Rend. Sem. Mat. Univ. Poi. Torino Fascicolo Speciale 1988 Hyperbolic Equations, (1987)

Tommaso Ruggeri

THERMODYNAMICS AND SYMMETRIC HYPERBOLIC SYSTEMS

Introduction

The aim of this paper consists to give a brief survey on some recent results concerning the relations between mathematical problems for a generic quasi-linear hyperbolic systems with a convex density of entropy and a new possible model of the non equilibrium thermodynamics in continuum theories.

In particular we show how the system of balance laws can be symmetrized by use of an appropriate choice of the field variables (main field). By using this kind of technique we obtain results on shock wave theory and moreover it is possible to construct in a naturai way new hyperbolic systems of fields equations for non equilibrium theories that are compatible with an entropy principle.

We consider a generic quasi-linear first order system of the type:

A° (u) bu/bt + ^ ( 1 1 ) 011/9^.=/(11)

for the unknown Af-column vector u(t, x) : u = (ul> u2ì ..., uN)T ; x belong-ing to Rn ; A0, A1 are real N X N matrices which are functions of u, the source term f (u) is also a vector ofRN, T indicates the transpose and the Einstein convention on the index is used.

By identifying the time t with x°, it is possible to write the system in the abbreviated form :

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(1) ^ a ( u ) 5 a U = f(u) (òa = Ò/ÒX°; X°=t; CK=0, 1, . . . ,» ; 1= 1, ..., »)

Definition of hyperbolicity:

The system (1) is hyperbolic in the ^-direction, if:

i) det A0 * 0 , ii) v€. Rn : || "v\\ = 1, the following eigenvalue problem

(Av-\A°)x = 0

has only real proper values X (chàracteristic velocities) and N linearly inde-pendent eigenvectors r (Av -A1 vi).

Definition of conservative system:

The system (1) is said to be "conservative" (or better: a system of balance laws), if there exist four vectors F a of RN, such that the matrices A01 are gradients of F a with respect to a suitable field u : Aa = 9F°73u, i.e. exhibit the forni:

(2) 3 a F a ( u ) = f(u) .

We observe that a system of type (2) is very common in Mathematical Physics, because it represents the locai form (when differentiability conditions hold) of an integrai balance. On the other hand it is well known that for this kind of system it is possible to consider a more general class of solutions: "the weak solution", for which continuity is not necessary. Shock waves, that are very important in some physical problems, belong to this class.

Definition of symmetric hyperbolic systems:

A system of type (1) is said to be symmetric if

i) A* = (A01)7 ; ii) A0 is positive definite; (from linear algebra any symmetric system is automatically hiperbolic).

The symmetric system plays an important role concerning the well-posing. of the Cauchy problem; in fact there exist some theorems ensuring existence, uniqueness and continuous dependence for the locai Cauchy problem. For

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instance we quote the result due to Fischer and Marsden [1] in 1972: "Any symmetric system, the initial data of which are chosen in a Sobolev space IP with s > 4 , has a unique solution u G / f , in the neighbourhood of the initial manifold".

The reader who is interested may also see, in a more general context, the important result by Volpert and Hudiaev [2].

Supplementary conservation law (entropy principle):

In 1971 Friedrichs and Lax [3] formulated the following question: which type of system (2) is compatirle with the following assumptions?

i) ali the solutions of (2) also satisfy another supplementary conservation law:

(3) a a * ° ( u ) = s(ii) ,

ii)by choosing u = F°, h° is a strictly convex function of u in a convex domain DCRN.

Under these hypotheses they proved that there exists a positive definite N X N matrix H (u), such that the new system

(4) H(u). {daF"(u)-f(u)} =0

obtained from (2) times H, is a symmetric hyperbolic one. The important result of these authors has, however, the following desadvan-

tages:

1) The conservative form of the new symmetric system (4) is lost; 2) We have not an explicit expression for the functional dependence of F01

with respect the field u .

The former restriction doesn't allow the use of the peculiar properties of symmetric systems concerning weak solutions and, in particular, shocks. In fact, many systems of evolution in the macroscopic theories of continua are written in the form (2), but the functional dependence of F a with respect to the field u is completely assigned only when the constitutive relations are known. One of the most important problems is to establish the general criteria for constitutive relations.

In the modem approach of thermodynamics the entropy inequality is regarded as a constraint on the constitutive relations and not as an identifi-

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cation of a privileged time orientation. First sùpporters of this point of view were Coleman and Noli [4]. Later, a more general principle was proposed by Mùller in [5].

The situation is substantially the following: both the entropy density and the entropy flux are thought of constitutive relations for the set of field variables and for ali thermodynamical processes i.e., for ali the solutions of the partial differential system, the following inequality must be identically satisfied

bt pS + 3,- <£>* = s > 0 (p is the mass density) .

In the limit of the constitutive functions that locally depend on the field, this problems is completely the same as the previous one, through the identifica-tions h°=-pS, and !?*= — & .

In this case it is important not only to have a symmetric system but also to give an explicit characterisation of functional dependence of F a and ha

with respect to the field u and therefore a complete characterisation of the compatible constitutive equation.

Now we quote a theorem that gives a different proof to the one by Frie-drichs and Lax. It is free from the previous objections and at the same time, permits a connection with the entropy principle.

Main field and Generators of a Symmetric System: "Entropy Theorem".

Theorem: "A necessary and sufficient condition for a conservative system

. 3 a F a = f

to be compatible with a supplementary conservation inequality

dah«=g<0,

is that the constitutive relations are ali the ones for which four scalars h,<x

and a privileged set of field variables u', exist such that:

(5) Fa = dh'a/òu' ; u ' - f = g < 0 .

Thè components of the "main field" u' are the "Integrai factors", such that:

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(6) u' 'd¥0i=dhot ,

and the (potentials h'01) are related to ha by:

(7) /b ' a=u'- F a - / ? a .

Moreover if it is possible to choose F° as field u, we have from (6)

(8) u' = òh°/òu ,

and if b° is a convex function of u = F° (or likewise, if h'° is a convex function of u'), then the originai system is a very special symmetric one in the field u':

(9) ( d 2 £ ' a / d u ' d u ' ) 3 a u ' = f .

Therefore the locai Cauchy problem is well posed". Substantially, this theorem guarantees that there exists a field u' which

symmetrizes the originai system without losing the conservative structure and with respect to this field, the system assumes the special form (9)-We observe that ali matrices in (9) are Hessian ones and h'° is the Legendre conjugate function of b° (see (7), (8) for a = 0).

The systems pf type (9) have been considered first by Godunov [6] in 1961, who observed that the system for a perfect fluid and the Euler-Lagrange equa-tions are susceptible to this form. Subsequently the same author noted that the systems of this form (9) always admit an equation (3) as a consequence [7].

The introduction of the field u' through the eq. (8) for a generic hyperbo-lic system is due to Boillat [8] in 1974.

A more general approach, in which u' is introduced in a covariant forma-lism and new properties appear is due to Ruggeri and Strumia [9] in 1981 and a general proof which include also non-hyperbolic systems was presented recently by Ruggeri in [10].

Definition of "Generatore":

The systems of type (2) with the supplementary conservation law (3) are completely determined if we know the 2ÌV + 4 objects: {u', h'a, f} . In

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fact from (5.1) we determine F a and from (7) and (5.2) ha and g. There­fore we cali these quantities "generators".

Now it is necessary to distinguish two different kind of problems.

Problem I

If one knows the supplementary conservation law and the constitutive rela-tiòns, it is very easy to evaluate the generators from (8) and (7) and therefore the symmetric form of the system. In this situation it is possible to use the special structure of the system (9) and prove some important theorems on shock waves.

Problem II

If we regard the supplementary law as an "entropy principle" the theorem becomes only qualitative, because "a priori" u' and h'a are not known. In other words we know that the system is in the form (9), or equivalently

(10) a a(d£ 'V3u') = f,

but what is not evident are the relations between the quantities u' and b'u

and the physical quantities appearing in the problem. In the next section, we present first some recent results on shock waves

related to problem I and then we present some possible approaches to the second problem.

Consequences for shock waves

Under the hypotheses of the previous theorem, it is possible to prove some interesting properties on shock waves :

i) the shock velocities s are proved to satisfy the inequalities:

(11) inf minX (*><s< sup MaxX(*> u £ D k uGD k

l

where X are the characteristic velocities. This theorem (Boillat and Ruggeri [11]), ensures that the Rankine-Hugo-

niot equations:

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[F«(u) ]0 a =O,

([ ] denotes the jump; $(xa) = 0 is the equation of the shock surface T; 0a = òQ/òx01 and »,- = 0/1 01, s = - 0 , / | v 0 l are respectively the unit nor-mal and the velocity of the shock surface), admit only the trivial solution [u] = 0 if (11) is not satisfied. Therefore non-vanishing shocks take place only if their speed is greater than the smallest characterstic speed and smaller than the greatest one.

A covariant version of this theorem is stated in [9] and as a consequence in a relativistic theory | s | < c , where e is the light velocity in vacuum.

ii) Formally the Rankine-Hugoniot eqs. are obtained from the field eqs. (2) througri the correspondence rule ba -* [ ] 0 a . However this rule does not work when it is applied to the supplementary eq. (3); in fact

(12) T7 = [/?a]0a on T

does not, in general, vanish. It is known that the positive signature on T? for a fluid implies the growth of thermodynamic entropy across the shock. This is the reason why the condtion 17 > 0 is often called in the literature the "entropy growth condition". Here, this criterion picks up the physical shocks among the solutions of the Rankine-Hugoniot eqs. instead of selecting the constitutive eqs., as in the differentiable case. Usually this growth is proved through artificial viscosity considerations [12]; a proof which uses only convexity arguments for ^-shocks is made in [13], [9].

iii) The mere knowledge of the scalar function 17 on T is enough to determi­ne the jump of the field u': 17 behaves as a sort of "potential" for the shock [13].

Example: Perfect Fluid

As a test of the previous considerations, we present the simple and well known example of the eqs. of a perfect fluid.

The eqs. in conservative form are:

òtp + div (pzT) = 0 (mass balance)

(13) bt (piT) + div T= 0 (momentum eq.)

òte + div {(e + p ) v] = 0 (energy balance)

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where T = pv ® v+pl; e = p(e + v2/2) and p, v, e, p are respectively the mass density, velocity, internai energy and pressure.

In this case the supplementary law is the entropy balance:

(14) òt (pS) + div (pSv) = 0 .

The system (13) and the eq. (14) are equivalent to (2) and (3), through the following identifications:

F°=(p,pv',e)T; F|, = ( p t ^ r M e + p ) * , , ) r ; f = 0;

h° =-pSyhi = -pSvi\ g = 0 .

In this case, simple calculations give the following generators:

u =—-(G-z> 2 /2 ,^", - l ) T ; h'0=p/9; bH = ptf/0 ,

where 0 and G are respectively the absolute temperature and the free enthalpy (the chemical potential: G = e — 6S + p/p).

The convexity of h° = — pS can be proved if G{p,B) is a concave fune-tion with respect to (p, 0), i.e.:

et > 0, (3 V/òp)d < 0 (K= 1/p, etì is the specific heat at Constant pressure.

These conditions are the usuai thermodynamic stability conditions at equilibrium.

Therefore the system (13) is symmetric in the main field u' and allpre-vious considerations hold (on this subject see [6], [14], [15]). Several other examples in this area are considered in the recent literature viz: hyperelastic solids [16]; relativistic fluids and magnetofluids [9], [17]; non linear electro-dynamics [18], [19], [20]; superfluids [21]; mixture of fluids [22].

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Construction of Systems of Balance Laws that satisifes the Entropy Principle: The Extended Thermodynamics

The physically significant examples examined so far were pertinent to the first problem, where the supplementary law was "a priori" known togethér with the associated constitutive equations. For instance, in the previous case of a perfect fluid we have followed the classical theory; in other words the Gibbs relation was supposed to hold and the concept of absolute temperature, entropy and internai energy was "a priori" accepted. The aim was to obtain anew symmetric hyperbolic conservative system in order to have qualitative information on the solution (well posedness of the Cauchy problem, some properties of shock waves and se on ...). Alternatively if we look at the pro­blem from the point of view of "rational thermodynamics", we don't accept these concepts as primitive ones and by considering the entropy inequality as a constraint òn the constitutive equations, the Gibbs relations emerges auto-matically as an integrability condition, and the absolute temperature as an integrating factor. Clearly this approach provides a qualitative jump from the view point of the axiomatic thermodynamics, even though it does not change the practical results. Moreover, sometimes the true equations are not known "a priori" because they represent new models of some physical phenomena. For instance, for a viscous, heat conducting fluid, the problem is ot determine the constitutive relations for the heat flux q and for the viscous tensor

<» = (<ty). In the classical approach these are the Fourier and Navier-Stokes equations

which, as it is well known, destroy the hyperbolicity of the field system with the consequence that an infinite wave propagation speed is obtained.

In order to eliminate the paradox and hyperbolize the system, in the lite-rature we have a lot of papers that starting from the pioneer work of Cattaneo [23] gives evolution equations for q and a by preserving the hyperbolicity of the entire system. Unfortunately these papers contain many unmotivated hypothesis from the physical point of view and several questions are unclear from the mathematical stand point. Starting from these observations the main ideas of a new procedure to construct hyperbolic system for dissipati ve problems were presented in the papers [24], [25], [26] and now this approach is called Extended Thermodynamics.

We give a brief survey of this subject and we explain the physical and mathematical motivations. In particular we present the important role of the previous entropy theorem to establish the new evolution equations.

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The Extended Thermodynamics

First of ali we quote several motivations that need hyperbolic system for these kind of problems :

i) propagation problems: for monoatomic classical gases we have experimen-tal data that gives finite velocities [27]; for relativistic gases we have of course the relativity principle that requests finite velocities.

ii) For monoatomic gases using the Kinetic Theory in the 13-moment theory we obtain a set of hyperbolic equations (Grad [28]).

iii) In the Grad system the equations for the heat flux and the viscous stress does satisfies the Objectivity principle (Mùller [29]) that is used as usuai principle for constitutive equations. This fact was interpreted by Ruggeri [30], [31] and Bressan [32] to mean the new equations are real new balance laws with the same role of the mass, momentum and energy equations.

iv) We have several situations in the continuum approach for the physics of low temperature as superfluid and phonons for rigid conductors where an hyperbolic theory is requested.

Previously we recalled the first example due to Cattaneo for a rigid conduc-tor. In fact in the classic case we have the energy equation:

e + div q = 0

supplemented by the "constitutive" Fourier Law:

that gives the well know parabolic heat equation:

In the Cattaneo approach the energy equations is supplemented by the rate equation:

r q + q = - x v #

that gives the hyperbolic Telegraphist equation:

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(e is the specific heat, ù is the absolute temperature, x *s t n e n e a t conducti-vity).

Starting from the previous considerations, the assumptions of the extended thermodynamics for classic fluids are the following:

1) The state of the system is identified by 13 fields, i.e.

(15) u = (p, e, v, a, q)T

2) The system for the unknown field u is a system of 13 Balance laws:

bp Ò T ~ + - T - T ( P ^ ) = O bt bxl

bpvi b

bt ÒX* KP % i V

(mass balance)

(momentum)

be b (16) — - + — — (e Vi + q{ - Uj Vj) = 0 (energy)

ÒW: Ò L + ^*::=P: bt bx* tJ J

dtyk) b (new flux balance)

bt +^rMm>'Nv*>

where ty -p 6/y — Oty is the stress tensor, o= \<JÌJ\ is the viscous deviator tensor (o^—oy,-; tr a - 0), e = p(e + v2/2) is the total energy and the brackets < > indicate traceless tensor.

3) The quantities that does appear in the list of variables u are thought as constitutive quantities

<17> vs(p. "/. % rv*„ Mm), pf, Nijk)) T

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that we suppose related through the constitutive equations to the field in a locai mànner, i.e.:

(18) V=V(u)

4) The restrictions on the constitutive quantities (18) come only from univer-sal principels, i.e.:

i) Entropy Principle: There exists a density of entropy - h° and an entropy flux - hx that are constitutive quantities

(19) h°=h°(u), & ' s # ( u ) ,

such that for ali therfnodynamic processes (solution)

òb° oh1

ii) Objectivity principle: the system (16) is Galilean invariant and the consti­tutive equations (18), (19) are independent of a generic observer.

iii) h° is a strictly convex function of the conserved quantities.

A brief remark about these assumptions: in the assumption 1) we extend the usuai state of the system with the non equilibrium variables q and a and therefore we extend also the balance laws with the condition 2): a yecto-rial equation in correspondence with the new variable q and a tensorial traceless equation in correspondence of a. Moreover, the hypothesis of locai constitutive equations guarantees that our system is a first order quasi-linear system of balance laws. The condition iii) ensures that the system is symme-tric from a mathematical point of view and also the stability condition of maximum of entropy in equilibrium. Therefore the problem is now an usuai constitutive one, i.e. the determination of the class of constitutive equations (18), (19) that are compatible with the universal principles stated in the assumption 4).

Of course even if the problem is very simple from the conceptual point of view it is very hard from the practical point of view. In this case it is very helpful to use the very special structure of the systems that are compatible with an entropy principle, i.e., the structure (10).

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To explain this technique in a very simple manner we give now the detail in the more simple case of a rigid conductor and we quote for the interested reader the following references for the previous case and the analogous cases: [10], [31], [33] for the basic problem, [25], [34], [35] for classic gases, [26], [36] for the relativistic case, [40] for solids and [10], [37], [38] for rigid conductors.

The Rigid Conductor

Following the previous scheme in this case the state of the system is identi-fied by the pair (e, q) and the balance laws and the entropy principle beco-mes in this case:

( bte + di q, = 0 (21)

(22) òtb° + 9 , ^ 0 .

The unknown constitutive equations are:

w. = w. (*, q) ; faj = $ij(e,q) (23) ' '

h° = h° {e, q); tt = b* (e, q); Pj = P- (e, q) .

In this cas , we have:

For the "Entropy Theorem" there exists the main field

u' = (£, Ay)

and the potentials h'°, h'% such that (5) and (7) hold. Therefore in this case we have :

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(25)

(26) òu1

(27)

(28)

F' = òu'

Òh'° e — -n

bh'° ii}. — 9A;.

n* f f c - n

òhH

ÒAj

• M'° bh'° . (29) l b°=l—r- + A / - ^ I h

ba = u' • F* - ti01 3£ ' ÒAj

(30) / é . . É _ _ + i S » ,

(31) u - f < 0 => s = -AjPj>0 .

The objectivity principle provided that h'° and b'f are isotropie functions of u' and then :

h'° = h'° «, z); bH = <t>(t 2) A1' ; (2 = A2 /2).

To go back to the usuai physical variables e, q it is necessary to solve a non linear invertibility problem.

In fact form (25), (27) it is possible, at least in principle, to obtain:

(32) £ = £(*, q); A = À(e,q),

and inserting (32) into (26), (28)-(31), we determine the acceptable costitu­tive equations (23). We note that invertibility is ensured by convexity argu-ments.

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In the more interesting case of a theory valid in a neighbourhood of the equilibrium state this problem can be resolved explicitly. In fact by definition equilibrium is the state for which the entropy source (31) s reaches the minimum value zero. After some easy arguments this implies that:

M E = 0 , È I E = - 1 / 0 .

Therefore in the neighbourhood of equilibrium, we put:

(33) ¢ = - 1 / 0 + /i; A = X,

where fi and X are small perturbations. Inserting (33) into (25)-(30), the map (25), (27) becomes linear and the constitutive equations (32) are easily obtained.

The final results are that the system (21) becomes :

(34) 8 te + divq = 0

(35) *t{q}H' &Y+*iKP) = -l'qjlX

where l(ù) is an arbitrary function of the absolute temperature. For the choice / (#): l'#2 =const., (35) is just a Cattaneo equation with an appro­priate relaxation time T(#):

7=const. x W d 2 .

Substantially this proof is contained in the paper [10]. In [37] the proof of convexity and consequences on wave propagation are established and finally in the paper [38] an appropriate choice of /(#) is made such that the wave velocities are in agreement with recent experimental data [39], [40].

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REFERENCES

[1

[2

[3

[4

[5

[6

[7

[8

[9

10

11

12

13

14

15

16

17

18

19

20

A. Fischer and D.P. Marsden, Comm. Math. Phys. 28, 1 (1972).

A.I. Volpert and S.I. Hudiaev, Math. USSR Sbornik 10, 571 (1972).

P.D. Lax and K.O. Friedrichs, Proc, Nat. Acad. Se. USA 68,1686 (1971).

B.D. Coleman and W. Noli, Arch, Rat. Mech. Anal. 17,1 (1963).

I, Mùller, Arch. Rat. Mech. Anal. 40, 319 (1971).

S.K. Godunov, Sov. Math. 2, 947 (1961).

S.K. Godunov, Russ. Math. Surv. 17, 145 (1962).

G. Boillat, C.R. Acad. Se. Paris, 278-A, 909 (1974).

T. Ruggeri and A. Strumia, Ann. Inst. H. Poincaré 34, 65 (1981).

T. Ruggeri, Suppl. B.U.M.I. Fisica Matematica, 4 (5), 261 (1985).

G. Boillat and T. Ruggeri, C.R. Acad. Se. Paris 289-A, 257 (1979).

P.D. Lax, in contribution to non linear functional analysis; Zarantonello (ed.) (Aca-demic Press, New York 1971).

G. Boillat, C.R. Acad. Se. Paris 283-A, 409 (1976).

D. Fusco, Rend. Sem. Mat. Modena 28, 223 (1979).

T. Ruggeri, Appunti di propagazione ondosa, 6a. Scuola Estiva di Fisica Matematica del CNR. - Dipartimento di Matematica - Università di Bologna (Sett. 1981).

G. Boillat and T. Ruggeri, Acta Mech. 35, 271 (1980).

T. Ruggeri and A. Strumia, J. Math. Phys. 22, 1824 (1981).

G. Boillat, C.R. Acad. Se. Paris 290-A, 259 (1980).

A. Strumia, Lett. Nuovo Cimento 36 (17), 569 (1983).

G. Boillat and G. Venturi, Il Nuovo Cimento 77-A (3), 358 (1983).

183

[21

[22

[23

[24

[25

[26

[27

[28

[29

[30

[3-1

[32

[33

[34

[35

[36

[37

[38

[39

[40

G. Boillat and A. Muracchini, ZAMP 35, 282 (1984); 36, 901 (1985).

N. Vigorpia and F. Ferraioli, Il Nuovo Cimento 81-B, 197 (1984).

C. Cattaneo, Atti Sem. Mat. Fis. Modena 3, 83 (1948).

T. Ruggeri, Acta Mech. 47, 163 (1983).

I-Shih Liu and I. Mùller, Arch. Rat. Mech. Anal. 83 (4), 285 (1983).

I-Shih Liu, I. Mùller and T. Ruggeri, Ann. of Phys. 169 (1), 191 (1986).

M. Greenspan, J. Acoust. Soc. Am. 28 (4), 644 (1956).

H. Grad, Comm. Pure Appi. Math. 2, (1949).

I. Mùller, Arch. Rat. Mech. Anal. 45 (1972).

T. Ruggeri, Rend. Sem. Mat. Univ. Padova 68, 79 (1982).

A. Morrò and T. Ruggeri, Propagazione del Calóre ed Equazioni Costitutive, 8a. Scuola Estiva di Fisica Matematica del C.N.R. (Pitagora, Bologna 1984).

A. Bressan, Lett. Nuovo Cimento 33 (1982).

I. Mùller and T. Ruggeri (Eds.), ISIMM Symposium on Kinetic Theory and Extended Thermodynamics, Bologna maggio 1987 (Pitagora ed. 1987).

I. Mùller, Thermodynamics (Pitman - Series ISIMM, Boston-London, Melbourne 1985).

G.M. Kremer, Ann. Inst. H. Poincaré 45, 401 (1986).

T. Ruggeri, Relativistic Extended Thermodynamics. Corso CIME (Noto 1987), Lecture Notes in Phys. Springer-Verlag (to be published).

A. Morrò and T. Ruggeri, Int. J. of Non-linear Mech. 22 (1), 27 (1987).

A. Morrò and T. Ruggeri, Nonequilibrium properties ofsolids through second sound measurements. J. of Phys. C: Solid State Phys. 21,1743 (1988).

H.E. Jackson, C.T. Walker and T.F McNelly, Phys. Rev 25, 26 (1970).

B.D. Coleman and D.R. Owen, Comp. Math. with Appls. 9, 527 (1983).

Tommaso RUGGERI - Dipartimento di Matematica Università di Bologna (Italy).