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 ) 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 ) ,
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
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 ) " ] .
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 / 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 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 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 ~~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 ; ~ + 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 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 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) 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).
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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
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)
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, 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
