability of the outcome of each microscopic state, i.e. of any individual spin configuration. The two most important physical systems whose macroscopic properties (thermodynamics) are expected to be deduced by the above models are the magnetic materials and the lattice gases. The lattice gas is a system of N particles: a dicotomic variable uz is defined at each site x by assuming uz = 1 if there is one particle at x and

gz

= 0 if there is none.

To deduce the macroscopic properties of the system out of the microscopic model just described, statistical mechanics relies upon a basic

Postulate The equilibrium state of the system with potential EA(u,h ) at temperature T is given by the Boltzmann-Gibbs state, in which the probability of a configuration u is

1 Here as usual /3 := - where k is the Boltzmann constant, and the normalization kT factor

UEA

is the partition function of the system. Moreover FA(P,h) := -PlnZA(P? h ) , is by definition the (Helmholtz) free energy. The specific free energy

Pf (P, h ) , or

free energy density, is

Thermodynamic stability is the property that the dependence of the specific free energy on the volume should disappear as the volume increases to infinity. This property holds if the specific free energy approaches a limit as A

+ Zd.

Therefore:

70 Definition 1

W e say that the thermodynamic limit exists if the free energy density converges to a limit as A

-+

Zd, namely

If the thermodynamic limit exists, the free energy

Pf(P, h )

yields everything,

namely all macroscopic quantities: magnetization, entropy, etc.

2

Phase transitions

The same magnetic materials may exhibit, at different temperatures, different responses t o the action of an external magnetic field. Paramagnetism is observed when the field-induced magnetization disappears when the field is removed; ferro-

magnetism is instead observed when the magnetization does not disappear with the field, and the permanent (or spontaneous, or residual) magnetization takes place. It is natural t o think at these phenomena as two different phases of the same substance, taking place at different temperatures, in the same way as ice and water describe two different phases of the same substance taking place at different temperatures. Unlike the macroscopic theories, statistical mechanics has been able t o account for this phenomenon (at least in some particular cases), and this represents its main achievement. To describe a magnetic transition, consider the (site dependent, in general) magnetization, defined as the expectation of the spin variable oxon the Gibbs state at the thermodynamic limit:

The limits have t o be performed in the stated order t o get a non trivial result. A very important particular case corresponds to the translation invariance of the coupling matrix: J,,y = JzPy.This case takes place, for instance, in crystals, but not in amorphous materials such as glasses. Then the magnetization is site-independent:

71

Then: 1. If lim lim mA(p,h ) = 0 , we have the paramagnetic phase. After removal h+O A+Zd

of the field the spins are randomly oriented, or, equivalently, there is no long range order.

2. If lim lim mA(P,h ) = m(P) h+O+ A-Zd

> 0, we have the ferromagnetic phase. The

spins remain on the average aligned along the direction of the external field even after its removal. Equivalently, there is long range order.

3. There is a phase transition at the critical point

temperature T * )if m(P) = 0 for P

< p*

p* (equivalently, at the

critical

and m(P)# 0 for ,B > P*.

4. The magnetization m(P) as a function of the (inverse) temperature is called

order parameter because its non triviality completely parametrizes the orderdisorder transition. The existence of phase transitions has been proved in many instances. The most classical examples where existence of a phase transition has been proved and the corresponding critical temperature determined are: 1. The Curie-Weiss (or mean field) model (Bragg and Williams, 1934). Here 1 d = 1, Jx,31= =. It provides a justification of the Curie-Weiss theory of the

magnetization curves. J , lz- Yl 5 1 0, b - Y l > 1 The Ising model is the standard model for short range, ferromagnetic interac-

2. The Ising model (Onsager 1944):Here d = 2, and Jz,y=

{

tions.

3

Amorphous magnetic materials (spin glasses)

In amorphous, or disordered, materials there is no underlying crystal structure; hence, a priori, no translation invariance of the couplings. It can happen (as observed in the 70's) that on any scale A:

72 In other words: the average magnetization may be arbitrarily small on any scale, but magnetism still exists and is not small (under the form, for instance, of magnetic currents). These materials exhibit therefore magnetic properties other than those of the ferromagnetic ones. The standard statistical models describing the amorphous magnetic materials have a potential of the form (1); this time, however, to simulate the intrinsic disorder of the material the couplings Jzyare specified only as random variables of specified probability distribution. Each one of this models describes a spin glass.

Most general (Gaussian) model of spin glasses Let M be a countable set; A

cM

be a finite subset, and IAl = N. To each element

i E A we associate a spin variable aiE (-1, +l}.Once again, a spin configuration in A is an element of EN = {-1, +l}N. The spin glass potential is

where

(q, = 0). Remark that X is a collection of multi-indices, so that as

X

varies in A

Jx describes the two-spin couplings, three spins couplings and so on. Each coupling

Jx is a Gaussian random variable such that Av(Jx) = 0 ,

(zero mean)

= A$

(variance).

Av(J;)

Moreover JXand J y are independent if X

# Y.

Particular cases of this general definition are the following standard models:

Examples: 1. The Sherrington-Kirckpatrick (SK) model: here M = 1 A x = 0 if # 2 and AX = otherwise.

1x1

N,A

= {1,2, ...,N},

N

2. The Derrida Random Energy Model (REM) and the Derrida-Gardner Gener-

alized Random Energy Model (GREM).

73 3. The p-spin model: Here M = N,A = {1,2, ...,N } , A x = 0 if 1 Ax = otherwise. Nz

1x1 # p

and

4. The Edwards-Anderson model: M = Zd, A is a cube. The nearest neighbor case is defined by A x

# 0 iff X

= ( n ,n’) and In - n’l = 1. This is to this day

the first and foremost model for realistic spin glasses. More generally one might consider an interactions rapidly decreasing with the distance among sites:

For the general Gaussiann model, one defines the random partition function:

Denoting Av the Gaussian expectation, one correspondingly defines the quenched free energy

FA := -PAv(lnZA(J)) , where the quenching operation means that the Gaussian average is performed after taking the logarithm. Equivalently, one may consider the quenched pressure

PA := Av (In Z ^ ( J ) ) , The random Gibbs-Boltzmann state is defined as

and consequently the quenched state is

< - >= Av(w(-)) . As in the deterministic case:

Definition 2 We say that the quenched thermodynamic limit exzsts if the quenched pressure density converges to a limit as A

-+ Zd,

namely

As above, the quenched pressure yields all the thermodynamical quantities.

74

Unlike the ferromagnetic materials, the basic mathematical understanding of the spin glasses is still far from being satisfactory; however there have been recently two major advances as far the existence of the thermodynamic limit and the validity of the Parisi solution of the Sherrington-Kirckpatrick models are concerned. For this last aspect we simply refer to Talagrand'.

As far as the thermodynamic limit is

concerned, the old problem of proving its existence for the Sherrington-Kirckpatrick model has been solved by Guerra and Toninelli

through the introduction of the

interpolation method. This technique, when applicable, also yields the very important property of the monotonicity in the volume of the approximating sequence for the pressure. It will now be recalled how this technique yields existence and monotonicity in the most general spin glass model as defined above. In particular, it yields the monotonicity for the Edwards-Anderson model, where existence was proved by Khanin and Sinai

4

long ago.

Thermodynamic limit

Consider first the above examples. 1. The Sherrington-Kirckpatrick (SK) model. The existence of the quenched ther-

modynamic limit for this model, according to Definition 2 above, has been proved by Guerra and Toninelli

through the interpolation method which also

yields monotonicity of the approximation sequence. 2. The Derrida Random Energy Model (REM) and the Derrida-Gardner Gen-

eralized Random Energy Model (GREM). Here the existence of the quenched thermodynamic limit and the monotonicity of the approximation sequence have been proved by Contucci e t a1 by adapting the Guerra-Toninelli interpolation argument

2;

3. The p-spin model: Here the existence of the quenched thermodynamic limit

and the monotonicity of the approximation sequence have been proved in

for

p even;

4. The Edwards-Anderson model: M

= Zd, A is a cube. Here the existence of

75 the quenched thermodynamic limit has been proved long ago by Khanin and Sinai

5; this

proof however does not yield the monotonicity in A

Let us now state the conditions for the existence and monotonicity of the thermodynamic limit in the general spin glass model. A potential is called stable if its variance grows like the volume, i.e. if there is c > 0 such that:

C A$

AV ( U ( J ,u ) ~ = )

5 NC

XCA

Theorem PA . If the potential is stable the sequence zs monotonically increasing and bounded:

I4

Proof. For the proof of this result, see

(l).

Monotonicity is a consequence of the

positivity and superadditiuity properties to be recalled below. POSITIVITY. The analogous of the first Griffiths inequality (see e.g.') holds;

Lemma 1

< JXUX > 1 0 SUPERADDITIVITY Consider a partition of A into n non empty disjoint sets As (clusters): n

A= UAs, s=l

For each partition the intercluster potential is defined as

from (3) we have that

(13)

76 where CA is the set of all X c A which are not subsets of any A,. Then: Lemma 2

The quenched pressure is superadditive, namely: n

PA

2 EpA. s=1

By a standard argument

the superadditivity entails that the sequence A

is monotonically increasing.

H

PA

-

I4

References 1. P.Contucci and S.Graffi, J.Stat.Phys., in press

2. P.Contucci, M.Degli Esposti, C.Giardin&, S.Graffi, Commun.Math.Phys. 236, 55-63 (2003)

3. F.Guerra and F.Toninelli, Commun.Math.Phys.230, 71-79 (2002) 4. M.MBzard, G.Parisi, M.A.Virasoro, Spin Glasses and Beyond, World Scientific,

Singapore 1987 5. J.Khanin and Ya.G.Sinai, J.Stat.Phys. 20, 573-584 (1979) 6. D.Ruelle, Statistical Mehcanics, W.A.Benjamin, New York 1969 (Second Edition: 1980) 7. B.Simon, Statistical Mehcanics of Lattice Systems, Princeton University Press 1991

8. M.Talagrand, On the generalized Parisi solution, preprint, June 2003

RIGID MOTIONS IN CELESTIAL MECHANICS. KEPLERIAN MOTIONS G. GRIOLI via X X V I I Luglio 54, 98100 Messina, Italy.

1

Introduction

In the past (see [l],[a], [3], [4]) I characterized some rigid motions that I called generalized precessions. They are rigid motions for which a vector defined by a certain function of the angular velocity during the motion belongs to the plane defined by two straight lines one of them fixed in the space and the other one fixed in the rigid body. In particular, I characterized the precessions of a rigid body by means of a relation between the components of the angular velocity with respect to a trihedral fixed in the body. Now I will apply that relation for studying the dynamical problem of the motion of a rigid body whose center of mass is moving in a fixed plane, K , and the body is subjected to a gravitational force due to a material point contained in K (for example, Earth and Sun). The mathematical model is constructed following an approximation usual in Celestial Mechanics and regards the case in which the body is a gyroscope. The aim is the characterization of the possible precessional motions. It is evident that the theory may be useful in some questions regarding the planetary motions. 2

Kinematical Premises

Let B be a rigid body, g its angular velocity and p i , (i = l , 2 , 3 ) , the components of g with respect to a right-hand rectangular trihedral, T = GJI ,& ,& , fixed in the body. G is the center of mass of B and Jl,J2,L are the unit vectors along the axes of

T. I consider the vector

where arsare constants, and denote by g a unit vector fixed in the body and by c another one invariable in the space. The following property subsists: Necessary and s u f i c i e n t condition that during the motion of B the vector Q belongs t o the plane determined b y the vectors g and -c applied in G is that the quyntities

77

78

and a function of the time O ( t ) satisfy the equalities

In the (2),(3) the dot denotes the derivative with respect to the time and is assumed Q x 14 # 0. If theT3) is satisfied, the vector c defined by

is invariable in the space. I called the correspondent motions generalized precessions. In fact, for a = 1 (a,, = drS) then Q = g and

w x e . g = 0,

0 = const.

That is the motion of B is a precession. In this particular case, assuming 21 = &, sin0 according t o ( 3 ) , the following equality:

subsists. Further, putting p =

# 0

(5)

and putting y = cot6,

d w ,the vector

is invariable in the space. The equality (6) characterizes the precessions whose axis of figure is g = while that of precession is parallel to c. Putting

L

79 Pl = PCOSP,

P2

= psinv,

the equality ( 6 ) is equivalent to (P=yp-p3.

It is interesting to observe that when the central ellipsoid of inertia is round and the motion is a precessional motion with 21 = & coinciding with the gyroscopic axis, then the angular momentum of the body with respect to G has a similar property: it belongs to the plane of the applied vectors Gc and G i 3 . 3

Dynamics

Now I assume that G is moving in a plane, 7r, fixed in a inertial space and denote by g a unit vector perpendicular to 7r. Let 0 be a fixed point of 7r and put

IOG( = r ( t ) > 0, OG = r f l , H 2 = 1. The variability of g in x may be characterized by the equality

H+gxH=ugx& where v(t) denotes the scalar angular velocity due to the rotation of

(10)

(11) in the plane

7r.

I assume that in the point 0 there is a mass m and that r ( t ) is very great in comparison with the dimensions of the body B. The moment vector, M,with respect to GI of the gravitational force due to the mass m acting on B , according to an usual approximation in Celestial Mechanics, may be expressed by

M = aH x a€€, where h denotes the Gauss constant. For the study of the motion of B with respect to a rectangular trihedral, TG, whose origin is in G and the axes have invariable directions in the inertial space, it is important to observe that the centrifugal forces and the Coriolis ones have the resultant moment with respect to G equal to zero because of the fact that the motion of the trihedral TG is a traslatory motion. Therefore, the dynamical equation for the body B in its motion with respect to TG is u& i g x ug = aH x

aE

(13) where a = Ilarsll denotes the inertia matrix of the body and the time derivative is made with respect to the axes fixed in the body. Together with the (13), the equations which characterize the motion of and the invariability of c in the inertial space are to be considered. They are [see (ll)]

80 The integral relations

-e . H = O

e2 = 1,

H 2 = 1,

are to be associated to the equations (13), (14). They are consequences of the differential equations and simplify the resolution of the analytical problem. On the basis of equation (13), the vector H ( t ) can be expressed as a function of the variables p i , p i . Therefore, the problem is limited to the study of a system of differential equations in the variables p i , ei, (i = l , 2 , 3 ) . In conclusion, the problem of the determination of precessional motions having c as axis of precession and u_ = & as axis of figure consists in the research of solutions which satisfy the condition (6). 4

Gyroscopic Case

I assume that the body B is a gyroscope having as gyroscopic axis GG. The inertia matrix may be put in the form (T

= ( A ) 000AOOOA3 ,

A

> 0,

A3

> 0.

Putting

the equation (13) is equivalent to the scalar equations @l

f pZp3 = aH2H3

-3732 + pip3 = aHlH3, p3 = p: = const.

It must be remarked that in every motion in which H3 = 0, the angular moment with respect to G is constant in the space. Therefore, according to a known property in the dynamics of the rigid bodies, the motion is always a precession. Therefore, in the following I will assume H3 # 0. Keeping in mind (8), from (18) follows 1 H1 = -[-x(Psincp aH3

+ pcoscpCp) + pcoscpp~], (19)

1

= -[x(Pcoscp - psincpd) +psincpps] aH3 The expression (7) of c, according to (8), becomes If2

c = sin O(cos cpil

+ sin pi2)+ cos O&

Therefore, the equality (15.3) is equivalent (H3 # 0) to

81 where

w = P[XYP - (1 + X)P31. The condition (15.1), according to (19) becomes

Further, projecting (14.1) on the axis G i , one has u sin 0 W . w - --a - 2 y ( l - -)xpp = 0 a P To sum up, in the hypothesis that a ( t ) and v ( t ) are prefixed, the analytical problem of finding the precessional motions of the body consists in the compatibility of the equations (23),(24) in the unknown function p ( t ) . If a solution p ( t ) exist,(9) determines the function p(t). Therefore a precession whose axis of figure is Gj and that of precession Gg exists. It is to be remarked that the equations (23),(24 in general imply a connection between a and v that is, according to (lO),(ll), between the trajectory of G in the

plane 7r and its temporal law. In particular, the solution p = const. (p = 0) exists when the trajectory of G is a circumference ( a = const.). In general, according to (22), one has 2i, = [2pxy - (1

+ x)p3]p.

(25)

Therefore, from (24) one deduces P[2x:yvsin0 - p3(l PbZY

+ x)] -_ -

- (1 + X)P31

a

Keeping in mind (22) and (26), the equation (23) becomes an algebraic relation between the variables p , a, a, v. Therefore p ( t ) is determined as a function of a , a, v that must satisfy the equation (26) which represents the final compatibility equation. In general, it is not difficult to integrate the equation (26), in particular, if u = const. and y=2xyvsin0-p3(l+x) # O

(27)

one finds

where 7 is an arbitrary constant and

I renounce to show that in the two hypotheses p = const.

or p = vsin0, the resolution of the problem is very easy.

(sin0 # 0 ) .

(30)

82 References

1. G.Grioli, Rend. Acc. Naz. dei Lincei, S.VII1, vol.XXXIV, f.6 (1963). 2. G.Grioli, Rend. del Seminario Matematico e Fisico di Milano, vol. XXXIV (1964). 3. G.Grioli, Corso CIME, Bressanone, 2-12 June, 1971. 4. G.Grioli, Archive for Rational Mechanics and Analysis, 110, n.2 (1990).

STABILITY SWITCHES ON FOUR TYPES OF CHARACTERISTIC EQUATION WITH DISCRETE DELAY MA ZHIEN Department of Applied Mathematics, Xi‘an Jiaotong University, Xi’an 710049, China E-mail: [email protected]. cn

LI JIANQUAN Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, China Telecommunication Engineering Institute, Air Force Engineering University, Xi’an 710077, China E-mail: jianq-li@263. net Delay differential systems are widely used in many different fields. It is important to determine the local stability of their equilibria. Discussing the corresponding characteristic equation is one of common methods. In this paper, the stability switches on four types of characteristic equation with discrete delay are introduced summarily. We show that there exist some essential difference between the characteristic equations with delay independent parameters and those with delay dependent parameters. When the characteristic equation has pure imaginary roots, systems with delay independent parameters must be ultimately unstable, and systems with delay dependent parameters may be one of ultimately stable, ultimately unstable, and alternate between stable and unstable. Applying our results, the ultimate stability of some characteristic equations with delay dependent parameters can be often decided directly and need not appeal to mathematic software, and two examples are given in the paper.

1

Introduction

It is well known that the delay differential systems can usually make the mathematical models established more real, and are widely used in many different fields such as biology, medicine, chemistry, economics, control theory and so 0n132,3. Equilibria and their stability are important problems t o be concerned by mathematicians and related scientists. For the local stability, the standard approach is to analyze the stability of the characteristic equation corresponding to the equilibrium. If all the eigenvalues have negative real part, then the equilibrium is locally asymptotically stable, or simply to say that the characteristic equation is stable; if there is at least one eigenvalue with positive real part, then the characteristic equation is unstable. Because the characteristic equation of differential system with discrete delay has infinite number of eigenvalues, the stability is not very easy to be determined. Moreover, the stability may be changed as the delay increases. Sometimes, when time delay is increasing, the stability of the characteristic equation may be changed alternately many times. This phenomenon is called t o be stability switches by Cooke and Grossman4. In this article, the stability switches on four types characteristic equation with discrete delay will be introduced summarily. These results were obtained by Cooke & van den Driessche5, Huang & Ma6,Beretta & Kuang7, and us8)’ respectively.

83

a4 2

+

Equation P(X) Q(X)e-XT = 0

In 1982, Cooke and Grossman were the first to study the influence of the delay to the changes of stability of the characteristic equation. In 1986, Cooke and van den Driessche investigated the stability switches on the characteristic equation

P(A) + Q(X)e-"

= 0,

(2.1)

where 'T is the delay; P(X) and Q(X) are analytic functions of 'T, or polynomials as a special case. The idea of their method is the following: Suppose that all the eigenvalues have negative real parts when 'T = 0, that is, all the eigenvalues are located in the left side of the complex plane when 'T = 0. The eigenvalues will vary when 'T increases. If an eigenvalue has positive real part when 'T = 7 , then the eigenvalue must pass through the pure imaginary axis at 'T = 'TI < 7;. So in order to study the change of stability, one should be first t o find when the eigenvalues will be passing through the pure imaginary axis, and to determine their crossing directions. For the above purpose, let X = iy(y > 0), substituting it into the equation (2.1) and separating its real part and imaginary part, we obtain P ~ ( i y +) Q ~ ( i y ) ~ ~ ~ ~ ~ + Q z ( i y=)0,s i n y ' ~ P~(iy)+Q~(iy)cos~'~-Q~~(= i ~0 ), s i n y ' ~ where PR(Y) and QR(Y) are the real parts of P(iy) and Q(iy) respectively, Pr(y) and Ql(y) are the imaginary parts of P(iy) and Q(iy) respectively, so that

then 2

F(y) := IP(iy)12 - IQ(iy)I = 0.

(2.3)

Basing on these equations, Cooke et a1 proved the following results. Theorem 2.15 Suppose that P(X) and Q(X) are both polynomials, they have no common pure imaginary roots; the degree of P(X) is higher than the degree of &(A); the positive roots of equation F(y) = 0 are y j , j = 1 , 2 , . . . , n , all these roots are simple, yj and corresponding T* satisfy the system (2.2). Then i y j ( ' ~ * ) must be a pure imaginary root of (2.1), and when 'T increases from T * , the crossing directions of the eigenvalue X ( T ) through the imaginary axis are determined by the following formula m

s k = signF'(yk),

i. e.,

s k = sign j=1,

(T& - r j ) ,

(2.4)

j#k

where r j = yj. The crossing direction is from left to right if s k = 1, from right to left if sk = -1. Theorem 2.25 Suppose that the characteristic equation (2.1) is stable at 'T = 0.

85 (1) If equation (2.3) has only one positive simple root, then the stability will be changed when the eigenvalue curve passes through the imaginary axis at some T = 71, and (2.1) will keep unstable for all T > 71. (2) If equation (2.3) has two positive simple roots, then the stability will be changed finite times as T increases, and must be ultimately unstable. 3

+

Equation P(A) Q(A)edXT

+ R(X)eP2"

=0

In many cases, the characteristic equation of some models is of the following type

P(A) + Q(A)e-"

+ R(A)e-2XT = 0.

(3.1)

For example, a larva-adult stage-structure population model proposed by Nisbet and Gurney lo is the following

I

st"-,

L ( t ) = wo + B(z)dz, dA-B (t - T ) - bA(t), qB(t - T)W(t)- 6B(t), t W t )= wo + Jt-T ")ldz.

gr

(3.2)

The characteristic equation corresponding to the positive equilibrium of model (3.2) is of the following form

x3 -+ (rx2(1 - e-AT) + ~

( 1 -e-XT)2 = 0 ,

(3.3)

which belongs to the type (3.1). Brauer and Ma l1 investigated the stability of equation (3.3). Using the similar idea presented by Cooke et a1 ', Huang and Ma extended the Cooke's results to the equation (3.1). Theorem 3.16 Let

G(Y) = IP(iY)12- IRGY)12,

F(Y)= G2 - [QR(PR- RR)+ Q r ( R - Rr)I2 -[QR(~I

+ RI)- Q I ( ~ R+ R R ) I 2 .

Suppose that P(X),&(A) and R(X) in equation (3.1) are all polynomials, they have no common pure imaginary roots, the degree of P(A) is higher than the one of &(A) and R(A),all the positive roots of F ( y ) = 0 are simple. If y* is a positive root of F ( y ) = 0, then there exists T* > 0 such that iy' is a pure imaginary root of (3.1) at T = T * , and when T increases from T * , the crossing directions of the eigenvalue A(T) through the imaginary axis are determined by

The crossing direction is from left to right if S = 1, from right to left if S = -1. Similarly, we also proved that if (3.1) is stable at T = 0, and equation F ( y ) = 0 has at least a positive root, then when T increases, the stability switches of equation (3.1) can be happened at most finite times and must be ultimately unstable.

86 4

Equation P(X,7 )

+ Q(X, r ) e P X r= 0

Many characteristic equations of discrete-delay differential systems obtained from Beretta and Kuang real problems have delay dependent parameters investigated the stability switch of the following characteristic equation 12i13,14,15116.

+

P(X,r ) &(A, 7)e-"

= 0,

(4.1)

where n

rn

and n , m E No,n > m, pn(7) = 1, and pk(.),qk(.) : R+o + R are continuous and differentiable functions of T such that P(O,T) Q(O,T) # 0 for any T E R+o. For example, the stage-structure population model with time delay 7 as the mature time

+

{

J ' ( t ) = s [ F ( t )- e-mJrF(t - T ) ] - m J J ( t ) , A'(t) = s e - m J T F ( t- 7 ) - mAA(t), F ( t ) = rnax(0,l - a ~ J ( t-) a A A ( t ) } ,

(4.2)

has the corresponding characteristic equation at the positive steady state X~

+ a~ + c + ( b +~d)eWxT= 0,

(4.3)

where

+ + +

a = mA mJ ~ J S , b = b ( r ) = (UA - a J ) s e - m J T , c = (mAmJ a J m A s ) , d = d(7)= ( U A ~ J aJmA)se-mJ'. Equation (4.3) is a special case of equation (4.1). For equation (4.1), Beretta and Kuang assume the following: (1) If X = iy, y E R, then P(iy,.r) Q ( i y , r ) # 0 , E~R+o; (2) F ( y , 7 ) := [P(iy,7)I2- (Q(iy,r)I2 for each r has at most a finite number of real zeros; (3) Each positive root Y(T) of F ( y , r ) = 0 is continuous and differentiable in r whenever it exists. Similar to the inference of section 2, Let X = iy(y > 0) in (4.1), then

+

hence,

If iy(y > 0) is a pure imaginary root of (4.1), then y and corresponding r must satisfy (4.4) and (4.5). On the other hand, if there are y and r satisfying (4.4) and (4.5), then iy must be a pure imaginary root of (4.1). Notice that the root, y, of (4.5) generally depends on the the delay r , denoted by y = y ( r ) , and corresponding

87 i y may not be the root of (4.1). Further, in order to find the pure imaginary root of (4.1) and corresponding delay value, substituting y = y ( r ) into one of (4.4) gives an equation of r , which has a root denoted by r*, then iy(r*) must be the pure imaginary root of (4.1). So the process of determining the pure imaginary root and corresponding r of (4.1) is different from that of (2.1). For the convenience of inference, by the periodicity of sine function and cosine ) (4.1) must satisfy function, r* corresponding to the pure imaginary root i y ( ~ * of the following equation:

+

8(r) 2nr =O Y (7) for some n E NO,where the angle function 8(r) E ( 0 , 2 r ) is determined by equations Sn(r):= 7 -

Therefore, Beretta and Kuang proved the following theorem: Theorem 4.17 Assume that y( r ) is a positive real root of F ( y , r ) = 0 and at some r* > 0, Sn(r*)= 0 for some n E NO. Then a pair of simple conjugate pure imaginary roots X = f i y ( r * ) of (4.1) exists at r = r* which is crossing the imaginary axis according to

The crossing direction is from left to right if S = 1; from right to left if S = -1 Beretta and Kuang especially studied the following specific case

P(X,7 ) = X2

+ a(r)X + ~ ( r ) &(A, ,

r ) = b(r)X

+d(r),

(4.7)

where functions a ( ~ b) (, r ) ,C ( T ) and c ( r )are all continuous differential in the interval R+o = [O, +m), and ~ ( r )d ( r ) # 0 for any r E R+o, equation (4.1) with (4.7) often arise in the stability study of ecological, epidemic and other models with delay dependent parameters. Since it is too complicated to solve the pure imaginary root and corresponding r from equation (4.1) with (4.7), so instead of analytic method Beretta and Kuang used numerical (geometric) analysis to determine the stability of equation (4.3) for some given specific parameters. According to Theorem 4.1, Their simulation shows that the characteristic equation with delay dependent parameters may be ultimately stable after a finite times of stability switches. This is an essential difference from the characteristic equation with delay independent parameters. But the general conditions under which the characteristic equation is ultimately stable or unstable are not obtained by them. On the basis of Beretta and Kuang's work, we did some progress on the equation (4.1) with (4.7) under some general assumptions, which are easy to be satisfied. For equation (4.1) with (4.7), we make the following assumptions: (Al) Functions a ( r ) ,b ( r ) ,c ( r ) and d ( r ) are all continuously differentiable in the interval R+o = 10, +m);

+

aa

+

(A2) C(T) d ( ~#) 0 for any 7 E R+o, which ensures that X = 0 is not a root of (4.1) and that C(T)and d ( 7 ) can not be zero simultaneously; (A3) P ( i y , T ) Q(iy, T ) # 0 for any T E R+o, which ensures that equations P(X,T ) = 0 and &(A, T ) = 0 have no common pure imaginary root; (A4) (4.1) is stable when 7 = 0. Notice that

+

+

+ [C2(T)- d 2 ( T ) ] = 0,

F ( y , 7) := y4 - [ b 2 ( T ) 2C(T) - U 2 ( 7 ) ] y 2

then assume that F ( y , T ) satisfies the following: (A5) For any T E R+o, equation F (y , T ) = 0 of y has at most one positive root y = Y ( T ) , which conforms to the common cases. And function c ~ ( T-) d2(.) has at most one zero on R+o. According to (A5), if function c2(7)-d2(7) has no zero on R+o,then the existent set of y = y(7) is interval (0, +co) when equation F ( y , T ) = 0 has just one positive root y = ~ ( 7 )if; function c2(.) -d2(7) has just one zero 7; E R+o, then the existent ) interval ( 0 , t ) or (7;,+co), and y(7) = 0. In the following, we set of y = y ( ~ is ) ( a ,p). denote all of the existent interval of y = y ( ~ as Denote

where y(7) is a positive root of equation F ( y , T ) = 0, tions

O(T)

is determined by equa-

Assumption (A3) ensures that O ( T ) # 0,27r, so that O ( 7 ) E (0,27r). For function S(T) we assume the following: (A6) For any n 6 NO = (0, 1 , 2 , . . .}, all the roots of equation S(T) = n,T E ( a ,p) are simple if they exist. This assumption ensures the pure imaginary root iy of (4.1) is simple. According t o Theorem 4.1, we have Theorem 4.2 If F ( y , T ) = 0 has no positive root for any T E R+o, or equation S(T) = n has no root for any n E NO and 7 E (a,p),then (4.1) is always stable for any T E [ O , + c o ) , that is, (4.1) is absolutely stable, it is, of course, ultimately stable. In order t o investigate the ultimate stability of (4.1) for other cases, we first prove the following Lemma. Lemma 4.3 Suppose that (4.1) has two pure imaginary roots: iy(~;),iy(~;) (Y(T;),Y(T;) > 0), where a < T ; < T; < p. (1) If S(T;) # S(T;),then iy(7;) and Z Y ( T ; ) are located on different branches of eigenvalue curves of (4.1); (2) If S(T;) = S(T;), and T;,T; are consecutive points such that S’(7;) > 0, S’(T;) < 0, then the pure imaginary roots iy(7;) and i y ( ~ ; ) are located on the same branch of the eigenvalue curves. Proof:

89

(1) Suppose that S(T;) = n l , S ( ~ ;= ) n2 and nl

y(7;)T; = O ( 7 : ) y(7;)7; = O ( 7 ; )

# n2, then

+ 2 n l ~E ( 2 n 1 ~2(nl+ , I)T), + 2n2r E ( 2 n 2 ~2(n2 , +l ) ~ ) ,

since O E ( O , ~ T ) . If iy(7;) and iy(7-z) are located on the same branch X(7) = z(T)+~Y(T),then from the continuity of x(~), y(7) and O ( - r ) , we know that Y(T)T will be changed continuously from O ( 7 ; ) 2 n l ~ to O ( 7 ; ) +2n27r as T varies continuously , 1 ) ~n () 2 n 2 ~2(n2 , 1 ) ~= ) from 7; to 7;. This is impossible because ( 2 n l ~2(n1+

+

4.

+

(2) R o m (A5), we have lirn y ( 7 ) ~= 0, so lim S(T) < 0 due to 0 # 0 , 2 ~ . 7+a+

r+a+

For a given n E NO,suppose that S ( T )= n(.r E ( a ,p)) has m-roots, denoted by T;k, 1 5 k 5 m, respectively, and rz1 < T , * ~< . . . < r7trn. Since T , * ~is the minimum root of equation S(T) = n(7 E ( a l p ) )and 77t2 is its consecutive root, we have S’(T&) > 0, S’(77t2)< 0 by the continuity of S(T) and assumption (A6). ) i y ( ~ , * ~are ) not located on the same branch of the eigenvalue If i y ( ~ , * ~and curves, by Theorem 4.1, there are two branches of eigenvalue curves, X 1 ( r ) and Xz(7). X ~ ( T ) crosses the imaginary axis at point ( O , ~ ( T , * ~from ) ) left to right, and Xz(7) crosses the imaginary axis at point (0,y(7;t2)) from right to left. But according to assumption (A4), all the eigenvalues of (4.1) are located on the left half plane when T = 0, then X 2 ( 7 ) must crosses the imaginary axis from left to right at some point (0,y(~;,)), where n’ = S(r:,) E No,T;,< 77t2 and T;, # 77tl. So S(r,*,)# S(77t2).Due to the conclusion (l),points (0,y(~;,)) and (0, y(r7t2)) are on different branches of eigenvalue curves of (4.1). The contradiction occurs. This contradiction implies that iy(r&) and iy(7z2) are located on a same branch. ~ )0 , 1 5 k 5 [ f ] ( [ xexpresses ] It is easy to see that S’(7;t,2kVl) > 0, S ’ ( T ; , ~< the integral part of x). Using the same inference as above, we know that i y ( ~ ; , ~ ~ and iY(7;,2k) are located on the same branch. Therefore, the conclusion (2) is true. The proof of Lemma 4.3 is complete. In the following, we give the theorem of ultimate stability for (4.1). Theorem 4.4 The following conclusions are true for characteristic equation (4.1). (1) When the existent interval of y(7) is finite, (4.1) must be ultimately stable; (2) When the existent interval of y(7) is infinite, (4.1) is ultimately stable if there is a T > 0 such that S(7) < 0 for all 7 > T , ultimately unstable if there is a T > 0 such that S(T) > 0 for all 7 > T , and the stability switches of (4.1) will appear forever as 7 increases if there is always a zero of S = S ( 7 ) for 7 > T ( where T is a arbitrarily positive number). Proof: (1) According to (A5), when the existent interval of y + ( ~ is ) finite, then ( a ,p) = (O,?). Since lim y ( r ) = 0 and O ( 7 ) # 0,27r, then lim S ( 7 ) < 0. 7’7-

From lim S ( 7 ) 7+0+

r-7-

< 0, we have that the curve S

= S(T),T E (O,?) has even

number of intersections with the horizontal line S = 0 if they intersect. In this case, we assume that the values of r of these intersections are r;, T ; , . . . ,rzm,respectively, where 7; < ~ ; + ~ , i= 1,2, ... , 2m - 1. By assumption (A6), we know S ’ ( T & ~ ) >

90 0 and S ’ ( T ~<~0, ) j = 1 , 2 , . . . , m. By Theorem 4.1 and Lemma 4.3, there is a branch of eigenvalue curves X ( T ) = X(T) ZY(T), which crosses the imaginary axis from the left to the right when T increases through 7zjPl and from right to left when T increases through T&. It implies that this branch X ( T ) enters into the right half-plane through (0, Y ( T & ~ ) ) and stay there as T ; ~ - < ~ T < T $ , and then comes back to the left half-plane through ( O , ~ ( T & ) ) . Hence, (4.1) is unstable in the interval ( T & ~ , T & ) , 1 5 k 5 m. In the following we will prove that (4.1) is stable in each interval (T&~,T&~) (1 5 j 5 m) where T$ = 0. Obviously, it is stable as 0 5 T < T i . We now prove first that it is stable when T E ( T ; , T ; ) . In fact, if the curve S = S(T) has no intersection with S = 1, then the result is clear. If S = S(T) intersects with S = i(k 2 1) in the interval ( T i , T ; ) , and let these intersections correspond to T&, T ; ~ ., . . , T ; , ~in~ order, then due to the monodromy of S(T) and (A6), we must have

+

S ’ ( T ~ , ~>~O- ,~S )’ ( T ~ < , ~0,~ ) m = 1 , 2 , . . . , 1.

By Lemma 4.3, Z ~ ( T ; , ~ ~ - ~and ) iy($zm)(l 5 m 5 1 ) are located on the same branch. And this branch will enter t h l right half-plane when T increases through 7:k,2m-1, and come back to the left half-plane when T increases through T ; , ~ thus ~ , as T = T; all the roots of (4.1) have negative real part except a pair of imaginary roots + Z Y ( T ; ) . Because S ( T ) < 0 as 7 E ( T ; , T $ ) , the curve S ( T )= 0 has not root in the interval ( T ; , T ; ) , hence (4.1) is stable as T E ( T ; , T ; ) . Repeating above process we know that (4.1) is stable as T E ( T & ~ , T ; ~ - ~ ) , unstable as T E ( T ; ~ - ~T,; ~ ) j , = 1 , 2 , . . . , m, TO* = 0. Because T&,, is the last root of S(T) = O(T E (O,?)), so (4.1) is always stable when T > T&, i.e. part (1) is true. (2) According to (As), when the existent interval of Y ( T ) is infinite, then ( a ,P ) = (0, +m) or (?, +m). Suppose that ( a ,p) = ( 0 ,+m), notice that lim S(T) < 0. If there is a T > 0 T’O+

such that S(T) < 0 for all T > T , the curve S = S(T) must have even number of intersections with the horizontal line S = 0 if they intersect. So the conclusion (2) may be obtained by the same inference as part (1). If there is a T > 0 such that S ( T )> 0 for all T > T, equation S ( T ) = 0 has odd number of roots in ( 0 ,+m) and T = T* is the largest one. Therefore, by the same inference, it can be obtained that (4.1) will become unstable when T > 7 % . Suppose that ( a ,p) = ( 7 ,+m), notice that lim S ( 7 ) < 0 due to lim Y ( T ) = 0 T’t+

r-t+

and O ( T ) # 0,27r. Since (2.3) has no positive root in the interval (0, ?I, then (4.1) is stable as T E (0,71. Hence, this case is the same as the case ( a ,P ) = (0, +m). When there is always a zero of S = S(T) for T > T ( where T is a arbitrarily positive number), the inference is similar to above. Summarizing above, part (2) is true. The proof of Theorem 4.4 is complete.

91

When equation F ( y , T ) = 0 of y has two positive roots, we have also found the range of corresponding 7 and theorem determining the ultimate stability of (4.1) with (1.7) 8. In the following, two examples are given to illustrate our results. Example 1 Suppose that mJ = m~ = m in (4.3), then, corresponding to (4.5), we have the equation

where

~ Z ( T )=

+

m’[(m ujs)’ - ( u A

-

~~)’s’e-’~~].

Notice that (4.3) is stable as T = 0. It is easy to see f l ( ~>) 0 as f z ( ~ )2 0, so (4.9) has no positive root as f i ( ~2)0. Since f ’ ( ~ > ) 0 for all T > 0 when ( m + a ~ s ) ’2 (uA-uJ)’s’, hence, (4.3) is always stable for all T > 0 by Theorem 4.2 when ( m U J S ) ~2 ( U A - UJ)’S’. If ( m UJS)’ < ( U A - UJ)’S’, then f 1 ( ~ < ) 0 as 0 < T < .r = In ( ~ ~ ~ ~ j , ’ : ’ and f l ( ~ > ) 0 as T > 7 , so (4.9) has only a positive root y = y ( ~ as ) 0 < T < 7. Therefore, (4.3) is ultimately stable by Theorem 4.4 when ( m UJS)’ < ( u A -

+

+

&

+

UJ)’S’.

Hence, the positive steady state of (4.2) must be ultimately stable when mJ = mA.

Example 2 We consider an SEIS epidemic model

{

+

S’(t) = ( b - d ) S ( t )- p S ( t ) I ( t ) y I ( t ) , E’(t) = p S ( t ) ~ ( t-) p e - d T S ( t - T ) I ( t - T ) - d E ( t ) , I’(t) = p e c d T S ( t - 7)1(t- T ) - ( d + a + y ) I ( t ) .

(4.10)

In model (4.10), S ( t ) , E ( t )and I ( t ) are the number of individuals, who are susceptible, exposed (i.e. in the latent period) and infectious, respectively. T is the period of latency. Here, b, d, P,y are positive constants and LY is a nonnegative constant. Since the first equation and the third equation in (4.10) do not contain E ( t ) explicitly, then in the following we will consider system

S’(t) { I’(t)

+

= ( b - d ) S ( t )- p S ( t ) I ( t ) yI(t), = pepdTS(t- T ) I ( ~- 7) - w ~ ( t ) ,

(4.11)

where w = d+a+y 2 d+y. When b > d, (4.11) has an unique endemic equilibrium Pe(SetIe), where W

w(b - d )

S e = F , I -- P[w- ye+‘]

’

It is easy to obtain the characteristic equation of (4.11) at Pe (4.12) where f ( ~=) ;%,g

-dr

=b

-

d > 0.

92 When

T

= 0, (4.12) becomes

X2

+ aw - Y x + wg = 0,

so P, is stable. Corresponding to (4.5), we have the equation y4

+ g 2 f 2 ( T ) y 2- W2g2[1- 2f(T)] = 0,

(4.13)

where 1 - 2 f ( ~ = ) +. When w 2 37, 1 - 2f(7) > 0 for all T > 0, so the existent interval of the ) 0 for positive root y = y ( ~ ) of (4.13) is (O,+m). When y < w < 37, 1 - 2 f ( ~ > T > 7 = d l n z , so the existent interval of the positive root y = Y ( T ) of (4.13) is (7,+m).

Since lim f ( ~=) 0, then, from (4.13) we can obtain lim

7++m

T'+CC

Y(T) =

fi.Due

to O(T) E [0,27~), we have lim S ( T )= +m. 7++CC

Therefore, by Theorem 4.4 the endemic equilibrium P, is ultimately unstable. 5

Equation P(X,T )

+ Q(X,

+

T)e--xT R(X,T)e--2Xr = 0

Corresponding to the equation (3.1), we may study the following equation with delay dependent parameters P(X,T )

+ &(A, T)e-Xr + R(X,T)e-2Xr

- 0.

(5.1)

In fact, for the model (3.2), if we consider the death of the larvae during their mature period, 7 , then the corresponding characteristic equation will belong to the type (5.1). Assume n

P ( x , T )= x P k ( T ) X k l k=O

k=O 71.

Denote

> maz{m,1},72 O,P,(T)

= 1,

93

+

F(Y,r ) = G2- [ Q R ( ~ I R I )- Q I ( ~ R+ RR)]’ - [ Q R ( ~ R- RR)+ Qr(pr - Rr)I’, then, for (5.1) the imaginary root X = iy(y equations

{

> 0) and

corresponding r must satisfy

sinyr = QR(PI+RI)-QI(PR+RR) 1 G cosyr = -QR(PR-RR)+QI(P~-RI) G

(5.2)

7

and F ( y , 7) = 0.

(5.3)

Suppose that there is the unique, positive, simple and continuously differentiable function y = y(r) determined by (5.3) in the interval (O,?), where ? = sup{r : T>O

Y(7) > 01. By the same idea with Section 3 and 4,we proved the following results under some popular assumptions, which are easy to be satisfied. Theorem 5.1’ The necessary and sufficient condition of existing the pure imaginary roots for (5.1) is that there exists some k E NO such that the straight line S = k intersects with the curve S = S ( 7 ) , r E (O,?), where S ( r ) is defined in the same form with (4.8). Thus, if r* E (O,?) satisfies S ( r ) = k E No, then A = *zy(r*)(y(r*) > 0) is a pair of pure imagine roots of (5.1). Moreover, when r increases from r*, the crossing directions of the eigenvalue X ( T ) through the imaginary axis are determined by the formula

V = sign[S’(r*)]. If V = 1, then the direction is from left t o right; if V = -1, then the direction is from right to left. Again, suppose that (5.1) is stable as r = 0, then, for (5.1) the results with respect t o stability and ultimate stability are given by following Theorem 5.2 and 5.3, respectively. Theorem 5.29 The following results hold. (1) If equation S ( r ) = k has no root for any k E NO,then (5.1) is stable for any r 2 0. (2) If ? < 00, then there exists the even number of zeros of S = S(7) in (O,?), which are denoted by rr ,r;, . . . ,rTm(r; < r; < . . . < respectively, such that = 0 , 1 , 2 , . . . , m , r ; = O , T $ ~ + , = +00), and (5.1) is stable if r E (~2;c,r2;E+~)(k ( 0, k 1, 2, . . . ,m - 1). unstable if 7 E ( T & + ~ ,~ ; ~ + ~ )= (3) Provided ? = +co. (i) If lim s u p S ( r ) < 0, then there exist even number of zeros of S = S(T) T-+W

in (0,+00), which are denoted by r;,r;,...,r;, (r; < 7; < . . +< rTm)respectively, such that D ( X , r ) = 0 is stable for r E ( r & , ~ & + ~ ) (=k 0,1,2;..,m,r; = 0, r;m+l= +00), and unstable for r E ( ~ 2 ; c + r&+’)(k ~, = 0, 1, 2, . . . , m - 1). If lim in f S (r ) > 0, then there exists the odd number of zeros of S = (ii) T’+W S ( r ) in (0,+00), which are denoted by r ~ , ~ ; * , . . . , r ; ~ +<~ (7;r ;< . . . < T ; ~ + ~

) ,

94

respectively, such that (5.1) is stable for T E (T&, ~ & + ~ )=( k0,1,2,. . . ,m,7; = 0), and unstable for T E ( T & + ~ , ~ $ + ~ ) = ( k0,1,2,. . . , m, T T ~ = + ~+m). (iii) If lim s u p S ( ~ ) > 0 and lim inf S(T) < 0, then there exists T-++CO

T++W

the infinite number of zeros of S = S(T) in (O,+m), which are denoted by T;,T;,...,T&,...(T: < 7; < . . . < T& < ...) respectively, such that (5.1) is stable for T E ( T & , T & + ~ ) ( ~= 0,1,2,...,m,...,~; = 0), and unstable for T E (T;k+l, T ; k + 2 ) ( k = 0,1,2,. . . , m , . . .). Theorem 5.3’ The followings are true. (1) When the existent interval of y ( 7 ) is finite, (5.1) must be ultimately stable; (2) When the existent interval of y ( ~ is) infinite, (5.1) is ultimately stable if there is a T > 0 such that S(T) < 0 for all T > T , ultimately unstable if there is a T > 0 such that S ( T ) > 0 for all T > T , and the stability switches of (5.1) will appear forever as T increases if there is always a zero of S = S(T) for T > T ( where T is a arbitrarily positive number). References 1. R. Bellman, K. L. Cooke in Differential-Difference Equations, (Academic Press, New York, 1963). 2. J. Hale in T h e o y of Functional Differential Equations, (Springer-Verlag, New York, 1977). 3. Y. Kuang in Delay Diflerential Equations with Applications in Population Dynamics, (Academic Press, Boston, 1993). 4. K. L. Cooke, Z. Grossman, J. Math. Anal. Appl. 86, 592 (1982). 5. K. L. Cooke, P. van den Driessche, Funkcial. Evac. 29, 77 (1986). 6. Q. Huang, Z. E. Ma, Ann. of Diff. Eqs. 6,21 (1990). 7. E. Beretta and Y. Kuang, S I A M J. Math. Analysis 33, 1144 (2002). 8. J. Q. Li, Z. E. Ma, Stability of some characteristic equation with delay dependent parameters, to appear. 9. J. Q. Li, Z. E. Ma, Stability switches on a type of characteristic equation with delay dependent parameters, to appear. 10. R. M. Nisbet, W. S. C. Gurney, Lecture Notes in Biomath. 54,97 (1984). 11. F. Brauer, Z. E. Ma, J. Math. Anal. Appl. 126, 301 (1987). 12. J. R. Bence, R. M. Nisbet, Ecology 70, 1434 (1989). 13. Y. Kuang, J. W. H. So, S I A M J. Appl. Math. 55, 1675 (1995). 14. W. D. Wang,Applied Mathematics Letters 15,423 (2002). 15. H. W. Hethcote, P. van den Driessche,J. Math. Biol. 40,3 (2000). 16. S. L. Yuan, Z. E. Ma, Journal of System Science and Complexity 14, 327 (2001).

ON THE BEST VALUE OF THE CRITICAL STABILITY NUMBER IN THE ANISOTROPIC MAGNETOHYDRODYNAMIC BENARD PROBLEM* IN HONOUR O F T H E 70TH BIRTHDAY O F PROF. S. RIONERO

MICHELE MAIELLAROt ABSTRACT. - We prove that the best value of the critical stability number for the linear BCnard problem in anisotropic Magnetohydrodynamics is equal to the best value of the critical stability number for the BCnard problem in Hydrodynamics. Keywords: Magnetohydrodynamics, Convective Instability A.M.S. Classification: 76315, 25. 1

Introduction

In Hydrodynamics and in isotropic Magnetohydrodynamics, the BCnard problem of the stability of the steady motionless thermodiffusive flows has been object of several investigations, in the past and in these last years, owing its importance from the mathematical point of view, as well as in laboratory experiments and for several industrial applications [l][2][3]. In the field of the anisotropic MHD, the influence of the electrical anisotropic ion-slip currents on the dynamic and the stability have been taken into account in [4].. . [22]. For instance, in [12], the stability of the anisotropic MHD plane CouettePoiseuille flows has been studied via the direct Liapunov method and has been shown that, at least when the perturbations are laminar and parallel to the embedding magnetic field, does not exist an instabilizing focus effect. The completely anisotropic MHD BCnard (MHDB) problem, in presence of both the anisotropic Hall and ion-slip currents, begin to be investigated in the papers [4][9]for the linear case and in [lo] for the nonlinear case. In particular, recently, in [19][21],among other things, conditions ensuring the validity of the principle of exchange of stabilities (P.E.S.) in the anisotropic MHDB problem have been obtained. In the present paper, the stability in the linear anisotropic MHDB problem is investigated, in order t o obtain the best value of the critical stability number. The plan of the paper is the following: In Section 2 we recall the basic equations and the linear MHDB problem when both the anisotropic electrical currents are taken into account [21]. In Section 3, by "normal mode" analysis and in the "stress-free" boundary case, we *THIS RESEARCH HAS BEEN SUPPORTED BY T H E ITALIAN MINISTRY FOR UNIVERSITY AND SCIENTIFIC RESEARCH (M.U.R.S.T) UNDER 60% CONTRACT, AND BY G.N.F.M. O F T H E I.N.D.A.M. tUNIVERSITA DEGLI STUD1 DI BARI, DIPARTIMENTO DI MATEMATICA, VIA E. ORABONA, 4 - 70125 BARI (ITALIA).

95

96

consider the spectral problem for the stability, from which, thanks to the validity of the P.E.S. proved in [21],we obtain the equation of the critical stability curves in a very simple and useful form for the sequel. The main property that appear from this equation is that these curves change at the variation of the physical transport parameters of the anisotropy in such a way that we shall state the following preliminary result: the critical stability curves, at the increase of the anisotropy, tend to superimpose uniformly to the critical stability curve of the classical hydrodynamic Bdnard (HDB) problem; therefore the critical stability numbers of the anisotropic M H D B problem 27 tend to the classical critical stability number -r4 of the hydrodynamic case. 4 In Section 4, we apply the rigorous Liapunov direct energy method to the linear stability problem, in order to give an exact critical value of linear stability and to compare this value with those obtained by "normal mode" expansion at the variations of the anisotropy. To this end, preliminarly, by an "ad hoc" variational inequality, we obtain an estimate which gives us the simple value i~' as a first critical stability number. This critical number, although is not the best one for the optimal stability, nevertheless it is interesting because it is valid whatever the anisotropies are (low or high). Then we solve the Euler-Lagrange equations of the maximum variational problem for the stability and, thanks to the above variational inequality, we obtain, finally, the following result announced in the Abstract: the best value of the critical number for the optimum stability obtained b y the Liapunov direct method is the same obtained - in the limit to infinity of the anisotropic currents - b y "normal mode" expansion. This value is equal to the best value of the critical number of the optimum stability in the classical H D B problem [3], that 27 is -r4. 4 2

Basic equations and linear BBnard problem

The evolution equations governing the dynamics of a thermal-electro- conducting fluid in the anisotropic MHD [24] [25] and in the Boussinesq approximation [3] are the following: Vt

I

1

= - v . VV - -VT P

+ V A ~+VAP H . VH + g[l P

-

a(T - To)]

H~=VX(VXH)+~,A~H+P~VX(HXVXH)+ +p2V[H x (H x V x H)]

Tt = - v . VT

+ kA2T

V.V=O ; V'H=O

+

where r = p p e H 2 / 2 is the total pressure; v, H, T are kinetic, magnetic and thermal fields, respectively, and g is the gravity. Moreover the positive constants p , v, pe, a, ve, To, k are density, kinematic viscosity, magnetic permeability, ther-

97 ma1 expansion, magnetic viscosity, reference temperature, thermometric conductivity, respectively. The positive constants P 1 and P 2 are two physical transport parameters which take into account of the anisotropic Hall and ion-slip currents, respectively. Of course, to complete a given problem, initial and boundary conditions must be added to the eq.s (2.1). Let us assume that the fluid fills the horizontal plane layer 0 5 z 5 d , of thickness d , embedded in the external constant magnetic field Ho = Hoe3, with e3 vertical positive upwards. Moreover, let the boundary walls z = 0, z = d be insulating and kept at fixed temperatures TOand T d respectively. As it is well known [l]the BBnard problem is the stability- instability problem of the following steady solution of (2.1):

{v

, H = Ho , T = To - (To - T d ) z / d , p = p 2 ( ~ }) (2.2) below (TO> T d ) , with p z ( z ) function of the second order in z ,

=0

in the layer heated obtained by (2.1)l. By the nondimensionalization

x = x*d ; t = t*d2/Ve ; u = u * q e / d ; h = h * H o ; 0 = O*(To- T d ) q e / k and after dropped all stars, we have the following linearized equations:

I

+ g 2 M 2 e 3 .V h + a2A2u+ R r 0 e 3 ht = e 3 . V u + Aph + PHV x (e3 x V x h) + P ~ O=,u3 + a2e ut = -VF

x

[e3 x (e3 x

V x h)]

(V.u=O ; V.h=O

and, for nonconducting boundaries, with on z = 0 , z = 1

u = h = (V x h)3 = 0 = 0

(2.4) which govern the evolution of "small" kinetic, magnetic, thermal and pressure perturbations { u ( x , t ) ; h ( x ,t ) ; q x , t ) ; P ( x ,t ) } respectively, on the flow (2.2). In the dimensionless system (2.3), g2 = u/q,, P," = V e / k and 1

M = Hod(Pu,/povVe)' ; RZ = agd3(To - Td)/Vek are dimensionless Hartmann and magnetic Rayleigh number, respectively. Moreover

PH = PlHOIVe

1

PI

=PZH;/V~

are dimensionless Hall and ion-slip numbers, respectively. We underline that

RT

= g2Ra

, R,

= agd3(To - T d ) / u k ,

with R, usual hydrodynamic Rayleigh number.

(2.5)

98

3

Critical Rayleigh curves

To investigate on the stability of the flow (2.2) by normal mode analysis, in the ”stress-free” case, we shall introduce the vorticity and the electrical current density fields, given by:

w=Vxu

, j = V x h .

(3.1)

On applying the operators V x and V x V X on the first equation (2.3)1, and V x on the second equation (2.3)2, and then on taking the third components of the obtained equations, we have, finally, the system:

2

=

a3u3-

with the usual boundary conditions [l]:

u3 = 0 = j 3 = h3 = d3w3 = d:3u3 = 0 on x3 = 0 , 1.

+

In the equations (3.2) A2 is the Laplace operator, A t ) = a:, is the plane Laplace operator and, as usual, differentiations with respect to the third variable is signalled by 83 and The analysis into normal mode is carried out by the perturbations

a:,.

( ~ 3 ~,

3 j 3, 1

h3, 0) = ( W ( z ) ,Z ( z ) , X ( z ) , K ( z ) , @ ( z ) )ex~[i (ai ~+azy)+A t ](3.3)

+

with X in general complex and a2 = a: a; wave number. On putting (3.3) in (3.2) and introducing the operator zeta-derivative D = d / d z , we have the spectral problem:

I

[A - (1 [A

-

+ p 1 ) ( D 2- a2) ] K + PHDX - D W = 0

a 2 ( D 2- a 2 ) ] 2- a 2 M 2 D X = 0

+ P I ) P + a 2 ] X- PHD(D’ - a 2 ) K - D Z = 0 (02 2 )[A o ~ ( D d~ ) ]- ~~ M ~ D -( 2D )~+ ~~ [A - (1 -

-

-

[PFA- (0’ - a’)] 0 - W = 0

~ = o2

(3.4) 0

99

In [21], among other things, it has been proved that the conditions (3.18) of that paper, that is

assure the validity of the P.E.S. in the anisotropic MHDB problem, when both the anisotropic currents are taken into account. It is important, for the sequel, to underline that this principle holds also with very high levels of the anisotropic currents (PH -+ 00,Pz + cm). Therefore, provided that these conditions hold, from (3.4), at the criticality (A = O), we obtain the following ten order differential equation for the study of the linear instability:

R r a 2 { & D 2 ( D 2 - a 2 ) + (1+,LIZ)[ ( D 2- a 2 ) ( ( 1+ pz)D2 - a 2 ) = o2(02

+~

- a'){ [ M ' D ~- (1 + P ~ ) ( D a2)'] ~ 2

~ -2P ;1 D

+ M 2 D 2 ] } W=

+ p z ) ~-2a2)+

[(02 - a 2 > ( ( 1

~ ( D-~a2)3}w

(3.5) with the boundary conditions W = D 2 W = 0 on z = 0, z = 1, that, with the equation (3.5) imply D2mW = 0 on the boundary, with m positive integer (cfr. 111). Let us introduce the (modified) Chandrasekhar number Q (i. e. - substantially - the external magnetic field), the wave number x and the (modified) wave number r as follows:

+

Q = M2/r2 ; x =a2/r2 ; r = 1 x (3.6) respectively. On taking into account of (3.4)6 and the vanishing of all even derivative D2"W on the boundary, by perturbations of wave form W = Asinnn-z, with A constant and n positive integer, from (3.5), after some suitable transformations, we have the equation of the critical Rayleigh curve which is marginal to the onset of stationary convectiona: R ~ H J )= n-4 (PI+!)-I (3.7)

where 4

I+!) =

Ckr4-' k=l 1

+ Pz + PH

c1 = c2 =

P;

cg =

(Q - Pz)(1 + Pz) - P i

~4 =

-Q(1+

aIt is easy to prove that the lowest curve is obtained for

2 -

TI

=1.

1

PI)

100

+ + Qcl); therefore, by the

We underline that it results 1c, = ( r - l ) ( b l r 2 b2r positions (3.6)2,3 that imply T > 1, it follows I) > 0. By introduction of the function

m = T4(1

+ z)/z

the equation (3.7) can be easily transformed in the following suitable form:

I

RLH1')= m ( r 2 + Q S N - l ) S

= r2

+ PIT + Q

N = S(l +/?I)

(3.8)

+Pi.

when both the two anisotropic currents are present, and in the critical Rayleigh curve (PI = 0):

I

RLH)= m ( r 2 + QUT-l) =

[sl,l=O

=

"I,,=o

(3.9)

in presence of the Hall current only. Of course, by (3.7) (3.8) (3.9) the Rayleigh curves of the isotropic MHD:

RLM)= m ( r 2 + Q )

PH =PI

=O

(3.10)

and of the hydrodynamics:

{

R,

= mr2

PH=PI=Q=O

(3.11)

are obtained and are well known [l]. By ( 3 . 8 ) ~(3.8)s ~ and (3.9)2, (3.9)3, clearly, it results

S
and

U
(3.12)

Consequently from (3.8). . . (3.12) - evidently and without any calculation - i t results: (3.13) Besides, from (3.8), we have that (3.14)

(3.15)

101

where

+ + 0)

C = Qr(r2 Plr

(3.16)

5

(3.17) with a1 =

1;

a2 = 2pr

; a3 = PF -

,&+ 2Q ; a4 = 2QP1

; a5 = Q2.

(3.18)

The positivity of the polynomial C ( r )is clear. The positivity of the polynomial E ( r ) is assured by the condition PI > /?H which holds from the validity of the P.E.S. [21]. Therefore the positivity of these polynomials assure the negativity of the ,Ll$derivative and the PI-derivative in (3.14) and (3.15), respectively; and so the (uniform) decreasing of the critical Rayleigh curves (3.8) at the increasing of the anisotropy is assured. Moreover, since PH and PI belong to the interval [0,03[, from (3.8) (3.9) (3.11) (3.13) (3.14) (3.15), at the increase t o the infinity of the parameters PI and/or P H , (because of the validity of the P.E.S. also in these limits) it follows that

RLHvl)--f R,

and

RLHH) -+ R,.

So, at the increase of the anisotropy the Rayleigh curves of the critical stability in the anisotropic linear MHDB problem, tends (uniformly) t o the Rayleigh curve of the critical stability in the classical HDB problem. Therefore we obtain the following statement: the minimum value of the linear stability, b y "normal mode" expansion, in the 27 anisotropic MHDB problem, tends to the minimum value -7r4 for the stability in 4 the HDB problem. 4

Linear energy stability

The rigorous Liapunov direct energy method gives sufficient conditions of stability [2] [3] [23]. Generally this method is employed in the nonlinear context, but - of course - it can be used also in the linear case [14] [23], to compare these values with those obtained by normal mode expansion, in order t o give exact critical Rayleigh values of linear stability. Now firstly, by this method and an ad hoc embedding inequality, we shall obtain, as a test, a meaningful linear energy stability estimate. To this end, we introduce a Liapunov functional E ( t ) , by the L2-energy perturbations. Let V be the usual period cell for perturbation and let 11 . 11 and < . > be the norm on L 2 ( V )and integration over V, respectively. Let us introduce the energy measure:

in which c2 is a coupling parameter to be chosen optimally later.

102 The time evolution of this measure is given by the time derivative:

dE - - < u . u ~ > + u 2 M 2 < h . h t > + c 2 P ~< 8 8 , > (4.2) dt On multiplying (2.3)1,2,3 by u, a2M2h,c20, respectively, and putting the obtained equations in (4.2), we have the following energy equation: dE d t = (R:+C2)Z-2)

(4.3)

where, as definition:

In order to obtain the announced estimate and, over all, to solve next the variational problem related with the maximum problem that will be derived later from (4.3) (4.4), we firstly need an embedding inequality that concerns the last term in ( 4 .4 ) ~.In fact we are interested in the minimum: (4.5) in the space 7-1 of admissible smooth solutions (with V . h = 0, hlc = j31, = 0). The Euler-Lagrange equation that follows from the related variational problem from (4.5), is given by:

V x l(e3 x j) x e31 - mh

= Vll,

with ll, a Lagrange multiplier. By the application of the operator O X to this equation, we have the spectral problem

{

$j3+mj3=0

onz=0,1

33=0

that, by the usual normal mode representation, j 3 = Asinnnz, gives m, = n2n2, with n positive integer, from which, with n = 1, we have the desired minimum m = n2, so that: 11% X V

X

h1I2 2 n211h112

(4.6)

Well, let us go now to obtain, primarily, the (linear) asymptotic exponential energy stability estimate. Let us introduce the following functionals:

Y1 = lu112; Yz”= lh112; Y,”= 11Q112. By (4.3) (4.4) and by Schwarz and Poincarh inequalities [3], on taking into account (4.6), we have:

dE

- < -@ dt -

103 with

+

+PI)&

@ =T ~ { u ~ u $2 M 2 ( 1

+ c’~y3”- [(RF + c2)/r2]y~y3}

The definite positivity of the quadratic form @, given by the condition

< c(2x2u - c) 0 < c < 2n2a Rr

(4.7)

assures, thanks t o the stability theorem given in 1261, the existence of a positive constant a , such that

E ( t ) 5 E(0)exp(-at) that is (linear) asymptotic exponential energy stability. To select optimally the coupling parameter c, in order to improve the stability condition (4.7), on maximizing over c the r.h.s. of (4.7)1, we have the optimal value c = a x 2 and, on recalling that by (2.6) it is R r = u2Rarwe obtain the following stability estimate

R, < x4. This result, about seven times lower than RZ = 27/4x2 of the hydrodynamic case, although certainly not the best one, nevertheless it is noteworthy because it is valid for any value of ,BH and PI, that is whatever (low or high) the anisotropic currents are. In order to investigate now on the best critical Rayleigh number by the related variational problem, let us consider the maximum problem and the Euler-Lagrange equations, on starting from the equation (4.3) by the standard energy method; and so defineb: 1 - max 2Rf7-1 D

z

7-l being the space of admissible smooth solutions. From (4.3) (4.4) (4.8) it follows that dE

- < [ ( R r + c2)/2Rf dt -

- l]D

Therefore, by the condition R:+c2<2Rf

(4.9)

with the PoincarC inequality and the minimum (4.6), we have, from (4.1) (4.3) (4.8) (4.9):

E ( t ) 5 E(0)exp{ -7[2RF

-

(Ram

+c2)]t}

that is linear exponential stability, where

bThe existence problem linked t o (4.8) follows the prescription of Rionero 1271.

(4.10)

104

To calculate Rf by the Euler-Lagrange equations which follow from (4.8), we write these equations:

V p = a 2 A 2 u+ RfBe3

i

+ PrV x [e3 x (e3 x Vh)] e3 + c2A2B

V+ = A2h 0 = Rfu

’

(4.11)

with -2q and -2a2M21C, Lagrange multipliers of the divergence constraints. The equations (4.11) are solved, for the critical Rayleigh number, as for the equations (2.3), by introducing again the vorticity vector w = V x u and the current density vector j = V x h and applying suitably the operators V X and V x V X . The result is established from the following equations:

I

A2w3 = O

(4.12) a2A2A2u3

+ RfAg’B

=0

By normal mode representation (like (3.3) with X = 0), we have the spectral problem:

( D 2 - a2)Z = 0

+ PI)D’ - a2]x= o R f W + c2(D2 a2)Q = 0 [(I

(4.13)

-

a 2 ( D 2- a2)2W- Rfa2Q = 0 with D Z = X = W = D ~ W = Q = O on

Z=O,I.

First we observe that, when the marginal conditions prevail, from (4.13)1,2, we have that the z-components of the vorticity and the current density vanish identically, as in the isotropic case ([l],pp. 32 and 166). From (4.13)3,4it follows the same characteristic equation for Q and W (like F):

( D 2- a2)3F= - [c; -RE L ] 2 a 2 F

(4.14)

with the boundary conditions F(0) = F(l)= 0 and with zero even derivatives on the boundary z = 0, z = 1. But in the classical HDB problem the result ([l], p. 26):

( D 2 - a 2 ) 3 F= -R2a2F

(4.15)

105

leads to the hydrodynamic critical Rayleigh curve (3.11) of Sect. 3, and so to the critical Rayleigh number RZ of the classical hydrodynamic stability case, given by, respectively I

R:

27 4 = m i n R 2 = -ir

4 Therefore, the comparison betwen (4.14) and (4.15) gives us

RE

= cuR

;

z

Rflc= min Rf

(4.16)

= CUR,

and, on recalling that R Z = u2Rarthe stability estimate (4.9) takes the form Ra

< f (c) =

c(2uRC- C) u2

(4.17)

The best choice of the coupling parameter c, such that 0 < c < 2uRc is given by c = uRc, for which we have max f (c) = RE, and on putting this value in (4.17), C we obtain finally the optimum stability (4.18) of the motionless state (2.2), in the energy measure (4.1). But this condition is the same condition ( 4 . 1 6 ) ~of the stability for the classical HDB problem [2][3],so:

the result obtained proves what has been declared i n the A B S T R A C T . References [l] S. Chandrasekhar - Hydroynamic and Hydromagnetic Stability - Dover Publ.,

N.Y. 1981 [2] D. D. Joseph - Stability of fluid motions - Voll. 1-11, Springer Tkacts 27, Springer-Verlag, 1976 [3] B. Straughan - The Energy Method, Stability and Nonlinear Convection - Appl. Math. Sc. 91, Springer Verlag, N.Y. 1992 [4] M. Maiellaro - L. Palese - Electrical anisotropic effects on thermal instability Int. J. Engng Sc. (4), 22, 1984 [5] M. Maiellaro - O n the stability of the thermodiffusive equilibrium - Proceed. Int. Conf. on Nonlinear Mech., Shanghai, 1985 [6] M. Maiellaro - Qualitative aspects of some differential equations in Mathematical Physics - ZAMM, (5), @, 1986 [7] M. Maiellaro - Some nonlinear effects on MHD stability - proceed. Int Conf. on Nonlinear Oscillations, Budapest Univ. of Techn., 1987 [8] M. Maiellaro - Nonlinear dynamics and stability in anisotropic MHD - Abstr. EUROMECH 229 on Nonlinear Applied Dynamics, Stuttgard (FRG), 1987 [9] M. Maiellaro - L. Palese - A. Labianca - Instabilizing - Stabilizing effects of MHD anisotropic cunents - Int. J. Engng Sc., (ll),27, 1989 [lo] M. Maiellaro - A. Labianca - Nonlinear energy stability estimates in the anisotropic MHD Bknard problem - Magnetohydrodynamics Int. J. (l), 3, 1992

106 [ll]M. Maiellaro - Nonlinear stability for a system of PDE’s in MHD: recent results and open problems - Proceed. Sixth Int. Coll. on Diff. Eq.s, Plovdiv (Bulgaria) 1995 [12] S. Rionero - M. Maiellaro - O n the stability of Couette-Poiseuille flows in the anisotropic MHD via the Liapunov Direct Method - Rend. Accad. Sc. Fis. Mat. Napoli, 62, 1995 [13] G. Mulone - V. A. Solonnikov- O n the solvability of some initial boundary value problems of magnetofiuidmechanics with Hall and ion-slip effects - Rend. Mat. Accad. Lincei, f3, 1995 [14] M. Maiellaro - A. Labianca - O n the nonlinear stability in anisotropic MHD with application to Couette-Poiseuille flows - Int. J. Engng. Sc., 40, 2002 [15] M. Maiellaro - ”Pagoda” energy partition functionals to some nonlinear hydrodynamic and hydromagnetic stability problems - Proceed. 3rd Nat. Congr. SIMAI, Salice Terme (Pavia), 1996 [16] A. Labianca - V. Mauro - Un funzionale di ripartizione energetica a ”pagoda” per moti fEuidi in generatori Hall di potenza - Proceed. 3rd Nat. Congr. SIMAI, Salice Terme (Pavia), 1996 [17] C. Erriquenz - V. Mauro - O n the Hagen flows and their stability in anisotropic MHD - Proceed. ”INCOWASCOM 97” - IX Tnt. Conf. on Waves and Stability in Continuous Media, Univ. Bari (Italy), 1997 - Rend. Circolo Mat. Palermo (2) 57,1998 [18] M. Maiellaro - On. the nonlinear stability of the anisotropic MHD Couette Poiseuille flows - Proceed. ”INCOWASCOM 97” - IX Int. Conf. on Waves and Stability in Continuous Media, Univ. Bari (Italy), 1997 - Rend. Circolo Mat. Palermo (2) 57, 1998 [19] M. Maiellaro - O n the linear Behard problem i n anisotropic MHD - 2nd part: the principle of the exchange of stabilities - Int. Report 33/99 of the Math. Dept. Univ. Bari - Italy, 1999 [20] M. Maiellaro - A. Labianca - O n the nonlinear stability i n anisotropic MHD Int. Rep. 31/99, Math. Dept. Univ. Bari, 1999. [21] M. Maiellaro - O n the principle of exchange of stabilities in the anisotropic magnetohydrodynamic BLnard problem - Ricerche di Matematica, (l),51,2002. [22] J. N. Flavin - S. Rionero - Qualitative Estimates for Partial Differential Equations - CRC Press, New York, 1996 (Sect. 8.3). [23] G. Mulone - S. Rionero - B. Straughan - Convection with temperature dependent viscosity and thermal conductivity: linear energy stability theory - Rend. Accad. Sc. Fis. Mat. Univ. Napoli, 6l,1994 [24] S. I. Pai - Modern Fluid Mechanics - Van Nostrand, 1981 [25] G. Sutton - A. Sherman - Engineering Magnetohydrodynamics - Mc Graw Hill, 1965 [26] S. Rionero, M. Maiellaro - Sull’unicitd e stabilitci universale in media nella dinamica dei fEuidi - Rend. Circolo Mat. Palermo (2), 27, 1978 [27] S . Rionero - Metodi variazionali per la stabilitci asintotica in media in magnetoidrodinarnica - Ann. Mat. Pura ed Appl. 78, 1968

A BGK-TYPE MODEL FOR A GAS MIXTURE WITH REVERSIBLE REACTIONS R. MONACO AND M. PANDOLFI BIANCHI Dipartamento di Matematica, Politecnico di Torino C. Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: [email protected], [email protected] In this paper we propose a BGK-type procedure which is applied to the full Boltzmann equation extended to a mixture of four gases undergoing a reversible bimolecular reaction. Based on a suitable modelling of the reactive cross sections, the kinetic equations are obtained in an approximated form preserving conservation laws. A numerical simulation, showing the influence of elastic collisions and reactive interactions, is proposed in order to test the trend to equilibrium of the model.

1

Introduction

Boltzmann-type kinetic equations for reacting gas mixtures are well known in literature since the beginning of sixties More recently, Rossani and Spiga' have proposed a general derivation of the Boltzmann equation for particles undergoing inelastic interactions. Starting from this paper, the same authors have applied such Boltzmann equation to the particular case of gases with reactions of bimolecular and dissociation-recombination type. Nevertheless, it should be observed that the mathematical structure of the said models7i8 is rather cumbersome, since the collisional operators, as usual in the full Boltzmann equation, are expressed by 5-fold integrals. Consequently direct numerical applications of these models present several computational difficulties. A manner to overcome such an obstacle may be that of approximating the integral operators with a BGK procedure, which is well-known for an inert gas '. Burgers lohas proposed this approach in the very special case of a gas undergoing the irreversible reaction A -+B , treating, however, the elastic and inelastic contributions in a similar way. In the present paper a BGK-type procedure is applied to a mixture of four gases, say A i , i = 1,. . . ,4,where a reversible reaction of type 112131415.

takes place. The principal aim consists in deriving a kinetic model where the unknown distribution functions of gas particles never figure inside the integral

107

108

operators, so that the right-hand-sides of the kinetic equations assume a form favourable for computations. At this end, we propose a new approximation procedure for what concerns the reactive terms of the kinetic equations, leaving the approximation of the elastic operators according to the standard BGK approach. This objective is basically obtained by taking into account the properties of reactive collisional frequencies derived in a recent paper l l . Finally, the influence of elastic and reactive interactions, towards the trend to equilibrium of the model, is tested through some numerical simulations in the spatial homogeneous case. 2

Kinetic equations and their properties

Consider four different chemical compounds undergoing reaction (1) whose molecular masses and internal energies are m l , . . . ,m4 and E l , . . E4, respectively. Because of mass conservation during the reaction, it must result ml m2 = m3 m4. Moreover, if the forward reaction A1 A2 -+ A3 A4 is settled as the endothermic one, then the quantity A E = E3 E4 - E l - E2 results to be greater than zero. The microscopic state of the gas is defined by the one-particle distribution functions

+

+

fi=fi(t,x,v),

+ +

i = l , ..., 4,

tEIR+,

xEIR3,

VER3.

+

(2)

Unless necessary, hereinafter the explicit dependence of fi on time t and position x will be omitted. The kinetic equations for the reactive gas mixture7 are given by afi

-+v.

at

where f = {fl Ji[f](v)

i = 1,...

141

(3)

, . . . , f4) and

= Gi[f](v) - Li[fl(v), Ri[fI(v) = G[fl(v) - -GEfI(v).

(4)

The quantities Gi and Gi are the gain terms due to elastic collisions and chemical reactions, respectively, while Li and ,Ci are the corresponding loss terms. In order to express the gain and loss terms in detail, let v , w denote the velocities before collision of the test and field-particles, respectively. Moreover let g = Iw - vI be the relative pre-collisional speed whose direction is given by the unit vector a.

109

The elastic collisional terms are well-known in literature l 2 when one considers the full Boltzmann equation for a mixture of four inert gases. In particular, they can be written as

where B;(g, p ) are the scattering cross sections and p is defined by p = a .a’, 0’ denoting the unit vector of..the relative post-collisional velocity. Because of microreversibility, we have Bij = B$. The post-collisional velocities preserving mass, momentum and kinetic energy, are given by vij =

miv

+ mjw - mjg0‘ mi

+ mj

1

W ‘”3

-

miv

+ mjw + rniga‘ mi

+ mj

.

(7)

According to paper 7, the explicit expressions of the gain and loss terms due to chemical reactions are given by

110

where M = m1m 2 . Computation of post-collisional velocities, preserving 71237714

mass, momentum and total energy, leads to v1

=T

v2

= T2V

v3 = T3V v4

+

+ + T1W - T3V0’, + - T 2-v s 1 ’ , + - rlvn’,

~ V T ~ W - T~VCV,

= T4V

+~ 3 V 0 ‘ + T4V0’ + + r2VO2/,

w1= T ~ V T ~ W

= T2V + TlW w3 = T3V T4W w4 = T4V T3W w2

+ +

T4W

T3W

TlTm

where

=

, i = l , ..., 4,

mi

m1+

m2

-

1 v = /z(g2+Tj:)

v = J-,

and m3m4

ml m2

Moreover in (8)-(9),C$ are the reactive cross sections which must obey to the microreversibility condition l3

H being the Heaviside function. Since the forward reaction is endothermic, production of particles (A3, A4) needs that the relative pre-collisional speed g must exceed the threshold value gs . This explains the presence of both gs in the post-collisional velocities and the Heaviside function in the microreversibility condition. However, detailed calculations of the collisional terms can be found in paper ’. For what follows it is also useful to recall that the conservation laws of mass, momentum and total energy imply the following properties

Rl(V)dV s,3

=

SR3

R2(v)dv = -

R3(v)dv = s,3

s,.

R4(v)dv

(12)

+

p(v) being alternatively equal to mi, miv, i m i v 2 Ei. In paper7, as well, it has been proven that Ji[f](v) = 0 (i.e. mechanical equilibrium condition) requires that the following equalities hold fi(vij)fj(wij) =.fi(v)fj(w),

vv,w,vij,wij i , j = 1 1 . . . , 4 ,

(13)

111

which are satisfied when the distributions f j are Maxwellians, namely

On the other hand, if the distribution functions are Maxwellians, it can be shown that condition Ri[f](v) = 0 (i.e. chemical equilibrium condition) is assured if one of the follo&g equalities also holds

The last conditions imply

which expresses the mass-action-law of chemical equilibrium 14. In (14) k B is the Boltzmann constant; the number densities n i , the mean velocity u and the temperature T are defined as usual by

where

112

3

A BGK-type approach

The BGK model’ of the Boltzmann equation for inert gases is based on the ansatz that the distribution functions appearing in the gain terms of the collisional integral are local Maxwellians. From a physical point of view this means that the gas is sufficiently near equilibrium so that particles after one collision only reach a local equilibrium. This kind of ansatz seems to be even more physically meaningful in the case of a mixture of reacting gases. In such a situation the gas-particles exhibit two relaxation processes: the first is represented by the time necessary for particles to join a mechanical equilibrium, the second by the time necessary to reach also chemical equilibrium. Experimental evidences l5 and well-known simulations l6 have shown that the time of mechanical relaxation of a gas is in general much smaller than the one due to chemical relaxation. Therefore, having in mind both the mechanical equilibrium condition, corresponding to the Maxwellian distribution (14), and the chemical equilibrium condition, defined by the mass-action-law (15), the BGK-type approach, presented here, will be based on the following assumption: The distribution functions of particles appearing in the gain terms (5) and (8) are Maxwellians with number densities ni not satisfying condition (151, necessary to chemical equilibrium. In order to show the proposed procedure, it is convenient to distinguish between elastic and inelastic contributions.

3.1 Approximation of the elastic terms The gain contributions (5) will be re-written in terms of the Maxwellians Thus, substituting there the f i by the and recalling that

5

the elastic gain terms read

Let now denote the last sum of integrals in (19) by

5.

113

This quantity can be explicitely computed once the elastic cross sections Bjj are known, as it will be shown in next section. It can be observed that 5; represents a collision frequency (the average number of collisions that a single molecule experiences in the unit time), so that l/Fi is the characteristic time of elastic collisions. Moreover referring to the integral operator appearing in the loss term (6), we define the collision frequency vi as 4

.i(V)

SfC J j=1

IR3

dwJ B:;(g,p)fj(w)dn’ S

Since elastic collisions preserve masses, momenta and energies at any point of the space x, the number density, momentum and kinetic energy of each species j at (t,x) before and after collisions must remain the same. This condition is expressed, as well knownI2, by

where p is alternatively equal to m j , mjw or $mjw2. On the other hand, due to the BGK-ansatz, it results that

Putting together the last two equalities, we get

which shows that the distribution function fj(w) and the Maxwellian &(w) must possess the same moments nj , n j u j and T . Thus, fj and will differ only for the part concerning the computation of moments of higher order. Therefore one can argue that the further approximation vi N 5i can be considered valid. In conclusion, the elastic collisional terms (5) and (6) in the kinetic equations (3) can be approximated by

5

3.2 Approximation of the inelastic terms Similarly to the procedure adopted for the elastic collisional operators, let us rewrite the gain terms (8) due to chemical reactions, in mechanical equilibrium.

114

We get

G I @ = & ( v ) = M3/R3 d w s , C ~ , ” ( g 1 ~ ) f 3 ( v i ) f 4 ( w l ) d a ’ (26)

Gz[B = &(v) = M3

&@I

- -

IR3

= G3(v) = 3 1

G4[3 = 54(v) = 3

d w l C,”,”(g,~).&(vz)&(wz)da’

/IR~

dw

(27)

34 9 ,P ) . E ( v ~ ) . L ( w ~ ) ~(28) ~’ s,W

SIR. s, dw

cJ‘g(gi p)5(V4)E(W4)da’.

(29)

The Maxwellians in (26)-(29) do not satisfy the chemical equilibrium conditions (15), and therefore cannot be re-written as functions depending on pre-collisional velocities v and w , as in the elastic case. However, the quantities Gi can be explicitely computed] once the cross sections C?: and Cij are known, as it will be shown in next section. On the contrary, for what concerns the loss terms of inelastic interactions] the procedure followed to deduce equations (25) for elastic collisions, is still valid. The only difference is that now ‘p is alternatively equal to m j , mjw or $mjw2 Ej, recalling that Ej are the internal energies. Thus the loss terms (9) will be re-written substituting the particle distributions fi (w) by the corresponding Maxwellians fi (w) so that

-

+

-

where

Integrals (31)-(34) will be explicitely computed] as well. The BGK-type approximation of inelastic gain and loss terms finally leads to the following expression

-

Ri[f](V)

Gi(V)

- ai(V)fi(V).

(35)

115

Consequently, the kinetic equations for a gas mixture with bimolecular reaction (1) assume the approximated form afi -+ v . V f j = Fi(i(v)[fi(v)- f; (v)]+ Gi (v)- Zi(~)fi(v), i = 1, . . . ,4. (36)

at

It is important to observe that, in general, equations (36) do not preserve mass, momentum and energy, since they do not satisfy properties (11) and (12). However we will show how this drawback can be overcome, considering suitable models of cross sections. 4

The BGK-type model

-

First of all, with the aim of providing explicit expressions to the integral operators Pi, Gi and Zi,it is necessary to specify both elastic and reactive cross sections. We assume, for elastic interactions, cross sections of Maxwell molecules which, as known 13, do not depend on incoming and outcoming velocities, i.e. ..

B!? '3 = aij where

aij

(37)

are constants. Thus it results

c 4

P i ( t , x) = 47r

aijnj(t,x).

(38)

j=1

Moreover the reactive cross section (73142, related to the exothermic reaction, will be assumed as the one proposed in17, that is C,',"(S) = P (1

-

f > H(g

- 0,

(39)

P being the Arrhenius parameter14 and E the threshold

velocity for the exothermic reaction. According to the microreversibility condition (lo), it turns out that the cross section C;; of the endothermic reaction can be expressed by

M ( g E22- 9:)

)H

(g

-

J%+,:i

.

(40)

116

-

By inserting expressions (39) and (40) into the integrals (26)-(29), after standard but rather cumbersome calculations 11, the reactive gain terms Gi read

where

In a similar way inserting expressions (39), (40) into the other integrals (31)(34), also the reactive collisional frequencies Zi can be explicitely computed, namely

51 = v(T)m, O3 = Y ( T ) n 4 ~

-

2.

Z4

= 17(T)n1, = y(T)n3.

(43) (44)

It is worthwhile to recall that expressions (41)-(44) depend also on the variables ( t ,x) through the macroscopic quantities 121, . . . , 124 and T. Concluding, the final version of the BGK-type model is straightforward obtained by casting expressions (38), (41)-(44) into equations (36). Thanks to the choice of the specific cross sections (37), (39) and (40), the proposed model assures conservation of mass, momentum and energy, as it can be easily verified taking into account properties (11) and (12). Moreover the mathematical structure of the model turns out to satisfy the requirements discussed in the introduction, and thus seems suitable for numerical applications. 5

A numerical test

In order to test the behaviour of the model with respect to its trend to equilibrium, in this section we propose a numerical application which shows the influence of elastic collisions and reactive interactions. In particular we will consider the bimolecular reaction

HzO

+

H

+

OH

+ Ha,

117 f3

Figure 1: Initial data for f3(0,v) and

fd(0,V)

versus V; f i ( 0 , ~= ) f2(O,v)

= 0, VV.

and take into account kinetic equations (36) in the spatial homogeneous case, setting as initial data bimodal distributions for particles A3 and A4 in absence of particles A1 and A2. Such distributions at t = 0 are shown in Fig.1. We examine the time evolution of the distribution functions, as stated by the model equations (36) with the collisional terms given in Section 4. In Fig.:! the distribution profiles are given during the transient at time 0.01. It can be observed that, thanks to the peculiar collisional process due to the BGK ansatz, the distribution functions of the chemical products (A1 and A2) have already assumed a ”Maxwellian” shape, whereas the reactants still present a shape analogous to that of initial data. At time t = 0.1, when the chemical process is completed since species A3 has vanished, also the reactant A4 has reached a ”Maxwellian” form, as it can be seen in Fig.3. This behaviour is essentially due to the presence of elastic collisions which drive the distributions to become Maxwellians. In fact, as well evidentiated by Fig.4, if the elastic collisional term is set equal to zero (aij = 0) in (36), the reactant A4 still exhibits the bimodal shape of initial datum even when the chemical process is completed (at t = 0.1). On the other hand, the species A1 and Az, produced by the collision process, assume the ”Maxwellian” shape. Conversely, if in Eqs. (36) the reactive terms are set equal to zero ( p = 0), then the distribution functions of species A3 and A4 (which in this case are no longer reactants) accomodate at equilibrium to ” Maxwellians” and the picture is the one of Fig.5. For the test here developed, a 4th-order Runge-Kutta routine has been employed with a discretization of the space velocity into 500 nodes, with a cutoff at lzll = 10000 mlsec.

118 t.0.010

fZ 8002

f l ,0004 nl=Q.384

-1QOQQ

u

,/,,

18000 -1000Q

u

lWQ0

u

10000

f3 .0004

nY.21Z

-10000

I

u

,n4=1.608

10000 -10000

,

Figure 2: Transient shape of distributions .fi(t,v) versus v at t = 0.01.

n1.0.597

-10000

nZ.8.589

u

loo00 -10000

u

10000

f4 ,0004

-10000

u

lOoG0 -1QQQQ

! ! L u 10000

Figure 3: Equilibrium profile of distributions j i ( t , v ) versus v at t

= 0.1.

119 t=0.100

*=a

nl=O.hW

u

-loo00

.594

10000 -18000

u

lQ0W

Figure 4: Reactive collisions only: distribution profiles of f , ( t ,v ) versus v a t t = 0.1.

t=o.100 f 3 .OW4

f4 .m104

I

n t = l.99t

-18000

u

1om8 -18008

Figure 5: Elastic collisions only: distribution profiles of f;(t,v) versus v a t t = 0.1.

120

Acknowledgments

The paper is partially supported by INDAM-GNFM and the National Research Project COFIN 2003 ”Non linear mathematical problems of wave propagation and stability in models of continuous media” (Prof. T. Ruggeri). References

J . Ross, P. Mazur, J. Chem. Phys. 35, 19 (1961). B. Shizgal, M. Karplus, J. Chem. Phys. 5 2 , 4262 (1970) M. Moreau, Phys. Fluids 8, 3423 (1996). N. Xystris, J.S. Dahler, J. Chem. Phys. 6 8 , 354 (1978). L.S. Polak, A.V. Khachoyan, Soviet J. Chem. Phys. 2, 1474, (1985). A. Rossani, G. Spiga, Transp. Theory Stat. Phys. 27, 273, (1998). A. Rossani, G. Spiga, Physica A 272, 563, (1999). M. Groppi, A. Rossani, G. Spiga, J. Phys. A : Math. Gen. 33, 8819, (2000). 9. P.L. Bathnagar, E.P. Gross, M. Krook, Phys. Rev. 94, 511, (1954). 10. J.M. Burgers, Flow Equations for Composite Gases (Academic Press, New York 1969). 11. F. Conforto, R. Monaco, F. Schiirrer, I. Ziegler, J. Phys. A: Math. Gen., (to appear in 2003). 12. M.N. Kogan, Rarefied Gas Dynamics (Plenum Press, New York, 1969). 13. J.C. Light, J . Ross, K.E. Shuler, in Kinetic processes in gases and plasmas, 281, (Ed. A.R. Hochstim, Academic Press, New York, 1969). 14. C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, Cambridge, 2000) 15. K.K. Kuo, Principles of Combustion, (John Wiley, New York, 1986). 16. G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, (Clarendon Press, Oxford, 1993). 17. P. Griehsnig, F. Schiirrer, G. Kiiger, in Rarefied Gas Dynamics: Theory and Simulations, (Eds. B.D. Shizgal & D.P. Weaver, AIAA, Washington, 1992).

1. 2. 3. 4. 5. 6. 7. 8.

STABILIZING EFFECTS IN FLUID DYNAMICS PROBLEMS G. MULONE Dipartimento d i Matematica e Informatica, Cittd Universitaria Viale A . Doria, 6, 95125, Catania, Italy E-mail: [email protected]

Dedicated to S. Rionero in occasion of his 70th birthday Many fluid-dynamic and convection problems exhibit stabilizing effects which are confirmed both by the experiments and the linearized analysis. Here we present some Rionero's contributions to show these effects in the nonlinear context and some new recent results.

1

Introduction

In fluid-dynamics there are many problems that exhibit stabilizing effects. This happens, for example, in convection problems; typical problems are Magnetic BBnard Problem (MBP), Rotating BBnard Problem (RBP) and BBnard problem for a binary fluid mixture heated and salted from below. In these cases the experiments and the (approximate) linear instability analysis results (see Chandrasekhar ') show the inhibiting effect of magnetic field, of rotation (Coriolis force), and of concentration of solute on the onset of instability (thermosolutal convection). In order to show these effects in the full nonlinear systems, many efforts have been made by researchers in the last 40 years. The contribution of Rionero (see Rionero 1967a, 196713, 1968a, 1968b, 1988a, 1988b, 1989, 1997, 1998, 2001, 2002 12) to solving this problem is fundamental. In fact, he was the first to study asymptotic stability in magnetohydrodynamics and in the magnetic B6nard problem and to show that, by using the classical energy of perturbation (energy method, see Serrin 1959, 13),the magnetic field has a non-destabilizing effect on the onset of convection. He wrote "... il metodo classico dell'energia prova che (almeno per il campo magnetico irrotazionale) il campo magnetico n o n p u b destabilizzare ", 2 , "... la richiesta di stabilita asintotica in media ii forte, ... le richieste di stabilita ottenute appaiono alquanto restrittiue confrontate con quelle che si ricauano dai metodi lineari" '. The results of Rionero, Joseph l 4 and successively Galdi and Straughan (1985), 15, 16, Straughan (1991),17, Flavin and Rionero, 1996, 'O, have given rise to many researches and papers devoted to proving stabilizing effects in nonlinear fluid dynamics problems and, in particular, to prove necessary and sufficient conditions of nonlinear stability in the BBnard problems. This last problem is intimately connected with the choice of a measure for perturbations to a given basic motion (energy-like Lyapunov functions). At this moment, there is not a general method to give an optimal Lyapunov function to proving, at least when a principle of exchange of stabilities holds, the coincidence of the critical linear instability and critical nonlinear stability thresholds. Here we recall some contributions of Rionero in this context and some new recent results.

'-

121

122

The plan of the paper is as follows. In Sec. 2, we set the stability problem for a dynamical system in a Hilbert space ‘H. In Sec. 3, we recall some stability results of Ftionero in the B6nard problems and give some new recent stability results. 2

The stability problem

Let ‘H be a Hilbert space endowed with a scalar product ( , ) and associated norm 11 . 11. Let UO E ‘H and t E [ O , o o ) . We consider in IFI the following initial value problem (perturbation equation t o a given basic motion)

U’=L U + N U U ( 0 ) = uo where L represents a linear operator and N a nonlinear operator with N ( 0 ) = 0 and ( N U , U ) 5 0, VU E D ( N ) . We assume (see Galdi and Rionero 18, Straughan ” ) the spectrum of the operator L consists entirely of an at most denumerable number of complex eigenvalues {on},n positive integer, with finite multiplicities and the following inequalities hold

The operator L is in general non-symmetric, although it allows a decomposition into two parts L1 and Lz such that

(c) L z is skewsymmetric . The eigenvalues {A,}

of L1 are real and A1 2 x 2

5 ...A, I . . .

(3)

The linear stability is reduced to studying the sign of s = re(ol). In general s will depend on the basic motion through a dimensionless parameter R, such as

Reynolds or Rayleigh numbers (and also on some other parameters). The value R, of R at which linear instability sets in is the least value of R for which re(o1) = 0. It is called critical value (of linear instability). We say that the principle of exchange of stabilities (PES) holds if re(o1) = 0 im(o1)= 0 . If PES holds, a secondary stationary solution prevails at the onset of instability. Otherwise, we have overstability. For problem (1) an energy equation holds:

*

123 Since ( N U ,U ) 5 0 we have

We assume that

(L1U,U) = RI(U)- D(U), where I(U)and D ( U ) are quadratic forms in U,D(U) positive-definite, and suppose ~ that there exists a positive constant c (PoincarB's constant) such that ~ l l U 1 15 D(U). From ( 5 ) it follows

d R I(') -IIU1l2 dt 52(RD(U) - 1)D(U) 5 2(RE

-

l)D(U)

where

and S is the space of the kinematically admissible fields. RE is the critical value of nonlinear stability. If R < RE, from last inequality, we obtain

2c lIU1/25 IIUol12exP[-(R RE

- RE)^], b't

> 0.

(7)

We note that the stability guaranteed by (7) is exponential. From the above arguments it follows that while the linear stability problem reduced to studying the eigenvalue problem associated with all of L , nonlinear stability according to the standard energy method involves the eigenvalues of the symmetric part of L1 only, (see Rionero 5 , Galdi and Rionero 1 8 , Galdi and Straughan 1 6 , Straughan 17). Moreover, whenever L2 = 0, the two eigenvalue problems coinnonlinear stability. But, in cide ( in this case RE = R,) and linear stability general, if the operator L2 # 0, RE < R, . If the operator L is symmetrizable the critical linear and nonlinear stability parameters coincide (see 1 6 ) .

*

From a physical point of view, a skew-symmetric linear operator L2 represents a stabilizing effect. This effect gets lost if we use the classical energy, in fact, now (L2U,U ) = 0.

Now the following questions arise

# 0 i s it possible to find a new scalar product (, )I and a n associated n o r m El := 11 111 such that RE < RE^ I R,?

1. i f L2

2. if Lz # 0 and PES holds, is it possible to find a norm 1(.1(1, eventually equivalent t o 11 . 11, such that RE^ = R,

124

Obviously, in the general cases, we expect that (NU,U)1 0 and the stability we obtain is conditional. In the case (NU,U)1 5 0, the stability is unconditional. The above questions (in particular question 2) are, for a general system (l), open problems . As concerns question 1, in Galdi and Padula, (1990), ‘l, a generalized energy method has been given and sufficient conditions to ensure the stabilizing influence of La, in the case of a weakly coupled system, has been proved. In the next section we recall some Rionero’s contributions to proving stabilizing effects, and in some cases RE, = R,, in the BBnard’s problems and some new results. 3

Nonlinear stability results in the BQnardproblems: Rionero’s contributions and some new results

Let us consider a layer of a fluid (or a binary mixture) heated from below, in the Oberbeck-Boussinesq scheme, bounded by two horizontal parallel planes. Let d > 0 , C l d = R2 x (-d/2,d/2) and Oxyz be a Cartesian frame of reference with unit vectors i, j,k respectively. Let us assume that the layer is parallel to the plane z = 0. The non-dimensional equations which govern the evolution disturbance to the velocity, temperature, magnetic field, concentration and pressure fields (u,h, 19,y,p2) to the basic motionless state in R1 x (O,co),where 0 1 = R2 x (-f, f ) are: i) in the magnetic BBnard problem:

I

Ut

+ u . VU - Pmh . V h = -Vpl+

v.u=o

R6k

+ Qhz + AU

+

U . Vh - h VU) = Q u + ~Ah V.h=O PT(6t U . v 6 ) = R W + A6

Pm(ht

+

ii) in the rotating BBnard problem:

{

Ut

+ u . VU = -Vpl+

v.u=o f

U .

RBk + ‘Tu x k + AU (9)

v 6 ) = RW4-A6

iii) in the case of a mixture heated ans salted from below:

By using the classical energies

125

global exponential stability has been obtained for

R2 < R;, where R; is the critical value in the simple BBnard problem for a homogeneous fluid (i.e. the BP without any stabilizing effect). Thus the stabilizing effects on the onset of convection are not achieved. The main ideas of Rionero are: i) in the papers 6-8: to introduce the essential variables and the admissible balances of the problem. To choose as Lyapunov function E = EO bEl, with EO Lyapunov function for the linearized problem while El must dominate the nonlinear problem, e.g. in the rotating case:

+

where C = (V x u) . k. The main results of Rionero et al., ', 6 , are: o The coincidence between the linear and nonlinear (conditional) stability parameters in the case P, < PT, Q2 < 101.18 has been shown under computable conditions on the initial data, in case of MBP. For the other values of Chandrasekhar number Q2 (related to the magnitude of magnetic field), the stabilizing effect of magnetic field has been proved and the results of Galdi l 9 have been improved. o The coincidence between the linear and nonlinear stability parameters for Prandtl numbers greater than or equal to 1 and for Taylor numbers less than or equal to 807r4, in case of RBP. For the other values of Taylor number T 2 (related to the magnitude of rotation), the stabilizing effect of rotation has been proved and the results of Galdi and Straughan l 5 have been improved. ii) in the papers ' - 1 2 : a) to introduce new field variables (which may depend on some Lyapunov parameters) which are in a one-to-one correspondence with the variables u, 29, y and transform the perturbations system in an equivalent one; b) to introduce a Lyapunov function (norm) equivalent to the classical energy norm; c) to choose the Lyapunov parameters in an optimal way (by maximizing or minimizing) in order to reach the coincidence of critical linear and nonlinear values. The Lyapunov function introduced in the case of MBP is

126

with

C‘”’

= h2, - hi,,,

r$ = RA10 + &Ahz,

x = R A r 0 +p6QAhz,

and 1 E l ( t ) = s[IIAuiIz PmIIAhI12 + PrIIA01121.

+

The Lyapunov function introduced in the case of RBP is

Eo(t)= ;[xIIvwl12 + lld4121+ 9 l l A l 0 l l 2 & ( t ) = lllV~Il2 prllA~112 + 11vC112+ llAw112 + l I V ~ l U l l 2 1 with r$ = R A l 0 - T & .

+

The Lyapunov function introduced in the case of BBnard problem for a mixture is

The results obtained are the following: The coincidence between the linear and nonlinear (conditional) stability parameters in the case P, < PT and any Chandrasekhar number has been shown under computable conditions on the initial data. 0 The coincidence between the linear and nonlinear (conditional) stability pal and any Taylor number has been shown under comrameters in the case PT putable conditions on the initial data. o Necessary and sufficient conditions of unconditional nonlinear stability in the case p 5 1, i.e., the coincidence between the linear and nonlinear stability parameters for any Rayleigh number of the the concentration of solute C2. Necessary and sufficient conditions of unconditional nonlinear stability in the case p > 1, and 0

>

C2<&.

Remark. We observe that in all above definitions of Lyapunov functions the more difficult step is the step a). How we have to choose the (best) new field variables?

A possible answer is to introduce an ordinary differential system associated in a natural way to the dynamical system U’ = LU + NU and to study the stability of zero solution of this system. This may suggest, at least in particular cases, a good choice of new field variables in our problem and we have ”operative” conditions under which the operator L is symmetrizable. For example, let us consider the Bknard problem for a binary fluid mixture heated and salted from below. Here we refer only to the simplest cases PT = PC = 1

127

and PT = Pc # 1 (where instability begins whenever R2 > C2 perturbation equations are

+

+ +

u . V U = -Vp2 (R6 -Cy)k 6 t + ~ . V 6= ) Rw+AI~ yt U . Vy) = CW Ay.

LI~

{I

+ Au,

+ R;).Now the

V .U= 0 (16)

+

It is simple t o check that, when R2 - C2 5 0, the basic solution is globally stable, thus we assume p2 := R2 - C2 > 0. Let us consider the nonlinear ordinary differential system: 5 1 = Rz2 x2

=Rzl

k3 = czl

-

+

f l (z1, 5 2 , z3) +fZ(zliz2,53)

- c23 - RBzl -RBz2

RBz3 + f3(zlrz21z3)

(17)

where fi(x1,x2,x3), a = 1,2,3, are nonlinear functions which vanish in the point ( 0 , O I 0). In order t o study linear instability of the zero solution, we compute the eigenvalues of the linear matrix

-RB

(;

R

-;B:iB).

The eigenvalues are

-RBI

Xz = - R B - p ,

X3=-RB+P,

The matrix Q of the corresponding eigenvectors and its inverse are given by

If we use the transformation y

= Q-lx, i.e.,

1

system (17) becomes

[

Y1 = -(-Cz2

P2

1 Yz = -+x1 2p 1 Y 3 = -+pz1 2P

+ Rz3) + (Rz2 - C Q ) ]

+ (Rzz - Cz3)],

128

+ f4(Yl,YZ,Y3) 9 2 = - ( R B + -)Y2 + fS(Yl,YZ,Y3) ?jl

= -RBY1

93

= -(RB - -)Y3

(19)

+ fS(yl,Y2,?/3)

+

+

From this it follows that R2> 72% C2,we have instability (if R2< R : C2 we have linear stability). We note that for particular values of fi (for example for f1 = -x?, f2 = 0, f3 = 0), it can be proved that the condition R2< Rg +C2 implies nonlinear stability in the norm 1

El(t)= $Y:’

+ Y; + Y3.

If we use the “energy”

+ + x;)

1

E ( t ) = s(x:’ x; we obtain

E ( t ) 5 E ( 0 )exp 2(R - R B ) t and the stability condition now becomes

R2< R;, then we loose the stabilizing effect of C. If we make the transformation of variables (18) in the system (16):

I

’p

*

1

= -(-CB

P2

1 =

-+PW

2P 1

+Ry)

+ (Rd - Cy)]

x=7 [-Pw + (RB 2P

we easily obtain the system

-

Cy)],

129

By introducing the “energy”

we have

and from this, by studying the associated maximum problem, we obtain the nonlinear stability condition R2 < Ri C2: the critical linear and nonlinear parameters coincide. Now we observe that

+

1

pw=i-x,

1

pz4=i+x.

By using the parallelogram identity

2[Ilxl12+ 11~1121= IIX + +112 + ll@- xl12 we easily obtain 1 El(t) = ~ I P 2 l l ~+I l1214112+ llYl121 where now 4 appears as a canonical field variable (while in (15) it has been introduced only in a heuristical way).

If PT

= Pc

# 1, we consider the L1associated”ordinary differential system:

where a = (PT)-’. Proceeding as before, we make the transformation of variables

I

1

y = -(-C8

+Ry)

P2

1

1c, = &2aP2w

x=

+ ((1- U ) R B + u)(R8- Cy)]

1 m[-2aP2w+((-1+a)RB

in the system (lo), where We obtain the equations:

+n)(Rd-Cy)],

P = d m ,u = JR i ( a - 1)2 + 4 a p .

(24)

130

As before we can easily obtain the evolution equation

Remark. We note that in order to eliminate the pressure in the previous evolution equation we must write the evolution equation of w. Now 1 w = -[((-I a ) R B a)$ ((-1 2a and the equation we obtain is (as it is obvious)

+

Wt

The presence of for 4.

+

+

+ a ) R B - u)x]

aP + 4 + Aw. + U . V W= -az

4 in the last equation force us to use also the evolution equation

In the general case PT # Pc, and for other fluid dynamics systems, the problem of finding the “best Lyapunov functions” is much more complicated and we are studying the question in a work in progress, 22.

Acknowledgments The author has been partially supported by University of Catania, the Italian Ministry for University and Scientific Research (M.I.U.R.) and by “Gruppo Nazionale della Fisica Matematica” of the “Istituto Nazionale di Aka Matematica”.

References 1. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Oxford: Clarendon Press, 1961). 2. S. Rionero, Ann. Mat. Pura Appl. 76, 75-92 (1967). 3. S. Rionero, Ricerche Mat. 16, 250-263 (1967) 4. S. Rionero , Ann. Mat. Pura Appl., 7 8 , 339-364 (1968).

131

5. S. Rionero, Ricerche di Mat. 17,64-78 (1968). 6. S. Rionero and G. Mulone, Arch. Rational Mech. Anal. 103,347-368 (1988). 7. S. Rionero in Energy stability and convection. Pitman Res. Notes in Math., vol. 168, (Longman: Harlow 1988). 8. G. Mulone and S. Rionero, J. Math. Anal. Appl. 144,No. 1, 109-127 (1989). 9. G. Mulone and S. Rionero, Continuum Mech. Termodyn. 9,347-363 (1997). 10. G. Mulone and S. Rionero, Rend. Mat. Acc. Lincei. s. 9, 9,221-236 (1998). 11. S. Lombardo, G. Mulone and S. Rionero, J. Mat. Anal. Appl. , 262, n.1, 191-207 (2001). 12. G. Mulone and S. Rionero, Arch. Rational Mech. Anal. 166,n. 3, 197-218 (2003). 13. J. Serrin, Arch. Rational Mech. Anal. 3,1-13 (1959) 14. D.D. Joseph, Arch. Rational Mech. Anal. 36,285-292 (1970). 15. G.P. Galdi and B. Straughan, Proc. Roy. SOC.London Ser. A 402, 257-283 (1985). 16. G.P. Galdi and B. Straughan, Arch. Rational Mech. Anal. 89,211-228 (1985). 17. B. Straughan, The Energy Method, Stability, and Nonlinear Convection. Springer-Verlag: Ser. in Appl. Math. Sci., 91 (1991). 18. G.P. Galdi and S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity, Springer-Verlag, Berlin, Lecture Notes In Mathematics, 1134 (1985). 19. G.P. Galdi, Arch. Rational Mech. Anal. 87,167-186 (1985). 20. J. Flavin and S. Rionero, Qualitative estimates for partial diflerential equations. A n introduction. Boca Raton, (Florida: CRC Press 1996). 21. G.P. Galdi and M. Padula, Arch. Rational Mech. Anal. 110,187-286 (1990). 22. G. Mulone, S. Rionero and M. Trovato, On the choice of Lyapunov functions in fluid dynamic problems in the presence of stabilizing effects (in preparation, 2003).

A N A L T E R N A T I V E K I N E M A T I C S FOR M U L T I L A T T I C E S MARIO PITTERI DMMMSA, Universitd di Padova Via Belzoni 7, 35131 Padova ITALY ernail: [email protected]

Dedicated to S. Rionero o n the occasion of his 70th birthday I present a n explicit framework for weak phase transitions in complex crystals which is alternative to th e one given by Ericksen3 and myselp. This approach seems better suited t o treat crystals with many atoms in t h e unit cell. As an example, we analyze th e model of quartz in Pitterig from t h e present viewpoint.

1 In t r o d u ctio n The recently renewed interest in the geometry and kinematics of multilattices, aiming at constructing a nonlinear model of the thermomechanical behavior of complex crystals, is acknowledged by Pitteri and Zanzotto", where background for this paper, more details and references are given. There, also some still unsolved problems are outlined, in particular the need for a unified kinematics of crystals of different complexity. Here, as in Pitteri8i9, this problem is not addressed, in that only essential multilattices are considered; that is, ones for which the lattice of translations is the maximal one. More work is needed to handle the general case, which is of both theoretical and applicative interest. The kinematics below is a variation of the one in Pitterig, so we only describe the differences, addressing the reader to that paper for more details and comments. 2

Preliminaries

As in Pitteri and Zanzotto'', I use the summation convention and 'running indices' without specifying their range; for instance in expressions like 'the lattice basis e,' instead of 'the lattice basis {e,, a = 1 , 2 , 3 } ' , or 'the function $(ear4,O)' instead of 'the function 4(el, e2, e3, d l , . . . , &,O)'. Also, the relations 5 and < between groups mean subgroup of and proper subgroup o i respectively. Let Z and R denote the integral and real numbers, respectively. Consider first the simplest triply-periodic structures, that is, simple lattices (or 1-lattices):

L = { W e a ,a

= l,2,3,Na E

Z}

= L(e,).

(1)

The lattice vectors (or lattice basis) e, are linearly independent in R3. Any basis e, uniquely determines the 1-lattice L(e,), but not vice versa: b

L(eL) = L(e,) w eL = m,eb,

m E GL(3, Z);

here GL(3, Z)is the group of 3 by 3 integral matrices with determinant H. The crystallographic point group (or holohedry), P(e,), of the lattice L(e,) is then defined as the group of all the orthogonal transformations mapping L(e,) to itself; 132

133

Figure 1. Equivalent bases for a given (planar) lattice

equivalently :

P(e,) = { Q E O(3) : Qe, = mb,eb}. Notice that the basis e, satisfies (3) if and only if the lattice metric

c = (cab),

Cab = ea ' eb,

(3)

(4)

is a fixed point of the map C H mtCm.

(5)

The conjugacy classes of the holohedries in O ( 3 ) correspond to the well known 7 crystal systems. By looking at the right-hand side of the equation defining P(e,) in ( 3 ) we introduce the lattice group L(e,) of the lattice L(e,), related to the basis e,:

L(e,) = {m E GL(3, Z) : mteb = Qe,, Q E P(e,)} = { m E GL(3, Z) : mtCm = C } .

(6)

By the last equality the lattice group depends on the basis e, only through the corresponding metric C, hence can be denoted by L ( C ) . The conjugacy classes of lattice groups in GL(3, Z) correspond to the well known 14 (Bravais) lattice types. 3

Multilattice kinematics

Real crystals (hexagonal metals, alloys, etc.) are not in general 1-lattices. Their geometry and kinematics can be described by means of multilattices, which are the union of a finite number of nontrivial translates of a 1-lattice. A simple, well known exampIe of 2-lattice is the hexagonal close-packed (hcp) structure, whose standard unit cell is sketched in Fig. 2. In general, an n-lattice M in 3-dimensional affine space can be defined as follows, in terms of an origin 0 fixed at will, and in Grassman notation:

M

= M ( e a , 4 , .. . , &) = U:==,

{O

+ L(%) + 6);

(7)

C(e,) is called the skeletal lattice of M , and each displacement vector d, gives the position, relative to 0,of a point in one of the simple lattices constituting M . Fig. 3

134

$"

Y

Figure 2. An elementary cell for the hcp lattice

Figure 3. Unit cells of the component 1-lattices of a planar 2-lattice

is a schematic picture of a 2-dimensional (planar) 2-lattice; if the atoms represented by filled circles are physically indistinguishable from the ones represented by open circles, the 2-lattice is called monatomic, otherwise diatomic. The multilattice descriptors (e,, d,.) =: 6,, 0 = 1,.. . ,n 3, (in terms of which we can write M = M(6,)) satisfy the following conditions guaranteeing the threedimensionality of M and the non-overlap of the constituent 1-lattices:

+

el.ezxes#O,

d , . # d s + l F s e a , r , s = l , ..., n,1FSEEZ.

(8)

An n-lattice M is called essential if its skeletal lattice contains all the translations mapping M to itself. In this case the lattice cell has minimum volume. The 'global symmetry group' of essential n-lattices expresses the indeterminateness in the choice of the multilattice descriptors:

M ( & ) = M(6,) where Gn+3 < G L ( n

H

6; = ~ : 6 u , v

E

Gn+3,

+ 3, Z) consists of the matrices of the form

for m E GL(3, Z), 1% E Z, a = 1,2,3, r,s = 1,..., n, and permutation matrix of the set { 1,.. . , n}. Explicitly

eL=mb,eb,

(9)

p

=

(p:)

d ~ = ~ P " , d s + l , " e ,a, , b = 1 , 2 , 3 , r , s = l ,...,n .

E O(n) a

(11)

135

The group Gn+3 represents changes of descriptors independently of the chosen origin 0. Moreover, the subgroup Tn+3of skeletal lattice translations, that is, of the matrices v for which

m = 1 E GL(3, Z), p = 1 E GL(n, Z), 1:

= 1; = . . . l z E

Z,a = 1, 2, 3,

(12)

is a normal subgroup of Gn+3. Here and below it is understood that only permutations of physically identical multilattice points must be considered. If we label by an index cy the distinct atomic species in the multilattice, cy = 1,.. . , k , and suitably reorder the component lattices, the permutation matrix ,O becomes a block matrix, the block, say, Pa being a permutation of atoms of the species a. It is useful to introduce the center of muss g of the displacements, and the barycentric displacements &:

d", = 4 - g .

g = n-lC",_lc,i.

(13)

This and (11)2 easily imply, in obvious notation,

-

-

d; = P;dS

-

c

+ l:ea,

= 1:

-

n - l C ns=1 1"s =+

-

CLG = 0.

(14)

6

Incidentally the are l/n-integers, unlike the 1," which are integers. It is convenient to denote by G;,, the analogue of Gn+3 for the integers 1," replaced by the l / nsatisfying (14)3. integers By the above relations two matrices v,v' E Gn+3 belong to the same Tn+3-coset, which is indicated by [v](= [v']) E Gn+3/Tn+3, if and only if they have the same m, p, and-1;; here and below the last assertion is a shorthand for l;, 1'; producing the same 1; through (14)2 and its obvious analogue for 1';. Among all changes of descriptors, particular importance have those which produce an affine isometry of the multilattice to itself. If, with respect to the chosen origin 0, we represent the element e of the space E ( 3 ) of isometries by a paira ( t ,Q ) , t E EX3, Q E 0 ( 3 ) , it must be

Ira

Qea = rnteb,

Q4+ t

= P;ds

+ l,"ea.

(15)

The changes for the center of mass and the barycentric displacements are then Qg

+ t = g + n-lCyz"=,l,"

ear

Q&

= PS,d",

+tea.

(16)

In particular, if 0 coincides with the center of mass, that is, if g = 0, then

t

= n-lC:=l

I,"ea.

(17)

For any essentially described multilattice M = M (6,) the isometries represented with respect to 0 by solutions ( t ,Q) of (15) depend on M itself and neither on its specific descriptors 6 , nor on 0, and constitute the space group S ( M ) < E(3) of M . The corresponding matrices v form an isomorphic 'arithmetic' group, say A(&), which depends on 6,. Replacing the descriptors 6, by an equivalent set changes A(6,) to a Gn+3-conjugate. The orthogonal maps Q appearing in (15) constitute the point group P ( M ) (or P ( & ) ) of M , which is a subgroup of the Grassmann notation, t = e ( 0 ) - 0 , or e ( 0 ) = 0

+ t.

136

skeletal holohedry P ( e , ) ; and the quotient L(6,) = A(S,)/Tn+3,is a finite group isomorphic t o P(6,). As for any element of Gn+3/Tn+3,all the matrices v representing the same element [v]of L(S,) have the same m, ,B, and i:. Assume the origin 0 to be one of the multilattice points, thus reducing one of the &, say dl, to 0 ; and introduce the shifts p , = &+l,r = 1,.. . , n - 1, and the n 2 multilattice descriptors q, = ( e a , p T ) . The global invariance group r,+, < GL(n 2,Z) and the lattice group A(&,) < r,+, of the multilattice M = M(E,), detailed by Pitteri and Zanzotto", are isomorphic to Gn+3/Tn+3 and L(&,), respectively. Therefore properties of the former can be translated into properties of the latter. In particular, the analysis of the invariance of the constitutive equations of a thermoelastic multilattice is considerably simpler if one restricts the attention to suitably small distortions: Proposition 1 A n y given set 6: of essential multilattice descriptors has a neighto be called a (E(3)-invariant) wt-nbhd of &,: which i s borhood N in IW3(,-f3), invariant under the m a p ( e a ,4 )H ( Q e, , Q& t ) f o r any Q E 0 ( 3 ) , t E EX3, and E N } i s disjoint f r o m N is such that the transformed set v ( N ):= (6; = v:6,16, i f and only i f u belongs to A(6:).

+

+

+

Therefore, in any E(3)-invariant neighborhood N' of 6: contained in N the global G,+S-invariance reduces t o the invariance under A(&,): for any 6, E N'

A(&,) 5 A(&:).

(18)

This result allows us to efficiently reduce the description of the invariance in the wt-nbhds, and to greatly simplify the classification of generic elastic bifurcations for essential multilattices. For simplicity, in this paper we consider a crystalline solid in equilibrium with a heat-bath of which we only control the temperature. One can extend this treatment to accommodate other controls, for instance pressure (Ericksen3) or shear stresses (Budiansky and Truskinovsky', Simha and Truskinovsky12, Fadda et a14). Here an appropriate thermodynamic potential is the Helmoltz free energy, which is assumed to have a density per unit skeletal cell; this is a sufficiently smooth function

4 = $( ear4 , ~=)i@,,e),

(19)

where B denotes the absolute environmental temperature, regarded as a control. The free energy density can be reasonably assumed to depend on the locatjon of the multilattice points at any given temperature. Therefore the functions 4 or must have the same value on any two equivalent sets of descriptors for the same configuration; hence, for any v E Gn+3, they satisfy the invariance conditions

6

d(mb,eb,P;dS

+ l",ealQ)

= &a,

&,Q),

4(Q,,Q)

= 4(6u,QL

(20)

respectively. In addition, for these functions Galilean invariance is equivalent to invariance under orientation-preserving affine isometries. In particular

&QG, Q& + t , e ) = &ea,

for

Q

E

sow, t E

(21)

hence, by ( 1 3 ) , &e,,&,B) = & e , , d " , , ~ = ) @(s,G~,&,,Q), for

(22)

137 Cab =

e,

. eb, d,,

-

= d , . ear s = sgn(e1 ' e2 x e 3 ) .

(23)

As the shifts in Pitteri and Zanzotto", the barycentric displacements 2, are multilattice descriptors which are unaffected by any overall translation, hence are a viable choice of variables in the potential which is alternative to the shifts. Let s ( v ) be the sign of det m ( v ) ,the latter being the submatrix m of v. By (14), (22), and the above invariance conditions on the function @ satisfies the equality

I,

@(s7 C a b , h

a , 0)

= @ ( s s ( v ) ,miCi3mjb,mb,(P:d,b

+ Cbipr),0)

(24)

for any m E GL(3, Z), any species-preserving permutation P, and any l/c-integers such that = 0, a = 1 , 2 , 3 . Notice that any such triple (m,0, l:) forms a matrix v' E GL+3 and acts according to (ll)', (14)1 on Wg x S , S c W3" being the subspace of barycentric displacements.

p,

C;==,

4 Phase changes in a wt-nbhd Consider a reference configuration of an essential n-lattice, described by vectors 6; = (e,",d:). The following analogue of (15) holds:

mb,ei = Qe:,

p",df

+ l:e:

= Qdf

+t,

(25)

for any ( t ,Q ) representing an element of S ( M ( 6 : ) ) , v being the corresponding element of A(@,). Consider also a wt-nbhd N of 6;, based on Proposition 1. Since: (1) we restrict the attention to n-lattices; (2) our results are local; (3) in the space of descriptors the nonessential multilattices correspond to smooth submanifolds of strictly lower dimension (see Pitteri and Zanzotto"), and (4) N is the union of disjoint neighborhoods N+ and N - of 6; and -6:, respectively, each one of which is invariant under orientation-preserving affine isornetries, we can assume, without loss of generality, that in N+ all descriptors (ear4 ) are essential, and of course s = so := sgn(eF. e i x e,"). Denoting by Sym the space of symmetric tensors, and by Sym' the convex cone of the positive definite ones, we normalize the orientation of the (deformed) skeletal lattices of the multilattices in N+ by choosing lattice bases of the form e, = Ue:, U E Sym>, and introduce the increments n, [a,] in the referential [referential barycentric] displacements by means of the equations:

4- = U ( d : +T,), d", = u(d;o+a,), a,= 7r, - n-'Cy=17rs =+

c;=L,lws = 0.

(26) (27)

The following is easily proved: Cab =

e," . c e f ,

c = u2, and

with an obvious analogue for d,, and

d,,

-0

= e," . C ( d ,

T,..Therefore

+ w,),

(28)

in N+, where s is fixed,

$(ear4 , e ) = d(ea,d",,0) = * ( C , r , , e ) = * ( C , w r , Q ) ,

(29)

for a suitable function a. We denote by P+(S;) the subgroup of positive-determinant elements of P(S:), and likewise for the m component of the matrices v

138

in A+(S;) 5 A(S;); and let L+(S;) 5 L(6:) consist of the cosets [ v ] such that v E A+(S;). For any Q E P+(S:) we have

*( C ,w r y 0 ) = *(Qt C Q ,P t Q t w s

3

e),

(30)

where /3 is the permutation matrix of any v satisfying ( 2 5 ) €or some t (remember that all such v have the same submatrices m and p). Indeed, for any v E A+(S;) and the corresponding isometry ( t ,Q ) given by (15),

*(C,Tr,e)=4(ea,4 , ~ ) =d(mb,eb,Prds l;eea,e)

+

(31)

=& UQe:, UQ[dY + PSTQtrs + Q"], Q ) , from which the conclusion follows by ( 2 2 ) and (26)-(29). Notice that the last expression in ( 3 1 ) s equals a( Q'CQ, Qt(p;7rs+ t ) ,e), thus providing the extension of ( 3 0 ) to arbitrary reference displacements 7 r r . At this point we follow a procedure of Ericksen2 (also in Pitteri and Zanzotto") to classify generic weak bifurcations in simple lattices: we introduce orthonormal bases Vk,k = 1,.. . , 6 , in Sym and c,, a = 1 , 2 , 3 , in R3,and the representations

C = 14-2 E ,

E

= xE=1Yk v k ,

Tr

=

Y:Ca,

(32)

so that, in particular, C is near 1 if and only if the nonlinear strain tensor E E Sym is near 0 or, equivalently, (y1, . . . ,y6) is near 0 E R6. Here and below we must keep in mind that the coordinates representing the barycentric displacements a, are not all independent because they must satisfy the equations (34)s below. Remember that we have conveniently denoted by S the 3 n - 3-dimensional linear subspace of R3" consisting of barycentric displacements. In the treatment of the reduced problems in 54 we will choose the basis (c1,c2, cg) to coincide with ( i , j ,k) introduced there, and the basis V1,. . . , is meant to be represented in the basis ( i , j ,Ic) by the matrices in formulae (47)(49) of Pitterig. The format chosen here is an alternative to the one proposed there and applied to the a-P phase transition in quartz, treated as the 3-lattice consisting of the Si atoms alone. As is shown below, we obtain here an equivalent description of the class of generic bifurcations considered there. w e put in a single list (yi) = (yl, . . . , y6,,:y . . .yz) the 6+3n coordinates in ( 3 2 ) , hence, for the displacements, a single-indexed coordinate yi represents the doubleindexed y,"; and denote by fi the coordinates related t o barycentric displacements, so that fi E R6 x S. We can correspondingly introduce a new free energy function $(yi, 0) = *( C ,r r ,0), which then enjoys the invariance

vf,

$(Yi, 0) = $(Yi,

e),

yi

= Qijyj +ti,

Q E 0 ( 6 + 3n);

(33)

here ti = (0,...,O , T ' , P ,..., T ~ ) six , being the zeros and n the copies of P = ( Qtt)" E R3, and for the displacements the matrix elements Qij express in a twoindex way the four-index p," Qk, the latter being the Cartesian components of the tensor Q. In particular, the affine transformation in ( 3 3 ) ~reduces to the linear action by Q on the subspace R6 x S. By ( 3 0 ) each matrix Q is made of a 6 by 6 and a 3n by 3 n blocks, each one itself orthogonal. In addition, modulo a suitable

139 reordering of the displacements, each atomic species determines itself an orthogonal block in the 3n by 3n submatrix. We denote by G the group of such orthogonal matrices Q corresponding t o elements [v]of L+(S:). In the absence of loads and in a convenient notation, for a = 1,.. . ,6, r = 1,...,n, and a = 1 , 2 , 3 , the equations of equilibrium in R6 x S in terms of '$ are

a9

Gym := -($,e)

= 0,

+

$y:(jv) k, = 0,

c;=lY,"= 0,

(34)

the k, being Lagrange multipliers corresponding t o the constraints (34)4. We assume conditions (34) to hold for B = Bo and yi = 0, the latterAgivingthe (reference) multilattice M(S:). Consider now the tensor L = (Lij) = (QyiYj(O,&)) of moduli a t the transition, and the second-order symmetric tensor i on R6 x S associated to the symmetric bilinear form ((Vi),

(mi))

++

LijViZjl (Vi), (mi) E R~ x S.

(35)

L is invertibleb the equilibrium equations (34) have one solution Yi = % ( S ) in R6 x S for 3! near 00, such that $(&) = 0, by the implicit function theorem. Also, by continuity and uniqueness, all such equilibria have the same symmetry as M(6:). Therefore symmetry breaking can only occur if the tensor i has a nontrivial kernel. The invariance (33) forces, by differentiation, the following identity among the second derivatives of Q a t (0,Bo): If

L = QtLQ

for any Q E G,

(36)

hence the eigenspaces.of L are invariant under the action of G: QtLQy = Ly = Xy

LQy = XQy.

(37)

The same is true for the eigenspaces of L, each one of which is obtained by intersecting an eigenspac: of L with R6 x S. Equivalently, invariance forces certain eigenvalues of L (or of L) to be equal. We restrict the attention to the case, called generic by Ericksen' (see also Pitteri and Zanzottoll), in which the only conditions to be imposed on derivatives of 9 are those guaranteeing that ( 0 , B o ) is a stable equilibrium at which bifurcation occurs, and those forced by invariance; for instance (36). In particular, the only eigenvalues of i that are equal are the ones that are forced to be so by invariance; or, the eigenspaces of i are irreducible invariant (i.i.) subspaces of R6 x S under the action y H Qy of the group G , and exactly one of them is the kernel of L. Then, the condition that a stable phase exists, say, for 8 > 00 forces all the other eigenvalues t o be strictly positive. We call reduced the action of G on each i.i. subspace, and also call reduced the group representing such action on that subspace. If one chooses the basis above aligned with a choice of i.i. subspaces, then each matrix Q is a block matrix, each orthogonal block corresponding to an i.i. subspace, and actually being an element of the reduced group on that subspace. Since the action of G does not mix the first 6 and the last 3n coordinates, and the permutation matrices ,B preserve each atomic species (see (30)), the set of i.i. bNotice that the kernel of L always contains the nontrivial space (0) x S*, where 0 E Rs and S* is the 3-dimensional orthogonal complement of S in R3".

140 subspaces of R6 x S necessarily contains those of either one of the forms V1

x {O}

or

(0) x Vz,

(38)

where V1 [Vz]is an i.i. subspace of R6 [of S consisting in nonvanishing increments of the reference displacements of multilattice points of a single atomic species] and 0 E S [0 E R6]. Case (38)1 corresponds to configurational transitions, in which the motif follows the deformation of the skeleton, at least in the beginning. Case ( 3 8 ) ~describes structural transitions which are driven by the deformation of a single species of the motif, followed by a suitable consequent deformation of the other species and of the skeleton. We then follow a classical procedure: we determine the i.i. subspaces of R6 [of S] for case (38)1 [(38)2], and consider the corresponding reduced problem; a description of these can be found in Golubitsky et al 5 , Ericksen', Toledano and Dimitriev13, Pitteri and Zanzotto". Details about the possible additional i.i. subspaces are given in Pitteri". 5

The case of @-quartz

At low pressures quartz exhibits two stable phases, called 'low' (or trigonal, or a-) quartz and 'high' (or hexagonal, or 0-) quartz; at room pressure, these phases are observed below and above about 574OC, respectively. Here we follow James7 (and Pitteri and Zanzotto") by assuming that in any configuration of the SiOz structure the positions of the Si atoms be compatible with the definition of a 3-lattice, and neglect the oxigens; thus we describe the crystalline structure of both quartz phases by a monatomic 3-lattice, whose points are the positions of the Si atoms in the SiOz lattice. This is a reasonable first choice in the description of generic phase transitions of &quartz because, as pointed out above, these include the ones taking place in an i.i. subspace of displacements of a single atomic species; in this case the species of Si. In the literature% -a-@ transition is attributed t o a suitable deformation of tetrahedra having the center at a Si atom, and the four nearest 0 atoms as vertices. This will be analyzed in Pitteri", by considering also the generic transitions taking place in an i.i. subspace of displacements of the 0 atoms. In both a- and @-quartzthe skeletal lattice type is hexagonal. A common choice of lattice vectors is the following: a &a e3 = (O,O,c), (39) 2' 2 in an orthonormal basis ( i l j ,k ) . The rotational subgroup of the corresponding hexagonal holohedry is el = (a,O,O),

e2 = (--

where RZ denotes the rotation by the angle w about the direction of the vector v. In the crystallographic literature the plane of el and e2 is called the basal plane, and the direction of e3 (and of k ) is called the (hexagonal) optic axis. One of the two possible (enantiomorphic) 3-lattice structures of Si atoms in = 1,. . . , 6 , where 0-quartz at the transition temperature 0'6 has descriptors a,:

141

A basal plane

I3 +

I C

Q -Ic

Figure 4. Projection onto the basal plane of the Si atoms in right-handed &quartz, and of the descriptors s", = (e:, d:, d i , d!) for the 3-lattice given by (39) and (41)with X > 0

the lattice vectors e," are given by (39) for suitable choices ao, C O , of a and c, and barycentric displacements can be chosen as follows:

Fig. 4 i s the projection of the 3-lattice M(6:) onto the basal plane ( e ," ,e ; )orthogonal t o the optic axis e:. We address to James7 (or Pitteri and Zanzotto", Pitterig) for the construction of this 3-lattice structure by means of suitable helices whose projection on the basal plane is shown in the lower-right part of Fig. 4. The point group P(6:) < SO(3) of this 3-lattice is the rotational subgroup ' H k (see (40)) of the hexagonal holohedry P(e:), and is generated for instance by R;l3 and RH.The associated crystal class is called hexagonal trapezohedral,is denoted by 622 in the International Tables6, and is the actual crystal class of ,&quartz, so that the monatomic 3-lattice M(6:) of Si atoms gives already a good approximation of the actual (geometric) symmetry of this quartz phase. The treatment of Case (38)1 is the same as in Pitterig and is not repeated here. Case ( 3 8 ) ~

We denote by P;I3 the submatrix ,B of any element of the coset [ v : / ~E] L+(S:) that corresponds to the rotation R;/3 according to (15), etc., denoting the 3 by 3 identity. Based on (15) we have

142

We now analyze the action induced by (33)2 on the 6-dimensional space S C Rg of ( w l , w 2 , m 3with ) , typical element (al,az,a3,bl,bz,b 3 , ~ 1 , ~ 2 , subject ~3) to the constraints ni bi ci = 0 , i = 1,2,3, and the related i.i. subspaces. These are

+ +

determined by intersecting S with the invariant subspaces under the linenr action of each Q in G on the whole of Rg. First of all, the linear action of any Q E G maps to themselves the subspaces

W1 = {(O,O, n3,0,0, b3,0,0, ~

3 ) ) and ~

(45)

w = ((nl,n2,0,bl,b2,0,clrc2,0)),

(46) of displacement components along the optic axis and along the basal plane, respectively. This is because the hexagonal group xk has the optic axis and the basal plane as (irreducible) invariant subspaces. The (reduced) group of G on W1 has order 6, and its action on the coordinates (u3, b3,c3) is generated by the matrices‘

This is the group of symmetries of the equilateral triangle in S with vertices at (1,0, -1), ( 0 , -1, l), (-l,l,O). In particular, R,“ is represented by the 3 by 3 identity matrix. Since the only nontrivial invariant subspace of W1 is orthogonal to S , W1 n S is a 2-dimensional irreducible invariant subspace of S , and consists of monoclinic 3-lattices, with axis k . To within a rotation of the coordinates on S , this reduced problem is the same as the one in item (3) of Case (38)1 in Pitterig. As there, the bifurcation diagram consists of three unstable transcritical bifurcating curves of orthorhombic 222 symmetry. For instance, the orthorhombic axes (besides k) are i and j for the choice ~ l = X k = -w3,

w 2 = 0,

x E R.

(48)

This corresponds to the 1-dimensional subspace

Y E B, of S , which is invariant under the identical actions of Rq and Rj”. ( n 3 ,b 3 , c3) = Y(L0,

-I),

(49)

The subspace W n S decomposes into the orthogonal sum of three i.i. subspaces, W2, W3, W4, the first two of dimension 1, the third of dimension 2. They are respectively generated by

4 -,o,--,o,o,

4

1

6

2

w2 = (-,

3

4 1 -,--,O), 6 2

CHereand below x means ‘represented by’ or ‘representing’.

143 1

4

w3 = (--,-,o,o,--,o,2

A

W4

= (-,

6

6

4 3

1 . 6

-,0 ) , and

2' 6

1 4 4 1 1 4 --, 0 , --, 0 , 0 , -, -, O), wi = (-, -, 2

3

6

2

2

6

0,0,

4 1 6 --, 0 , - -, -, 2

3

6

0).

(52)

The reduced group on W2 [W3]is (1, -l}; for instance 1 M Rq [l M RT] and - 1 M R:l3 M RC. (53) Therefore, as is known, the bifurcation diagram is the standard pitchfork. A fourthorder polynomial energy is sufficient to capture the qualitative features of a (supercritical) second-order bifurcation, while a subcritical first-order one, as in the case of quartz, requires a sixth-order polynomial (see for instance E r i ~ k s e n ' ,or ~ Pitteri and Zanzotto"). In W2 [in W3] the crystal class of the bifurcating multilattices is trigolzal trupezohedrul (32 in the International Tables'), with k as 3-fold axis; the additional generator of the point group is RT [is R;]. The reduced group on W4 n S = {zw4 ywi} has order 12, and the changes in the coordinates 3: and y are generated by the matrices

+

f

:= (-10

01 )

x R; (also, -

this is the symmetry group of a regular hexagon in R2 with center at the origin and a vertex on the 3: axis. This reduced problem is the same as the one in item (4) of Case (38)1 in Pitterig. As there, the bifurcating branches consist in two triples of symmetry-related pitchforks, all of monoclinic 2 crystal class. Only one of the triples can be stable. We now analyze in detail the two trigonal trapezohedral subspaces W2 and W3. The reference displacement increments corresponding t o W 2 are

wt = X(1, &,o) a,+ = X(-Z, O,O) = X'e,",

= -X'(e,"

+ e;),

m3+= X(1,

-&,

(55)

0 ) = X'ei,

(56) X,X' being real parameters. Equivalently, by (26), denoting by d,f the present shifts, with d,. = Udf, e, = Ue:, and X a real parameter, dT = dl - X(e1

+ ez),

4'

=6

+ Xel,

d.$ = d3

+ Xe2.

(57) The displacement increments are directed radially from the center of mass, outwards for X > 0 (see Fig. 5); they represent deformed P-quartz for X = 0, while for X # 0 they give the M ( 6 , f )3-lattice model for trigonal trapezohedral a-quartz proposed by James7, and used also by Ericksen3 and Pitteri and Zanzotto'l. Indeed, if the origin is put on the lattice point displaced by d: the shifts are, in obvious notation, f-

+

pl - 4 - d: = PI

+ X(e1-t 2 4 ,

+

+

p$ = 4 ' - d: = p2 X(2el e2). (58) We address to James7 (also Pitteri and Zanzotto", Pitterig) for a description of how the a-quartz structure can be obtained by deforming the helices mentioned above for ,&quartz. The projection of the M(S:) 3-lattice onto the (basal) plane of el and e2 is sketched in Fig. 6 for X > 0. For fixed A, the other possibility is given by the

144

Figure 5 . Displacement increments mJ,f and mh producing the two trigonal trapezohedral quartz phases for the same X > 0

A

basal plane

(3

+

c

0 -Ic

Figure 6. Projection as in Fig. 4 for the right-handed a-quartz structure, and of the descriptors 6: = ( e a ,d:, d z , d:) for the 3-lattice given by (39) and (57) with U = 1 and X > 0

Dauphine' twin M(6,), where the 6; = (e,, d,, and displacements vectors e, as d, = dl

+ X(el+ e2) ,

%-, dy) have

4- = a& - Xel,

the same lattice

d; = d3 - Xe2,

(59)

with the same X as in (57). Pictorially, in Fig. 6 one has to perform on the displace: a rotation RE with respect to the center of mass. This easily implies ments d that the twin multilattice descriptors 6 , can be obtained from 6: by means of the rotation RE, of order 2 (see also (53)). The two twinned configurations correspond t o symmetry-related points on the bifurcated branches of the pitchfork in the Wz subspace mentioned above. We address the reader to Pitteri and Zanzotto" for more details on Dauphin6 twins.

145

The reference displacement increments corresponding to elements of the subspace W3 are, in terms of real parameters ,G,v’,

mi = p(0, -2,O) = -v’(Ze,” + e,”), Equivalently, by

wi = 0(&,1,0)

(as),denoting by d; the corresponding

= v’(2e:

+ e,”).

(61)

present displacements,

with p a real parameter. The displacement increments are azimuthal, oriented clockwise for p > 0. We have deformed @-quartzfor p = 0, while symmetry-related points on the bifurcated branches of the pitchfork in this subspace correspond to opposite values of p # 0 in the displacements given by (62). The related multilattices, say M ( 6 / , )and M(6:), are another example of shufjle twins, very similar to the Dauphin6 twins described above. In particular, the twin multilattice M(6:) can be obtained from M ( 6 & )by means of the same rotation RZ,of order 2 about the center of mass, which relates the Dauphin6 twins (see (53)). This can be obtained pictorially from Figs 5 and 7: by replacing each a; with its negative, the corresponding d; becomes its symmetric with respect to the line of d:, and the symmetric displacement is the Rl-transform of a displacement equivalent to 4; for r = 1 this is di el e2, etc. By putting the origin on the lattice point displaced by d ; , we obtain the shifts describing this quartz phase in Pitterig:

+ +

As there, one can describe this low-symmetry phase and the twins in terms of deformation of the reference /?-quartz helices. Now the radius of those helices shrinks, hence neighboring helices do not intersect anymore. Fig. 7 shows one of the possible arrangements of the actual helices. Looking at the hexagon drawn in the lower right corner, the other possibility - which gives the twinned configuration - is obtained by exchanging the occupied and the nonoccupied helices in that hexagon, and then coherently in the whole structure. We address the reader to Pitterig for some details on how the trigonal subspaces above are used by Ericksen3 in his bifurcation analysis of the a - /? transition.

Acknowledgments This work is part of the research activities of the EU Network ‘Phase Transitions in Crystalline Solids’, and is partially supported by the Italian M.I.U.R. through the project ‘Mathematical Models for Materials Science’.

References 1. B. Budiansky and L. Truskinovsky, J . Mech. Phys. Solids 41, 1445-1459 (1993). 2. J. L. Ericksen in Microstructure and phase transition, IMA Volumes in Mathematics and its Applications, n.54, ed. J.L. Ericksen, R.D. James, D. Kinderlehrer and M. Luskin (Springer-Verlag, New York, etc., 1993).

146

W A basal plane

EI +

c

Q -l_c

Figure 7. Projection as in Fig. 6 for the right-handed quartz structure with displacements given U = 1 and p > 0

by (62) for

3. J. L. Ericksen, J. of Elasticity 63, 61-86 (2001). 4. G. Fadda, L. Truskinovsky and G. Zanzotto, Phys. Rev. B 66, 174107 1-10 (2002). 5 . M. Golubitsky, D.G. Schaeffer and I.N. Stewart, Singularities and groups in bifurcation theory, Vol 11, Springer Verlag, New York, etc., 1988. 6. International Tables for X-ray Crystallography, Volume A , ed. T. Hahn, Reidel Publishing Company, Dordrecht, Boston, 1996. 7. R. D. James in Metastability and Incompletely Posed Problems, IMA Volumes in Mathematics and its Applications n.3, ed. S. S. Antman, J. L. Ericksen, D. Kinderlehrer and I. Muller (Springer-Verlag, New York, etc., 1987). 8. M. Pitteri in Proceedings of WASCOM 2001, Porto Ercole, Italy, ed. R. Monaco, M. Pandolfi and S. Rionero (World Scientific, Singapore etc., 2002) 9. M. Pitteri, On weak phase transformations in multilattices, TMR network ‘Phase Transitions in Crystalline Solids’ Preprint n. 100, also Rapport0 Tecnico DMMMSA n. 88, 2/12/2002; appearing in J. of Elasticity. 10. M. Pitteri, Full kinematics of P-quartz and its generic weak phase transitions, preprint. 11. M. Pitteri and G. Zanzotto, Continuum models for phase transitions and twinning in crystals, CRC/Chapman & Hall, Boca Raton, London, etc., 2002. 12. N. K. Simha and L. Truskinovsky in Contemporary research in the mechanics and mathematics of materials, ed. R. Batra and M. Beatty (CIMNE, Barcelona, 1996). 13. P. Toledano and V. Dmitriev, Reconstructive phase transformations: in crystals and quasicrystals, World Scientific, Singapore etc., 1996.

ON CONTACT POWERS AND NULL LAGRANGIAN FLUXES PAOLO PODIO-GUIDUGLI Dipartimento di Ingegneria Civile, llniversitb d i Roma “Tor Vergata”, Via Politecnico i, 0013.9 Roma, Italy E-mail: ppgQuniroma2it GIORGIO VERGARA CAFFARELLI Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Clniwersitb di Roma ‘Za Sapienza”, Via Scarpa 10, Roma, Italy E-mail: vergaraQdmmm.uniromai.it The general issue here discussed is what ‘stresses’ and ‘contact forces’ are t o be expected when either the material response is not simple or the body part of interest has a nonsmooth boundary, or else the two circumstances combine. In particular, for materials whose stored energy is a null lagrangian, certain stress/contact-flux identities are derived.

1 Introduction Standard particle physics considers one type of body interactions, namely, interactions at a distance, modeled as a force vector field which is generally obtained from a potential scalar field. Contact interactions are typical of continuum physics, where they are thought of as accounting for the short-range distance interactions between neighboring particles; also typical of continuum mechanics is a concept of stress. Both a continuous body and its environment and two body parts are presumed to have distance and contact interactions. How these force fields relate to the stress field is a well understood matter in the classical case of simple Cauchy bodies with smooth boundaries: a) the assignment of a balanced system of distance and contact forces - that is to say, of a force system expending null power for whatever admissible rigid motion - is equivalent to the assignment of a corresponding stress field; b) contact interactions manifest solely as a vector field (with the dimensions of force per unit area) over the regular part of a body’s boundary, while no concentrated contact interactions arise at either edges or vertices, if any. We here study a fairly general notion of contact power, with a view to better understand what ‘contact forces’ and ‘stresses’ are to be expected when either the material response is not simple or the body part of interest has a nonsmooth boundary, or else the two circumstances combine. We leave out such complications as fractal boundaries, bodies whose elements have not a persistent material identity, bodies without an interior, etc.; bodies and body parts with finite perimeter are general enough for our present purposes. We move from the observation that, for a simple Cauchy body, the contact power expended in a motion equals the contact flux; and that the latter both serves as a convenient weak notion of contact force and provides the basic information to construct the stress. Firstly, we consider the easy instance of elastic Cauchy bodies, simple or not; as is well-known, these bodies are constitutively described by one primary mathe147

148

matical object, their stored-energy mapping: notions like contact interactions and stress are secondary. We suggest general notions for the distance interaction and contact flux associated with a given stored energy, and we determine their forms for materials of grade-1 (i.e.,simple), whose stored energy depends only on the first gradient, and of grade-2, whose stored energy depends on the second gradient as well. From these results, we induce that, for material bodies of grade-2 or higher, be they elastic or not, quite complex contact interactions are to be expected, consisting not only of contact forces but also of contact couples per unit area, as well as of edge forces per unit length, vertex forces, and more. Secondly, we consider stored energies that are null lagrangians, i.e., are such that their volume integral, whatever the deformation and the body part, equals the surface integral of an energy density having the form of a flux. Consequently, the distance interaction associated to a null lagrangian is identically null, and the contact flux has two alternative representations. We are then in a position to derive, both for grade-1 and grade-2 null lagrangians, certain stress/contact-flux identities that may help to further clarify the general issue of producing mutually compatible pairs of ‘contact forces’ and ‘stresses’. 2

2.1

Contact Interactions and Stress in Simple Cauchy Bodies Standard Doctrine

For a simple Cauchy body, (i) a motion @It) fblt) is a time-dependent family of deformations f (., t ) from a given reference shape, a region B in a three-dimensional Euclidean point space, into a current shape f ( B ,t ) where distance and contact interactions are required to be partwise balanced; (ii) contact interactions are represented by one field s ( . , ., .; f ) over B x R x U,the stress vector, depending functionally on f up to the order N = 1 (here U is the sphere of all unimodular elements of the three-dimensional vector space V ) . Remark. For any fixed time t , we say that a field cp(.,t)defined over B depends functionally up to the order N on the deformation f ( . , t ) if there is a mapping @ such that

cpb,t ) = 9&t , {iVf}Nblt ) ) ,

P E B, where {,Vf}P is the list of the iterated gradients o f f , from 0V f = f to lVf E Of up to N V f , a Whenever, as is the case here, time plays the role of a parameter, we do not display time dependence, and write p(.;f ) to signal that ‘p depends functionally on f : e.g., for each n E U fixed, we write the stress-vector field over B a s s ( . , n ;f ) , leaving the dependence on t tacit.

Parts of a Cauchy body are subbodies, hence regions in space themselves, with almost-everywhere-smooth boundary under form of a closed surface. Let P be a aAt times, but not in this paper, it is convenient to interpret {iVf}r as the list of the histories up t o time t of the iterated gradients of f.

149

body part, and let 6'P be its boundary surface, positively oriented by the outer unit normal n. When evaluated at a regular point p E 6'P, the stress-vector field delivers the force s ( p , n @ ) ; f ) per unit reference area exerted at p either by the environment (if p E aP n 3B) or by an adjacent part (if p E aP n B ) . Given the stress-vector mapping s ( p , .; f ) from U into V at a point p , and chosen any three linearly independent vectors n(i) E U,the construct

yields the stress-tensor field at p ; conversely, the linear action of the stress tensor over the normal n to a plane through p gives the stress vector

4%n ; f) = Sb;f b

(2)

on that plane. Thus, and this is main thrust of Cauchy's Stress Theorem, either one of the mappings s(p, .; f ) and S(p;f)[.]over the unit sphere conveys the same information about contact interactions. In addition, the distance-force field balancing the contact interactions associated with a given stress tensor field is d ( p ;f ) := -Div S(p;f).

2.2

(3)

Contact Power, Distance Power, and Contact Flux

For a specified collection of velocity fields w over BUaB, the contact power expended on part P is the linear functional

7rc(P)[v] := the distance power is

s,, s,

7rd(P)[w] :=

s.

v;

(4)

d .v

(5)

We note that, due to (2) and the divergence theorem, the contact power can be given the equivalent forms

r c ( P ) [ v= ]

lp s, Sn . w =

Div (STv);

we call the vector field c(P; f"1

:= ST@;f)v

(7)

the contact f i ~ x for ~ the > ~ velocity field w . It is easy to check, with the use of (3) and (6), that contact and distance powers add up to equal the stress power T ~ :

7rc(P)[w]+ 7 r d ( P ) [ W ] = rs(P)[w]:=

JP

s . ow.

(8)

150

2.3

Weak Notions of Contact Force and Stress

One might think of regarding the contact power as a weak notion of contact force, and the stress power as the corresponding weak notion of stress. However, unlike the stress-vector mapping and the stress-tensor mapping, these two notions are not generally equivalent, as relation (8) makes clear. It is instead the contact flux that provides us with an appropriate, alternative weak notion of contact force, because

for all body parts and for all test velocities. Interestingly, if one bases a mechanical description of contact interactions on a suitable notion of contact flux, without postulating ab initio the form (7) for it, then it is reasonable to hope to find a setting within which the stress mapping could be not only proved to exist but also constructed from a given contact flux, for fairly general classes of material universes. This research program was initiated by Gurtin & Martins7 and continued, among others, by Si1hav$l5J6, Gurtin, Williams, & Ziemer' and, more recently, by Degiovanni, Marzocchi, & M u s e ~ t i ~ and Marzocchi & Musestig; it has the added attraction that, as relations (3)-(8) suggest, the resulting stress has divergence measure, as required to weakly balance the distance interactions. In this paper we pursue the same program, except we focus on universes of Cauchy bodies that are not simple, because their contact interactions depend functionally up to an order N > 1 on their deformations. These material bodies, which are often called of grade N , exhibit an interesting p h e n o m e n ~ n :at ~ ?those ~ ~ portions of their boundary which are not smooth - say, at edges, cusps and vertices - force and hyperforce fields appear that are absent when N = 1. 3

Contact Flux and Stress(es) Associated with an Energy

Let F be a specified collection of admissible motions, and let A(.; f ) be a scalarvalued mapping over B . The variational derivative b,A(.; f ) of A(.; f) at f E 3 is defined as follows:

for all E E No, an arbitrary open neighborhood of 0 in R,for all C r test vector fields h , and for all body parts P. Moreover, for each p E B, consider the linear mapping c X @ ; f)[.]of V into itself defined by

d for w@, t ) = -f@, t ) the motion velocity at time t and for all body parts P. Given dt a physical context where A(.; f) is interpreted as energy stored per unit volume, the vector fields 6fA(.; f) and .A(.; f)[v]can be respectively interpreted as the distance interaction and contact flux for the velocity w associated to A(.; f). We give below two examples of increasing complexity.

151 3.1

Grade-I Elasticity

The class of simple Cauchy bodies which are elastic obtains by taking

X ( . ; f ) = g ( . , F ) , F zz Of. One finds that, for each p E B and 21 E V , the 'distance interaction' is

(12)

f) = -Div

(13)

+X@;

(dF.b,

Vf@)))

and the 'contact flux' is c x b ; f"1

= (~F.@,vf(P)))T~'

(14)

with the 'stress'

S @ ;f) := d F d P , V f ( P ) ) . (15) In elasticity, the concept of stress is derived from the one primary object, which is energy. With a view towards applying the format embodied into (Q), we identify the contact power expended in a motion on a typical part P of B with the contact flux for the motion velocity:

(note the difference with definition (4)); consequently, since

s,

s,cx@;f)iwl."=

4P,n;f).v,

(17)

for the stress vector we set

f

s@, n ;

:= S b ; f )n,

(18)

with S given by (15); note the formal coincidence of this definition with (2). 3.2

Grade-2 Elasticity

To characterize elastic grade-2 Cauchy bodies,6,10,11,12,13,17,18 we choose A(.;

f)

o ( . ,1 F , 2 F

1

),

1F

F

= V f, 9 12V f = V(Vf).

(19)

For is&;

f) := d i F f l b ,vfb),2vf(P))

(2

= 1, 2,

(20)

the 'stresses' associated with the given energy,b we find that b f X k ; f ) = -Div (1Sb; f) - DivzS@; f)), f)[4 = lST@; f"1 + 2 S T ( P ; f"4.

ex@;

(21)

Paralleling the line of reasoning in Subsection 2.3, we are then driven to regard (22) bNote that the third-order tensors zF' and 2 s have t he same algebraic symmetrim, namely, ( f l a ) b = (2Fb)a for all vectors a , b. Note also that 2ST is defined by the identity 2 S T [ A ] ,a = aS[a]. A for all ( A ,a ) E Lin x V .

152 as the contact power expended in a motion. To see what form the associated contact interaction should have, we manipulate the right side of the last relation in the manner of the Appendix to h f . 13. We then let r ( P ) be the (possibly empty) finite collection of all edge curves laying on the boundary surface d P of part P. At a point of an oriented edge curve, where the unit tangent to the curve is the vector t , the outer normal to d P has a jump [ n ] ,and thus so does the unit vector m := t x n. What we find, with the use of a standard surface-divergence identity, is that

1,

2Sn.Vw = -

s,,

(("Div (2Sn)+2H(zSn)n).w+(~Sn)n,d*w)+ [ (2Sn)m].w,

J,

where H is the mean curvature at a regular point of dP. Thus, the contact interaction consists of three vector fields: two of them,

+ 2H(zSn)n),

:= 1Sn - ('Div (2Sn)

1s := ( z S n ) n ,

(23) interpreted as, respectively, the contact force and the (first-order) contact couple (both per unit area), are defined over the smooth part dP' of d P ; the third, 0s

f E := I ( 2 s n ) m ] ,

(24)

interpreted as the edge force per unit length, has support on r ( P ) . 3.3 Gradient Theories in General

In the case of elastic grade-3 Cauchy bodies13J9, the changes due to the contact interactions of type (23) and (24) are those that one would expect, given that thirdorder gradients and stresses are now in order, as is the appearance of second-order contact couples 2s per unit area over dP*; the facts that there may be tensorial contact couples T E per unit length over r ( P ) and concentrated vertex forces f V are perhaps a bit more surprising. The variational examples just considered give us guidance as to what virtualwork formulation of the basic balance law we should lay down to obtain a general mechanical theory of grade-N Cauchy bodies for N > 1. We do not find it necessary to do it here.

4 Contact Flux and Stress(es) Associated with Null Lagrangians

A scalar-valued mapping

u b ;f

) is a null lagrangian if

for all E E No,for all C r test vector fields h, and for all body parts P. Clearly, if u is a null lagrangian, then its variational derivative vanishes identically over 3 (and conversely); in addition, in view of (ll),

Therefore, when an energy is a null lagrangian, the associated distance interaction is null and the contact flux measures the time rate of change of the total energy of

153 a given body part. As to the contact flux,more can be said, as we shall quickly see, by exploiting the fact that null lagrangians have a well-known alternative form as surface potentials (see Refs 1 and 14, and the literature quoted therein). 4.1

Grade-1 Null Lagrangaans

These null lagrangians have the form 4 P ; f ) = ab, f ( p L V f @ ) )

(cf. (12)); their variational derivative,

Sfa =

- Diva,,,

must be identically null over 3:

dfa-Divdp

5

0;

moreover, they have the alternative representation

with T ( P , n;f ) = 4.23 f @), Of b))‘ n @ ) , ( 30) where, for each p E B and f E F fixed, the mapping u @ ,f ,.) from Lid into V must be such that the third-order tensor U = &u@, f,.) have the following skew symmetry:

(( Uc)b)u= -(( Uu)b)c for all vectors a , b , c.

(31)

Remarks. 1. Let the the major transpose UT and the left minor transpose tU of the third-order tensor U be defined by, respectively,

(( U T C ) b ) U := (( U a ) b ) c

(32)

and

( ( W c ) b ) a:= (( Uc)a)b for all vectors a , b, c.

(33) If U is skew-symmetric in the sense of (31), then UT is skew-symmetric as well; moreover,

%Y[A]= 0 for all symmetric second-order tensors A . (34) 2. The reason for requiring that the mapping ub,f , Of) have the property (31) is that, otherwise, the consistency condition a h ;f ) = D i v u b ; f ) (35) could not be satisfied. Indeed, if u@,f , O f ) would not enjoy that property, then we would have that Divu = d p u

+ a f u .FT + d

~ ugT, .

(36) an expression involving the second gradient of f , from which c does not depend

154

Combining (26)-(30) we find that

((6'fu)v

+ a ~ u ( V v ] )n,.

whence

for all parts P and all velocity fields v. Decomposing the gradient of v into its normal and tangential parts, we have that

because UT is skew-symmetric. By the same token, we have from a standard consequence of Stokes theorem that

( U T n ) ."Vw = L P

But, since

Lp

"Div ( U T n ) = ("Div

v . "Div ( U T n ) .

U)n + tU[.On],

and since the curvature tensor is symmetric, we conclude with the use of (34) that

v . ("Div U ) n = -

v . (Div U ) n

(38)

(the divergence and the surface divergence of U differ by (a, U ) n ,but (a, U ) n = 0 because the third-order tensor a, U inherits the skew symmetry of U ) . Relation (37) then takes the form

1,

( d F a ) n .v

=

s,P

((8jzd)v. n - v . (Divdpu)n),

(39)

for all parts P and all velocity fields v . We localize this statement at an arbitrary interior point of B by exploiting the arbitrariness inherent t o the quantification. The local statement is: i 3 ~ u= ( d f ~ -) Div ~ a~u,

(40)

an identity over F. Interestingly, and with no surprise in view of (29), the right side of this relation is the variational derivative of the surface potential (30). Indeed, if we set

we find

Sfr = (6ju)n, with b f u := ( a f u ) T- DivdFu, (42) an expression which we refer to as the lagrangian derivative of the vector field u . If we then regard &a as 'stress', by analogy with (15), we may read the identity (40) as the assertion that the 'stress' associated t o a grade-I null lagrangian is the lagrangian derivative of the corresponding contact flux.

155 4.2

Grade-2 Null Lagrangians

These null lagrangians have the form

and the alternative representation

where the contact-flux field u depends functionally on f up to the order N = 2. We set:

OU := d f u , 1U := d g u , and 2U := 8,u. For the compatibility condition (35) to hold, the fourth-order tensor 2U must satisfy the following symmetry relations:

+ ((((2Ud)b)u)c+ (((2Uc)b)d)u= 0

for all vectors a,b , c , d . (45) With a view to deducing the result corresponding to the stress/contact-flux identity (40), we begin by taking the variational derivative of the two integrals in (44). For the left one we find: (((2Uu)b)c)d

P

where it is understood that, once a typical interior point p of B has been selected, any such part P can be chosen as to have p E aP and to be flat up to the order N = 2 at p , in the sense that the first two gradients of the normal field are null at p ; moreover, to localize, the test fields w will be chosen so as to have support shrinking to p itself. The variational derivative of the right integral has a rather cumbersome expression, in which, however, no term involving second normal derivatives of the test field appear, due to the symmetries specified by (45). We find: F

JP

F

+

4

t(lU - 2DivzU)[n @ n] . d,w (oUT - DivlU

+ Div (Div 2U)) n . w

P

(47)

( V t ( l U - 2 D i v z U ) [ n @ n ~ n n ].)w , +

JOP

On equating the right sides of (46) and (47) and exploiting the arbitrariness inherent to the choice of both the test fields and the tangent plane through p , we find that the volume and the surface densities m and 7 of a grade-2 null lagrangian of type (43)-(44) must be such as to satisfy identically the following two relations, which must be identically satisfied over 3’:

1s - 2 D i v 9 = OUT - DivlU + Div(Div2U)

(48)

156

and ( 2 s - t(lU

- 2Div2U))[A] = 0 for all symmetric second-order tensors A . (49)

The first relation is the appropriate generalization of (40), the second is peculiar of the grade-2 case; on cancelling the grade-2 terms 9 and z U , (40) is recovered from (48), while (49) reduces to (34). are associated to a grade-2 null lagrangian. The Two ‘stresses’, S and 9, second of these stresses has the symmetry detailed in footnote b. For this reason, were (49) regarded as an equation for $3, it would completely determine it; but then (48) would effectively determine 1s.All in all, we may read off the system of (48) and (49) the assertion that the ‘stresses’ associated to a grade-2 null lagrangian are determined b y the corresponding contact flux. Acknowledgments This work has been supported by Progetto Cofinanziato 2002 “Modelli Matematici per la Scienza dei Material?’ and by TMR Contract FMRX-CT98-0229 “Phase Transitions in Crystalline Solids”. References 1. S. Carillo et al., in Rational Continua, Classical and New, ed.s M. Brocato and P. Podio-Guidugli (Springer, 2002) 2. M. Degiovanni et al., A.R.M.A. 147, 197 (1999). Quad. Sem. Mat. Brescia 27, (2002). 3. M. Degiovanni et d., 4. M. Degiovanni et al., Meccanica 38,369 (2003). 5. A. DiCarlo and A. Tatone, in AIMETA 01, Proc. 15th AIMETA Congress

(2001). G. Grioli, Annali Mat. Pura Appl. 50, 389 (1960). M.E. Gurtin and L.C. Martins, A.R.M.A. 60, 305 (1976). M.E. Gurtin et al., A.R.M.A. 92, 1 (1986). A. Marzocchi and A. Musesti, to appear in Rend. Sem. Mat. Univ. Padova 109, (2003). 10. R.D. Mindlin, A.R.M.A. 16, 51 (1964). 11. R.D. Mindlin, Int. J. Solids Structures 1 , 417 (1965). 12. R.D. Mindlin and H.F. Tiersten, A.R.M.A. 11,415 (1962). 13. P. Podio-Guidugli, TAM 28-29, 261 (2002). 14. P. Podio-Guidugli and G. Vergara Cdarelli, A.R.M.A. 109, 343 (1990). 15. M. Silhavj, A.R.M.A. 90, 195 (1985). 16. M. Silhavj, A.R.M.A. 116, 223 (1991). 17. R.A. Toupin, A.R.M.A. 11, 385 (1962). 18. R.A. Toupin, R.A., A.R.M.A. 17, 85 (1964). 19. C.H. Wu, Quart. Appl. Math. 50, 73 (1992). 6. 7. 8. 9.

MODELS OF CELLULAR POPULATIONS WITH DIFFERENT STATES OF ACTIVITY M. PRIMICERIO AND F. TALAMUCCI Dipartimento d i Matematica “U.Dini” Universith d i Firenze, Viale Morgagni, 67/a, 50134 Firenze, Italy E-mail: [email protected], talamucci0math.unifi.it The study of a biological system made of several populations of cells and different constituents is performed. Especially, we refer to the growth of a tumoural mass in the avascular state. A general scheme of the corresponding mathematical problem is obtained and some of the most representative models in literature are discussed. We propose a new approach to the problem, based either on nonlocal interactions among the constituents or on the existence of a chemical potential driving motion of intercellular fluid.

1

Introduction

The process of tumour growth is the result of the interaction of several phenomena of chemical, biological and mechanical type, strictly coupled t o each other. An appropriate approach t o the problem requires with no doubt a double attention both to the microscopic scale (cellular level) and t o the macroscopic one. The medical and scientific literature pointed out a sequence of stages of tumour growth, which can be essentially summarized as follows: a single genetically mutated cell (cancerous cell) proliferates giving rise t o a small avascular node (primary tumour); the nodule increases its mass by consumption of nutrient in loco or transferred by diffusion (avascular phase); in a more advanced state the tumour is able to deliver chemical agents which stimulate the formation of a capillary network (angiogenesis) transporting nutrients and inducing a new growth of the tumour mass (vascular phase); cancer cells are transported by the blood circulation system (intravasation); a new colony of cancer cells (metastasis) initiates t o grow in a distant site from the original tumour and a second neoplasia initiates to develop according to the listed sequential steps. An important feature of biological systems in which tumours evolve is the different state of activity of the cells: necrotic, quiescent and proliferating cells can usually be observed in a formed tumoural spheroid. The dynamics of transfer from one class to another is governed by a certain number of chemical factors, or by a natural decay of part of the populations. The phenomenon of growth of tumoural systems is at the present time studied not only by researchers in medical, biological and biophysical sciences, but also by mathematicians and computer scientists. 157

158 A full mathematical description of the whole process is indeed a difficult task: models in literature concentrate the attention on a specific phase in the tumoural evolution (immune system-mutated cells competition, avascular growth, angiogenesis, metastasis,. ..) . Our specific interest consists in modelling the dynamics of cellular proliferation, which is a crucial step in studying tumour growth and the possibility of controlling its speed. The present paper intends to continue the study undertaken in 1 1 , where we confined our discussion to a spatially homogeneous medium. Even referring to the simplest situation, the mathematical model has to take into account that cells can have different states of activity with respect to replication. Moreover, a total mass balance should include, besides of cellular mass, the material which can be used to construct new cells and the material that is “useless” under this aspect. The theoretical approach by mathematicians produced a series of models, appeared in the literature, which are based on specific assumptions adopted in order to face the problem. A conceptual idea that is often used consists in assuming that the medium is a continuum system where a number of different components coexist: by such an optics, the medium is considered as a multiphase system where processes of modification and migration of the constituents take place. Generally speaking, the starting point consists in writing the mass balance for each constituent involved in the process (Section 2). The specification of the dynamical processes of transfer, production or destruction of the various constituents and their motions in the mixture establishes a particular model of the process: this is discussed in Sections 3 and 4. Finally, in Section 5 we propose a different approach to the problem, trying to overcome some drastic assumptions existing in literature. 2

Mass conservation

The starting point consists in selecting the quantities involved in the biological process and writing the mass balance for each of them. We study the evolution of a population of cells and in the spirit of continuum mechanics we assume that a function m ( x ,t ) exists such that the cellular mass Mv contained in any domain V at time t is given by

Mv

=

1

m ( x ,t)dz.

V

We will use sometimes the term “cellular concentration” to denote function m. There is no doubt that the coexistence of more than one state of the cells (highly proliferation, dormancy, prenecrosis, ...) plays an important role in the tumoural growth. This can be taken into account by introducing N subclasses of cells and defining a concentration mi, i = 1,.. . ,N , for each of them so that

m = ml

+ . . . + mN.

(1) Remark 2.1 If the number N is large enough, we can introduce, instead of the (‘compartments”m l , . . . ,m~ a partition index a ranging from 0 to 1 and a partition

159

function of cellular activity cp(a,x , t ) with the property

i 0

cp(a,x,t)da= 1, v t 2 0 ,

and such that in any region V the quantity

corresponds to the mass of cells having index a E ( a l , a a ) and contained in the volume V at time t . In order to make the discussion of the model more clear, we confine to the discrete compartmental model. Following the arguments we used in the extension to the continuous distribution is straightforward. The intercellular space is occupied by: - molecules which provide cells with nourishment necessary to metabolism and

with material to be synthetized for mitosis, - “waste” material, where we include all the intercellular substances not taking

part the cellular synthesis. Let us denote by p the density of molecules of the first group and by q the density of waste products. We also define functions yj(x,t ) ,j = 1,.. . , K , representing any quantity that can influence the process (e. g. temperature, radioactivity index etc. ) but does not take part in the mass balance. Of course, we can also include in this family density of chemical substances that do not affect relevantly the mass balance but play a role in regulating cellular activity. In many cases, partial pressure of oxygen can be one of the ~ j ’ s . The conservation equations for each population (see, e. g. , ’) can be written as follows: dmi -+V.Jmi=Imi, i = l , . . . ,N, (3) at aP V . J p = I,, (4) at

+

aq

+ V . Jq = Iq. (5) at Equations ( 3 ) - ( 5 ) hold for x E 0, which is the region where the process takes place, and t 2 0 a and J is the flux of each constituent ( J ’ n corresponds to the amount of mass passing through a unit surface with normal n in a unit time) and I is the rate of production or loss (increase or decrease of the mass of the constituent per -

OIn principle, we might introduce within the material formed by p a number of subclasses p i , with different rate of production and different way of acting on the metabolism of the system. However, it is likely to think that internal exchanges of mass among the p i ’ s are absent, so that the introduction of them in the scheme is nothing but a formal complication.

160

unit volume and per unit time). In particular, the terms Imi must incorporate the dynamics of transfers from one class to another. In our point of view, the constituents mi, i = 1,.. . ,N , p and q are those (and only those) which take part in the general mass balance: the increase or reduction of a single constituent occurs only at the expenses of the other ones (for a different approach that singles out some components of the mixture considered as an open system, see 9). Hence, the total mass must be conserved:

5

(Irni+ I,

+ I q )dx = 0.

(6)

i=I

Assuming that (6) holds for any volume contained in R, i. e. assuming that all processes are localized, we write: N

(7)

Therefore the overall mass balance equation for the multi-component continuum formed by cells, by “useful” material and by “waste” is written as

If the system occupies the entire available space (saturation), we can write: N

(9)

q=Xqrni + q p + q q = 1 i= 1

with

specific volume of each constituent (volume fraction) and e, specific density (mass of the constituent in a unit volume occupied by the same constituent), which is assumed to be constant for any a. In terms of specific volumes, the overall balance is

where the first term in the left-hand side (time derivative) is zero whenever (9) holds. In the special case of equal specific densities @a= @,

a = m l , . .. , m N , P , q (12) we have that (11) coincides with (8). Moreover, if (9) holds, q is constant and (11) reduces t o

=0 i=l

161

corresponding to volume conservation. Of course, the mass balance equations can also be written by considering the number of elements (cells, molecules, ...) in a REV as the reference variables. Assume that each element of the constituent a occupies a volume V, and let n, be the number of elements per unit volume. Then the following relations hold (see (10)):

a = n,e,Va = @a% Thus, if both of elements:

e,

(14)

and V, are constant, any of eqq. (3)-(5) writes, in terms of number

-

where J, = J,/e,V, is the number of elements passing through a unit surface per unit time, I, = I,/e,V, is the number of the elements produced (or lost) in a unit volume per unit time. Obviously, the sum of the right-hand sides of (15) (written for each population) generally is not zero, but, according to (7)

Modelling the problem

3

To avoid irrelevant formal complications we assume (12), so that the saturation assumption (9) takes the form

+

The set of N 2 equations (3)-(5) with the constraints (7) and (17) contain the N K + 2 unknown quantities mi, p , q, -yj for i = 1 , . . . ,N , j = 1 , . . . ,K . We also assume to know the K equations governing the evolution of the quantities ' y j . At this moment, to complete the model we have to give the constitutive assumptions in order to specify:

+

1. the production/destruction terms Imi, I,, Iql 2. the mass fluxes J m i ,J,, J,.

We are going to examine the two aspects separately. Xl

Production and destruction

As to point 1, the quantities Imi, i = 1. . . ,N , I, and Iq have to take into account at least the following main biological processes: ( i ) proliferation of cells by mitosis: this requires necessary elements which are supplied by p ,

162

(ii) death of cells, which can either be recycled as available material (material p ) or become waste material y,

(iii) metabolism of the living cells, at expenses of molecules p ; the “burned” material appears as waste material q. (iu) transitions from one class mi to another.

Each of these processes can be stimulated or inhibited by the factors y j . Generally speaking, we expect that Imi is a function of r n l , . . . ,m N of p and y (for istance in case of catabolism), of the factors y j , j = 1,.. . , K , which may affect the rate of reproduction or decay of cells, of the position x and of time t. In a more general context, we may assume that the state of the system in a position x is affected by a neighbourhood of the point: nonlocal effects will be discussed in Section 5. To be more specific, assuming that the transition from one class mi, i = 1,.. . ,N , to another is istantaneous, we write: N

Im; = C V i , l & 1=1

N -

Pi

+ C(Tl+-i

724)

(18)

1=1

where

4denotes the proliferation of class 1 (new cellular mass originating from proliferation of cells of the l-th class per unit time and unit volume, at time t and position x): part of the newborn cells belongs to class i according to a distribution function V ~ J .Obviously, it is

- each

for each 1 = 1,.. . , N ; in a situation where each class produces only cells of the same class, we have simply vi,l = & J , with 6 i , ~Kronecker’s symbols. Moreover, following a philosophy of “mass action” law, we may set (as it is often the case in literature)

- The quantity pi refers to cell death in the i class (loss of mass per unit time

and volume in ( z , t ) ) ,due either t o necrosis or apoptosis, or t o the action of some factor yj;in analogy with (20), one could set Pi = CLi(p,q,7l,’”,yKIz,t)mi.

-

(21)

71+i is the rate of cellular mass transfer from class 1 t o class i; such terms describe the internal dynamics within the population rn; it is clear that 71+i must vanish for rnl going to zero: its simplest form (with respect t o r n l ) will be

n+i = Ai,l(P,q,71,”‘,7K,z,t)ml

(22)

163 In a more general case, we could assume that fj, m j and Xj+i depend also on the concentrations mk , k # j . Finally, recalling points (i)-(iii),the production and loss terms I p and Iq can be written

where the first sum in (23) corresponds to the molecules necessary t o mitosis of cells, the second sum takes into account the fraction wi (0 5 w i 5 1, i = 1 . . . ,N) of the mass of dead cells which can be recycled, while Gi is the rate at which molecules p are burned (in metabolic processes) by the cells of the i-th class. The meaning of the terms in (24) is evident. 3.2

Mass fluxes

Point 2 introduced at the beginning of this Section represents indeed a difficult step in modelling the process. Actually, t o formulate constitutive laws modelling the fluxes of mi, i = 1 , . . . ,N , p and q in terms of the concentrations and possibly of their derivatives appears a complex task. Let us denote by a any of the “populations” mi (i = 1 , . . . , N),p , q, so that (3), . . . , (5) write

aa + V . J,

-

at

= I,.

In analogy with the approach commonly used in the theory of mixtures (see ’), one could write:

J, = CYV, -

c

Da,BVa

a

where v, would represent a drift (or convective) velocity while the second term denotes diffusion. Thus, the balance equation (3), . . . , (5) writes

aa

-

at

+ Q . (av,) - V .

Using (26) extensively in the model of cellular dynamics seems not completely appropriate. For instance, one could postulate the presence of an inert intercellular liquid in which the populations p and q diffuse (more disputable would be assuming a relevant role of diffusion in the motion of the cells); but in such case the concentrations whose gradients drive the diffusion would be the concentrations of p and q relative to the intercellular liquid, i. e. p(l - m)-’ and q ( l - m)-’, respectively. In any case, whenever (26) is assumed, condition (11) (possibly, in its particular form (13)) gives a condition that has t o be fulfilled.

164 4

Closing the problem

At this point, the strategy that can be pursued can follow one of the lines: (a)completing the set of equations with the niomentum balance, ( b ) introducing some specific assumptions, that allow t o describe movements of the constituents of the system.

4.1

Momentum balance

Point ( a ) requires a complete description of the dynamics of the system. We have t o write for a generic population a:

5, is the flux of momentum of the population a and I, is the rate of production of the momentum density av,. One usually writes (see, €or istance, ’):

-where the second-rank tensor

where in (29) 8 is the diadic product of the two vectors, T, is the Cauchy partial stress tensor, due to the co-presence of the other constituents, in (30) b refers t o the body forces and Qa is the momentum supply referring to the mutual interactions of the constituents. The last term in (30) is the momentum supply which corresponds to the production of mass from one constituent t o the other. Furthermore, it is required that

(Q, + L v a ) = 0

(31)

a

Note that, owing to (25), Eq. (28) (with assumptions (29) and (30)) can be written also in the following way:

a

(”-at

+ V . (v, 8 v , )

The problem consists at this point in linking the quantities appearing in (32) with the external forces and the stress tensor. For istance, in the “growing porous media” model of (where N = 1 and q =_ 0) material p is assumed to behave like a liquid moving in a porous material formed by the cells. Chemical factors -yj are assumed to diffuse with respect to the moving liquid medium *. The mathematical problem consists in Eqs. (25) for m and p , diffusion equations (27) for ~j and the two momentum equations (32) for m and p , where inertial and external terms are neglected t o express J, = av,. Moreover, constitutive equations for the T , and Qa are to be assumed. In the system is considered as an elastic viscous fluid, with the constraint (13), written only for the two constituents m and p and assuming (9).

+

bOr, rather, they diffuse with respect t o the system liquid cells since yj represent the concentrations with respect t o the total volume and not to the volume of the liquid

165 4.2 Specific assumptions A. Consider the special case N = 1 and assume that m occupies a constant volume fraction (i. e. q, constant, see(lO), or m 5 mo, constant). This corresponds t o claim that at any point 5 wherever cells are present there is a fixed number of cells per unit volume. Equation (3) becomes V . J, = I,. If the population does not diffuse, we have

v . v,

= I,.

(33) Equation (33) expresses the obvious fact that the volumetric increase of any region in which m has the given concentration mo is determined by the proliferation rate. Assume I,,, is given (possibly as a function of position and/or concentration of nutrients or inhibitors, see, for instance, 3 ) . Then, if spherical (or cylindrical, or planar) symmetry is postulated, (33) gives the growth rate of the spheroid occupied by proliferating cells. Note that in this approach no role is played by material p that is assumed to be always present in the quantity needed to fulfill mass balance. B. Still referring t o a symmetry assumption, eq. (25) reduces to:

do -+--

at

1 d rn-l ar

(rnP1Ja)= I ,

(34)

( n = 1 , 2 , 3 for planar, cylindrical and spherical symmetry respectively) where J , = J, . errwith e, unit vector along the relevant direction. In model where the evolution of tumoural cords is studied, three population ml (viable cells), m2 (dead cells) and p (intercellular material) are assumed to saturate the medium. The internal dynamics is defined by the death rate of cells ml thus passing t o population m2 (owing t o spontaneous death, t o chemical agents and to treatment by radiation), by cell proliferation ( p towards m l ) and decay of dead cells (m2 towards p ) . The concentration of a subsance y (oxygen) with negligible mass determines the behaviour of the tumoural cells: threshold values of y establish the boundaries of regions of fully proliferating, quiescent and dead cells, or a mixture of them. Such a point of view is assumed also in several models where the analytical problem (typically, reaction-diffusion equations for y in domains with free boundaries) is studied (see, among others, 6 , '). The constitutive assumptions are vml = v,, and qp constant (hence qn constant, by virtue of (9)). By means of (12), one finds:

v . (qmvm + q p v p ) = 0

(35)

+

where 17, = qml qmz and v, = v,, = v,, . The spherical symmetry of the problem allows t o determine the velocities by using (35) and (34), which reduces t o

with em = ern, = ern,. Nevertheless, the problem presents in general cases relevant difficulties since I,, and I,, exhibit in practice nontrivial dependence om ml and m2 and the concentration of oxygen.

166

C . In the model l 2 N = 1 and q 0. The evolution of a nutrient y (with negligible mass) affects the dynamics of the process. The living cells m and the recycled material p are assumed t o saturate all the available space ((9) holds with 7, and 77,). Eqs. (15) are considered and the terms I,, 1, are modeled by assuming specific relations between the volume of one cell of m and one molecule of p . The drift velocities in (26) are assumed to be equal: v, = v p = v.

Moreover, a diffusion for p and y according to Fick’s law is postulated. The drift velocity v can be determined by means of (see (11) and (15))

V . v = V . (D,Vp)

+ V,f, + V,Ip

where D, is the diffusivity of material p and V,, V, are the volume of a living cell and of a molecule of basic material. It is assumed that a total volume of XV, of cellular material is required for mitosis. Assuming spherical symmetry of the model provides the closure of the problem. The tumoural spheroid is assumed to expand at the velocity v and a Robin-type boundary condition (according t o which the flux of material p at the boundary is proportional to the jump of concentration) is assumed t o hold. 5

A different approach

In the schemes we just described two facts are evident:

(i) the expansion of the tumoural mass is ascribed t o the velocity of cells mi, i = 1,...,N , (ii) threshold values for quantities not entering mass balance define moving/free boundaries which determine the regions of specific type of cells (in full activity, quiescent, ...). The models link the velocities of such boundaries t o the velocity fields of the constituents (for istance, decide whether a boundary is a material surface or not). Following a different idea and allowing mi, p and q to depend on x and t , we may think of models incorporating in the balance equations a mechanism of expansion. We will pursue this goal by proposing two alternative ways:

(i) introduce (spatially) nonlocal effects in the dynamics of the cells m , (ii) relate the movement of intercellular material to the gradient of a “chemical potential”. For the sake of simplicity, let us consider only one class of cells m ( N = 1) for tumoural cells, one type of a diffusing chemical factor y ( K = 1) and assume that the system is saturated, satisfies (12) and presents planar simmetry. As we anticipated, these assumptions make the presentation more transparent but could be easily relaxed. We write again the balance equations (see (34), n = 1):

167

aa at

-

dJ, += I,, ax

a = m,p,y

with the constraint m+p+y = 1,or, equivalently (see (13)), Jq = -(J,+ where h is determined by means of the boundary conditions. Let us examine points (i) and (ii)in detail.

Jp)+h(t),

5.1 Nonlocal interactions During the process of mitosis, the duplicated cell has to seetle itself in some space adjacent t o the generating cell. The “search of space” of a living cell m corresponds to the consumption of material p in some neighbourhood of the cell. The birth of a new cell in position x depends on the availability of material p in x and on the presence of cells m in the vicinity. Thus, we model the proliferating term I , in the following way:

I , = P T K ( x ,6,t m m ,

P b , t , m,P , 4 )

(37)

--M

where K is a positive function with compact support (say K ( x ,<) 0 for 1x-(1 2 6 for some 6 > 0) and 3 goes to zero with m. This approach is similar to the one proposed for spread of infections in spatially heterogeneous regions ( l o ) . Let us take J , = 0 and J p = 0: we thus assume that cells movement does not play any role in the mechanism of expansion, which is on the contrary incorporated in the proliferating term for m. Following such an idea, we write the mathematical problem (3)-(5) in the following form:

--M

t-M

-m

y = 1- ( m

+p )

In order to treat system (38) mathematically, one may start by following the same approach as in 4 , 8, by writing the Taylor’s series of the function 3 (w. r. t. x) and reduce system (38) t o its diffusive approximation (nonlinear parabolic equations).

5.2 Chemical potential An alternative approach that allows to by-pass the momentum equations (without introducing a free boundary separating a region in which there are no cells from another in which the number of cells per unit volume is constant) consists in postulating that the intercellular material formed by p and 4 can be assimilated to a fluid that moves under the effect of a chemical potential a.

168

It seems to be natural to think that a more intense reproduction activity of cells m attracts a larger number of molecules p , which are needed for reproduction and survival. Actually, cells “drain” from their environment the molecules they need to synthesize proteins and other macromolecules. Thus, it seems reasonable to assume that @ is proportional to the proliferation rate defined as the first term in (18). Moreover, we postulate that the gradient of acts on p and q without distinction:

J,

+ J,

= KV@.

(39)

Equation (39) is a sort of Darcy’s law and actually K plays a role of a permeability function. It is reasonable to assume that it drops to zero if m exceeds a threshold value f i :

K ( m ) > 0 for 0 5 m < T?L, K ( m ) = 0 for T?L < m 5 1,

(40) (41)

Assumption (40) corresponds to postulate that crowding of the cells m inhibits the supply of p . If v is the local speed of the intercellular fluid, we have

J, = p v ,

J, = q v (42) and we have that Eq. (39) (together with (9) and (12)) allows to close the system with no need of postulating any symmetry, since, from (13), one gets (assume N = 1):

V . J, = -V . ( K V @ ) .

(43)

Indeed, for N = 1 we find the following system for m and p :

_am_

v .(KV@= ) F-+v.(AKv@) aP =-FfWCL ,Ll

at

at

1-m

+

where the functions K , @, F and , ~depend l on m, p and q = 1 - ( m p ) . If more than one class of cells m is present (N > l ) , one may postulate that @ is proportional to the sum of the proliferation rates Fi, i = 1,.. . , N (see (18)). On the other hand, it can be assumed (similarly to (42))

Jmi = miu,

i = 1 , . . . ,N

(44)

so that only two drift velocities u (for the cells) and v (for the intercellular material) are introduced in the model. Assumption (44) allows to close the problem also in this case, since the balance equation for the populations mi, i = 1 , . . . ,N writes:

-ami -V.(-KV@ mi) = I , , , at m

(45)

N

where m =

C mi. Of course, this is not enough to claim that the problem has one i=l

and only one solution, but the analysis of the full model is beyond the scope of this

169

presentation and will be discussed in a future paper. Nevertheless, we can see how the solution looks like in very special simple cases. We assume planar symmetry (in order to keep the equation formally simpler), we take N = 1 and q 0. Therefore we have m = 1 - p and (see (43)):

=

d dm J , = -K-@(m,p) = -@(m)dX dX with 1 - m ) - -(m, aP and the equation for m(x,t ) has the form

1 - m)

Eq. (47) is a nonlinear parabolic equation degenerating for m 2 7iz (see (40)) and possibly for m = 0, according t o the specific form of the function @. This fact may allow for a finite speed of the line m = 0 in the ( x ,t)-plane and at m = riz where the “crowding” effect inhibits the percolation of intercellular liquid. Thus the problem is reduced to solving a degenerate parabolic equation with suitable initial and boundary data. It is easy to see that assumption q 0 is not crucial and can be released. The presence of diffusing substances ~j can also be taken into account without affecting the structure of the mathematical problem. Difficulties increase if N > 1, i. e. when cells can be encountered in different states, because in this case @ is determined by the sum of the proliferating rate of each “population” and the degeneration of the i-th equation may depend not on i but on mi. Leaving this question aside we present a numerical simulation of (47) in order to emphasize what we sketched above.

=

Ci

5.3 A numerical simulation The undertaking study of the nonlinear process started by a simple numerical simulation for Eq. (47) which generates the graphs of Figure 1 for the cells m. The input data are:

W m , p )= F ( ~ , P=)mp, p(m,p)= 0 , 1 ~ ( m=) ( m - -)41tl(i- 4m) f o r o 5 m 4

dm -(O,t) dX

dm

= -(l,t) dX

=0

fort

(48)

5

1,

(49)

2 0.

In (49) and (50) ‘B is the Heaviside step function. The threshold riz (see ( 4 0 ) )is while (see (46)) 1 4

Q = ( m - -)4(1 -

am)

for

o 5 m 5 riz,

Q =o

f o r 7iz < m 5 1.

1 4,

170

I

0.8

rn(x.t) for increasing times

__

-

0

0.2

0.4

0.6

0.8

1

Figure 1. A simulation for Eq. (47) with the specified data. T h e lower function corresponds to m(z,O). T h e increasing profiles of m(z, t ) are plotted for 10 sequential times.

The initial profile (50) is under the threshold h for any z E [0,1]. The boundary conditions (51) correspond to the absence of flux at z = 0 and z = 1. The sequence of profiles of m ( z ,t ) is shown in Fig. 1, for different values of t , suggests a finite speed of the front m = 0. When m exceeds the threshold h,the flux of cells stops (due to crowding) but their concentration still increases, tending to the saturation m = 1. Note that in this case equation (47) does not degenerate for m = 0, because of the choice (48). References 1. D. Ambrosi, L. Preziosi, On the Closure of Mass Balance Models for Tumour

Growth, Math. Mod. Meth. Appl. Sci. 12, 5, 737-754 (2002). 2. J. Bear, Dynamics of Fluids in Porous Media, Elsevier (1972). 3. H. M. Byrne, M. A. J. Chaplain, Free boundary value problem associated with the growth and development of multicellular spheroids, Eur. Appl. Math. 8, 639-658 (1997) 4. V. Capasso, Modelli matematici per malattie infettive, Quaderni dell’Istituto di Analisi Matematica di Bari (1980). 5. A. Bertuzzi, A. Fasano and A. Gandolfi, A mathematical model for the growth of tuor cords incorporating the dynamics of a nutrient, in Free Boundary Problems: Theory and Applications 11, N.Kenmochi, ed., Math. Sci. Appl. 14, Gakkotosho, Tokyo, 31-46 (2002).

171

6. S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumours, J . Math. Anal. Appl. , 255, 636-677 (2001). 7. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biology 38,262-284 (1999). 8. F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM Reg. Conf. Appl. Math. 20 SIAM, Philadelphia (1975). 9. J. D. Humphrey, K. R. Rajagopal, A constrained mixture model for growth and remodeling of soft tissues, Math. Mod. Meth. in Applied Sciences 12 n. 3, 407-430, (2002). 10. D. G. Kendall, Mathematical models for the spread of infections, in Mathematics and Computers Science in Biology and Medicine, HMSO London (1965). 11. M. Primicerio, F. Talamucci, Discrete and continuous compartmental models of cellular populations, Math. Models and Methods in Applied Sciences 12 n. 5, 649-663 (2002). 12. J.P. Ward, J.R. King, Mathematical modelling of avascular-tumour growth, IMA J . Math. Appl. Med. Biol. 14, 39-69 (1997)

FLOWS OF A FLUID WITH PRESSURE DEPENDANT VISCOSITIES BETWEEN ROTATING PARALLEL PLATES K.R. RAJAGOPAL AND KANNAN, K. Department of Mechanical Engineering, Texas A BM University, College Station, T X

77843-3123 E-mail: [email protected]. d u We investigate two classes of flows of fluids with pressure dependent viscosities between rotating parallel plates. First, we consider the flow of such fluids between two parallel plates rotating with the same angular spread about non-coincident axes, namely the flow in an orthogonal rheometer. This flow is a motion with constant principal relative stretch history and has been studied in great detail within the context of several fluids. Second, we study torsional flows. For both these flows we establish explicit exact solutions for a variety of relationships between the viscosity and pressure. We find that gravity can have a very pronounced effect on the flow field. In view of gravity, the pressure and hence the viscosity varies allowing for the development of pronounced boundary layers not at both boundaries but adjacent to just one of the plates.

1

Introduction

Stokes l2 recognized that the viscosity of a fluid could depend on the normal stresses and was careful to delineate the type of flows in which the viscosity could be assumed to be a constant. Stokes states “Let us now consider in what cases it is allowable to suppose u , to be independent of the pressure. It has been considered by Du Buat from his experiments on the motion of water in pipes and canals, that the total retardation of the velocity due to friction is not increased by increasing the pressure. ....I shall therefore suppose that for water, and by analogy for other incompressible fluids, p is independent of the pressure”. While for pipe flows under normal conditions the viscosity might not vary with pressure, there are several technologically significant problems wherein the viscosity can change by several orders of magnitude due to the changes in the pressure. An example of such a situation is provided by elastohydrodynamics wherein the viscosity can change by such a factor. That the viscosity would indeed depend on the pressure is obvious if one stops to think about the friction between adjacent layers near the surface of the Pacific Ocean versus adjacent layers near the bottom. The viscosity which is a consequence of this friction will be widely different. There has been a considerable amount of research concerning the response of both fluids and solids under high pressures (see Bridgman 3 ) . Andrade developed an expression for the viscosity as a function of both the temperature and pressure, and in his model with regard to the pressure, the viscosity varies exponentially. This variation of viscosity with pressure has been corroborated by numerous experiments (see Hron et al. for a detailed discussion of the recent experimental literature). In general, the viscosity of the fluid can depend on the shear rate, the temperature and the normal stresses. Here, we shall merely consider incompressible 172

173

fluids whose viscosity depends on the mean normal stress. The fact that the viscosity depends on the mean normal stress implies, within the context of the models considered that it depends on the Lagrange multiplier that enforces the constraint of incompressibility. In this sense, the model is strikingly dissimilar from models used within the context of classical continuum mechanics wherein the extra stress is assumed to be independent of the constraint response. Antman and Antman and Marlow have studied the possibility of the extra stress depending on the constraint response, a possibility that was considered by Stokes in his seminal paper of 1845. Mathematical issues concerning the flows of fluids with pressure dependent viscosity have been studied in some detail recently. Renardy lo and Gazzola established some results concerning existence that are local-in-time of a smooth solution under unrealistic assumptions concerning the variation of the viscosity with presP (PI + 0 a s p + 00, while experiments clearly show that sure. They assumed that P the viscosity does not vary in such a manner. Recently, Malek et al. ', Franta et al. have established existence of solutions to three dimensional flows that meet periodic boundary conditions and Dirichlet boundary conditions, respectively, under reasonable assumptions concerning the variation of viscosity with pressure. Hron et al. have carried out numerical simulations concerning the flows of such fluids. However, few boundary value problems have been solved concerning the flow of fluids with pressure dependent viscosity and even fewer explicit exact solutions have been established. In this paper, we shall consider specific boundary value problems concerning the flow of fluids with pressure dependent viscosity and establish explicit exact solutions. An interesting feature of our study is the inclusion of the body force field in the equations of motion, a factor that is usually ignored. We show that the presence of the body force field is critical in establishing the solutions. We consider flows between two parallel plates rotating about non-coincident axes or the torsional flow between a common axis. This development of boundary layers in torsional flow takes place even when inertial effects are ignored. Due of the presence of gravity, the pressure varies with respect to the depth. This has the effect of producing pronounced boundary layers adjacent to the plate at the top. 2

Equations of Motion

We shall consider the flow of a fluid in which the Cauchy stress T is related to the velocity in the following manner: where A1

T = -PI + /I (p) Ai, is the symmetric part of the velocity gradient

(1)

+ grad^)^] ,

(2)

A1 = [(gradv)

a

where grad denotes the eulerian spatial derivative -, and p l denotes the spherical ax stress due to the constraint of incompressibility. As we remarked in the introduction, the model 1 is markedly different from the classical models in which the extra

174

Y

Figure 1. Schematic of an orthogonal rheometer

stress depends on the constraint response p . Since the fluid is incompressible it can only undergo isochoric motion and thus

trAl = divv= 0 Substituting (1) into the balance of linear momentum div T+pb=p-

dv dt

(3)

(4)

and using the constraint (3) leads to -gradp+p (P) Av

+ [All (gradp) + pb = P-dv dt

(5)

We are particularly interested in incorporating the effect of gravity and thus we shall assume that the body force b is given by

b = -grad4

(6)

i.e., a conservative body force field. We shall find that the presence of the body force field is critical to the solutions that are established. We shall suppose that the two parallel plates are such that the normal to them is along the z-axis (see Figure 1).

2.1

Flow between two plates rotating about non-coincident axes.

We shall suppose that the velocity field has the form u = x = -R [y - g (Z)],

175

w=z=o, (9) where u, v and w are the x, y and z components of the velocity, respectively. The above flow corresponds to planes that are parallel to the plates rotating rigidly , focus of the centers of rotation being defined by about the point ( f ( z ) , g ( z ) ) the the equation x = f ( z ) and y = g ( z ) . The above flow is a motion with constant principal relative stretch history and has been studied in great detail (see Rajagopal 9). The flow domain corresponds to that in an orthogonal rheometer, an instrument that is used t o characterize the material moduli of non-Newtonian fluids. The normal forces and the torque acting on the plates axe correlated with the expressions for them to determine the material moduli. Here, we are not interested in the consequences of the flow being one of constant principal relative stretch history or in characterizing the material moduli of the fluid. Instead, we are merely interested in establishing the solutions to the velocity field for the flow under consideration and highlighting the development of boundary layers even when inertial effects are ignored. A straightforward calculation leads to the following equations of motion

a [Rp (p) g’] = -pR2 [. -2 - f (.)I ax + aZ a -2 dy + 8.2[-Rp ( P ) f’]= -pa2 [y - g ( z ) ], 3

where the prime denotes derivative with respect to z. In deriving the above equation we have assumed that gravity acts along the negative z-axis. We shall now seek a special solution for the pressure field

P =P k ) , (13) and we shall consider slow flows and thus ignore inertial effects. Under such conditions, the equations 10-12 reduce to

(PI 9’1’ = 0,

(14)

P P (P) f’l’ = 0,

(15)

It follows from 16 that P = Po

- Pgz

where po is the pressure at z = 0. It then follows trivially from 14 and 15 that

(17)

176

1 z pgh Figure 2. Variation of -g(-) versus 4 when p ( p ) = apn. Here, @ = -. a h h Pa

where the constants of integration are determined by enforcing the boundary conditions

f(h)=O, f(-h)=O,

a

-a

g ( h ) = 2 , g(-h)=- 2

(20)

In determining 20 we have assumed that the fluid adheres to the rotating plates. Depending on the form of the viscosity-pressure relationship we can evaluate the integrals (18) and (19) either exactly or numerically. In the absence of gravity, the pressure field p is a constant and thus the viscosity is a constant and we find that the locus of the centers of rotation is a straight line joining the centers about which the top and bottom plates are rotating. (i) p ( p ) = a p , a-constant

,

a> 0.

In this case, it is trivial to show that

f ( 4 = 0,

where Pa is the atmospheric pressure.

():

Z

versus - is depicted in the figure 2 (n = 1) and we notice The variation of -g a h a departure of the curve from linearity (the case for a fluid with constant viscosity) though it is not that marked. A simple calculation shows that the magnitude of

177 Non-dimensionalized votic'#y (dh.1)

IWPI

Figure 3. Non-dimensionalized vorticity as a function of

h

when p ( p ) = cup".

the vorticity JIwJJ is given by

llwll = PI

m.

(23)

We notice from figure 3 that while the vorticity magnitude adjacent to the top plate is larger that that near the bottom, it is not significantly different. This is because the variation of the viscosity with pressure is not that profound. We will see in the cases that follow, significant departures occur from the classical NavierStokes solution. (ii) p ( p ) = AeaP ; A, a-constant,

A > 0 , a> 0.

It follows from (18), (19) and (20) that

f (4 = 0, a

+

= exp (-aP,) - exp [-a (Pa 2pgh)l

We conclude from Fig. 4 that for the case of a fluid wherein the viscosity varies exponentially, marked departures occur in the locus of the center of rotation from the constant viscosity case. The flow rotates essentially as though it were a rigid z body when -1 5 - < 0.2 and undergoes significant shear in the rest of the d e h main. The magnitude of the vorticity is once again given by the Eq. ( 2 3 ) and we find that in the region of extreme shear the magnitude of vorticity is significantly

178

WhYa

1 z Figure 4. Variation of -g(-) versus 5 when p(p) = Aexp(ap). a h h

IbVlQl

Figure 5. Non-dimensionaliaed vorticity as a function of

5 when p ( p ) = A exp(ap). h

greater than elsewhere (see Fig. 5). Unlike the development of boundary layers in the Navier-Stokes theory wherein we have a confinement of vorticity in a narrow region adjacent to a solid boundary past which the fluid flows, due the effect of inertia, the magnitude of vorticity being negligible outside this narrow region, here we have vorticity everywhere, it being much higher adjacent to the plate. We thus

179

yet have a boundary layer effect and even when inertial effects are being ignored. (iii))30.11

= apn , a,n-constant

, a> 0,n-integer,n # 1.

f (4= 0,

(26)

(27)

1 z The variation of -g versus - is portrayed for various power-law exponents n a h in Fig. 2 , and the manner in which the magnitude of vorticity, which as before is given by Eq. (23), varies in a manner portrayed in Fig. 3.

2.2 Torsional Flow In a torsional flow, the velocity field has the form

v=*zx,

u=-+zy,

w=o,

(28)

where u,v and w are the x , y and z components of the velocity field, respectively and II, is a constant. The above flow field has been studied extensively (see Rivlin 11, Pipkin 8, and is a viscometric flow. Such a flow is possible in a wide class of fluids, provided the inertial terms are neglected. In such a flow, planes parallel to the 2 - y plane rotate as rigid discs, the angular speed varying linearly with the z-co-ordinates of the planes. Here we seek a generalization of the flow field (28):

I).(

- [n

21 =

Y,

w = 0,

(29)

(31)

where the angular speed n ( z ) is not necessarily linear in z. We once again seek a pressure field that depends only on z and ignore the inertial terms in the equations of motion to obtain

dP - pg = 0. -_ dz

We find that

(34)

180

ld4Qh1

Figure 6. Non-dimensionalized centerline vorticity as a function of f when p ( p ) = a p n and h 0 - h = 0.

and the two equations 29 and 30 collapse to the determination of R(z) through

with

(-h) = n - h , (h)= o h . (37) We are enforcing the ”neslip” boundary conditions in order to arrive at the Eq. 37. (i) p ( p ) = ap, a

> 0-constant.

It immediately follows that the magnitude of vorticity, unlike the previous problem, depends on all the three coordinates x, y and z and is given by llwll = J(x2

+ y”)(R’)2 + 402.

(39)

Thus, at the centerline, llwll =

A

m.

(40)

z plot of the non-dimensionalized centerline vorticity (magnitude) versus - is

h

181

I4/P,I

Figure 7. Non-dimensionalized centerline vorticity as a function of 1 when p ( p ) = crpn and h = 2Rh.

0-h

0-h 0-h provided in Figs. 6 and 7 for the cases = 0 and = 2, respectively (the

(ii) 1.1 ( p ) = AeaP, A

0(Z)

=

(nh

ah

o h

plots corresponding to n = 1.

> 0, a > 0, A, a-constant.

-a-h)

1 - exp (-2apgh)

{exp (-apg ( h - 2)) - exp (-2apgh))

+0-h.

(41)

The variation of the magnitude of vorticity at the centerline for the two cases 0-h &!! = 0 and = 2 can be found in the Figs. 8 and 9. Once again we see

o h

o h

a marked difference between the solution for constant viscosity versus the solution when the viscosity depends exponentially on the pressure.

(iii) 1.1(p) = Apn, A

> 0, n-integer(n # l), A-constant.

(42) The variation of the magnitude of the vorticity at the centerline can be found in 0-h 0-h Figs. 6 and 7 for the two cases = 0 and = 2. oh

nh

182

Figure 8. Non-dimensionalized centerline vorticity as a function of R-h = 0.

Figure 9. Non-dimensionalized centerline vorticity as a function of n-h

= 2Rh.

5 when &J) h

= A exp(crp) and

5 when p(p) = Aexp(ap) and h

We have not plotted how n(z) varies. Suffice to say that as the magnitude of vorticity at the centerline and In(a)I varies in a manner similar to $ IIwII.

183 References 1. S.S. Antman, Atti della Accademia Nazionale dei Lincei, Rendiconti, Classe di Scienza Fisiche, Matematiche e Naturali 70, 256 (1982). 2. S.S. Antman and R.S. Marlow, Arch. Rational Mech. Anal. 116, 257 (1991). 3. P.W. Bridgman, The physics of high pressure (The Macmillan Company, New York, 1931). 4. M. Franta et al., On steady flows of fluids with pressure and shear-dependent viscosities, 1-24 (Submitted). 5. F. Gazzola, 2. Angew. Math. Phys. 48, 760 (1997). 6. J. Hron et al.,Proc. R. SOC.Load. Ser. A 457, 1603 (2001). 7. J. Malek et al., Arch. Rational Mech. Anal. 165,243 (2002). 8. A.C. Pipkin, Quarterly Appl. Math. 26, 87 (1967). 9. K.R. Rajagopal, Arch. Rational Mech. Anal. 79, 29 (1982). 10. M. Renardy, Commun. Partial Differ. Equations 11, 779 (1986). 11. R.S. Rivlin, J. Rational Mech. Analysis 5, 179 (1956). 12. G.G. Stokes, !l!rans. Cambridge Phil. SOC.8, 287 (1845).

CONTROL ASPECTS IN GAS DYNAMICS PASQUALE RENNO Faculty of Engineering. Dept. of Mathematics and Appl, via Claudio 21, 80125, Naples, Italy. E-mail: [email protected] A non linear third - order strictly hyperbolic equation, typical of several dissipative models, is considered. Some basic properties of the explicit fundamental solution related to the principal part of the operator are outlined. In order to estimate some control aspects, the signaling problem is analyzed and maximum properties, together with continuous dependence, are obtained.

1

Introduction

In mathematical modelling of evaporators & , a basic problem is the control of the superheating temperature at the outlet of & . Usually, this control is achieved by means of prefixed variations of the refrigerant mass flowrate and pressure at the inlet of the evaporator. After the phase transition, in the superheated region, the onedimensional flow can be modelled by the equations of inviscid Gas Dynamics. As for the energy balance, the Newton's law for the heat flux at the wall can be applied. The related heat- transfer coefficient is obtained by experiments and so typical features of inverse problems are present too. To characterize control aspects, a research on the behavior of the refrigerant mass flowrate u ( z ,t ) along the evaporator is important. By means of simplifying hypotheses, this behavior of 'LL could be modelled by the following strictly hyperbolic equation

where f is a non linear prefixed function of it's variables, while co and c1 are characteristic speeds depending on thermodynamic properties of the specific refrigerant employed. Usually, it is ci < cf, so that the operator L, is typical of wave hierarchies. li2 Further, E > 0 is a characteristic dissipation parameter. The equation (1.1)describes also many other dissipative models. According t o the meaning of the source term f , typical examples can be found in Dynamics of relaxing gases, Thermochemistry, Heat- transfer and Viscoelasticity ( see references from [l]to [15] and [21]). In the linear case ( f = f (5, t ) ) , the equation (1.1) has been studied in depth in previous papers by Renno. Also when the number of space dimensions is two or three, explicit fundamental solutions of L, have been obtained and various boundary - value problems have been explicitly solved. Further, several basic aspects of wave behaviour and diffusion have been estimated, together with maximum principles and asymptotic properties as t + 03. 13120

184

185 This analysis has cleared up the roles of the highest order waves ( with speed . For instance, as for the signaling problem, the speed at which the main signal travels differs from the speed c1 of the wave - front. Indeed, the dissipation gives rise to the diffusion of waves which is connected with the lowest speed co and represents, at large t , the main part of the disturbance. Moreover, when singular perturbation problems for E + 0 are examined, attention must be paid t o boundary or interior layers which can appear in dependence of the various boundary - value problems which one deals with. Further, when E -+ 0 , a singularity as t -+ co might be too. In the linear case, all these aspects have already been examined in various papers l6 - 2o and pointwise estimates uniformly valid for all t > 0 have been achieved too. Aim of this paper is to apply some results obtained by the author 2o to the qualitative analysis of two propagation problems for the non linear equation (1.1). Maximum properties of the solutions and continuous dependence upon the data are obtained. c1 ) and the lowest order waves ( with speed co )

2

Basic properties of the fundamental solution

In this section, we recall some properties of the explicit fundamental solution of the operator .C, already obtained by Renno. 2o Let

c2 = cg/c: < 1,

r = lzI/cl,

p2

= c 2 / ~, and

i

R E { ( r , t ): 0 < r < t , 0 < t < T }

Y - { ( s , t ) : z E ! 3 t R , O < t< T } .

Further, if

r2= (1 + c2)/2&,k 2 = (1 - c ' ) / ~ E ,

7 = k2(t - ~ ) / 2 ,

5 = 2 c [ 2 r q / ~ ] ~w /=~ ,k 2 ( t 2 - r 2 ) ' / ' , consider the following function K ( r ,t ) defined in 0:

186

where Ik denotes the modified Bessel function of first kind. More, let v ( t ) the step-function and

Then, denoting by S the class of rapidly decreasing functions, the following theorem holds:

THEOREM 2.1 - T h e functional

is a tempered positive distribution which represents the only fundamental solution of L, with support contained an Y.

AS for the function K = -&K, defined in (2.1), it’s possible to prove that:

THEOREM 2.2 - T h e kernel K ( r ,t ) has the following properties:

187

Moreover, to obtain maximum principles for (l.l),the following properties of the kernels K and K1 are basic. THEOREM 2.3 - The C"(n) positive value junctions K and K1, everywhere in fl,are such that

3

Linear evolution and explicit behavior

The importance of explicit fundamental solutions is well known. In the linear case ( f = f(z,t ) ) several boundary value problems can be explictly solved and various aspects of wave propagation and diffusion can be evaluated in details. When f is not linear, appropriate estimates can be obtained by integro - differential equations (n.4). At first, let's consider two typical examples of linear evolution by means of the initial - value problem P and the signaling problem 3t .

If the initial conditions related to (1.1)are

a,u(z,o) =

let

f;:

and

fi(Z)

i = 0'1'2

zE

3-2

f the classical mean values of the data given by

and let

(3.3)

188

Then, the problem P is solved by THEOREM 3.1 - When the source term f ( x , t ) E C 2 ( Y )and the initial data f , ( x ) E C3-'(%),then the initial value problem P has a unique regular solution given by

The mean values fi, f are typical of pure wave behavior, while K is a rapidly decreasing function whose properties (th.2.2 and 2.3) imply directly estimates and maximum properties. For instance, let

(3.5)

with 0 5

T

5 t and hl and h2 such that

Then, the solution (3.4) of the problem principle:

u- 5

U(X,t)

5

P (

u+

with f

= 0 ) verifies the maximum

V(X,t) E Y

Other estimates can be found in the paper by Renno.

2o

Consider now the signaling problem 7-l for (1.1)defined in

Y + = { ( q t ) : x E R+, 0 < t < T } ; the initial-boundary conditions are:

(3.7)

189

When f = 0 , the evolution is caused by p ( t ) and it results:

where KO is the C" ( R ) kernel defined in ( 2 . 5 ) ~ . When cp = 0 , the share due to f(x,t ) is given by

on the condition that f( -9, *) = - f( y, '). The kernel and is such that

ar ~

K1

~ t( ) 0= ,- ( E c 2 ) - l e - P Z t .

is defined in (2.2)

(3.11)

So, the solution of the problem ?isi given by

",( q t ) = ",+, Further, for

+

"f

( G t ) E Y+.

(3.12)

X , estimates similar to (3.7) hold too. For instance, as KO > 0 and

Ko(r,t)dt = 1 - e-kZr,

( r = ./C)

(3.13)

it results

(3.14)

190

Moreover,when E --t 0, the properties of the kernels K , K O ,K1 , imply rigorous estimates also for singular perturbation problems .

4

Estimates for non linear problems Consider now the non linear problem 7 i

and assume that the function f (x,t ; u , p , q ) is continuous and bounded on the set 23 = { ( Z , t , % P , d : (G)E y + ,('LL,p,q) E !R3

1.

Then, by means of (3.9) - (3.10), one can see that the problem (4.1) admits a unique bounded solution u(x,t)which is the unique solution with bounded continuous derivative of the integro - differential equation

where, for concision, F ( x ,t ) denotes the non linear function

(4.3)

and p(t) E C2([0,T]), with

(0) = 0 ( i = 0 , 1 , 2 ) .

Moreover, one has: KO 2 0, KI 2 0 and

191 r+r

O<

I-,

Kl(1 y - T

1]7)d y

= 27 - 2&(1- e - - r q .

(4.4)

As consequence, if

by (3.14), (4.2), (4.4) it results:

Owing to the properties of kernels K , KO, K 1 , (th. 2.2 and 2.3), similar estimates hold also for the derivatives of the solution 2~ . The previous results allow to deduce also the continuous dependence upon the data (‘p, f ) . In fact, let u l , u2 the solutions of problem 3t corresponding to the data ( ‘pi,fi ) ( i = 1 , 2 ) and let

Then, the foregoing estimates imply

where the positive constant C depends on T . Similar analysis can be applied also to the non linear initial value problem P If suffices t o apply Theorem 3.1 and the basic properties of Theorem 2.2.

.

References 1. G.B. Whitham , Comm. on Pure and Appl. Math. Vol. XII, 113 (1959). 2. G.B. Whitham Linear and Non Linear Waves (J. Wiles & Sons 1959). 3. J. F. Clarke and M. McChesney Dynamics of Relaxing Gases ( Butterworts 1976).

192

H. Kolsky Stress Waves in Solids (Dover, N.Y. 1968). C. Hunter, Progress in Solid Mechanics Vol. I, 1 (1964). J. A. Morrison, Quarterly in Applied Mathematics . X V I , 153 (1956). D. GrafEi, Euromec 127 on Wave propagation in viscoelastic media 52, 1 (1982). G. Crupi, Boll. Un. Mat. Ital. (3) Vol. 12,439 (1957). A. Donato and D. Fusco, Atti Acc Peloritana dei Pericolanti Classe I Sc. Fis Mat. e Nut. 59, 149 (1981). 10. B. T. Chu, Journal de Mecanique Vol. I 4, 439 (1961). 11. B.J. Matkowsky and E. L. Reiss, Archive for Rational Mechanics and Analysis Vol. 42, 194 (1971). 12. F. Mainardi and G. Turchetti, Mech. Res. Comm. 2 , 107 (1975). 13. P. Renno, Rend. Accademia Nazionale delle Scienze detta dei X L Vol IV 98, 43 (1979). 14. P. Renno, Mechanics Research Communications Vol 8 (2), 83 (1981). 15. J. L. Lions, Perturbations Singulieres duns les Problemes aux Limites et en Controle Optimal (Lectures Notes in Mathematics n 323 Springer-Verlag 1973). 16. P. Renno, Atti Acc. Naz. Lincei Rend. Vol L X X I, 6 (1981). 17. P. Renno, Waves and Stability in continuous media - Catania 1981 (1982). 18. P. Renno, Atti Acc. Naz. Lincei Rend 75, 6 (1983). 19. P. Renno, Waves and Stability in countinuous media -Cosenza 1983 (1983). 20. P. Renno, Annuli d i Matematica pura e applicata Vol CXXXVI, 355 (1984). 21. M. De Angelis et all Mathematical and Computer modelling (2003). 4. 5. 6. 7. 8. 9.

A FUNCTIONAL FRAMEWORK FOR APPLIED CONTINUUM MECHANICS G . ROMANO AND M. DIACO Dipartimento di Scienza delle Costruzioni, Universita di Napoli Federico 11. E-mail: [email protected] We present an abstract formulation of the mechanics of continuous bodies in which the kinematic description is given as basic and the statics is derived by duality. A proper definition of the functional spaces based on finite decompositions of the reference domain allows to include in the theory all the powerful tools usually adopted in the analysis of engineering structures. In this respect general proofs are provided for the Virtual Work Theorem, the abstract CAUCHY'S Theorem and the Theorem of Kinematic Compatibility. A decomposition formula of special relevance in homogenization theory is derived as a direct application of the previous results.

1 Structural model Let us consider a bounded domain 0 of an n-dimensional Euclidean space with boundary 8 0 and closure = 0 U 8 0 and the space L2(0;V) of square integrable functions in 0 with values in the finite dimensional inner product space V . The LEBESGUEmeasures in 0 and on 8 0 will be denoted by d p and d u . To provide an abstract definition of a continuous structural model let us preliminarily provide some basic mathematical tools. 0

0

The pivot HILBERTspaces ( spaces identified with their dual ) N(0)= L 2 ( 0 ;W) and H(0 ) = L2 (0;V) of square integrable tensor and vector fields in 0 with inner products (( . . )) and ( . . ) . The SOBOLEV space of order H m ( Q V) of vector fields with square integrable distributional derivatives of order up to rn (see e.g. [2], [S]): H m ( O ; V ) = {v E L 2 ( 0 ; V ) I

Dpv E L 2 ( 0 ; V ) , V l p l 5 rn, rn integer 2 0 }

where the derivatives

are taken in the sense of distributions. 0

The linear space D ( 0 ; V) = C T ( 0 ; V) of test vector fields which are infinitely differentiable in 0 and have compact support in 0 . The support of a field u E C ( 0 ; V) , denoted by supp( u) ,is the smallest closed subset of 0 outside which the field vanishes.

The space ID(0;V) is endowed with the pseudo-topology induced by the following definition of convergence.

193

194

D(R;V) is said to converge to u E D(R; V) if there exists a compact subset K c R such that supp(u,) c K and DPu, -+ DPu uniformly in 0 for any vectorial multiindex p . The vectorial multiindex p is a list of p scalar multiindices each formed by n non negative integers to denote the order of partial differentiation with respect to the corresponding coordinate. The symbol I p I denotes the sum of the integers in p .

W A sequence {un} E

The dual of D ( Q V) is the linear space D’(R; V) of p-distributions on R,formed by the linear functionals which are continuous on D(R;V) . The value of T E D’(R; V) at cp E D ( 0 ; V) is denoted by ( T , cp ). The space D’(R; V) is in turn endowed with the pseudo-topology induced by the following definition of convergence. 0

A sequence of distributions {T,} E D’(0; V) is said to converge to a distribution T E D’(0; V) if ( T, , cp ) -+ ( T Icp ) for any test field cp E D(R;V) .

A continuous structural model is characterized by the definition of a kinematic operator B which is a linear differential operator of order m . The general form of an m th-order differential operator B : H(R) -+ D‘(0; W) is

(Bu)(x) : =

C

Ip

Ism

5 A%(x)Dpui(x), x

E 0,

i=l

where A2 (x) is a regular field of n x n matrices in R. P Any m-times differentiable kinematic field u in R is transformed by B into the corresponding strain rate field. The characteristic property of the kinematic operator is that the strain rate field vanishes if and only if the parent kinematic field is rigid. In the linearized theory of structural models, displacement fields are treated as kinematic fields and the corresponding strain rate fields are called linearized strain fields (or for shorthand simply strain fields). In structural mechanics it is however compelling to consider more general kinematic fields. The motivation is twofold. The first request is a technical one and consists in the need for the complection of the kinematic space, with respect to the mean square norm, to render the nice properties of HILBERTspaces available. The second request is that discontinuous kinematic fields must be allowed to be dealt with in the analysis of structural models. This latter demand stems from the mechanical principle which we refer to as the axiom of reproducibility. The basic idea is the one expressed by the classical principle and CAUCHY. The axiom of reproducibility states that the of sectioning due to EULER kinematic space must include discontinuous fields capable of describing regular relative motions between the elements of an arbitrary finite subdivision of the base domain 0. Generalized functions (distributions)are needed since the mathematical modelling leads to analyze vector fields that, being discontinuous, are not differentiable in the classical sense. Here regularity means that the strain rate field associated with a kinematic field must be a distribution representable by a square integrable field in each element of the subdivision.

195 A continuous structural model is characterized by a distributional differential operator

B : H(R) -+D’(R; W) which provides the distributional strain rate field Bv E D’(R; W) corresponding to a square integrable kinematic field v E H ( R; V) . A kinematic field v E H(R) is piecewise regular if the corresponding distributional strain rate field Bv E D’(R; W) is piecewise square integrable in 0 . It is then convenient to consider the regular kinematic fields to be defined on a subdivision of R into subdomains P on each of which they belong to H m ( P ;V) . Let us then consider a decompositions I(R) of R into a finite family of non-overlapping subdomains R C R with boundary 8 0 , ( e = 1,. . . ,n ) which realize a covering of The closure = 80,U 0,is called an element of I(R) and the following properties are assumed to be fulfilled:

n.

eK

n

O,nRp=O

afP

for

and

une=n. e=l

A field v on R is said to be piecewise H m ( R ;V) if its restriction vie to each element Qe of a suitable decomposition 7 ( R ) belong to Hm(Re;V ) ,

The kinematic space V , C H ( 0 ) of piecewise GREEN-regular kinematic fields on R is then defined by assuming that for any v E V , , there exists a decomposition I, ( 0 ) such that the distributional strain (Bv)],, restriction of Bv E D’(R; W) to R e , defined as:

v 4 E D(fl;W)

(Bv)Ie(4) : = (Ev)(4)7

: S ~ P cP Re ~

1

is square integrable over the element Re of 7,(R) . Then

The regular part of the strain distribution Bv E D(R;W) , denoted by Bv E ‘H(R) , is defined to be the list of square integrable strain fields (Bv)~,E ‘H(Re) ( e = 1,.. . , n ) . The space VR is a pre-HILBERT space when endowed with the inner product ( u , v ) ~ ,: =

s R

u.vdp

s

+R

B~:BV~~=(~,V)+((BU,BV)),

and the induced norm

which is equivalent to the sum of the norms, since

196 The kinematic operator B E BL { V,, X(0 ) )is a bounded linear map from V , into X(0) which provides the regular part Bv E X(0)of the distributional strain Bv E D’(0; W) corresponding to the kinematic field u E V n . The space F, of force systems is the topological dual of V , . The kinematic and the equilibrium operators B E B L { V , , X ( 0 ) } and B‘ E B L { X ( 0 ) , F n } are dual counterparts associated with the fundamental bilinear form b which describes the kinematic properties of the model: b ( v , u ) : = (( u ,Bv))= ( B‘v, v),

u E X ( 0 ) ,v E V,.

The formal adjoint of B E BL { V,, X(Q)} is the distributional differential operator Bi : X(0)+ D’(0) of order m defined by the identity (

B ~ u v,)

: = ((

0,

BV )) ,

VV

E D ( 0 ; V)

, V U E %(a).

The space S, of piecewise GREEN-regular stress fields on 0 is then defined as the linear space of stress fields u E X(0)such that the corresponding body force distribution BLu E D’(0; V) is representable by a piecewice square integrable field on 0 :

s,

: = {u E

X ( 0 ) I31,(0)

:

(Bbu)l, E H ( Q e ) } .

The regular part of the body force Bbu E D’(0; V) ,denoted by B’,v E H ( 0 ) ,is defined to be the list of square integrable fields (Bbu)le E H ( Q n , )( e = 1,.. . ,n ) . The space So is a pre-HILBERT space when endowed with the inner product

and the induced norm

which is equivalent to the sum of the norms:

The body equilibrium operator Bb E BL { S o ,H ( 0 ) ) is a bounded linear map of the stress fields u E S, into the regular part Bbu E H(0) of the distributional body force Biu E D’(0; V) .

197

2 Green's formula In mathematical physics, and in particular in continuum mechanics, a fundamental role is played by the classic GREEN'Sformula which is the fundamental tool for the formulation of Boundary Values Problems. Let us consider a kinematic field v E V Q and the stress field a E So and let I,, (52) = I,( 0 )V 7, (Q) be a decomposition finer than I, (52) and I,( 0 ). The GREEN'S formula for the operator B E BL { V Q ,X ( Q ) } can be written (( a ,Bv )) = ( Bba,v )

+ (( Na , r v )) ,

V v E V Q , V u E So,

where by definition

s

( ( a , B v ) ): =

o:Bvdp,

(Bba,v) : = / B b a . v d p ,

R

R

and the duality pairing (( N a , I'v )) is the extension by continuity of the following sum of boundary integrals over aIV,(52) = U d o e , e = 1,.. . , n : Na.I'vdcr. OTv,(Q)

The operators I' and N are differential operators of order ranging from 0 to rn - 1 defined by the rule of integration by parts.

3 Bilateral constraints A basic constraint in mechanics is the requirement of piecewise regularity of kinematic fields. Let V = V ( I ( 0 ) )C VR and S = S ( 7 ( Q ) ) SR be the closed linear spaces of kinematic and stress fields which are GREEN-regular in correspondence to a given subdivision 7(52) . The spaces V and S are HILBERTspaces when endowed with the topology inherited by V Q and SR. We shall further denote by 3 = T ( I ( 5 2 ) ) the space of force systems in duality with V(I(52)). Once a regularity subdivision 7(52) has beeen fixed, the boundary operators appearing in GREEN'S formula can bequalified as bounded linear operators between suitable functional spaces. Let us denote by aV = aV('T(f2))the linear space of boundary fields that are traces of fields in V , so that aV : = I'V. The space aV is an HILBERTspace when endowed with the topology of the isomorphic quotient space V / K e r r [8]. The flux boundary operator N E BL {S,aF} takes its values in the dual HILBERT space of boundary forces 8.F = a F ( I ( 5 2 ) ) .

198

The operator N yields the boundary tractions Nu E 6 3 due to the stress fields S . The operator I' yields the boundary traces I'v E aU of the displacement fields v E U . The operators I? E BL {U,dU} and N E BL { S ,aF} are surjective: I m r = dU and I m N = 6 3 . Moreover K e r r is dense in H and KerN is dense in 3-1 181. Since Im I' = i3V, by the closed range theorem the dual operator I" E BL { 3 7 , F} is injective being Ker I?' = [ Im I?]' = (0) . Affine constraints are usually considered in mechanics so that admissible kinematic fields belong to a closed linear variety U, = Ua('T(Q)) 5 V ( l ( Q ) )defined by U, : = w C where w E V ( l ( Q ) )and C = C(T(Q)) C U ( l ( Q ) ) is the closed linear subspace of conforming kinematisms. The linear space F, = F L ( 7 ( Q ) )of active forces is the topological dual of the HILBERT space L c U endowed with the topology inherited by U . It can be proved that there exists an isometric isomorphism between the space F, and the quotient space F I L L [8]. To derive the main result concerning the existence of a stress field, it is convenient to introduce the following pair of dual operators:

u

E

+

0

0

the conforming kinematic operator B, E BL {C, 3-1} , defined as the restriction of B E B L { U , X } to c , the conforming equilibrium operator B> E BL { S ,T I C ' } ,defined by the position BLu : = B / u + C ' .

The kernels and the images of these operators are given by Ker B, = Ker B n C ImBL=BL,

,

Ker B> = (B')-'C'

= (BC)'

,

ImB;=(ImB'+C')/C'.

The mechanical property of firm, bilateral and smooth constraints is modeled by requiring that the constraint reactions must be orthogonal to conforming kinematisms:

R ='C

1

= {r E F ( r , v ) = 0

vv E

c}

The closed linear subspace U R I G : = Ker B, C U of conforming rigid kinematisms has a special relevance in structural mechanics since its elements appear as test fields in the equilibrium condition of a system of active forces:

The elimination of the rigidity constraint is the central issue of continuum mechanics and is performed by a technique of LAGRANGE multipliers originarily envisaged by GABRIO PIOLA in 1833 [I]. The issue will be discussed in the next sections.

199

4 Korn’s inequality In continuum mechanics the fundamental theorems concerning the variational formulations of equilibrium and compatibility are founded on the property that, for any closed linear subspace of conforming kinematisms, the corresponding conforming kinematic operator has a closed range and a finite dimensional kernel. It can be proved [7] that this property is fulfilled if and only if the kinematic operator B E BL { V ,3-1) meets an inequality of KORN’Stype:

where H m ( I Q)) ( is a SOBOLEV space of order m subordinated to the subdivision I (Q) . If KORN’Sinequality holds, the space V ( I ( 0 ) )endowed with the norm

is isomorphic and isometric to H m ( I ( Q ) ;V) . KORN’Sinequality is equivalent to state that for any conforming subspace C reduced kinematic operator B, E BL { C, 3-1) fulfils the conditions:

c V the

dim KerB, < +co,

II Bv IIH 1 CB II v IIC/KerBt

Vv E C w ImB,

closedin 3-1,

where cB is a positive constant [ 7 ] . The well-posedness of the structural model requires that for any conforming subspace C V the fundamental form b be closed on S x V . This property is expressed by the inf-sup condition [6]

c

The reduced kinematic operator B, E BL {C, ‘Ft} and the dual reduced equilibrium operator B i E BL (3-1, F,} have both closed ranges and meet the equivalent inequalities

for all u E

X.

200 5 Basic theorems Making appeal to BANACH'S closed range theorem [2] we get the proof of the following basic theorem [8] which provides a rigorous basis to the LAGRANGE multipliers method in [ 11. applied by PIOLA Proposition 5.1. Theorem of Virtual Powers.

Given a system of active forces I E

[ Ker B,]

in equilibrium on the constrained structure M { 0 ,C, B} there exists at least a stress state u E H such that the virtual power performed by C E Ker B,] for any conforming kinematic field v E C be equal to the virtual power performed by the stress state u E H for the corresponding tangent strainjeld B v E 3-1, i,e.

1

C E [KerB,]

I

LF,

*3

u ~ :H ( C , v ) = ( ( a , B v ) ) , V V E C .

Proof. Since the kinematic operator B E BL { V ;3.1) meets KORN'Sinequality, we infer from BANACH'S closed range theorem that I m B i = ( KerBL)' where B i E BL {H; 3,) is the dual of B, E BL {C; 'I} . The equilibrium condition reads then C E Im B i and this ensures the existence of a stress state a E H such that B i u = C ,i.e. (( u , BV )) = (

B,u , v

) =(

C,v ), Vv E C.

The statement has been thus proved.

0

According to this approach a stress state is introduced as a field of LAGRANGE multipliers suitable to eliminate the rigidity constraint on the conforming virtual kinematisms. Uniqueness of the stress field in equilibrium holds to within elements of the closed linear subspace of self-stresses, defined by

S~EL : =F{ a E H : ( ( u , B v ) ) = O , t " v E C ) = ( B L ) ' = = { a €KerBi : ((Nu,I'v))=O, V V E C } = = {a E

S : Bba = 0 ,N a E [rC]*} = KerBb n C ,

where C : = { u E S : ( B b u , v ) = ( ( ~ , B v )V)V E L } =

= { u E S :( ( N a , r v ) ) = O V V E C } , is the space of conforming stress fields, a closed linear subspace of S .

201 6 Boundary value problems Boundary value problems are characterized by the fact that constraints are imposed only on the boundary trace of 7(0)-regular kinematic fields v E V ( l ( 0 ) ). It follows that in boundary value problems all the 7(0)-regular kinematisms with vanishing trace on 8 7 ( 0 ) are conforming, a property expressed by the inclusion

As we shall see hereafter in proving an abstract version of CAUCHY theorem, this property is essential in order that variational and differential formulations of equilibrium condition be equivalent one another. The presence of rigid frictionless bilateral constraints on the boundary W ( 0 )can be spaces { A , A’} and {P,P’} and the described by considering the pairs of dual HILBERT bounded linear operators L E BL { 8V, A‘} and II E BL {P,8 V } . The operators L and II provide respectively implicit and explicit descriptions of the boundary constraints. We assume that L and I’Ihave closed ranges so that, denoting by L’ E BL { A , 8F}and II’E BL { 8 3 , P’} the dual operators, BANACH’Stheorem tell us that Im L’ = ( Ker L)’ and Im II = ( Ker II’)’ [8]. The closed linear subspace of conforming displacement fields is then characterized by

C = {v E V

I rv E

I m I I = KerL} ,

In boundary value problems the orthogonality property

Rc ( K e r r ) ’ =

R = 13’

yields the condition

Irnr’,

where I” E BL { 8F, F} is the dual of r E BL { V ,8 V } . Hence there exists a boundary reaction p E 8F such that r ’ p = r that is

This means that constraint reactions consists only of boundary reactions which are elements of the subspace

8R = { p E 8F I (( p , rv )) = 0 Vv

E

L} = (rC)’

= Im L’ = K e r n ’

Uniqueness of the parametric representations of C and 8R requires that Ker XI = {o} and KerL’ = {o} respectively. It is now possible to provide a simple proof of an abstract version of CAUCHY’S fundamental theorem for boundary value problems in the statics of continua.

202 Proposition6.1. CAUCHY'S Theorem. Let us consider a constrained model M { 52, C , B} with kinematic constraint conditions imposed on the boundary d 7 ( 52) of a subdivision I ( 5 2 ) . Then a system of body and contact forces {b,t } E H x 8.F and a stress state u E 'H meet the variational condition of equilibrium ( b , v ) + (( t , W)= (( 6 ,Bv )),

u E 7.1, Vv E C

ifand only ifthey satisfy the CAUCHY equilibrium equations

Bbu = b , Nu = t where u E S and p E

body equilibrium,

+p

boundary equilibrium,

[I'C]' is a reactive system acting on d l ( 5 2 ) .

Proof. Let the variational condition of equilibrium be met and assume as test fields the kinematisms cp E D(7( 52); V) C Ker l? C C 5 V . From the distributional definition of the operator Bk : 'H H D'(I(52); V) we get the relation

( b , cp) = ( ( aBPI) , =(Bb, cp),

vcp ED(7(52);V).

It follows that u E S and Bbu = b and GREEN'Sformula can be applied to prove that (( U ,Bv )) = (BLu,V )

+ (( N U , r v )) ,

VV

E

C, u E S .

From the variational condition of equilibrium we finally get (( t , r v )) = ( ( N U ,

r v )),

6

E

s

vv E

c,

or equivalently

NU

- t E [rC]' = dR.

On the other hand, if CAUCHY'S equilibrium conditions are met, observing that

( ( p , r v ) ) = o , V P E [ ~ L ] ~V ,V E L , the variational condition of equilibrium is readily inferred from GREEN'Sformula. Theclosedness of Im B, = BC and the definition SsELF : = (BC)I yield the equality BL = (BL)" = SiELF which provides another basic existence result in structural mechanics and leads to the variational method for kinematic compatibility stated below.

Proposition6.2. Let M { 52, C,B} be a constrained structure and let { E , w} E 7.1 x V be a kinematic system formed by an imposed distorsion E E 7.1 and an impressed kinematism w E V . Then we have the equivalence ((~,E-Bw))=O V U E S ~ E L F u ~ u E w + C :E = B u .

203 ' I

Proof. By BANACH'S closed range theorem we have that Im B, = ( Ker BL) E - BW E SkELF is equivalent to E - Bw E I m B L .

. Hence 0

The result in proposition 6.2 leads also to the following decomposition property.

Decomposition of the space 'H . The linear subspace BL of tangent strains which are compatible with conforming kinematisms and the linear subspace S ~ E L Fof self stresses provide a decomposition of the HILBERTspace 7-l of square integrable tangent strain fields into the direct sum of two orthogonal complements

where the symbol idenotes the direct sum and orthogonality has to be taken in the mean square sense in R , that is according to the hilbertian topology of the space 'H . The theory developed above allows us to establish a number of useful results which could not be deduced if a more nayve analysis were performed. Among these we quote several new representation formulas which are relevant in the complementary formulations of equilibrium and compatibility and in the statement of primal and complementary mixed and hybrid variational principles in elastostatics [3], [4],[7]. From the basic orthogonal decomposition of the space 3-1 another decomposition formula which plays a basic role in homogenization theory (see e.g. [5] and reference therein) can be directly inferred. To this end let M, E BL {'H; W} be the averaging operator which provides the mean value in R of fields E E 7-1. It is easy to see that I m M , = W and that the adjoint operator ML E BL {W; 'H} maps D E W into the constant field E(X) = D Vx E R . By the closed range theorem we have that Im M, = (Ker Mh)',

Im Mh = ( Ker M,)'

.

We have the following result.

Proposition 6.3. Let M { R, L, B} be a structural model such that the space B L of conforming strains includes the constantjelds: I m M h c BL. Then the following decomposition into the direct sum of orthogonal complements holds:

'H = Im Mh where

I BL n

ImML Ker Ma

+ BL n Ker M, + (BL)'

.

constant jelds, zero mean conforming strain jelds, zero mean selfequilibrated stress jelds.

204

Proof. The result follows from the formula B C = ImM’Jz+BLn(ImMh)’= and from the equivalence Im Mh C BL

I m M h + B C n KerM,, (BC)’ C Ker M,

.

0

In periodic homogenization theory the closed linear subspace of conforming kinematisms is defined to be C(C) : = Im Mh CpER(C).

+

c

Here is the periodicity cell and C(,C) periodic kinematisms defined by C(,C)

: = {V E

is the closed h e a r subspaceof GREEN-regular

V ( C )I B v ~ 6 L2(K;V)},

being K any compact neighborhood of the periodicity cell C and vtl the extension by periodicity of the kinematism v 6 V ( C ). It is easy to see that CpER(C)c Ker M, . Hence C(C) is closed being the sum of two orthogonal closed linear subspaces. References 1. G. PIOLA, La meccanica dei corpi naturalmente estesi trattata col calcolo delle variazioni, Opusc. Mat. Fis. di Diversi Autori, Giusti, Milano, 2,201-236 (1833). 2. K . YOSIDA,Functional Analysis, Fourth Ed. Springer-Verlag, New York (1974). 3. F. BREZZI,M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer, New York, (1991). 4. J.E. ROBERTS,J.-M. THOMAS, Mixed and Hybrid Methods, Handbook of Numerical Analysis, Ed. P.G. Ciarlet and J.J. Lions, Elsevier, New York, (1991). 5. P . BISEGNA,R. LUCIANO, Variational bounds for the overall properties of piezoelectric composites, 1.Mech. Phys. Solids, Vol44, No 4,583-602, (1996). 6. G. ROMANO, L. ROSATI,M. DIACO,Well Posedness of Mixed Formulations in Elasticity, ZAMM, 79,435-454, (1999). 7. G. ROMANO, On the necessity of Korn’s inequality, Symposium on Trends in Applications of Mathematics to Mechanics, STAMM 2000, National University of Ireland, Galway, July 9th-l4th, (2000). Theory of structural models, Part I, Elements of Linear Analysis 8. G . ROMANO, - Part 11, Structural Models, UniversitA di Napoli Federico 11, (2000). 9. G. ROMANO, M. DIACO,Basic Decomposition Theorems in the Mechanics of Continuous Structures, 15th AIMETA Congress of Theoretical and Applied Mechanics, Taormina, Italy (2001).

GLOBAL EXISTENCE, STABILITY AND NON LINEAR WAVE PROPAGATION IN BINARY MIXTURES OF EULER FLUIDS TOMMASO RUGGERI Department of Mathematics and Research Center of Applied Mathematics ( C I R A M ) University of Bologna, Via Saragozza 8, 40123 Bologna, Italy E-mail: [email protected]

Dedicated to S. Rionero in the occasion of his 70th birthday We present the onedimensional system governing the processes of a binary mixtures of ideal Euler fluids. First we discuss the existence of global smooth solutions and the stability of constant state, then in the case in which the difference between the molecular masses is negligible, we present some recent results concerning acceleration and shock waves.

1

Introduction

Mixtures of fluids exhibit a huge amount of diverse phenomena. The first rational treatise of mathematical model of homogeneous mixture of fluids was given by Truesdell in the context of Rational Thermodynamics The compatibility of the model with the second principle of thermodynamics was discussed by Muller and the mixture theory belong t o the Rational Extended Thermodynamics theory In4 it was observed that the case of a binary mixture of Euler fluids can be rewritten in the form of a single fluid with extending field variables describing behavior of the mixture as a whole (density p , velocity v and temperature T ) with variables describing behavior of one constituent (concentration variable c and diffusion flux vector J). Here we shall analyze the one-dimensional case 5 :

’.

’.

4 9

w+

52

pc(1 - c)

1

w +-J)

= 0.

cy

Governing equations consist of balance laws of mass, momentum and energy of the mixture (Eqs. (l)’, (1)3 and (1)s) and balance laws of mass and momentum of one constituent (Eqs. ( 1 ) ~and ( l ) 4 ) . In (1) p is the total pressure, E the total internal energy of the mixture, u = p l is the pressure of one constituent while l/a is the difference of enthalpy. 7 and -pJ represent the mass and momentum exchange between constituents. If there are not chemical reaction ~ ( pc,, T) E 0. We shall

205

206 assume that constituents obey thermal and caloric equation of state of ideal gas p , = (ks/m,)p,T, E, = p,/(p,(y, - I)), a = 1 , 2 , k B = 1.38. lOWZ3J/Kthe Boltzmann constant, ma the molecular masses and ya are the ratio of specific heat of the constituents. These constitutive assumptions, which are in accordance with entropy principle, give to the system (1)the structure of a quasi-linear hyperbolic system of balance laws. Detailed informations on the constitutive assumptions can be see in the recent paper of Ruggeri and Simib 5 . 2

Qualitative Analysis

The system (1) is a particular case of an hyperbolic system compatible with a convex entropy principle in which a block of equations are conservation laws and another one are balance laws: &U

+ &F(u) = f(u),

where u , F and f are RN vectors with the productions f first A4 components ( g E R N - M ) .

(2)

= ( O , g ) T have null the

2.1 Local well posedness The existence of a strictly convex entropy function permits t o put the original system in a symmetric form introducing a privileged set of field variables (the m a i n and is a basic condition for the well-posedness. In fact if the flux F field u') and the production f are smooth enough, in a suitable convex open set D E R", it is well known that system ( 2 ) has a unique local (in time) smooth solution for lo. smooth initial data However, in the general case, and even for arbitrarily small and smooth initial data, there is no global continuation for these smooth solutions, which may develop singularities, shocks or blowup, in finite time, see for instance 1 1 , 12. On the other hand, in many physical examples, thanks t o the interplay between the source term and the hyperbolicity there exist global smooth solutions for a suitable set of initial data. This is the case for example of the isentropic Euler system with damping. Roughly speaking, for such a system the relaxation term induces a dissipative effect. This effect then competes with the hyperbolicity. If the dissipation is sufficiently strong t o dominate the hyperbolicity, the system is dissipative, and we aspect that the classical solution exist for all time and converges t o a constant state. Otherwise, if the dissipation and the hyperbolicity are equally important, we expect that only part of the perturbation diffuses. In the latter case the system is called of composite type by Zeng 13. 6i

*I

2.2

'3

T h e Kawashima condition

In general, there are several ways t o identify whether a hyperbolic system with relaxation is dissipative or of composite type. One way is completely parallel t o the case of the hyperbolic-parabolic system, which was discussed first by Kawashima and for this reason is now called the Kawashima condition l4 or genuine coupling 15 :

207 I n the equilibrium manifold any characteristic eigenvector is not an the null space of Vf(u). 2.3

Global Existence and stability of constant state

For dissipative one dimensional systems (2) satisfying the K-condition it is possible to prove the following global existence theorem due to Hanouzet and Natalini 14: Theorem 1 Assume that the system (2) is strictly dissipative and the K-condition is satisfied. Then there exists 6 > 0 , such that, i f I \ U ’ ( X , O ) ~ ~5~ 6, there is a unique global smooth solution, which verifies U’ E

c0([o, 0 0 ) ; P ( Rn )cl([o, CO); H ~ ( R ) ) .

Moreover Ruggeri and Serre l5 have proved that the constant states are stable: Theorem 2 Under natural hypotheses of strongly convex entropy, strict dissipativeness, genuine coupling and ”zero mass” initial f o r the perturbation of the equilibrium variables the constant solution stabilizes IIU

2.4

(t)l12= 0

(t-)

.

Kawashima Condition for the mixture

In the following text the basic results about Kawashima condition are presented for the case ml zz r n 2 = rn but the results remain the same also in the general case. In this case the characteristic eigenvalues X and the right eigenvectors d were evaluated by Ruggeri and SimiC , choosing the field u = ( p , c , v ,J , T ) . In particular in equilibrium state uo _= (pota,O,O, TO)we have: = -co,

A?) = 0,

=

i”);

-Pa ;

d: =

co,

01

(3)

208 0 1 Po COD

--

di =

where k = k s / m , Co = wave) and y such that

0 1 0

d

m (sound velocity), COD= --1 Y(C)

--

-1

71

(adiabatic sound

+-712--c1 '

C

-1

Evaluating the matrix Bo = Vfl,: 0 0 0 0 0 r; r: 0 0 r$

where the index denotes differentiation to the arguments and the 0 denotes the equilibrium state. We obtain soon:

Bod: = Bod: 0 0 0.

0 0 0

We can deduce immediately that in the case of chemical reactions the system satisfies the K-condition. While in the case without chemical reaction the system is of

209 composite type because the eigenvectors corresponding to the sound velocities and the contact wave d:, dg, di belong t o the null space of Bo. Taking into account that for the entropy principle requirement we have that r can be proportional to the difference of the chemical potential of the two constituents (3) :

the differential system is strictly dissipative in the sense of Boillat and Ruggeri thermodynamical definition 16. Therefore for the previous general theorems, we conclude that mixtures of Euler fluids with chemical reaction have global smooth solutions that converge to an equilibrium state of the Euler single fluid provided the initial data are sufficiently smooth. While, without chemical reaction the global existence remains an open problem. 3

Acceleration Waves

Assuming m l M m2 and neglecting chemical reaction, the amplitude of the highest speed acceleration waves, which propagate along characteristic $(z, t ) = z - Cot = 0 , Co = satisfies the transport Bernoulli equation. As shown by Boillat17 (see also Ruggeri") it governs the behavior of weak discontinuities for all hyperbolic systems of balance laws. It can be showed that critical times for the formation of shocks in horizontal and vertical direction, respectively 5 :

d a ,

have the same form as in the single fluid case la. Go = [vt](0)= [v,++]$,(O> and g denotes the initial acceleration jump and the acceleration of gravity respectively. 4

Shock Waves

Analysis of shock waves in binary mixture of Euler fluids will be based upon the solution of Rankine-Hugoniot equations which govern the jump of field variables across the wave front:

[pu2

+

+

pc(1J 2- c )

pm2-2Ju+-+v

JPC2

I

=0 ;

1

=O; 52

pc(1 - c)

'I

U--J

a

=O,

210 where u = s - v and s is the speed of shock. Our attention will be restricted to k-shocks - weak shock waves which bifurcate from trivial solution of (6) where the speed of shock corresponds to the characteristic speed. Along with the search for nontrivial solutions of Rankine-Hugoniot equations, a question of shock admissibility will be raised. Apart from the classical case of genuine nonlinearity (VX . d # 0 for all u), where Lax condition XO < s < X and entropy growth criterion 71 > 0 could be well applied, and the linearly degenerate case (VX . d = 0 for all u) where s = XO = X and 77 = 0, we shall also encounter the case of local exceptionality: VX . d = 0 for some u. In this case the condition of genuine nonlinearity is violated on the critical manifold. Consequently, Lax cons(u0, dition has to be substituted by more general Liu condition '': s ( ~ , for all xo j i 5 x where x denotes the strength of the shock. In this last case the entropy growth criterion is not sufficient for the admissibility and it is necessary to add a new principle of superposition of shocks 'O. Taking into account that 77 = -s[ho] [h]where -ho and -h denotes the entropy density and the entropy flux, we have: 77 = [pus] - [Q] with

x) <

<

x)

+

k

S = ___ In ( L ) - k {cln c + (1 - c) ln(1 - c ) } ; Y(C) - 1 p7W

1

72 -

4.1

1'

(7)

We can distinguish the following three cases.

Sonic Shock

This case is characterized by the absence of jumps of concentration variable c = Q and diffusion flux J = JO = 0 and the following nontrivial solution of RankineHugoniot equations:

where Mo = uo/C0 is sonic Mach number and p 2 ( c ) =

y(c)--l,po = ~ ( c o ) .It Y(C) + 1

bifurcates from the characteristic speed Xf) - speed of sound, and obeys the same properties of shock admissibility as in the single fluid case: it is genuinely nonlinear and admissible for MO > 1.

21 1

1.01

5.0

1.00

0.29 0.30

0.31

0.32

0.33

c

0.70

c

x10: 1.015 1.010 1.005 ..................................

I

0.67

0.68

0.69

0.70

0.67 0.68

Figure 1. Lax condition and entropy growth in diffusive shock for and y1 = 1.35, yz = 1.40

4.2

~0 =

0.69

0.3, ~0 = 0.5,

Q

= 0.7

Diffusive Shock

In this case nontrivial solution of the system (6) can be expressed in terms of concentration c as shock parameter x. It is governed by the solution of biquadratic equation:

ao(c)w;

+ al(c)w; + a2(c) = 0 ,

212

where wo = uo/CoD,is diffusive Mach number, and reads:

J = Po COD wo(c) ( c -

%I,

where the explicit form of coefficients ao(c), a l ( c ) , a z ( c ) , a(.) and b(c) will be omitted (see 5 ) . It can be shown that solution (9) bifurcates from the second sound eigenspeed A.): This case is particularly interesting since it obeys the property of local exceptionality. Namely, by a straightforward computation one can prove:

so that the equilibrium critical manifold is the hyperplane c = 112, w = 0, J = 0. This result strongly influences admissibility of diffusive shock in such a way that there exists c such that (see Figure 1): 0

if co

< 1/2 shocks are admissible for co < c < c;

0

if

> 1/2 shocks are admissible €or F < c < CO;

0

if co = 112 there are no admissible shocks.

Numerical computation of Liu condition and entropy growth across the shock confirms this assertion and the Liu admissible conditions coincide in this case with the (generalized) Lax region (see Figure 1). These results are similar with the discussion of second sound phenomena in rigid heat conductor given by Ruggeri, Muracchini and SecciaZ1.

4.3 Characteristic Shock The final case is obtained for s = = 0 = 0, and corresponds t o the characteristic shock in the single fluid case. From the set of the Rankine-Hugoniot equations (6) it is easy to obtain the following result nontrivial solution: w = wo = 0;

c = CO; p

= po;

J

= JO = 0;

p

-

arbitrary.

(13)

213

Acknowledgments This paper was supported by fondi MIUR Progetto di interesse Nazionale Problemi Matematici Non Lineari di Propagazione e Stabilith nei Modelli del Continuo Coordinatore T. Ruggeri, by the GNFM-INDAM, and by Istituto Nazionale di Fisica Nucleare (INFN).

References 1. C. Truesdell, Sulle basi della termomeccanica, Rend. Accad. Naz. Lincei 8 , 158 (1957). 2. I. Muller, A new approach to thermodynamics of simple mixtures, Zeitschrij? fiir Naturforschung 28a,1801 (1973). 3. I. Muller and T. Ruggeri, Rational Extended Thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy 37,Springer-Verlag, New York (1998). 4. T. Ruggeri, The binary mixtures of Euler fluids: A unified theory of second sound phenomena in Continuum Mechanics and Applications in Geophysics and the Environment, Eds. B. Straughan, R. Greve, H. Ehrentraut and Y . Wang, p. 79, Springer-Verlag, Berlin (2001). 5. T. Ruggeri, S. SimiC, Non linear Wave Propagation in Binary Mixtures of Euler Fluids. Contin. Mech. Thermodyn. 15,No.7, (2003). 6. G. Boillat, Sur 1’Existence et la Recherche d’Equations de Conservation SupplCmentaires pour les Systhmes Hyperboliques. C.R. Acad. Sc. Paris, 278 A 909 (1974). Non Linear Fields and Waves. In CIME Course, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics 1640, 103-152 T. Ruggeri Ed. Springer-Verlag (1995). 7. T. Ruggeri, A. Strumia, Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincare‘, 34 A 65 (1981). 8. K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA, 68 1686 (1971). 9. S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. SOC. Edimburgh, 106A 169 (1987). 10. A. E. Fischer, J.E. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. Commun. Math. Phys. 28, 1 (1972). 11. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer Verlag, NewYork, 1984. 12. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer Verlag, Berlin (2000). 13. Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 no. 3, 225 (1999). 14. B. Hanouzet, R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal. 169 89 (2003).

214

15. T. Ruggeri, D. Serre, Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quarterly of Applied Math. 52, ( l ) , 163 (2004). 16. G. Boillat & T. Ruggeri, Hyperbolic Principal Subsystems: Entropy Convexity and Sub characteristic Conditions. Arch. Rat. Mech. Anal. 137 305 (1997). 17. G. Boillat, La Propagation des Ondes, Gauthier-Villars, Paris (1965). 18. T. Ruggeri Stability and Discontinuity Waves for Symmetric Hyperbolic Systems in Nonlinear Wave Motion, Ed. A. Jeffrey, pp. 148-161,Longman (1989). 19. T.-P. Liu, The Riemann Problem for General System of Conservation Laws, J. Differential Equations, 18, 218-234 (1975). 20. T.-P. Liu, T. Ruggeri, Entropy Production and Admissibility of Shocks, Acta Math. Appl. Sin., Engl. Ser. 19,No.1, 1-12 (2003). 21. T. Ruggeri, A. Muracchini, L. Seccia, A Continuum Approach to Phonon Gas and Shape Changes of Second Sound via Shock Waves Theory Nuovo Cimento D 16,15 (1994).

EXCHANGE OF STABILITIES IN POROUS MEDIA AND PENETRATIVE CONVECTION EFFECTS B. STRAUGHAN Department of Mathematical Sciences, University of Durham, DH1 3LE, U.K. E-mail: brian. straughan Odurham. ac. uk A review is given of recent results in penetrative convection in porous media. Attention is focussed on interesting questions regarding when the growth rate is real in the linear theory, or when a weaker principle of exchange of stabilities holds.

1 Introduction

The perturbation equations for convection in a porous medium with variable gravity a,nd heat source effects may be written

ui = -

+ RH(z)Bki,

~ , i

ui,i = 0,

(1)

(2)

B ,+ ~ uie,i = R N ( ~+ )AO. ~

(3) These equations hold on the region R2x ( 0 , l ) x { t > 0}, see e.g. S t r a ~ g h a np.~ ~79. , The functions ui c (u, v ,w), 0 are velocity and temperature perturbations, R2 is the Rayleigh number, k = (O,O, 1) and standard notation is adopted. The function H ( z ) arises due to a gravity which depends directly on the depth coordinate z , or possibly on a nonlinear buoyancy law p = p ( T ) . In the latter case equation (1) contains additional nonlinear terms in 8. The function N ( z ) arises due to a heat source which may depend on z . This induces a steady temperature profile which is nonlinear in z and hence yields the z-dependence in N . The linear instability equations which arise from (1) - (3) due to a time dependence like eUt are ~i

=-

+ RH(z)Bki,

~ , i

uij = 0,

(Te= R

N ( ~+)AO. ~

(4) (5) (6)

It is a very interesting question to ask whether (T may be complex and so give rise to oscillatory convection. S t r a ~ g h a np.~ 85 ~ notes that if H or N is constant, or if H 0: N , then exchange of stabilities holds in the (strong) sense that (T E R. Herr~n~ gives ' ~ ~arguments ~ to support the assertion that (T E R if H N 2 0,Vz E [0,1]. In this work we report on an instance where (T is found to be complex. The instance where (T E C arises in penetrative convection. The term penetrative convection is used to describe a situation where a part of the saturated porous layer R x (0, l) tends to be unstable and the instability penetrates into the rest of the layer (the stable part), cf. Veronid3'. Penetrative convection has been studied rcently by many writers, see e.g. C a r r 2 ~ 3 ~ 4 ~Chasnov 5 ~ 6 , & Tse7, 215

216

Hil116J5, Krishnamurti19, Larson20v21,Mahidjiba22, Normand & AzounP4, Payne & S t r a ~ g h a n S~ t~r, a ~ g h a n ~Straughan ~ , ~ ~ , & Walker37, Tse & C h a s n ~ v Zhang ~~, & S c h ~ b e r t ~In~ particular, ,~~. the topic of penetrative convection in a fluid or in a fluid saturated porous medium is covered in some depth in chapter 17 of the book by S t r a ~ g h a n ~ ~ . 2

Energy stability: theory and development

In this work we report instances of penetrative convection where o is found to be real and others where o E C. The articles concerned use linear instability and nonlinear energy stability techniques. It is pertinent at this point t o recall fundamental contributions to the theory of energy stability by Professor Salvatore Rionero. Rionero has made many substantial contributions in the field of nonlinear energy stability theory. We highlight three areas. The first is his fundamental work Rionero26127y28,29, in which he demonstrates the existence of a maximizing solution to a ratio, I I D . The subject hinges on the maximizing solution and his work is a cornerstone in the field. He later, Rionero3', Rionero & Galdi31~32,led the way with a weighted energy technique to yield uniqueness and continuous dependence results for the Navier-Stokes equations on an unbounded spatial region. This method has subsequently been exploited by many writers, and extended to elasticity, thermoelasticity, viscoelasticity, and several other areas of continuum mechanics, see e.g. the books by Flavin & Rionero8, Galdi & Rionero14, Ames & Straughan', and the references therein. Finally, we mention work of Flavin and R i o n e r ~ ~ , Mulone & R i ~ n e r oand ~ ~ ,Rionero & M a i e l l a r ~in~ which ~ they show how one may construct a natural energy functional to derive unconditional nonlinear energy stability results in magnetohydrodynamics, or in such problems as when the diffusion coefficient depends nonlinearly on a variable such as temperature or concentration.

3

Energy stability and porous convection

Hil116915has studied, in the porous medium context, an interesting model of penetrative convection first introduced by Kr i~hnamur ti lin~ a fluid. A nonlinear analysis of Krishnamurti's model is given by S t r a ~ g h a n ~Hill ~ , develops ~~. his analysis in the context of porous media by using Darcy's law, Brinkman's law, linear and quadratic heat source constitutive equations, and a buoyancy law in which density p definitely depends on temperature, T , and may depend on concentration, C. Very intersting results are derived in each case. He employs linear instability theory and nonlinear energy stability theory, his nonlinear energy stability thresholds being very close to the linear instability ones. Curiously, he discovers that the growth rate o of linear instability theory is real, except in a small parameter range when p = p(T,C). The results derived by Hill certainly merit attention. Carr & Straughan6 study penetrative convection in a viscous fluid overlying a porous medium. the coupled system of equations employs Darcy's law in the lower porous layer and the Navier-Stokes equations in the upper fluid layer and is somewhat complex. They derive a careful (numerical) linear instability analysis by adapting a Chebyshev tau D2 method. A surprisingly complex array of streamline

217

patterns is found. Again, the growth rate u is curiously found to be real. Straughan3' studies penetrative convection in a porous medium using a combination of a heat source and a nonlinear density model. The model analysed is analogous t o the equivalent model in a viscous Auid developed by Normand & AzouniZ4. Straughan3' finds for the porous model that there is a parameter range where u E C , at least for certain wavenumbers. Unlike the fluid case, however, S t r a ~ g h a nfinds ~ ~ that even though u may be complex, instability will commence as stationary convection. In addition to a linear instability analysis, Straughan3' develops an unconditional nonlinear energy stability analysis. The nonlinear critical Rayleigh number thresholds are sufficiently close to the linear instability ones as t o be practically useful. A surprising Rayleigh number behaviour is found in the linear and nonlinear situations. Acknowledgments This work was supported by the Leverhulme Research Grant number R F & G /9/2000/226. References 1. K.A. Ames and B. Straughan, Non-standard and Improperly Posed Problems, (Academic Press, San Diego, 1997). 2. M. Carr, Convection in porous media flows, (Ph.D. Thesis, University of Durham, 2003). 3. M. Carr, Math. Models Meth. Appl. Sci. 13, 207 (2003). 4. M. Carr, Continuum Mech. Thermodyn. 15, 45 (2003). 5. M. Carr and S. de Putter, Continuum Mech. Thermodyn. 15, 33 (2003). 6. M. Carr and B. Straughan, Advances in Water Resources 26, 263 (2003). 7. J.R. Chasnov and K.L. Tse, Fluid Dyn. Res. 28, 397 (2001). 8. J.N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations, (CRC Press, Boca Raton, 1995). 9. J.N. Flavin and S. Rionero, Rend. Matem. Acad. Lincei 99, 299 (1997). 10. J.N. Flavin and S. Rionero, J. Math. AnaLAppl. 228, 119 (1998). 11. J.N. Flavin and S. Rionero, Q. J. Mech. Appl. Math. 52, 441 (1999). 12. J.N. Flavin and S. Rionero, Continuum Mech. Thermodyn. 11, 173 (1999). 13. J.N. Flavin and S. Rionero, J . Math. AnaLAppl. 281, 221 (2003). 14. G.P.Galdi and S. Rionero, Weighted energy methods in fluid dynamics and elasticity, (Springer, Heidelberg, 1985). 15. A.A. Hill, Penetrative convection in porous media flows,(Ph.D. Thesis, University of Durham, t o appear). 16. A.A. Hill, Continuum Mech. Thermodyn. 15, 275 (2003). 17. I. Herron, SIAM J. Appl. Math. 61, 1362 (2000). 18. I. Herron, Int. J. Engng. Sci. 39, 201 (2001). 19. R. Krishnamurti, Dynamics of Atmospheres and Oceans 27, 367 (1997). 20. V.E. Larson, Q.J. Royal Meteorological SOC.126, 145 (2000). 21. V.E. Larson, Dynamics of Atmospheres and Oceans 34, 45 (2001).

218

22. A. Mahidjiba, L. Robillard and P. Vasseur, Int. J. Heat Mass Transfer 46, 323 (2003). 23. G. Mulone and S. Rionero, Arch. Rational Mech. Anal. 166, 197 (2003). 24. C. Normand and A. Azouni, Phys. Fluids A 4, 243 (1992). 25. L.E. Payne and B. Straughan, Stud. Appl. Math. 105, 59 (2000). 26. S. Rionero, Ann. Matem. Pura Appl. 76, 75 (1967). 27. S. Rionero, Ricerche Matem. 16, 250 (1967). 28. S. Rionero, Ann. Matem. Pura Appl. 78, 339 (1968). 29. S. Rionero, Ricerche Matem. 17, 64 (1968). 30. S. Rionero, O n the use of weighted n o r m s in stabilty questions o n exterior domains, (Conf. Sem. Matem. Univ. Bari, volume 157, 1979). 31. S. Rionero and G.P. Galdi, Arch. Rational Mech. Anal. 62, 295 (1976). 32. S. Rionero and G.P. Galdi, Arch. Rational Mech. Anal. 69, 37 (1979). 33. S. Rionero and M. Maiellaro, Rend. Acc. Sc. Fis. Mat. Napoli 62, 315 (1995). 34. B. Straughan, Dynamics of Atmospheres and Oceans 35, 351 (2002). 35. B. Straughan, Resonant porous penetrative convection, Manuscript (2003). 36. B. Straughan, The Energy Method, Stability, and Nonlinear Convection, 2nd edition, (Springer, New York, 2003). 37. B. Straughan and D.W. Walker, Proc. Roy. SOC.London A 452, 97 (1996). 38. K.L.Tse and J.R. Chasnov, J. Computational Physics 142, 489 (1998). 39. G. Veronis, Astrophys. J. 137, 641 (1963). 40. K.K. Zhang and G. Schubert, Science 290, 1944 (2000). 41. K.K. Zhang and G. Schubert, Astrophys. J. 572, 461 (2002).

EFFECTS OF ADAPTATION ON COMPETITION AMONG SPECIES D. LACITIGNOLA AND C. TEBALDI Department of Mathematics, University of Lecce Via Provinciale Lecce-Arnesano, 1-731 00 Lecce, Italy. E-mail: [email protected] - claudio. [email protected] We study the properties of a n2-dimensional Loth-Volterra system describing competition among n species, n 2 3, with adaptive slcills and with one species made ecologically differentiated with respect to the others by carrying capacity and/or intrinsic growth rate. In both cases the symmetry properties of such a system and the existence of a certain class of invariant subspaces allow the introduction of a 7-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the complete system. The reduced model is effective also in describing the time dependent regimes for a large range of parameter values. Ecological relevant questions, as species survival/exclusion, have been analysed focussing on the role of adaptation. Moreover interesting forms of species coexistence have been found (i.e. competitive stable equilibria, periodic oscillations, strange attractors)

1

Introduction and the Model

Nonlinear dynamical system theory provides relevant theoretical tools for the study of population dynamics 12. In this field, Lotka-Volterra systems have still a central role because of their historical importance as well as the many generalizations they have given rise to. Among such generalizations, Lotka-Volterra systems with adaptation have gained a special place because of the importance of adaptation inside many biological contexts. In particular, in the eighties’, attention has been devoted to a two-species adaptive Lotka-Volterra system, i.e. in which the interaction coefficients between two populations depend on the past hystory of these two species l3 14. More recently ’, these kind of systems have been considered when more than two species are involved in competition. This is certainly a relevant step towards a better understanding of some population phenomena through the analysis of LotkaVolterra systems. Recalling also the role of competition as one of the principal factors in the building of animal or plant communities, we focus our attention on adaptive competition phenomena starting from the following general system:

1I i,j I n, j

#i

Eq. (1.a) is just the n-species classical Loth-Volterra model depicting competition 219

220 among n different species. More precisely, the positive function N i ( t ) denotes the density of the i-species at time t; the positive constant ri can be interpreted as the intrinsic growth rate coefficient for the i-species whereas the positive constant ICi is its carrying capacity. The positive continuous function aij represents the interaction coefficient between the j and i species. In agreement with Noonburg 13, we assume

s r

K T ( u ) ~= u 1. where KT is an integral delay kernel satysfying The competition rate between the j and i species is therefore proportional to the average number of encounters of the two populations over the past and depends on e-t/T

K T . The weak exponential delay kernel KT = -is chosen because it provides T a reasonable effect of memory. Considering (2) in its differential form, Eq. (1.b) is obtained. As a consequence, the positive parameter Ti controls the adaptation rate of the i species, i.e. how species learn to compete by encounters: the smaller Ti, the faster the adaptation rate; Ti + 00 implies that adaptation is completely lost. The n-species system (1) has been extensively investigated under the assumptions ri=1

Ici=IC

Ti = T

'di = 1,.. . ,n

which are biologically oversimplified because competition is considered among identical species. In this symmetric case, i.e. when all the n species are characterized by the same ecological parameters, the existence of a family of invariant subspaces has been shown and a 4-dimensional model introduced2, with n as a parameter . Such a reduced model gives full account of existence and stability of the equilibria in the complete system '. Correspondence between the reduced model and the complete one has been found for a large range of parameter values also in the time dependent regimes, even in the presence of strange attractors. Moreover it is also shown that such a correspondence is lost because of a crisis4. Such a study, even if mathematically relevant and with striking reduction results, is however rather special. That is why, in the next sections, we want to mathematically analyse and biologically discuss two new cases in the direction of more realism from a biological point of view. The first one concerns competition phenomena in which one species, say the first, is made ecologically disadvantaged with respect to the others: such a feature is mathematically obtained by perturbing the species's carrying capacity. The second case concerns a further ecological differentiation among the species involved which are now made different on the ground of both the two characterizing parameters: carrying capacity and intrinsic growth rate. In particular, we analyse the effect of differentiation obtaining the coexistence/exclusion conditions and evaluating the role of adaptation in such more realistic biological situations.

221 This extension is also of strong mathematical interest because the related perturbations break a symmetry responsible for the existence of the low dimensional reduced model that, in the symmetric case ', captures most of the features of the complete system. Such a reduced model has turned out t o be very useful in studying competition phenomena with a large number of populations involved and is therefore important to investigate its persistence under more realistic ecological conditions.

2

Differentiation among species by carrying capacity

We consider model (1) with the biologically simplifying assumptions

ri=1

ki=k

Ti

=T

Qi

= 1,. . . , n

and decide to distinguish the first species with respect t o the others acting on its carrying capacity. Namely, by using the following scale changes

c = -1 k

A,.-% 2J - k

X i = Ni

V i , j = 1,. . . ,n

that formally simplify ( l ) ,we consider the following the system:

T

where the positive parameter E measures the size of a perturbation in the carrying capacity of the first species, which is therefore made ecologically disadvantaged with respect to the remaining ones. As in the completely symmetric case, also for system (3) we recognize the subspace Aij = A j i , 1 i,j n i # j , as invariant, attractive and thus able to describe the time asymptotic behavior of the system . Moreover, a low-dimensional reduced model also exists which captures most of the features of (3) (equilibrium points, their stability and bifurcations). More precisely, taking into account the new symmetries of the system, the existence of a family of (n - 1) invariant subspaces has been shown and the following 7dimensional reduced model introduced:

<

<

222

with n as a parameter. The correspondence between this reduced model (4) and the complete one (3) is also found in time dependent regimes. As in the symmetric case, any equilibrium of (3) with one or more components Xi = 0 is unstable for all the parameters values showing that also in this case, strictly speaking, competitive exclusion cannot occur. However, we stress that certain interesting behaviors in time dependent regimes have been found to be related to the presence of such unstable fixed points. By using the reduced model (4) and the symmetry properties, we can have a complete insight of the number and features of the fixed points of (3). An equilibrium R exists for which the first species has a density somewhat lower than the others, all at the same density level. Two more equilibria S and S’ exist, characterized by the dominance of the first species on the others, dominance that is much stronger in the case of S. In the same way, the ( n - 1) equilibria Bi’s and the ( n - 1) B:’s (2 5 i 5 n) are characterized by the dominance of the i-species, more enhanced for the former ones. Numerical investigations have shown that the number of interior fixed points depends on the value of c considered as well as on the size E of the perturbation. In particular, for fixed n, we have found a critical value of the parameter E , E = separating two different scenarios. Concerning stability, the parameter T is responsible for the equilibria destabilization, which always occurs because of Hopf bifurcation. However, the equilibria S* and Bf are unstable for every value of T. We stress the features of two different types of coexistence: one is represented by the equilibrium R for which all the species, except XI,have exactly the same density and the other is given by the S and Bi’s for which the first species or the i-species is strongly dominant on the others, depending on initial conditions. Equilibria for

223 which the dominance of one species is not so strong are always unstable, S' and

B5 . In the time dependent regimes, we have found that coexistence appears also in the form of periodic oscillations. More precisely, we have analysed time dependent behavior for the case of adaptive competition among 4 species in which the first species is disadvantaged because of its lower carrying capacity. In particular, when this species is not too disadvantaged, E < &,,it, fast adaptation for all the species leads to coexistence in the form of equilibria, which one occurring determined by the ecological conditions. Slower adaptation leads to periodic oscillations that can either exhibit the strongly dominance of one species, Fig.1, or an interesting mechanism of species alternance in dominance, Fig.2.

Figure 1. Coexistence with the dominance of the first species.

Even slower adaptation gives rise to coexistence in the form of complicated pattern, Fig.3 When a given species is much more disadvantaged than the others, E > €,,it, it can never be dominant. In favorable ecological conditions, fast adaptation leads to coexistence in the form of equilibria, whereas slower adaptation leads to periodic oscillations with strong dominance of one species. On the contrary, unfavorable ecological conditions always lead to the stable competitive equilibrium R, independently of the adaptation rate, see D.Lacitignola - C.Tebaldi for details.

224

l

,

,

Figure 2. Coexistence with species alternance.

time

I

2.51

lime

Figure 3. Coexistence with species complicated behavior.

3 Differentiation among species by carrying capacity and intrinsic growth rate With the aim to gain more realism with respect to the previous cases, we have further differentiated the first species on the ground of the two characterizing pa-

225

rameters: the carrying capacity and the intrinsic growth rate. We then consider model (1) with the following assumptions:

ri=r ki=k

vz=2, k=2,

...,n ...,n

A A

ri=rl ki=kl

z=l

i=l

The scale changes

lead to the model

lIi,j
which has the first species differentiated with respect to the remaining ( n- 1) ones. In particular the first species can result ecologically advantaged or disadvantaged with respect to the others (i.e. c1 respectively lower or higher than c) and slowly or fastly reproducing with respect to the others (i.e. r1 respectively higher or lower than r ) ; moreover the remaining n - 1 species can themselves be in good or bad ecological conditions according to the values of r and c respectively. We turn now to search for the existence and stability of the equilibria of system (5). At this aim, also in this case, we show the existence of a lower-dimensional reduced model that captures the many features of system (5) (equilibrium points, stability properties, bifurcation phenomena involved) and then we use it to investigate the structure and properties of the equilibria. The formal derivation of this model is made taking into account the symmetry properties of the system and investigating its invariant subspaces. One easily recognizes that the subspace

A 23, . - A3.% .

l
i#j

invariant and attractive, can be used to describe the time asymptotic behavior of the complete system Looking for different invariant subspaces, we choose arbitrarily h and k , h # k, 1 < h , k < n, h , k , i # 1 . By linear combination between the evolution equations of the variables X h , Xk and considering

226

one proves that

is an invariant subspace for system (5). The index i is fixed and can assume any value between 2 and n, so that the existence of a family of ( n- 1) invariant subspaces is proven for system (5). In subspace (6), system (5) becomes

which is a system of seven ordinary differential equations where n is a parameter. We will refer to (7) as to the 7-dimensional reduced model.

As shown in D. Lacitignola - C. Tebaldi ', we are allowed t o use such reduced model for a complete investigation on the existence and stability of the equilibria in system ( 5 ) . At first, we stress that critical points of ( 5 ) with one or more components X i = 0 are here unstable for all the values of the parameters. We search then for the interior fixed points of the system. Recalling that, at the equilibrium, Ali = X i X i Alh = X l X h Aih = xixh Ahj =

xi

we observe that, on subspace (6), the interior fixed points are the real and strictly positive solutions of the following set of equations:

Numerical investigations allow to state that the number of interior critical points for model (7) and thus for system ( 5 ) , depends critically on the intrinsic growth rates and on the carrying capacities of the species. In the following, we assume T = 1 and observe that the condition T I = 1 means thus equal reproduction rates for all the species whereas ~1 < 1 or T I > 1 indicate that the first species reproduces respectively more slowly or faster than the remaining ones. According to the chosen parameter values, in the reduced model we have at most five interior fixed points, namely

By using the system's symmetry properties we can assert that any solution of the reduced model (7) corresponds to ( n - 1) such solutions in the complete system (5), except for the ones that are invariant with respect the symmetry rules of the system, more precisely S, S* and R . In the complete system, we have the three equilibria R, S and S*

228

the (n - 1) equilibria Bi

(X!’, bi,X?,. . . ,X ? ) , ( X p , X ? , b l , ... , X ? ) , . . . , (X,bl,X?, . . . ,X?,bl) and the ( n - 1) equilibria Ba

( X P ,b2,X?,. ., , X?), (X?,X?,b2, where 2

,X?), ,. . , (X,bZ,XF,,. . ,x$,b 2 )

< i < n.

The number and the features of such interior fixed points depend sensitively on the first species level of differentiation as well as on the ecological conditions of the remaining species. As in the previous section, we consider the case of adaptive competition among four species. We consider the system parameters varying so that

The first three ranges settle the degree of advantage or disadvantage of the first species on the others and the general ecological conditions whereas the last one qualifies the species’ level of adaptation. According to the specific combination between the carrying capacity of the first species and the carrying capacity of the remaining ones, we have found three relevant scenarios. When the first species has an environmental advantage on the others given by its carrying capacity, we can distinguish between the case c1 << c and c1 < c. In the former case, which represents a strong ecological advantage of the first species on the others, the two-parameter bifurcation diagram in Fig.4 well summarizes the full picture of equilibria in the system. We have divided Fig.4 in four regions, according to the first species reproduction rate and the carrying capacity of the remaining species. For each region, we indicate the different fixed points of the system which we discuss also for their biological implications: the mathematical stmcture of these fixed points is the same previously introduced whereas their biological characterization can depend on the different regions. To better understand the diagram we recall that at lines separating different regions, tangent bifurcations take place. We notice that the fixed points S* and B; are always unstable and therefore not biologically meaningful. We discuss therefore only the others equilibria. The equilibrium S is of exclusion type (i.e. the first species will exclude the others) whereas the equilibria R and Bi’s are, strictly speaking, equilibria of coexistence. However, the Bi’s are characterized by the strong dominance of the i-species on the others. In Regionl, according to the initial values of species’s density, all the equilibria are possible results of competition: this is due to the balance among the reproduction disadvantage of the first species, its ecological advantage and the good ecological conditions of the remaining species.

229 Cl <
ClS).Ol

-'-I

D

S

Regiolu

0.2

0.3

0.4 B

0.5

0.6

0.7

0.8

C

Figure 4. Two parameters bifurcation diagram for the case c1

<< c, i.e.

c1 = 0.01

In Region2, the ecological advantage of the first species by both its carrying capacity and intrinsic growth rate together with the good ecological conditions of the remaining species, allow either S or the Bi's to be possible result of competition: according to the initial conditions, competition can lead to exclusion of all the species different from the first one or to a type of coexistence with strong dominance of the i-species on the others. In Region 4,the balance between the reproduction disadvantage of the first species and its ecological advantage, together with the unfavourable ecological conditions of the remaining species, prevents the occurence of coexistence with strong dominance of such species: according with the initial density of the different species, they can coexist all at the same density level except for the one of the first species which is slightly lower than the others, R, or the first species will exclude the remaining ones, S. In Region 3, the strong ecological advantage of the first species due to its carrying capacity and intrinsic growth rate together with the unfavourable ecological conditions of the remaining species, makes the equilibrium S the only possible output of competition. We want to observe that, differently from the cases in the previous section, here exclusion become again a possible output of competition and namely it is the only possibility when the first species is in position of strong advantage on the others (cfr. Region3). It is also remarkable that in such a situation, adaptation is not able to preserve the survival of the more disadvantaged species.

230 Decreasing the level of ecological advantage of the first species, one finds the scenario represented in Fig.5. One consequence, is the appearence of two new regions for very low values of r1 which are represented in Fig.6. A discussion and interpretation of the different outputs in each region can be performed following the same arguments of the previous case. cl < c

Cl$).l

Figure 5. Two parameters bifurcation diagram for the case c1

< c, i.e. c1 = 0.1

In Region5, the first species is strongly disadvantaged with respect to the others because of the very low values of T I ; such a disadvantage it is not balanced by a particular ecological advantage (because of the values of c and c1). According to the initial conditions, the possible outputs of competition can thus be R, coexistence with all the species at the same density and the first one at a lower density level, or the Bi’s which are here exclusion equilibria: the first species will be extinct. In Region6, the only possible output is the equilibrium R which is a coexistence equilibrium characterized by a remarkable disadvantage of the first species whereas all the others are at the same density level. Depending on the initial species’ density, as for Regionl of Fig.4, also in Regionl of Fig.5, all the equilibria are possible results of competition: if the system approaches the equilibrium S, then the first species will exclude the others; if the system approaches the equilibrium R or one of the Bi’s then there will be coexistence: in particular, in the latter case, coexistence occurs with a strong dominance of the i-species on the others. In Region2 of Fig.5, where the first species is very advantaged because of the high

231

Figure 6. Zoom of the two parameters bifurcation diagram for the case

c1

< c, i.e.

c1 = 0.1

values of r1 and the low value of q , competition can result in the exclusion equilibrium S. Nevertheless, the good ecological conditions for the remaining species, allow also coexistence with a strong dominance of the i-species on the others, Bi’s. It is interesting t o note that, moving towards Region3, the exclusion equilibrium S changes progressively into a coexistence equilibrium with strong dominance of the first species whereas in the coexistence equilibria Bi’s the dominance of the i-species on the remaining ones become gradually less pronounced. In Region3, the advantage by carrying capacity and intrinsic growth rate of the first species and the unfavourable ecological conditions of the remaining ones, justify S as the unique possible output. Nevertheless, S can be defined of mixed type being a coexistence equilibrium with strong dominance of the i-species for those parameter values near the borders with Region2 or Region4 and an exclusion equilibrium elsewhere in the region. In Region4, the balance among the ecological advantage of the first species, its reproduction disadvantage and the sufficiently good ecological conditions of the others species allow coexistence with a pronounced dominance of the first species, equilibrium S, or coexistence with all the species at the same density and the first one at a lower density level, equilibrium R. We conclude by observing that in this case situations of proper exclusion are less frequent than in the previous case and occur only for selected combination of the parameter values. When the ecological advantage of the first species is not too pronounced, c1 < c, adaptation seems to be more effective in avoiding the extinction of the disadvantaged species. Fig.7 shows what is obtained by furtherly increasing the value of c1 until the first species is ecologically disadvantaged with respect to the remaining ones but has variable reproduction rate. The related scenario is relatively poor: we have only two

232

I

c

Figure 7. Two parameters bifurcation diagram for the case c1

> c, i.e.

c1

= 1.2

regions and the possibility of the first species to exclude the others or to dominate the remaining species is here completely absent. In fact the equilibrium S is never an output of competition. In Region1 we have, as possible output, the equilibrium R representing coexistence with an advantaged or disadvantaged first species according to the values of r1. Moreover, according to the initial conditions, also the equilibria Bi’s can be the results of competition and represent coexistence with dominance of i-species on the others. In Region2, the equilibrium R is the only output of competition and represents formally a coexistence equilibrium with all the species at the same density and the first species at lower or higher density level according to the values of T I . We conclude this description by noticing that this case, phenomenologically less interesting than the previous ones, is also the one in which adaptation seems to be more effective because here the exclusion of the disadvantaged species is never occuring. We also recall that for T I = T = 1 the results of the previous section in the case E > Ec,.it are completely confirmed. 4

Conclusions

Studying a competitive adaptive n-species Loth-Volterra system in which one species is differentiated with respect to the others by carrying capacity and/or intrinsic growth rate, a 7-dimensional reduced model is introduced, where n appears as a parameter, which gives full account of existence and stability of equilibria for the complete system. Investigations in the case of four species also confirm the relevance of the reduced model on time dependent regimes. In the case of differentiation by carrying capacity, differently from the classical competitive Loth-Volterra systems, competitive exclusion does not happen even

233 when a fixed species is more disadvantaged than the others. In principle this could suggest that adaptation gives a certain advantage to the less favourite species, avoiding their extinction. In certain regions of the parameter space, coexistence also occurs in the form of periodic oscillations which appear in a variety of types. Increasing the level of realism for the model, namely differentiating one species from the others also by its intrinsic growth rate, we have found different scenarios according to the size of ecological advantage or disadvantage of the selected species. In some of these scenarios, we have recognized exclusion as one of the possible outputs of adaptive competition. In principle we can argue that adding realism to the model has put also the role of adaptation in a more realistic perspective: adaptation is not able to preserve species survival when one species is strongly advantaged than the others because its carrying capacity and intrinsic growth rate; adaptation is instead more effective in the other cases. Investigations of this case also in the time dependent regimes are still in progress in order to better clarify the role of adaptation and to study if, according to how fast or slow the species adapt, coexistence can appear in ways different from the equilibria, i.e. periodic oscillations, complicated patterns.

References 1. E. Barone and C. Tebaldi, Math. Meth. Appl. Sci. 23, 1179-1193 (2000). 2. C. Bortone and C. Tebaldi, Dyn. Cont. Impul. Sys. 4, 379-396 (1998). 3. J.M. Cushing, Siam J. Appl. Math. 3 2 , 82-95 (1977) 4. C. Grebogi and et al, Physica D 7,181-200 (1983). 5. J. Guckenheimer and P. Holmes in Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1997) 6. J. Kozlowski Oikos 86, 185-194 (1999) 7. Y.A. Kuznetsov in Elements of Applied Bifurcation Theory (Appl. Math. Sci. 112, Springer 1995) 8. D. Lacitignola and C. Tebaldi, Int. Jour. Bif. Chaosl3, 375-392 (2003) 9. R. Levins, A m . Nat. 114, 765-770 (1979) 10. M. Mange1 and C.W. Clark in Dynamic Modeling in Behavioral Ecology (Princeton University Press, 1992) 11. L.D. Mueller Ann. Rev. Ecol. Sys. 28, 269-288 (1997) 12. J.D. Murray in Mathematical Biology 1-11, (Springer, 2002) 13. V.W. Noomburg, J.Math.Bio1. 15, 239-247 (1982) 14. V.W Noomburg, J.Math.Biol24, 543-555 (1986) 15. H.L. Smith, A M S Math. Sum. and Monographs41, (Providence, 1995) 16. U. Sommer, Limonol. Oceanogr. 30, 335-342 (1993) 17. P. Waltman in CBMS-NSF Regional Conference Series in Applied Mathematics 4 5 , (SIAM, Phyladelphia, 1983)

WASSERSTEIN METRIC AND LARGE-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS J.A. CARRILLO Departament de Matemcitiques - ICREA, Universitat Autbnoma de Barcelona, E-08193 - Bellatema Spain E-mail: carrilloOmat.uab.es

G. TOSCANI Dipartimento d i Matematica, Universitd d i Pavia, 27100 Pawia Italy E-mail: toscaniOdimat.unipv.it We review here various recent applications of Wassertein-type metrics to both nonlinear partial differential equations and integrdifferential equations. Among others, we can describe the asymptotic behavior of nonlinear friction equations arising in the kinetic modelling of granular flows, and the growth of the support in nonlinear diffusion equations of porous medium type. Further examples include the approximation of nonlinear friction equations by adding viscosity, and the asymptotic behavior of degenerate convectiondiffusion equations.

1

Introduction

In recent years, due to its increasing importance both in the treatment of mass transportation problems and gradient flows 14716, Wasserstein metric became popular fifty years after its introduction into probability theory 1 3 . While applications in this area are widely known " , further possibilities of application are presently not well established. The connection between probability theory and nonlinear evolution equations comes through two main properties of solutions, namely positivity and mass preservation. Suppose we are considering the initial value problem for the evolution equation -a f ( 5 7 t )- Q ( f ) ( z , t ) ,

at

(5 E

R,t > 0),

where Q denotes here an operator acting on f which preserves positivity and mass,

s, f

(z, t>dx =

s,

fo(x) dx.

(2)

Then, given a initial datum which is a probability density (nonnegative and with unit mass), the solution remains a probability density at any subsequent time. Let F ( z ) denote the probability distribution induced by the density f (z),

Since F is not decreasing, we can define its pseudo inverse function by setting, for p E ( O , l ) , F-'(p) = inf{a: : F ( z ) > p } . Among the metrics which can be

235 defined on the space of probability measures, which metrize the weak convergence of measuresz5, one can consider the LP-distance of the pseudo inverse functions

As we shall see later on, dz(F, G) is nothing but the Wasserstein metric 'O. Metrics (4) can be fruitfully used to obtain results on uniqueness and large-time asymptotics of the solution every time equation (1) for f(z,t ) takes a simple form if written in terms of its pseudo inverse F - l ( z , t ) ,

a F - 1 ( p 7 t ) = Q*(F-')(p,t),

at

( p E ( 0 , 1 ) , t> 0).

(5)

This strategy has been recently applied to nonlinear friction equations arising in the modelling of granular gases '115, to nonlinear diffusion equations of porous medium type 6, and to degenerate convection-diffusion equations '. 2

Extremal distributions and Wasserstein-type metrics

Denote by Mo the space of all probability measures in R and by

M,

=

{F

E

M O:

1

( ~ \ ~ d F< ( z+)m , p 1 0 ,

(6)

the space of all Bore1 probability measures of finite momentum of order p , equipped with the topology of the weak convergence of the measures. On M , one can consider several types of metrics 25. Among them, an important class is given by the socalled minimal metrics. Let IT = II(F,G ) be the set of all joint probability distribution functions H on Wz having F and G as marginals, where F and G have finite positive variances. Within IT there are joint probability distribution functions H* and H , discovered by F'rkchet and Hoeffding l o which have maximum and minimum correlation. Let z+ = max(0, z} and z A y = min{z, y}. Then, in II(F,G) for all (2, y) E Wz,

H * ( z ,y) = F ( z ) A G(y) and H*(lc,y)

+ G(y) - 11'.

= [F(z)

The extremal distributions can also be characterized in another way, based on certain familiar properties of uniform distributions. If X is a real-valued random variable with distribution function F , and U is a random variable uniformly distributed on [0,1],it follows that F - l ( U ) has distribution function F , and, for any F, G with finite positive variances the pair [F-'(U),G-l(U)] has joint distribution function H* 24. Let

T,(F,G) =

inf

H E n ( F, C)

(7)

Then Tklpmetrizes the weak-* topology TW, on M,. For a detailed discussion, and application of these distances to statistics and information theory, see Vajda

'.

236 We remark that T:/' is known as the Kantorovich-Wasserstein distance of F and G l 3 y Z 0 . In this case

&(F, G)'

= T2(F,G) =

inf H€II(F,G)

s

(x- ~ ( ~ d H y)( x=,

I

10

- ~ ( ~ d H *y). ( x ,(8)

In fact, if the random vector ( X ,Y )has joint distribution function H with marginals F and G , and E ( . ) denotes mathematical expectation,

1 1%

+

- yI2 d H ( x ,y) = E [ ( X - Y)'] = E ( X 2 ) E ( Y 2 )- PE(XY).

(9)

+

Since X and Y have marginals F and G respectively, the quantity E ( X 2 ) E ( Y 2 ) remains constant for H E II(F,G). On the other hand, thanks to a result by Hoeffding lo

E ( X Y ) - E ( X ) E ( Y )= / [ H ( x , Y) - F(s)G(y)ldxdy I

Subtracting the constant quantity 2 E ( X ) E ( Y )on both sides of (9), we obtain (8). Recalling now that [ F-l (U),G-'(U)] has joint distribution function H* 24, we conclude that the Wasserstein distance between F and G can be rewritten as the L2-distance of the pseudo inverse functions

The previous result can be generalized to any convex cost instead of the quadratic cost as pointed out in 23. Therefore, dp(F,G)P = T,(F,G) for any 1 p < co. Moreover, Wassertein distances T;IP form an increasing sequence in p by Holder inequality and thus we can always define the Wasserstein distance for p = co as

<

d,(F,G)

= lim P+W

T;IP= lim dp(F,G)= [IF-' P+W

- G -lI I~ m(O, l).

(11)

Let us remark that this co-Wasserstein distance has not been used up to our knowledge to study supports of the probability densities, despite of the clear relation to them. Let us make several remarks to this respect: 0

0

Assume f(x) is compactly supported and g(z) verifies that d,(F,G) then g ( x ) is necessarily compactly supported.

<

00,

Assume f(z)verifies that its co-Wasserstein distance with respect to the Delta Dirac centered at 0 is finite, then f(z)is necessarily compactly supported.

We will make use of these properties in the next section.

237 3

Porous medium equations

Let us consider the Cauchy problem for the porous medium equation

ft = (f"),, ~ ( x , o=) f o ( x )2 o

EL~(IW n L)

~(IW)

(12) (13)

with 11 foIILl(w) = 1. It is well-known that this Cauchy problem has a unique solution in a strong sense uniformly bounded in time and space 2 1 ~ 2 2 .Mass conservation then implies that the solution f (., t ) is a timedependent probability density on R. The discovery of similarity solutions t o equation (12), the so-called Barenblatt-Pattle solutions, and its finite speed of propagation produced a great interest in studying the time evolution of the support of solutions. By means of heavy analytical tools, it was possible to show that the long time asymptotics and the velocity of expansion of the support was given by the similarity solution Here we will address the finite speed of propagation property from a completely different perspective. First, we will show formally that the flow for this equation is a contraction for all Wasserstein distances of order 2n. Second, we will deduce from this result the speed of propagation of the support. By an induction argument one can easily show that moments of the solution at any order are bounded if so are for the initial data. Precisely, given zlill,lz.

we have that

d -mz,(t) = 2 4 % - 1)m2,-Z(t), dt for any n 2 0, and thus, mzn(t) = O(tn). Now, one can compute the equation satisfied by the pseudo-inverse distribution F-l ( p , t ) function obtaining

It is quite natural now to wonder how the Wassertein distances between two solutions f,g of the equation evolve in time. It is straightforward t o derive by formal integration by parts that

, 2 1 , is decreasing. Note that the boundary for any n 2 1 since the function x - ~ m term vanishes due to the compact support of the solutions, which implies

lim (F;')-' p-+o+

= lim (F;I)-' p4l-

Therefore, we deduce that

dz,(F(t), G ( t ) )5 dz,(Fo,Go)

= 0.

238 for any t 2 0 and n 5 1 and thus, d,(F(t),

G(t)) 5 dm(F0, Go)

for any t 2 0. This fact has been done rigourously and exploited further for bounded supported initial data in to show an estimate of the growth of the support of the solutions.

Theorem 3.1 Given two postive initial densities f0,gO for the porous medium equation in L'(R) n Lm(R), with the same mass and compactly supported, then the solutions f(t),g(t) remain compactly supported and their supports verify the following estimates:

I inf{supp{f (t))) - inf{supp{g(t)))l I dm(F0, Go) I SUP{SUPP{f (t))) f o r any t

- sup{s?lpp{g(t)))I

5 dm(Fo, Go)

> 0.

The main consequence of the previous theorem is that we can use as g(t) the to explicit self-similar solution of Barenblatt-Pattle translated in time (see recover that the support of any solution f ( t ) grows in time as the support of the Barenblatt, which is known to be of order ti/(mt'). 22y4),

There are two simple extensions of the previous results: 1. General nonlinear diffusion equations:

ft = (@(f)Ixx f(x,o) = fo(x) 2 o

E L ~ ( Rn ) L"(R)

with 11 follLl(w) = 1. Here, the diffusion function @ : [O,m)-+ [0, m) is nondecreasing with @(O) = 0. Computing formally the equation satisfied for the pseudo-inverse distribution function, one finds

and thus, Wassertein distances of exponent 2n between any two solutions of these equations are non increasing in time. 2. Nonlinear Fokker-Planck equations:

f ( x , o )= fo(z) 2 o

E L ~ ( Rn ) L,(R)

where V(t,x) is a smooth strictly uniform (in x and in t ) convex potential ( g ( t , x ) 2 X > 0) and @ as above. In this case the equation for the pseudoinverse distribution function reads

239

and thus, Wassertein distances of exponent 2n between any two solutions of these equations are decaying exponentially fast as t -+ 00 at a rate f x t . Moreover, in this case also we have that for any two solutions

d m ( F ( t ) ,G ( t ) )I d,(Fo, Go) e-xt and therefore, supports of solutions are converging to the support of the stationary solution. This argument gives a very simple proof of the finite speed of propagation and convergence of the support of the solutions when the unique steady state is compactly supported. 4

Nonlinear friction equations

In this section we illustrate shortly a further application of the Wasserstein metric to the study of uniqueness and large-time behavior of the equation -a= f (-v l t )

at

a

aV

I

[ f ( U , t ) ) v - wI'(v

w

-

1

w)f(w,t)dw ,

where the unknown f(.,t ) is a time-dependent probability density on R, and y > -1. This equation, called nonlinear friction equation, arises in the study of granular flows, and has been introduced in 1 8 , in connection with the quasi-elastic limit of a model Boltzmann equation for rigid spheres with dissipative collisions and variable coefficient of restitution (see for the y = 1 case). The variable v E R represents the velocity of particles. The nonlinear friction equations exhibit the main properties of any kinetic model with dissipative collisions, like conservation of mass and mean velocity and decay of the temperature. Since the mean velocity is conserved in time, without loss of generality one can assume as initial values only probability measures with expectation equal to zero. The equilibrium state is given by a Dirac mass located at the mean velocity of particles. In addition, these equations exhibit similarity solutions, which are in general of noticeable importance to understand the cooling process of the granular flow. The passage to the pseudo inverse function shows a remarkable simplification. A direct computation shows that, if the probability density f ( v ,t ) satisfies (15), F - l ( p , t ) solves '1'

aF-l(p, t ) = - I ' I F - l ( p , t ) - F - l ( p , t ) l Y ( F - ' ( p 1 t ) - F - l ( p , t ) ) d p .

at

(16)

In view of (16), it becomes natural to look for the time evolution of the square of the Wasserstein metric. This evolution is easily found to satisfy

240

where

H ( p , p ) =: F - l ( p ) - F - ' ( p ) ;

K ( p , p ) =: G - l ( p ) - G - ' ( p ) .

(18)

The functions H and K are deeply linked to Wasserstein metric any time we are considering distribution functions with the same momentum. In this case in fact one can rewrite Wasserstein metric in the following form

Owing to this definition, one can easily handle the right-hand side of (17) to obtain the evolution equation for the Wasserstein metric ', which reads

Equation (19) can be studied in details, to obtain both uniqueness and asymptotic behavior of the solution to (15). While uniqueness is a direct consequence of the fact that the Wasserstein metric is non increasing, the study of the large-time behavior requires a deeper analysis. This has been done in ', and it can be resumed as follows

Theorem 4.1 Let F(w,t),G(w,t) E C1(R$,M2) be two solutions to the initial value problem f o r equation (15), corresponding t o the initial distributions Fo(v),Go(w) E M z , respectively. Then, i f 0 < y < 2 , the Wasserstein distance of F(w,t ) and G(w,t ) is monotonically decreasing with time, and the following decay holds

Moreover, if -1 < y < 0 and the initial density fo(w) has bounded support, S u p p ( f 0 ) = L < 00, the support of the solution decays to zero in finite time, and the following bound holds

Furthermore, i f both initial densities f o ( w ) ,go(w) , have bounded supports, the Wasserstein distance of F(w,t ) and G(w,t ) decays to zero at finite time, and the following time-decay holds

where L denotes the m a x i m u m of the supports.

24 1 5

Nonlinear friction equations with viscosity

An interesting application of Wasserstein metric comes out from the problem of the numerical approximation of nonlinear friction equations 15. As explained in the previous section, the solution t o this equation converges t o a Dirac mass located at the mean velocity of particles. The use of spectral methods in this situation is not allowed. To overcome the problem of the approximation of a Dirac mass, it is convenient t o add a small diffusion. The new equation to be considered is -af(v't)

at

-2 [ f ( v , t ) / Iv-wl'(v-w)f(w,t)dw dV w

]

+ E - a2f

dv2 '

(22)

where E > 0 denotes a small parameter. The solution t o (22) has an equilibrium which is a smooth function 3 , and the new problem can be well approximated by spectral methods 15. The question now is to understand if the correction due t o the small diffusion added can be controlled in time. Denote by fe the solution t o equation (22). Coupling the computations of Sections 3 and 4, we obtain that the pseudo inverse function F;' ( p ) satisfies

Let f ( t ) be the solution t o the nonlinear friction (15). Then, owing t o Theorem 4.1 we can bound the time evolution of the Wasserstein distance between f and f c l obtaining

Integrating by parts the last integral we get

Indeed, the boundary term is equal t o zero, due t o the fact that

Moreover we deduce from (24) the non negativity of the derivatives of F - l ( p ) and F ; l ( p ) . Thus we showed that the distance dz(F,,F) satisfies the differential

242

inequality

d 1 -dz(Fe(t), dt F ( t ) ) 2F - S & ( F e ( t ) , G(t))2+7 26.

+

(25)

Let D solve --

27-

1

D2+7+2& = o

Then, if at some time t = to it holds d z ( F , ( t o ) , F ( t o ) ) 5 D , we conclude from ( 2 5 ) that the maximum value for d 2 ( F E ( t ) , F ( t )at ) any subsequent time t can not overcome D . We proved l5

Theorem 5.1 Let f ( w , t ) , f E ( w , t ) E C1(Rt,M2) be the solutions t o the initial value problems for equations (15), (22) respectively, corresponding t o the same initial distribution fo(w) E M2. T h e n , if 0 < y < 2, the Wasserstein distance of f ( t ) and f e ( t ) i s uniformly bounded in time, and the following bound holds Pl(Z+Y)

d z ( f , ( t ) , f ( t ) )F 6

(26)

Degenerate convection-diffusion equations

Our find application of these Wasserstein metrics will be done in the case of equations which are perturbation of nonlinear diffusions. For instance, some degenerate convection-diffusion equations. Let us consider the initial value problem for u,

+ (u”), -

U(S

= 0) = fo

=0

2o

(27)

E L ~ ( Rn )L~(R)

+

with m 2 1 , q > m 1. We set IIuoll~l(~) = 1. Note that mass conservation holds. We refer to for the discussion of the well-posedness and asymptotic behavior for this problem. In order to study the asymptotic behavior of this equation, it is convenient to rescale the variables and the solution

’

y = zR(s),

t

=

1

- In (1 + As),

x

f (2, t ) = R(s)u,

where R ( s ) is defined as ~ ( s := ) (AS

+ I)+ ,

R(t) = et

+

and X = m 1. We emphasize that this rescaling leaves the initial data fo = uo unchanged and preserves the L1-norm Ilf(s)ll1 = IIu(s)II1 for all s 2 0. The corresponding Cauchy problem for f (z, t ) becomes

+ R(t)-6(f”z - (f%z - (zf)l: = 0 f ( t = 0) = f o 2 o E L ~ ( R n ) L~(R) ft

with 6 = q - m - 1.

(28)

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By mimicking the procedure in previous sections we derive the formal equation verified by the pseudo-inverse distribution function that in this case reads as:

As shown in ’, this problem can be considered in this range of exponents as an asymptotic perturbation for large times of the corresponding porous medium equation. In fact, a naive approach to previous equation implies that the convection term should give an asymptotic negligible contribution. In fact, this intuition can be made precise by using the L1 - LO3 smoothing effect of these equations. It is indeed possible t o prove that the new term coming from the convection is negligible for large times and in fact, the Wasserstein d2 distance between any two solutions converges exponentially to zero as t + co for the rescaled equation. We refer for the details to in which the following result is proved: Theorem 6.1 Given two initial data f o , go for the convection-diffusion equation for equation (28), in L1(R) n L”(R) with with the same mass and bounded second moment, then the Wasserstein distance of the solutions f ( t ) ,g ( t ) verifies

lim d z ( F ( t ) ,G ( t ) )= 0 ,

t’cc

and i n fact, it decays asymptotically as ePat with a = min($,1).

7 Conclusions We briefly described here some recent applications of Wasserstein metrics to onedimensional nonlinear diffusion equations and to nonlinear equations arising in kinetic theory of granular gases. The common features of these models is that, in consequence of mass conservation and positivity of the solutions, they can be easily reformulated in terms of the pseudo inverse function. These applications enlighten the power of Wasserstein metrics in deriving uniqueness, asymptotic behavior and approximations of solutions to nonlinear problems. Similar studies are in progress to understand if these applications can be easily extended to higher dimensions, where the representation of these metrics in terms of the pseudo inverse function fails. Likewise, we believe that, even in one-dimension of space, the use of these metrics allows for further applications to nonlinear Fokker-Planck type equations. Acknowledgments The authors acknowledge financial supports both from the project HYKE, “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis’’ financed by the European Union (IHP), Contract Number HPRN-CT-2002-00282, and from the bilateral project Azioni integrate Italia-Spagna, “Models of diffusion in partial differential equations for thin films, viscous fluids and semiconductors”. G.T. acknowledges

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support from the Italian MIUR project “Mathematical Problems of Kinetic Theories”. J.A.C. acknowledges the support from the Spanish DGI-MCYT/FEDER project BFM2002-01710. References 1. D. Benedetto, E. Caglioti, M. Pulvirenti, MZAN Math. Model. Numer. Anal. 31, 615 (1997). 2. D. Benedetto, E. Caglioti, J.A. Carrillo, M. Pulvirenti, J. Statist. Phys. 91, 979 (1998). 3. J.A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granu-

lar media and related equations: entropy dissipation and mass transportation estimates, Revista Mat. Iberoamericana (in press). 4. J.A. Carrillo, G. Toscani, Indiana Univ. Math. J. 49, 113 (2000). 5. J.A. Carrillo, K. Fellner, Long-time asymptotics for degenerate convectiondiffusion equations via entropy methods, (work in preparation) (2003). 6. J.A. Carrillo, M.P. Gualdani, G. Toscani, Wasserstein metrics and the propagation of the support in nonlinear diffusion equations, (work in preparation) (2003). 7. M. Escobedo, E. Zuazua, J . Funct. Anal. 100, 119 (1991). 8. M. FrBchet, Ann. Univ. Lyon Sect. A 14, 53 (1951). 9. Hailiang Li, G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal. (in press) (2003) 10. W. Hoeffding, Schrifien des Math. Inst. und des Inst. Angewandte Mathematik der Universitut Berlin, 5 , 179 (1940). 11. S. Kamin, Israel J. Math., 14, 76 (1973). 12. S. Kamin, Arch. Ration. Mech. Anal., 60, 171 (1976). 13. L. Kantorovich, Dokl. A N SSSR, 37 , 227 (1942). 14. R.J. McCann, Adv. Math. 128, 153 (1997). 15. G. Naldi, L. Pareschi, G. Toscani, MZAN Math. Model. Numer. Anal. 37, 73 (2003). 16. F. Otto, Comm. Partial Differential Equations 26 , 101 (2001). 17. S. Rachev, L. Ruschendorf, Mass transportation problems, Vol. 11: Probability and its applications, Springer Verlag, New York, 1998. 18. G. Toscani, M2AN Math. Model. Numer. Anal. 34 (2000), 1277-1292. 19. I. Vaida, Theory of statistical Inference and Information, Kluwer Academic Publishers, Dordrecht 1989. 20. L.N. Vasershtein, Probl. Pered. Inform., 5 , 64 (1969). 21. J.L. Vtizquez, Bans. Amer. Math. SOC.,277, 507 (1983). 22. J . L. Vtizquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ. (in press) (2002). 23. C. Villani. Topics in optimal transportation, AMS, Providence, (in press) 2002. 24. W. Whitt, Ann. Statist., 6, 1280 (1976). 25. V.M. Zolotarev, Theory Prob. Appl., 28, 278 (1983).

OPENING TALKS

Prof. PAOLO FERGOLA Presidente del Comitato Organizzatore del Convegno

Consider0 un grande privilegio dare inizio ai lavori di questo convegno dedicato ai settant’anni di Salvatore Rionero, a1 quale mi legano sentimenti di stima sia scientifica che umana, nonch6 di affettuosa amicizia. Quest’ultima ancor piu solida perch6 nata a valle di una serie di Lkontrapposizioni”,di natura squisitamente universitaria, verificatesi circa trentacinque anni fa (mi riferisco agli anni 67-70) tra due gruppi di ricerca “antagonisti”, interni alla Fisica Matematica napoletana di quel periodo. A posteriori devo dire che queste contrapposizioni, a quel tempo imputate a serie ed inconciliabili divergenze caratteriali ed ideologiche erano, pih plausibilmente, dovute alla diversit&di progetti locali di sviluppo universitario. Ricordo, tuttavia, che tutto cib non ci ha mai impedito , incrociandoci nei corridoi del vecchio Istituto di Matematica di via Mezzocannone, di salutarci , freddamente, ma con reciproco rispetto. Successivamente una lunghissima consuetudine di comune lavoro universitario ha via via rafforzato questa stima reciproca. A1 di lh, tuttavia, della partecipazione ad attivith scientifiche seminariali della nostra comune area disciplinare, i nostri campi di ricerca sono stati sempre distinti, l’uno nell’ambito delle equazioni differenziali alle derivate parziali, l’altro in quello delle equazioni differenziali ordinarie. Ne11’87 comincib a profilarsi una concreta possibilit& di una nostra pi6 diretta interazione scientifica e fu quando Salvatore mi invitb a presentare a1 WASCOM di Taormina, una comunicazione su un problema di stabilith concernente un sistema di O.D.E. interessante la biomatematica. Fu quella una delle prime occasioni in cui veniva ufficialmente riconosciuta alla biomatematica una legittima cittadinanza nell’area fisico-matematica. WASCOM infatti sta per “Waves and Stability in Continuous Media” (e lo precis0 per i presenti non addetti ai lavori) B un importante Convegno internazionale biennale, dedicato a questioni di stabilith interessanti la meccanica dei mezzi continui e fondato d a S. Rionero insieme con A.Greco, T.Ruggeri e M.Anile nel lontano 1979. A riprova di questo interesse di Salvatore per le fenomenologie biologiche citerb due circostanze. La partecipazione, in qualit&di docenti, di esperti di biomatematica, presso la Scuola Estiva di Fisica Matematica di Ravello, d a lui diretta sin dal 1976 . Quindi l’invito , che mi rivolse qualche tempo dopo, a parlare di dinamica delle popolazioni, in una riunione a Nola , organizzata in Suo onore presso il “Circolo Giordano Bruno”, per festeggiare il suo ingresso, in qualith di socio straordinario, a1l”Accademia dei Lincei. Risale ai primi anni ’90 una Vera e propria collaborazione scientifica con Salvatore su alcuni problemi concernenti questioni di stabilith per alcuni sistemi generalizzati di Loth-Volterra o di tip0 chemostato, conclusasi con la pubblicazione di tre lavori in collaborazione anche con C. Tenneriello. Ulteriori lavori, recentemente pubblicati, su altre questioni di biomatematica testimoniano che questo suo interesse non si B a tutt’oggi esaurito. Naturalmente non & questa l’occasione per entrare nel 245

246 merito di questi lavori n6 , pi6 in generale, della sua cosi , ampia produzione scientifica. Nd tocca a me ricordare i fondamentali contributi che Salvatore ha dato nell’ambito della teoria della stabilitk non lineare delle equazioni differenziali alle derivate parziali interessanti la fisica matematica. Vorrei piuttosto ricordare qui i nomi dei suoi allievi e collaboratori che ricoprono attualmente posizioni di rilievo in ambito sia universitario che scientifico : M. Maiellaro, P. Galdi, M. Padula, R. RUSSO,G. Mulone, C. Bortone, G. Guerriero, F. Capone, M. Gentile, nonchd i pih giovani : LTorcicollo, B. Buonomo, M.Vitiello, F. Perrini che hanno conseguito a Napoli il Dottorato di Ricerca in Matematica (del Consorzio delle Universith di Napoli, Salerno, Caserta e Basilicata) di cui Salvatore b coordinatore d a circa quindici anni. La stretta collaborazione con Michele Maiellaro nasce nel 1968 con la chiarnata di Salvatore, giovanissimo vincitore di concorso, presso la cattedra di Meccanica Razionale all’universiti di Bari. Qui Salvatore ha operato fino alla fine dell’ottobre del 1971, quando b rientrato a Napoli per ricoprire la cattedra, di cui B tuttora titolare, di Meccanica Razionale presso la nostra Facolt& di Scienze. Di alta reputazione scientifica internazionale sono tutti i ricercatori stranieri con i quali Salvatore ha collaborato. Tra questi ricorderb i nomi di B. Straughan e J. Flavin, qui presenti, e di P. Christiansen, R.Knops e D.D.Joseph. Probabilmente verranno d a loro, direttamente, altre notizie su queste collaborazioni. Mi limiterb qui ad aggiungere che quella con J. Flavin si protrae ormai d a molti anni e che si b concretizzata nella pubblicazione di diversi lavori e di un volume dal titolo “Qualitative Estimates for Partial Differential Equations. An introduction.”. Essa ha portato Salvatore a lavorare per lunghi periodi in Irlanda, presso 1’Universitk di Galway che, nell’estate 2001, gli ha conferito la Laurea Honoris causa con il titolo di Doctor in Sciences. Tra gli ospiti stranieri b qui presente il prof. Z. Ma della Xi’an Jiaotong University di Xi’an (Cina) con cui la nostra Universitk ha in atto una collaborazione scientifica nata circa 10 anni fa. Nel 2000, per iniziativa di Salvatore, Ma e del sottoscritto, nell’intento di proporre anche ad altre Universith italiane una concreta e stabile occasione di interazione con i matematici cinesi, b stato organizzato a Napoli , all’interno del IV Convegno Internazionale su “Matematica e Ambiente” , il Primo Colloquio Italo-Cinese su “ New contributions on Mathematical Modelling of Diffusion Problems ”. La seconda edizione di questo Colloquio (CICAM 2) si svolgerk nel prossimo autunno a Xi’an. L’esiguitk del tempo a disposizione mi consente a questo punto solo due osservazioni finali. La presenza di tanti autorevoli colleghi con i quali Salvatore ha collaborato e collabora per ragioni didattiche e scientifiche, costituisce una testimonianza di stima che gli perviene dai pi6 qualificati settori del mondo universitario e scientifico. Tra questi mi limiterb a nominare G.Trombetti per la nostra Universith, G. Grioli ed L. Salvadori per 1’Accademia dei Lincei, T. Ruggeri per il Gruppo Nazionale di Fisica Matematica A. Figk-Talamanca per 1’Istituto Nazionale di Alta Matematica, C. Sbordone per 1’Unione Matematica Italiana, M. Capaccioli per 1’Accademia delle Scienze Fisiche e Matematiche della Societk Nazionale di Scienze, Lettere ed Arti, L.M. Ricciardi per il Dottorato in Matematica Applicata e Informatica, V. Coti-Zelati per il Consiglio del Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” , F.De Giovanni per il Consiglio di Corso di Laurea in Matematica della nostra Universiti. Sono sicuro che tutti questi colleghi e i molti altri qui presenti potranno convenire

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nel riconoscere a Salvatore’ a1 di 1& della sua indiscussa autorevolezza scientifica e didattica, alcune doti umane del tutto rare quali una dedizione pressocchk totale alla professione universitaria, in tutte le sue molteplici articolazioni, una resistenza e capacith di lavoro veramente enormi, una grande generosith. Quest’ultima sempre pih accentuandosi in una persona che nonostante l’aumentare degli anni b ancora giovanissima, oltre che nella curiosit&e nell’entusiasmo della ricerca scientifica, anche nel fisico e nella mente.

Prof. GUIDO TROMBETTI Rettore dell’universit&. degli Studi di Napoli Federico 11

I1 piacere di portare il saluto dell’Ateneo in quest’occasione b immenso. Ho pensat0 in questi giorni cosa preparare, poi ho maturato il convincimento che Salvatore non meritasse un discorso “infiocchettato” , cioit troppo formale, poichb quello si riserva alle occasioni che hanno una valenza solo formale. Car0 Salvatore, i tuoi sforzi per nascondere il compleanno sono falliti, come vedi, per la malvagitii di Paolo Fergola che i: a tutti nota. Salvatore ha provato ad inventare le strategie pih sottili. Ad un certo momento, ha messo in giro la voce che nell’hteneo vi fosse un omonimo, e che quindi questo festeggiamento fosse frutto di un errore materiale. Conosco Salvatore Rionero da tantissimi anni e credo, non vorrei sbagliare i calcoli, che il primo incontro con lui sia stato nel ’67 o nel ’68, quando ho sostenuto l’esame di Meccanica Razionale; mi i! rimasto impress0 come uno degli esami pih intelligenti che ho subito, assolutamente irrituale nella struttura, dal quale ho tratto motivo di grande soddisfazione. Salvatore ed io abbiamo collaborato in tantissime occasioni, nella vita dell’Istituto e poi del Dipartimento, del quale Salvatore 5 stato il primo Direttore; Dipartimento quindi, si pub dire, che ha fondato, ha organizzato. Con Salvatore abbiamo collaborato in posizioni di carattere nazionale nell’ambito della comunitii matematica, per esempio all’INdAM; collaborazione, a mi0 avviso, interessante perche il contributo di Salvatore Rionero b sempre stato, in tutti i momenti gestionali ed organizzativi, un contributo di buon senso e di grande equilibrio. In un paese governato dall’alluvione delle norme, per cui rispetto ad ogni decisione non c’b che da scegliere tra una norma ed un’altra che dice il contrario, l’unica Vera bussola it il senso dell’equilibrio, che i: una cosa che “chi non ce l’ha non se la pub inventare”. Salvatore questa virtti l’ha sempre avuta, gliela ho riconosciuto gii in privato, e ora desidero riconoscergliela anche in pubblico. Lui ha, diciamo, per certi aspetti, un caratteraccio. Quello che conta, perb, b il segno della derivata, ed in tutto il suo agire la derivata b sempre stata positiva. Salvatore ha partecipato alla vita organizzativa dell’Ateneo; ha fatto parte della Commissione Ricerca della Federico 11, un tempo denominata Commissione Scientifica; ne 6 stato Presidente, e anche in quell’ambito ha cominciato a lavorare quando si affacciavano nella comunitii alcuni sostantivi ai quali non si era abituati: valutazione, autonomia .... Salvatore era 11, a portare il contributo della sua grande qualita scientifica, ed k questo poi l’aspetto fondamentale della sua vita accademica, aspetto del quale io sono stato spettatore ammirato. La sua capacitii di produrre attivita di ricerca, e la sua capacitii di formare allievi, ne fanno in sostanza un esempio di come l’universitii dovrebbe funzionare. Salvatore ancora oggi b instancabile: organizza convegni, workshop, viaggia ...., spende ...., costa. Allora ci si domanda: come porsi dinanzi ad una struttura di riforma dell’organizzazione universitaria, che sostanzialmente tende ad operare una rivoluzione copernicana, spostando il baricentro dell’organizzazione della struttura, dall’idea del legame a tempo indeterminato all’idea del legame a tempo determinato? Su questo non voglio esprimere giudizi, bisogna riflettere. Salvatore perb i! 248

249 un controesempio: se per caso dovesse maturare l’idea che precarieta equivale efficienza, ebbene noi avremmo un controesempio. Si pub lavorare ad alto livello con serieta, producendo, organizzando, amministrando, anche senza l’ansia della precarieth. Con questa affermazione gli faccio i miei pih affettuosi auguri, e lo ringrazio di avermi consentito di fare questa piccola confessione pubblica.

Prof. ALBERT0 DI DONATO Preside della Facolt& di Scienze Matematiche, Fisiche e Naturah dell’Universit& degli Studi d i Napoli Federico II

Caro Salvatore, ti porgo un saluto personale e a nome di tutta Facolt&di Scienze Matematiche Fisiche e Naturali. In questa occasione vorrei sottolineare come oggi, e lo dico come un paradosso, in un ottica di disgregazione la grande Facolt& di cui tu ed io facciamo parte B quasi un controsenso: le discipline vanno sempre pih su percorsi autonomi e trovano pochi momenti di interazione. Ebbene, la tua carriera scientifica e didattica i: un esempio della volont&di recuperare il valore di Facolt&come unione e sintesi di saperi , ed i: per questo che siamo fieri di averti con noi, come professore, nel quotidiano rapport0 con gli studenti, e come scienziato, sebbene non possa essere io a tessere le lodi dei tuoi meriti scientifici. Abbiamo avuto occasione, in passato, di collaborare in alcune commissioni di ateneo, dove io ho avuto mod0 di apprezzare la sensibilith e la capacitB di mettere in pratica quelli che sono dei principi di equilibrio. Ma certamente penso che un altro aspetto da sottolineare in questa occasione i: che tu sei secondo me l’esempio del fatto che la precarieth non w o l dire necessariamente qualit&. Leggo che ben 44 anni fa, nel ’56, tu sei diventato assistente ordinario dell’universitk; sono passati 44 anni, e tu come tanti altri come noi, sei stato inamovibile nel nostro Ateneo, hai lavorato duramente scientificamente e didatticamente. Quindi non i: una condizione necessaria essere precari per produrre. Tu come tanti altri in questa sala, in questo ateneo, negli atenei italiani, hai svolto un lavoro che B il quotidiano lavoro svolto dai docenti, mentre invece oggi sui giornali compaiono le eccezioni. Questo B quello che secondo me il sistema deve rifiutare: improntare una ipotesi di riforma del sistema universitario basandosi sulle eccezioni, invece di riconoscere quelli che sono i meriti del sistema. Cerio, l’esistenza delle eccezioni significa che abbiamo il dovere di riflettere su comportamenti e tendenze, e su questo dobbiamo lavorare, non opponendoci in maniera acritica a quelli che sono i cambiamenti che la societ& ci richiede: far parte del tessuto sociale, adeguarci e dare il nostro contributo per come siamo capaci di darlo, i: un nostro dovere e sono certo che il tuo percorso accademico ha dimostrato questo. Quindi permettimi di rinnovarti gli auguri della nostra Facolt&, che B felice ed B contenta di avere un membro di caratteristiche scientifiche e didattiche come le tue, e che ha certamente rappresentano un onore per tutti noi. Grazie.

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Prof. CARL0 SBORDONE Presidente del’uninone Matematica Italiana

Cari Colleghi, sono qui per il saluto affettuoso dell’UMI a Salvatore Rionero e per godere di questo intenso convegno organizzato per celebrare i suoi 70 anni. L’occasione di prendere la parola in questo consesso mi i: particolarmente gradita perchi: mi capita, a tale livello , per la prima volta, di farlo nella mia citta e di cib sono grato agli organizzatori. Mi i: spontaneo qui ricordare i suoi meriti nell’UMI di cui B socio sin dal marzo 1959, anno particolare per la Matematica Napoletana (i: l’anno in cui Caccioppoli decise di porre fine alla sua esistenza) ma anche per la Matematica Italiana che, in occasione del VI Congresso UMI tenutosi proprio qui dall ’11 a1 16 Settembre, vide una svolta nella politica dell’Unione con una contrap posizione generazionale che ebbe molti effetti positivi. Membro della Commissione Scientifica dal 1985 a1 1988, ha fatto parte, nel 1998 della Commissione per il premio Caccioppoli ed 6 stato Presidente del Comitato Organizzatore del XVI Congresso UMI svoltosi a Napoli nel settembre 1999. Come collega di Facolt&non posso non ricordare che egli i:stato il primo direttore del Dipartimento di Matematica e Applicazioni “R. Caccioppoli” nel triennio 84-86. La sua intelligenza, la sua capacita organizzativa, la sua abiliti nel rendere semplici le pratiche difficili, il suo senso del dovere, ne hanno fatto un Direttore illuminato e, per la prima volta, un punto di riferimento per la comunita matematica napoletana in un’epoca in cui, Carlo Miranda era appena scomparso e Carlo Ciliberto era stato ad essa sottratto da impegni Rettorali. La sua attiviti scientifica, le sue doti di Maestro e le cariche a livello nazionale, per esempio in ambito GNFM, o INdAM, verranno delineate da altri colleghi, ma qui vorrei ricordare che egli i: stato il primo, alla fine degli anni ’70, ad invitare a Napoli matematici della levatura di James Serrin e Daniel Joseph e fu promotore, insieme ad A. Canfora e G. Trombetti, del primo rilevante convegno matematico tenuto a Napoli dopo il gia menzionato Congresso UMI del59, il IV SAFA a marzo del 1980. Cari amici, vi sono pochi uomini che per la loro grande carica umana, per la loro saggezza e la statura intellettuale riescono sapientemente a dividersi tra i familiari, gli amici e i colleghi di lavoro in un equilibrato rapporto. Salvatore Rionero B uno di questi uomini e in quanto tale appartiene ugualmente alla sua famiglia con cui ha un legame tenerissimo, agli amici e ai colleghi. Dobbiamo, noi e soprattutto i giovani qui presenti, trarre profitto e prendere esempio dalla sua prestigiosa attiviti per proseguire con lui quest’opera nell’interesse comune della Scienza e della cultura.

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Prof. ALESSANDRO FIGA TALAMANCA Presidente dell ’Istituto Nationale d i Alta Matematica lLFrancescoSeveri” (INdAM)

10 sono qui come presidente dell’Istituto Nazionale di Alta Matematica, del quale Salvatore Rionero B il vicepresidente vicario. Penso, perb, che Salvatore s a p pia bene che sono qui principalmente come suo amico. Devo dire che mentre il Rettore Trombetti pub vantare una conoscenza e un’amicizia con Salvatore che risale a1 1967, io non posso dire altrettanto. Certo Salvatore mi era noto come uno dei capi di questo mondo misterioso dei fisici matematici italiani, i quali, in qualche modo, costituiscono una repubblica autonoma, all’interno della repubblica dei matematici. Cominciai ad apprezzarne le sue doti di lealt&e chiarezza in tempi abbastanza recenti, forse il 1996, quando 1’INdAM decise di attivare presso la sezione di Napoli alcuni corsi di sostegno del dottorato di ricerca. A quell’epoca e dopo, Salvatore era coordinatore del dottorato in matematica di Napoli. Ma la conoscenza diretta che ha dato luogo alla nostra amicizia risale solo all’estate del 1998. Eravamo in un momento cruciale della vita dell’INdAM. Si stava prendendo una decisione che ha cambiato l’Istituto, io penso in meglio, ma che avrebbe potuto cambiarlo in peggio. La decisione difficile e coraggiosa che prendemmo allora fu di sottrarre a1 Consiglio Nazionale delle Ricerche, i gruppi nazionali di ricerca. La valutazione che facemmo B che il CNR si stesse definitivamente allontanando dal mondo universitario e che i gruppi non avrebbero potuto sopravvivere a1 suo interno. Era importante, in quel momento, avere il consenso della cornunit&matematica, sarebbe bastata un piccola minoranza dissenziente a far saltare il progetto. Era importante in quel contest0 avere un’intesa, veramente leale, con le persone che rappresentavano la matematica. L’intesa con Salvatore Rionero B stata immediata e basata su una delle caratteristiche che avevo gi& imparato ad apprezzare: la lealt&e la chiarezza nelle decisioni. Una volta arrivati ad un’intesa che comprendeva una garanzia di autonomia dei gruppi e dei lor0 consigli scientifici, all’interno dell’istituto, non c’erano da temere riserve, dubbi, ambiguith, e c’era soprattutto la sicurezza che l’opinione espressa e la parola data non sarebbero state ritirate. Assieme a Salvatore siamo andati dal Ministro e abbiamo ottenuto il passaggio dei gruppi dal CNR all’INdAM. Poco dopo, nel 1999, Salvatore i: stato eletto nel Comitato Direttivo, e, dal Comitato Direttivo, eletto alla carica di vicepresidente vicario e membro del Consiglio di Amministrazione. Allora, naturalmente, ho iniziato ad apprezzare le altre sue qualit&e principalmente la generosith, cioB la disponibilit&a lavorare intensamente, con passione, e a pensare a cose che non riguardavano lui personalmente nB la sua disciplina o il suo gruppo, ma l’interesse generale. La sua collaborazione B stata determinante sia a livello propositivo, sia a livello esecutivo, in questo nuovo programma di borse di studio per studenti universitari, che B il fiore all’occhiello del nostro Istituto. Fu lui, assieme ad altri membri del Comitato Direttivo, che prevalse sulle mie esitazioni in merito alla fattibilitk di una prova nazionale decentrata in oltre venti sedi universitarie. Fu lui ad offrirsi di affrontare, assieme ad una commissione da h i presieduta, le innumerevoli difficolti pratiche nella selezione dei candidati.

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253 Penso di non dover aggiungere rnolto di piu, sul piano personale. Naturalmente porto il saluto dell’INdAM, del suo Comitato Direttivo, ed anche dei dipendenti, i quali, debbo dire, erano assolutarnente stupiti che si celebrasse il settantesirno compleanno del professor Rionero. Vorrei ribadire perb che la principale ragione per la quale sono venuto i: stata il desiderio di esprimere la soddisfazione di aver avuto la conoscenza e il piacere di lavorare con una persona del suo livello in questi anni.

Prof. TOMMASO RUGGERI Direttore det G T U ~Nazionale ~O per la Fisica Matematica detl’INdAM

Come B ben noto, uno dei famosi paradossi della Relativita B quello dei due gemelli: l’uno che si mantiene giovane viaggiando molto velocemente, mentre l’altro invecchia sulla terra. Adesso finalmente abbiamo una prova tangibile della validitii del paradosso, perch&quello che stiamo festeggiando i: di fatto il gemello anziano, mentre in questo momento il vero Salvatore giovane e baldanzoso contempla la bellezza dei viaggi alla velocita della luce. A1 gemello vecchietto voglio porgere un breve pensiero in un triplice mio ruolo: di direttore del GNFM di appartenente a1 medesimo gruppo di ricerca, e infine come amico fraterno da oltre un trentennio. Come direttore del GNFM porgo qui il saluto mio e del Consiglio Scientifico (molti membri sono qui presenti), e sono sicuro di poter interpretare il pensiero di molti Colleghi, nel porgere a Salvatore i nostri pih sentiti ringraziamenti per quanto ha fatto per la Fisica Matematica Italiana. Ricordo che h i i: stato per molti anni membro del Consiglio Scientifico del Gruppo, anche nel ruolo di presidente, ed i: attualmente membro eletto e vicepresidente dell’INdAM. F’ra i tanti meriti voglio rammentare quello di essere stato, insieme a1 Consiglio di allora, promotore della Scuola estiva di Fisica Matematica, che per merito di Salvatore ha trovato una sistemazione stabile nella magnifica e splendente Villa Rufolo di Ravello. La Scuola, di cui Rionero i: stato sempre il direttore, i: oggi il fiore all’occhiello del nostro Gruppo, e penso di tutta l’INdAM, sia perch&ogni anno vengono i migliori fisici matematici di tutto il mondo a tenere corsi, ma anche perch6 ha permesso a tantissimi giovani, talvolta provenienti anche da sedi periferiche, di avere l’occasione di conoscere l’avanzare del fronte della ricerca in Fisica Matematica, di confrontarsi con gli altri colleghi, e di potersi mettere in contatto con gruppi di ricerca in primissimo piano. Tutto questo ha avuto la sua ricaduta in una generazione di fisici matematici qualificati e soprattutto sprovincializzati. F’ra l’altro Ravello sembra sia un luogo speciale per Salvatore; molti sanno infatti che proprio a Ravello ha sposato la Sua Giuseppina, e da li , B nata la cosa pih importante, la sua famiglia, cui lui B affezionatissimo. Da molti anni Salvatore e io siamo accomunati dall’appartenenza a medesimi gruppi di ricerca, che coinvolge, su tematiche di propagazione non lineare e stabilitk, molti ricercatori di diverse sedi universitarie. Voglio ricordare che nel sen0 di questi progetti di ricerca, che per anni sono stati coordinati da Salvatore, si B realizzata un’altra iniziativa importante che i: stata ideata da h i : un congress0 internazionale a cadenza biennale, ‘‘Waves and Stability in Continuos Media”, conosciuto come WASCOM, che B ormai un punto di riferimento internazionale per i ricercatori che lavorano nel settore delle onde e della stabilita non lineare nei mezzi continui. In Salvatore come ricercatore, ho sempre apprezzato la sua grande capacitii di riunire le metodologie tipiche della Fisica Matematica classica, in cui vi B il gusto della costruzione del modello su basi fisiche precise, con l’uso di strumenti moderni e raffinati dell’analisi funzionale, risentendo sicuramente delle grandi scuole di Napoli: quella della Fisica Matematica classica di Tolotti e Stoppelli, e quella dei giganti dell’analisi quali Caccioppoli e Miranda. Infine un breve comment0 piu personale. Ho l’onore e il piacere di 254

255 essere uno dei pih cari amici di Salvatore. La nostra conoscenza risale a pib di trenta anni fa, quando io stavo ultimando i miei studi universitari a Messina, ed il compianto professor Carini invitb un giovanissimo vincitore di concorso a cattedra a tenere una brillante conferenza sulla stabilitk in magnetoidrodinamica . Anche per i motivi prima menzionati, la nostra frequentazione i: stata sempre assidua e ho sempre goduto della sua compagnia. Probabilmente, la nostra comune origine di appartenenza a1 regno delle due Sicilie fa si che, oltre ai nostri interessi scientifici, ci ha sempre accomunato quel fondo di ironia e di allegria che fa prendere la vita nel verso giusto, tipico della gente del sud. Cosi , le nostre chiacchierate finiscono sempre con piccole prese in giro reciproche su temi di politica, di donne o altro. La giacca che oggi indosso, B l’effetto di una nostra serata veramente spassosa passata in un negozio di F’rancoforte, gestito da turchi che vendevano giacche a 86 Euro ciascuna. Vaglio finire questo mio breve intervento dicendo ad entrambi i gemelli, sia a quello che sta li , su che a questo qua, che li consider0 persone dotate di grande simpatia e intelligenza, ma soprattutto quello che in loro ammiro di pib B la loro grande generositk, onesth e umanith. Gemelli Rionero, 100 di questi giorni!

Prof. MASSIMO CAPACCIOLI Direttore dell’Osseruatorio Astronomico d i Capodimonte, Napoli

Cercherb di essere estremamente breve anche perch&,come vuole il teorema test& enunciato dal Preside Di Donato, man mano che l’ordine dei relatori si snocciola, gli argomenti nuovi si vanno assottigliando in numero e consistenza. Desidero fare soltanto due considerazioni. Viviamo in un tempo di grandi cambiamenti. Non sono sicuro che cib sia per il meglio, ma & comunque certo che qualcosa di importante sta succedendo. In questi tempi di grandi incertezze, fa piacere constatare come restino ancora dei punti fermi: nell’Accademia la celebrazione, attraverso un convegno scientifico, di un collega che ha ben meritato nei confronti della scuola, degli studenti e della cultura in generale. A mio avviso tutto cib it molto bello. E mi auguro che i cambiamenti che abbiamo di fronte, a cui il Magnifico Rettore faceva riferimento, non vogliano modificare questo antic0 rito che con il corporativismo non ha niente a che spartire; ma che ha invece a che vedere con il rispetto del lavoro che facciamo, con grandissimo piacere prima ancora che pel bisogno di lavorare. Ho conosciuto Salvatore Rionero soltanto 10 anni fa, subito dopo il mio arrivo a Napoli. Ci siamo incontrati alla Societa Nazionale delle Scienze Lettere ed Arti, e per lo pih ci siamo visti e frequentati in quelle austere sale. I1 Prof. Grioli, invece, it stato mio professore di Meccanica Razionale all’UniversitBdi Padova. it lui che, tanti anni fa, mi costrinse ad accettare una cosa che ancora oggi non riesco a comprendere: perch&la natura si faccia descrivere cosi , bene dalla matematica. Siccome ho studiato per il resto della mia vita la natura, quella di cui ci parlerk proprio il Prof. Grioli nella prima relazione odierna, mi sento in debito con loro che, come l’amico Rionero, continuano a fornirci gli strumenti per leggere galileianamente quel libro che & meraviglioso leggere, attraverso una lingua che 6 difficile da sviluppare, che & difficile da gestire e che - ripeto funziona anche se io non capisco bene perchit. ~

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Prof. LUIGI MARIA RICCIARDI Vicepresidente del Polo delle Scienze e delle Tecnologie dell ’Universitci degli Studi d i Napoli Federico 11

Particolarmente lieto di essere qui oggi, desidero rivolgere un saluto di duplice natura a1 Professor Salvatore Rionero. I1 primo, doveroso ma peraltro molto sentito, nasce dalla mia presenza qui in sostituzione del Presidente del Polo delle Scienze e delle Tecnologie dell’universita di Napoli Federico 11, Professor Filippo Vinale, il quale mi ha esternato il suo vivo rincrescimento di essere impossibilitato ad essere qui tra i presenti. A1 Professor Rionero io porgo dunque il saluto del Polo delle Scienze e delle Tecnologie. Con l’occasione, vorrei anche sottolineare quanto particolarmente nota ed a p prezzata sia nell’ambito del Polo l’attivita scientifica del Professor Rionero non soltanto per gli aspetti di eccellenza, che ovviamente non spetta a noi valutare, ma particolarmente per lo sforzo che egli ha sempre dimostrato, negli anni, essere rivolto ad attivita scientifiche ed organizzative spesso a carattere marcatamente trasversale a molteplici discipline afferenti proprio a1 Polo delle Scienze e delle Tecnologie. Ben nota & l’attivita scientifica del Professor Rionero nell’area della Fisica Matematica, delle equazioni alle derivate parziali, dei problemi di stabilith; ma forse non tutti sanno che contributi molto rilevanti egli ha dato negli scorsi anni, insieme con i suoi collaboratori, anche a problemi tipici di settori molto diversi, ad esempio d’interesse ecologico, di dinamica di popolazioni e in generale di sistemi complessi aventi valenza davvero molto ampia, decisamente a1 di fuori della portata di chi non padroneggi strumenti matematici molto avanzati e non possegga aggiornata ed approfondita conoscenza delle problematiche specifiche. In aggiunta, il Professor Rionero associa la rara virth di possedere quella che io definirei una Ll . ’ visione dall’alto” del cosiddetto “libro della Natura” a1 quale poc’anzi faceva riferimento il Professor Capaccioli. Invero, a1 di 1& della quantificazione e della sistematizzazione matematica dei problemi che si pongono, non si deve dimenticare che, quando davvero significativi e di reale interesse speculativo o tecnico, tali problemi sono in ultima analisi riconducibili proprio all’attenta osservazione della Natura. B quindi essenziale che vengano sviluppati strumenti tecnici e metodi che, se pur possiedano intrinseca valenza, non siano finalizzati soltanto ad eleganza e a perfezione matematica, ma che possano trovare utilizzazioni anche di tip0 applicativo in contesti vari. Anche per cib il Polo apprezza molto la tenacia dimostrata proprio in tal senso dal Professor Rionero e dai suoi collaboratori con conseguente riconosciuto ampio successo; a1 contempo, il Polo ribadisce qui per mio tramite la ferma intenzione di promuovere e sostenere attivit&siffatte non soltanto per il lor0 indubbio merito scientifico, ma anche nell’auspicio che esse conducano in tempi brevi ad offerte didattiche trasversali, quindi non limitate ai soli corsi di studio nei quali ciascuno di noi B direttamente coinvolto, ma che possano essere fruite da una molto pib ampia comunita nell’ambito delle tre Facolta afferenti a1 Polo delle Scienze e delle Tecnologie. Questo B dunque il primo dei saluti, che porgo a nome del Polo, a1 Professor Rionero.

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258 I1 secondo saluto glielo porgo, invece, in qualit&di collega. Vorrei subito dire il Professor Rionero certo non lo ricordera - che la prima volta che ebbi mod0 di conoscerlo fu nel lontano 1960, quando, in occasione di qualche assenza del Prcfessor Carlo Toltoti, titolare dell’insegnamento di Meccanica Razionale, comparve in aula, in sostituzione, questo allora giovanissimo matematico. 10 sedevo nell’aula insieme con vari attuali nostri Colleghi. Ancora oggi k rimasta in me impressa la chiarezza e la luciditi espositiva che egli univa ad una estrema precisione alla quale non ero aduso, forse per la mia provenienza dagli studi classici. Questo mod0 di presentare la materia faceva a noi tutti ritenere che oramai non vi fosse pih spazio alcuno per ulteriori raffinamenti o estensioni: tutto era talmente chiaro e cosi, ben precisato talchk non ci restava che metabolizzare cib che ci veniva detto, senza speranza alcuna di poter un giorno noi contribuire ad una qualche sia pur piccola aggiunta o chiosa alla materia. Naturalmente in quell’epoca, giovane studente del secondo anno del corso di laurea in Fisica, non immaginavo quali orizzonti e quali mari si sarebbero poi aperti alle possibilit& conoscitive successive, come del resto credo sia accaduto un pochino a tutti noi. Anche se a differenza di altri in questa Sala io non sostenni l’esame di Meccanica Razionale con il Professor Rionero, ho voluto con lui in questa occasione condividere questi miei tutt’altro che sfocati ricordi risalenti, come ho detto, al lontano anno 1960. Poi k “volato” un ventennio, al termine del quale, esattamente nel Novembre 1981, io ebbi il privilegio e l’onore di aderire all’allora Istituto Renato Caccioppoli, poi diventato l’attuale Dipartimento di Matematica e Applicazioni: da allora, per circa, ventitre anni, il Professor Rionero ed io abbiamo avuto mod0 di conoscerci sempre pik approfonditamente, di interagire e di collaborare su tante iniziative. Desidero ricordare che si k lavorato molto spesso e bene insieme anche nel period0 in cui io fui Presidente del Consiglio del corso di laurea in Matematica durante il quale realizzammo una riforma interna che, su mio stimolo, diede maggior peso alla Fisica Matematica nel contest0 del curriculum generale. Successivamente abbiamo collaborato in tante altre occasioni che sarebbe qui troppo lungo elencare. Molto lavoro fu in comune svolto, ad esempio, in occasione del trasferimento del Dipartimento a Monte Sant’Angelo. Certo ricorderai, car0 Salvatore, quante volte con le planimetrie in mano abbiamo studiato che cosa si dovesse fare, che cosa si dovesse modificare, come potesse realizzarsi quel trasferimento che da molti, diciamolo pure, era guardato con grande sospetto e con grande timore. E credo che non poco del nostro tempo k stato anche speso nello studiare insieme aspetti e azioni, probabilmente da qualcuno riguardati come marginali, miranti a definire e consolidare quelle condizioni che sono poi risultate efficaci ed appaganti per la vita e lo sviluppo del Dipartimento. Desidero qui sottolinare come il Professor Rionero, nella mia memoria cosi, come nella mia attuale considerazione, sia persona di estrema, moderazione, grande competenza, eccezionale equilibrio. Termino quindi qui questo mio secondo saluto con l’augurio che per molti, molti anni ancora egli voglia condividere con noi la sua saggezza e la sua dottrina. -

Prof. VITTORIO COT1 ZELATI Direttore del Dipartimeato d i Matematica ed Applicazioni “Renato Caccioppoli” dell’llniversith degli Studi di Napoli Federico 11

Anche io esprimo la mia felicitk per essere qui in questa occasione a portare gli auguri del Dipartimento e miei personali a Salvatore Rionero e a dare il benvenuto a tutti i partecipanti a1 Convegno. Conosco Salvatore da un tempo relativamente breve, in particolare se confrontato con quello dei relatori che mi hanno preceduto, ma mi ha subito colpito la sua capacit& di organizzare la ricerca scientifica: il settore di Fisica Matematica, e in particolare la sua scuola, danno un contributo fondamentale alle attivita di ricerca del nostro Dipartimento. Da quando sono Direttore del Dipartimento, ed in particolare durante le lunghe discussioni in merito alla nuova riforma e ai problemi relativi a1 reclutamento, ne ho molto apprezzato la franchezza e la capacitk di mediazione. Lo ringrazio sinceramente, a nome del Dipartimento e mio personale, per tutto cib che ha fatto e mi auguro che potr& ancora collaborare per molto molto tempo. Grazie Mille.

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Prof. PASQUALE RENNO Decano del settore Fisico Maternatico della Facolt& d i Ingegneria d i Napoli dell’Universitk degli Studi d i Napoli Federico II

La personalitb scientifica di Salvatore Rionero e la sua intensa attivita di gestione di Enti Nazionali di ricerca sono state gii ricordate da illustri Colleghi. 10 mi limiterb a delineare solo alcuni aspetti significativi dell’impegno che egli ha profuso nell’ambito della Scuola matematica napoletana, a partire dalla seconda meti degli anni cinquanta. Nel lontano anno accademico 1956-57, frequentai il Corso di Esercitazioni di LLMeCCanicaRazionale con elementi d i Statica grafica ” tenuto dal giovanissimo “Assistente” Salvatore Rionero. Con orgoglio, posso quindi affermare di conoscere Salvatore da oltre quarantasei anni. Come se fosse ieri, ricordo ancora alcuni argcmenti delle sue lezioni e l’estrema chiarezza di esposizione. Successivamente, negli anni sessanta, ebbe inizio la mia carriera universitaria ed iI lungo period0 di amicizia con Salvatore. Egli, allora, era “ Libero Docente ” ma sopratutto era libero come uomo perchi: riservava ai giovani assistenti la stessa cordialitb e rispetto che aveva nei confronti dei cattedratici. Negli anni settanta, dopo aver vinto il concorso a cattedra presso 1’Universitb di Bari, rientrb a Napoli, dove trovb un ambiente ben diverso da quello che aveva lasciato, a causa della contestazione che si era manifestata in quegli anni, in tutte le sedi universitarie. Nonostante cib, operando con grande equilibrio e disponibiliti, riusci a dare notevole impulso alla ricerca di molti giovani, creando cod una Vera Scuola. Per quel che mi riguarda, devo dire che non c’i: mai stata alcuna collaborazione di ricerca tra me e Salvatore, perchi: la mia attivitb si i: sviluppata interamente nell’ambito della Facolta di Ingegneria. Tuttavia, il nostro rapport0 di amicizia si B rafforzato sempre pih, essendo caratterizzato da stima e fiducia reciproca. Anche quando vi sono state alcune ombre causate da vicende strumentali esterne, Salvatore mi ha dimostrato grande lealta, aiutandomi a superare qualche rnomento difficile. Inoltre, i:doveroso ricordare il ruolo svolto da Salvatore Rionero nella istituzione e sviluppo del Dipartirnento di Matematica e Applicazioni (1984), nel quale confluirono gli Istituti di Matematica delle Facoltb di Scienze ed Ingegneria, localizzati in sedi tra loro lontane. Afferirono oltre centoquaranta docenti con impegni didattici e potenziali di ricerca del tutto differenti. La nascita e la programmazione di funzionamento di un organism0 cosi articolato imponevano la scelta di un Direttore capace di grande impegno ed energia, con doti di equilibrio e rigore. All’unanimitk, venne eletto Salvatore che riusci ad organizzare e porre le giuste premesse per lo sviluppo di un Dipartimento di cui oggi tutti noi siamo orgogliosi. I1 prestigio scientific0 di Salvatore Rionero i: ben noto a livello internazionale. Nell’ambito della produzione pih recente, mi piace ricordare la monografia “Qualitative Estimates for Partial Differential Equations, scritta in collaborazione con il Professore James N. Flavin. Con la sua ampia bibliografia e la rassegna di problemi e metodi attuali, essa costituisce anche una guida eccellente per la ricerca

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dei giovani. Inoltre, la sua lettura mi ha permesso di individuare alcuni aspetti degli interessi scientifici di Salvatore che non conoscevo. In tale occasione, ebbi l’idea di proporgli una questione che non riuscivo a risolvere sulla buona posizione di un problema con frontiera mobile per un operatore parabolic0 di ordine superiore. In poco tempo egli dimostrb un teorema di unicith, confermando cosi’ la validith delle condizioni a1 contorno suggerite dalla realth fisica. Una ulteriore riflessione i: poi la seguente. Sono convinto che il principio di azione e rearione rende comprensibili anche alcuni aspetti del comportamento umano; ad esempio, l’ingratitudine ( o peggio ) quale reazione ad un beneficio ricevuto. Ebbene, le numerose esperienze e vicende di Salvatore hanno rappresentato un osservatorio speciale per la definitiva conferma di questa mia convinzione. ~

Infine, vorrei concludere questa testimonianza ricordando due confidenze che un nostro Maestro, il Professore Dario Graffi, ebbe a farmi in occasioni diverse: li Ho a w t o mod0 di ascoltare, per la prima volta, u n a conferenza di Rionero e devo dire che sono stato davvero colpito dal rigore e dalla estrema chiarezza; t stata senz’altro tra le conferenze piu’ limpide che io ricordi.”

‘‘ Mi ritengo soddisfatto della m i a lunga attivita’ di ricerca e d i alcuni risultati conseguiti. P e r queste ragioni n o n ho m a i provato invidia per alcuno. I1 solo motivo di invidia t l’idea di Rionero sulle applicazioni della funzione peso.” In conclusione, desidero esprimere e ribadire a Salvatore Rionero grande stima ed amicizia, insieme agli auguri piu’ belli.

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ACKNOWLEDGEMENTS

Deeply moved, I thank sincerely those who attended the conference, in particular Prof. G. Trombetti, the Rettore Magnifico of the University of Naples Federico 11, Prof. A. Di Donato, the Dean of the Faculty of Science of the University of Naples Federico 11, Prof. C. Sbordone, the President of U.M.I., Prof. A. Figb Talamanca, the President of I.N.d.A.M., Prof. M. Primicerio, the President of S.I.M.A.I., Prof. L. M. Ricciardi, the vice President of Polo delle Scienze of the University of Naples Federico 11,

Prof. M. Capaccioli, the Director of Capodimonte Astronomical Observatory, Prof. V. Coti Zelati, the Head of the Department “R. Caccioppoli” of the University of Naples Federico 11, those who gave talks, Italian and foreign, and all my colleagues. The esteem and affection shown to me were entirely due to their generosity.

I am indebted to the Institutions which provided the financial support and in particular to University Federico I1 - Naples(Italy), Accademia di Scienze Fisiche e Matematiche of Naples, National Group of Mathematical Physics of “Istituto di Alta Matematica” Department of Mathematics and Applications “R. Caccioppoli”. A special thank is addressed to Prof. P. Fergola and to all the Conference organizers.

S. Rionero

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