Proceedings ''WASCOM 2003'' 12th Conference on Waves and Stability in Continuous Media

31 (a) 3S(x)

= T — limSn(x),

for all x £ Ao, which is a *-derivation

of

A0; (b) 5-K, the *-derivation induced by IT, is well-defined and spatial; (c) if H is the self-adjoint operator which implements Sn, if < (Hn — H)£o>£ >—> 0 for all £ £ D^ then Hn converges weakly to H. This result can be seen in the following way: even if the sequence {Hn} of operators which implement {Sn} does not converge, t h e sequence Sn(X) may converge in the topology r . This is w h a t happens, for instance, to mean field spin systems, when X is a local operator 1 2 . Therefore, this limit defines a derivation 6 which, under the assumptions of Theorem 3.2, is spatial. This means t h a t an effective hamiltonian H***, in t h e sense of 7 , can be denned, and this simplifies t h e successive analysis of t h e finite time evolution of the system 5 . We recall here t h a t a model admits an effective hamiltonian in the ^ r e p r e s e n t a t i o n IT in the sense of 7 , if there exists a self-adjoint operator Htf* in HKO Vn) with the property n(S(A)) = i [HeJf\ir(A)] , Vv4 6 A0. This equation is understood in the following weak sense: < TT(5(A))4>,

ij> > = % {< 7r(i4)& EeJsi)

> - < H^ff, TT(A*)^

>} ,

for all 4>,if>e D(H%ff) and for all A € A0. T h e conclusion, therefore, is t h a t Theorem 3.2 produces a sufficient condition for a model to admit an effective hamiltonian, so t h a t all the results in 7 concerning t h e integration of a derivation can be applied here. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

R.Haag and D.Kastler, J.Math.Phys. 5, 848 (1964). F. Bagarello, J. Math. Phys. 39, 2730 (1998). F. Bagarello, Publ. RIMS, Kyoto Univ. 37, 397 (2001). F. Bagarello and C. Trapani, J.Stat. Phys. 65, 469 (1991). F. Bagarello and C. Trapani, II Nuovo Cimento B 108, 779 (1993). F. Bagarello and C. Trapani, Ann. Inst. H. Poincare 6 1 , 103 (1994). F. Bagarello, C. Trapani, J. Math. Phys. 37, 4219 (1996). F. Bagarello, C. Trapani, J. Math. Phys. 4 3 , 3280 (2002). D.A. Dubin, G.L. Sewell, J. Math. Phys. 11, 2990 (1970). G. Lassner, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30, 572 (1981). G. Morchio and F. Strocchi, J. Math. Phys. 28, 622 (1987). F. Bagarello and G. Morchio, J.Stat.Phys. 66, 849 (1992). W. Thirring and A. Wehrl, Commun.Math.Phys. 4, 303 (1967). S. Sakai, Operator Algebras in Dynamical Systems, Cambridge Univ. Press, Cambridge, 1991. 15. F. Bagarello, A. Inoue, C. Trapani, Derivations of quasi *-algebras, submitted to Int. Journ. Math, and Appl. Math.

CHAOS IN SOME L I N E A R KINETIC MODELS

J. BANASIAK School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, South Africa e-mail :banasiak@nu. ac.za

1. Introduction Contrary to the common knowledge, chaotic behaviour occurs not only in nonlinear systems but it also can happen in the linear ones, however, for this they must be infinite-dimensional. This phenomenon has been recently investigated in a series of papers 1 2 , 7 ' 1 1 ' 3 , 2 . Below we recall below the definition of topological chaos that was formulated in 8 and that is relevant to our considerations, introducing first some basic notation. Let (x(t, -))t>o be a continuous dynamical system in a metric space (X, d). By 0(p) = {x(t,p)}t>o we denote the orbit of (x(t, -))t>o, originating from p. We say that (x(t, -))t>o is topologically transitive if for any two non-empty open sets U,V C X there is to > 0 such that a; (to, U) n V ^ 0. A periodic point of (x(t, -))t>o is a n Y point p £ X satisfying x{T,p) = p for some T > 0. Definition 1.1. 8 Let X be a metric space. A dynamical system (x(t, -))t>o in X is said to be (topologically) chaotic in X if it is transitive and its set of periodic points is dense in X. Devaney's definition was introduced with nonlinear dynamical systems in mind but it turned out that it is very closely related to the property of iterates of linear operators investigated in e.g.9 that is called hypercyclicity: a bounded linear operator A o n a Banach space X is called hypecyclic if for some x 6 X we have {Anx}n>o = X or, rephrasing this in the language of dynamical systems, there is an orbit of the discrete dynamical system generated by A that is dense in X. In 9 the authors proved a theorem that A is hypercyclic if and only if it is topologically transitive, and this theorem can be easily generalized to continuous systems 7 ' 3 . Thus, we can rephrase

32

33

Devaney's definition saying that (x(t, -))t>o is chaotic if and only if it has an orbit that is dense in X and its set of periodic points is dense. In this note we discuss a generalization of results of7 to cover the case when the semigroup is chaotic only in a subspace of the original space. This result is then used to find the chaoticity space of a system coming from extended kinetic theory, that describes interactions of particles with the background in which particles can only lose energy,9'4. A more exhaustive discussion of this topic can be found in the forthcoming papers 1,5 . 2. Desch-Schappacher-Webb criterion and its consequences Possibly the most widely used set of conditions ensuring that a strongly continuous (Co) semigroup {T(t))t>o generated by an operator A is chaotic is given in the following theorem. Theorem 2.1. 7 Let X be a separable Banach space and let A be the infinitesimal generator of a strongly continuous semigroup (T(t))t>o on X. (T(t))t>o is chaotic if the following conditions are satisfied: 1. The point spectrum of A, o~p(A), contains an open connected set U such

that U n iR ^ 0; 2. There exists a selection U 9 A —> x^ of eigenvectors of A such that the function F$>{\) =<<&,x\> is analytic in U for any $ G X*; 3. F$ =0 on U if and only if <3> = 0. A close look at the proof of this theorem shows that if x\ is analytic in some open connected U, then Span{x\,\ e U'} is the same for any U' having an accumulation point in U. Thus, operators with analytic selection of eigenvectors may generate chaotic semigroups in a subspace of the original space X. Precisely, we have 5 Theorem 2.2. Assume that the point spectrum o~p{A) of the generator A of a Co-semigroup (T(t))t>o contains an open connected subset U of C on which there exists an analytic selection X —> x\ of eigenvectors of A. Denote Y — Span{x\,X £ U}. Then (1) ifiM.f]U^9, then (T(t))t>0 is chaotic in Y, (2) ifUc C± with iR (~l dU ^ 0, then the dynamics is unstable in the sense that an arbitrarily small perturbation ± e / of the generator, e > 0, makes the system chaotic in Y.

34

It is important to rule out certain classes of operators from being chaotic in X, or even in any subspace of X. Firstly, for sets M C X and N C X* we denote M x = {/ e X*; = x

0,Vx G M }

iV = {x e X; < / , x > = 0,V/ G AT}.

Then we can formulate a straightforward modification of a result from7. Theorem 2.3. Let (T(t))t>o be a Co- semigroup generated by A in a Banach space X, having a dense orbit in Xh C X. Then the adjoint A* of A and the dual semigroup (T* (t))t>o have the following properties: (1) Let 0 ^ G X*. / / the orbit {T*(t)}t>o is bounded, then ^ e l (2) If <j> is an eigenvector of A*, then £ X^ . Denote E* —

0

E\, where E\ is the eigenspace corresponding to A.

Then we have Corollary 2.1. If codim

E* < +oo,

then the semigroup can not be chaotic in any subspace of X. In particular, if (Jp{A*) = 0, then (T(t))t>o is not chaotic in X. Proof. In this case Xh would be finite-dimensional, which excludes chaos in it. • 3. Chaos in linear kinetic models We shall illustrate the above considerations on an example of a linear kinetic equations describing inelastic collisions in the extended kinetic theory. Here we consider a gas of test particles of mass m endowed only with translational degrees of freedom propagating through a three-dimensional host medium of much heavier particles that may have a quite complicated internal structure and thus non-negligible internal degrees of freedom. Test particles collide with host medium and here we assume that they can only lose energy in each collision4; the discussion of a more general case can be found in 1 . The particle distribution function in the space homogeneous case is governed by the linear Boltzmann equation, see e.g.9

g=C7,

(3-D

1

,

35

where [C* / ] ( w ) = nx ^ut /"2CT(u+, u; • u/)/(v+u/)du/ Js -

nif(vu>)H(v

- S) [

a{v,u>-u>')du>',

(3.2)

is the inelastic collision operator. Here, v — vu) is the velocity variable, with modulus t; and direction u> € S2 (the unit sphere in R 3 ), u+ = \Jv2 + S2, 52 = 2AE/m. Also, a is the inelastic collision frequency for the endothermic process, ni is the number density of the particles in the ground state (which are assumed constant) and H is the Heaviside function. To be able to describe precisely the properties of the chaotic behaviour of the system, we shall consider a homogeneous medium and, to simplify notation, we put n\a = 1. We shall also use the normalized kinetic energy £ = v2/2, normalize the energy jump to 1 and introduce the flux F((, w) = V£f(C,w). Thus, we get dF — (S,v,t)

f = J F(£ + l,u',t)du>'-4irH(C-l)F(C,u>,t). s

(3.3)

2

To analyse the dynamics of (3.3), let us consider the eigenvalue problem related to (3.3). However, it is advantageous to write this problem in the form of an infinite system of equations by introducing the reduced energy £ 6 [0,1[ and defining Fn(£, w) = F(( + n,u>). Note that the norm changes now to \\F\\x = J I J P C \Fn(M\)

dA du>.

(3.4)

Thus we get J Fifo^du'=

\F0(Z,u,),

s2

jFn+l(£,u,')du,' s2

- toFn{£,u>) = \Fn(£,u),

Let us decompose Fn = Fno + Fni,

(3.5)

36

where Fn0 = (47r) _1 J Fndw and f Fnidu> = 0. This splits X into the s2 s2 direct sum and therefore the above system is equivalent to two systems. However, the system for (iJ1ni)neN does not produce anything interesting. Thus, we shall focus on F\o = A i*oo

Fn+1,0 — Fno = A Fno • • •> where A' = \/4ir.

Solving, we see that we can take Foo arbitrary

(with / F00du) ± 0 and Fn0 = (1 + A')n_1A'.Foo- It is clear that if s2 FQ{£) € Li([0,1]), then all the relevant properties of ey = (Foo, A'i*bo> (1 + A')A'i
F$(A) =<$, e A >= aQ + J2 an(l + A')" - ^'.

(3.6)

n=l

To simplify the considerations we denote z = 1 + A', thus we have oo

oo

F$(z) = a 0 + ^ a „ z " - ^ a „ z " - 1 . n—l

(3.7)

n=l

Using the properties of power series, this function is identically zero in \z\ < 1 if an only if the coefficients satisfy an — an+i = 0 for n > 0, which gives an = oo for any n > 0. Thus, condition 3. of Theorem 2.1 is not satisfied. Moreover, clearly the found point spectrum only touches the imaginary axis for A' = 1. Thus we cannot claim that the dynamics generated by (3.1) is chaotic in the whole space. However, condition 2 of Theorem 2.2 is satisfied and from the above analysis we see that the perturbed system ^ F ( C , u , t ) = JF(<; + 1, w',t)du' -

4TTF(C

- 1)F(C«,t)

+

aF((,u,t),

s2 (3.8)

37 with arbitrarily space of L i ( R 3 ) t h a t the system the generator A (remember t h a t

small a > 0 will generate a chaotic dynamics in the subconsisting of functions t h a t annihilate constants. To show cannot be chaotic in the whole space, we find the adjoint of of (3.8). Since the generator is bounded, simple calculation we work in Li(S2 x [0, oo))) gives

(i4*G)(C,w) =

ff(C-l)

JG(C-l,u')du>'-4TrH(C-l)G((,u,)

+ aG(C,u>).

2

s and we immediately find t h a t G((, OJ) = 1 is an eigenfunction corresponding t o the eigenvalue X = a. Thus, by Corollary 2.1, (3.8) cannot be chaotic in the whole space. References 1. J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics in some linear kinetic models, submitted. 2. J. Banasiak and M. Lachowicz, Topological chaos for birth-and-deathtype models with proliferation, Math. Models Methods Appl. Sci., 12 (6), (2002), 755-775. 3. J. Banasiak and M. Lachowicz, Chaotic linear dynamical systems with applications, in: C. Kubrusly, N. Levan, M. da Silveira (Eds), Semigroups of Operators: Theory and Applications, 2nd International Conference, Rio de Janeiro, 2001, Optimization Software Inc., Publications, New York Los Angeles, 2002, 32-44. 4. J. Banasiak, G. Prosali, G. Spiga, Asymptotic Analysis for a particle Transport Equation with Inelastic Scattering in Extended Kinetic Theory, Math. Mod. Meth. Appl. Sci., 8 (5), 1998, 851-874. 5. J. Banasiak, M. Moszyriski, A remark on the Desch-Schappacher-Webb condition for chaotic linear semigroups, submitted. 6. J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99, (1992), 332-334. 7. W. Desch, W. Schappacher and G.F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17, 1997, 793-819. 8. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edn., Addison-Wesley, NY, 1989. 9. C. R. Garibotti, G. Spiga, Boltzmann equation for inelastic scattering, J. Phys. A: Math. Gen. 27 (1994), 2709-2717. 10. G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic manifolds, J. Funct. Anal, 98, (1991),229-269. 11. D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory, 28 (1992), 93-103. 12. V. Protopopescu, Y. Y. Azmy, Topological chaos for a class of linear models, Math. Models Methods Appl. Sci., 2(1), (1992), 79-90.

C H A R A C T E R I Z A T I O N OF T I M E P E R I O D I C B E H A V I O U R B Y PROPER ORTHOGONAL DECOMPOSITION

M. BASURTO Universita di Catania, Dip. Matematica e Informatica, Citta Universitaria, Viale A. Doria, 6, 95125 - Catania E-mail: [email protected] C. T E B A L D I * Universita di Lecce, Dip. Matematica "E. De Giorgi", Via Provinciale Lecce-Arnesano, 73100 - Lecce E-mail: [email protected]

The Proper Orthogonal Decomposition (POD) is used as a very efficient method to find low-dimensional models that accurately describe the periodic dynamics of a iV-dimensional system. Starting from a reduced model obtained by P O D , the possible existence of an unstable equilibrium, "related" to the orbit, can be investigated, the appearance of the periodic behaviour by Hopf bifurcation clarified and the equilibrium loosing stability found. T h e procedure can also be a substantial step toward center manifold and normal form calculations of the Hopf bifurcation.

1. Introduction In the last years the study of many complex systems has taken strong advantage from the theory of dynamical systems and, in particular the ones connected with turbulence in fluids, from the recognition of the existence of coherent structures, i.e. strongly persistent spatiotemporal structures [1,2]. The introduction of the Proper Othogonal Decomposition (POD) has revealed itself as an efficient technique to find, describe and model coherent structures, separating a given data set into orthogonal spatial and temporal modes which most efficiently absorb the data variability. Briefly, the POD consists in the orthogonal decomposition of the correlation associated with the state variable of the system and its representation, optimal in the least square sense, with respect to the orthonormal, complete set of 'corresponding author.

38

39 the eigenfunctions of the correlation operator. Differently from Fourier decomposition, the eigenfunctions obtained from POD can efficiently display possible spatial localizations. Moreover in the experiments POD provides a kind of data assimilation and noise reduction on data [3]. The POD, in connection with Galerkin method, can also be the technique leading to useful low-dimensional ODE approximations of PDE models and allows to take advantage of dynamical system theory. For example, Sirovich and Rodriguez apply POD to Ginzburg-Landau equation [4], Aubry, Holmes, Lumley and Stone study a turbulent boundary layer model [5,6], Glavaski, Marsden and Murray use POD in a compression system model [7]. More recently, Lopez and Hernandez-Garcia provided a low-dimensional system to analyze periodic moving eddies in the western Mediterranean sea [8]. 2. The method The POD technique is generally used to analyze experimental or numerical data with the view of extracting their dominant features, which will typically be patterns in space and time. In input, one has a data field v(x, t) (x G ft space, t > 0 time), to which the temporal average has been subtracted, on output the method provides a set of orthogonal functions which are the eigenfunctions of the covariance matrix of the data. For a more straightforward comparison with the usual procedure, we suppose to deal with a function U = U(x, t), expressed with respect to a complete orthonormal system { $ ( ( x ) } ; g N of (possibly) complex vectorvalued square summable functions in il. In practice the expansion of U is finite and we write: N

U(x,t) = J2XiM*i(x)-

(!)

If [0, T] is the time interval in which U is observed, the time average of U is defined as

(U(x,t)> = i / U(x,t)dt = J2 <*'(*)> *'W 1

Jo

i=i

(2)

!

so the state variable v is v(x,*) = U ( x , * ) - ( U ( x , 0 >

E £ i (Mt) - W ) » * J M = EL M*)*/(x),

(3)

40

i.e. the function U adjusted to zero-mean. Starting from v the two-point spatial correlation N

K(x,y) = (v(x,t)V(y,*)) = £

(bm(t)bn(t))

# m ( x ) • #*n(y)

(4)

171,71 = 1

is formed and the related eigenvalue problem K ( x , y ) - V ( y ) d y = AV(x) (5) n is then considered. The eigenfunctions V are assumed to have the form /

JV

V(x) = ]>> f c # f c (x) .

(6)

k=l

Introducing (6) into the problem (5) we have the corresponding matrix eigenvalue problem C • P = AP ,

(7)

where C = {(bm(t)bn(t))) is a JV x iV real symmetric and non-negative defined matrix and P = (p\, ...,£>#)*. The eigenvalues Ai > ... > Ajy > 0 of C coincide with those of K(x, y) and the eigenvectors P i , . . . , PAT, with P ; corresponding to A;, are orthonormal. At this stage the eigenfunctions V,(x) = £ P f c | , # f c ( x )

(J = 1,2,...,JV)

(8)

fe=i

are completely determined: they form a new orthonormal system of square summable vector functions on f2. The functions V; are referred to as (vector) coherent structures. In terms of (8) U can be expressed as JV

U(x,t) = 5 > ( t ) V , ( x ) +

(9)

with N

r at{t)

=

/ V(x,t) • V,(x) •^

=

J2 Pm,lbm{t) , m=l

(10)

or a(t) = Q*b(t), in which Q = ( P I , P 2 , . . . , P J V ) is an orthogonal matrix, a(f) = (oi(*),...,OA,(<)t and b(t) = {b1(t),...,bN(t))t. The expansion (9)

41 defines the POD of U(x, t), for which some important properties hold. First, POD is optimal in the least square sense, that is M

M

v(x,t)-53o,(t)V,(x)

<

v(x,t)-^^(*)W,(x) 1=1

L*l

(11) £2/

with di(t)=

/ v(x,*)-W,(x)dx JQ,

for any system { W 1 , W 2 , , . . , W J J } of mutually orthogonal vector functions on ft with M
a2S(||v(x,0lli2)=f>?) = X> i=i

;=i

and, from (10) and the definition of C, it immediately follows that N

(aiaj) = X I

Pm,i

(bmbn)Pn,j = P* • (C • Pj) = Xj5i}j .

(12)

m,n—l

Property (12) states that the amplitudes in (9) are uncorrelated. If U represents a fluid velocity field, a2 equals the average energy (E) of v modulo some multiplicative constants. Finally, the eigenfunctions calculation can be posed, in principle, in variational form as a hierarchy of maximum problems • A i = m a x { ( K f , f ) | ||f|| = 1} = ( K V b V i ) , A2 = m a x { ( K f , f ) | ||f|| = l , f ± V i } = ( K V 2 , V 2 ) , XN = m a x { ( K f , f ) | ||f|| = l , f ± V , (/ < N)} =

(13)

(KVN,VN)

related to the Fredholm type operator (Kf)(x)= [ K(x,y)-f(y)dy Jn wich is non-negative and hermitian. In practice, the number M in (11) is chosen in such a way that the approximate expansion M a

U(x, t)*Y, i=i

'(«)Vj(x) + (U(x, t))

(14)

42

captures the variance up to a given percentage. Consequently, it turns out that U(x, t) mainly evolves along V i , V2,..., V M in the time interval [0,T] and the square roots of A; can be regarded as a measure, on average, of the time evolution of v along the "direction" of V;. In [9] the method has been tested in the case of a fluid governed by the incompressible Navier-Stokes equations on a torus with a time periodic forcing. In that paper POD has been performed on a 98-mode model and an attractor, i.e. a stable time periodic solution, has been approximated by the first 6 eigenfunctions. The method has been extensively and succesfully used also in the context of image processing [10]. 3. P O D for finite—dimensional dynamical systems When the fields describing the system under study are represented with respect to a complete orthonormal set, as we have actually considered before, the Galerkin method provides a dynamical system for the time dependent amplitudes governing the dynamics. The implementation of the method turns out to be much less heavy. In the case of finite dimension an ODE system X = F(X,/i)

(XeR^/zeR*)

(15)

is given together with /x = /Uo, a vector of external parameters, and an initial condition X(0) = Xo. Let X = X(t) be the particular solution for tG[0,T],

the usual time average and x = X-(X(t)>

(16)

the state variable associated with X. Prom (16) the correlation matrix N

C = (X • X*) = J2 (X*Xi) ei • e 5 = ((xixi))i,j=l,...,N

(17)

is obtained, where {ej,..., e ^ } is the canonical basis of RN. The matrix C is real, symmetric and non-negative defined, therefore it admits N eigenvalues Ai > ... > AJV > 0 and, correspondingly, N orthonormal eigenvectors that form the vector system V = {Pi,..., PJV}. With respect to V we can write x(t) = Q • a(t) <* X(i) = Q • a(t) + (X(t)> (POD of X(t)) a(i) = Q* • x(t) *> a(t) = Q* • (X(t) - <X(t)»

V

'

43

where t G [0,T] and Q = ( P i , P 2 , The reduced expansion

...,PJV)

is an orthogonal matrix.

X(«) = Q M • a(i) + (X(i)> ,

(19)

where the N x M matrix Q M is obtained considering only the first M columns of Q, captures the most of the variance of X by the first M eigenvectors of C, provided that M is properly chosen. Taking into account (19), system (15) becomes a = QtM-F(QMa+(X),M)=f(a,/x) ,

(20)

that is a reduced system in the phase space R M for which the reconstruction of X(t) can be checked. When the particular solution X(i) of the originary system can be obtained with desired accuracy, if we let /i vary in a suitable neighbourhood of [IQ one can expect the reduced low-dimensional model as representative, to a some extent, of the dynamics in the full model. This is what we are going to show in the next section. 4. Applications to Navier—Stokes equations Let us consider the Navier-Stokes equations for an incompressible fluid on the torus T 2 = [0,2ir] x [0, 2TT] du — + (u • V) u = - Vp + f + i/Au, divu = 0,

I

(21)

ucbc = 0, u(t = 0) = uo

where u is the velocity field, p the pressure, v the kynematic viscosity and f a volume force. All the involved functions can be expanded in Fourier series

u(x,t) = E^(*)Sr e *' x ' f(x,i)

^

^

. „ >

pfr*) = £**(*)eik"x' . r ,., k

Mt)Wl+h(t)lk]]

eik-x

k/0

where k = (kx,ky) is a vector of integer components ("mode") and the divergence-free condition for the velocity field has been taken into account.

44

Imposing the reality condition 7_k = —7k; equation (21) yields the following ODEs system

7k = - | k | 7 k - ^

£

2| k 1 |lk 2 ||kl

ki+k 2 +k=0

'

i !

^ ^ + ^

W

' "^ ' '

Taking those modes with |k| < 8 (see [13]), the system (22) is a truncated model with N = 98 complex differential equations, which become 98 real equations on a Af-dimensional invariant subspace where the amplitudes are either real or purely imaginary (see again [13]). Such a system can be written as N

Xl = -\kl\2Xi+

E

Ml>m,nXmXn

+h

{I = 1,2,...,98),

(23)

m,n=l

with M^m,n a constant matrix and // = SgjR, where R is the Reynolds' number having set v = 1. In compact form, (23) becomes X = F(X, R)

(24)

in the phase space R 9 8 . For R = Ro = 200 the above system admits a periodic orbit with period T = 0.27 as an attractor. POD has been performed on the numerically integrated data and the first 10 eigenvalues of the correlation matrix are reported in Tab.l, together with the associated average energy percentages.

Table 1. T h e first 10 eigenvalues of C together with the energy percentages (R = 200). n

An

(En) (%)

n

A„

<£n> (%)

1

28.67149

60.019 %

6

0.07640

0.159 %

2

17.55377

36.746 %

7

0.72133 • 10" 2

0.015 %

3

0.70675

1.479 %

8

0.67613 • 10- 2

0.014 %

9

0.69104 • 10-

3

0.144- 10- 2 %

0.65827 • 10-

3

0.137 • 10- 2 %

4 5

0.66852 0.07815

1.399 % 0.163 %

10

We notice that, considering the first M •= 10 eigenvalues, the variance
45

accurately obtained. Since the relevance of the first four eigenvalues, we have attempted a further reduction coming to a 4-terms POD expansion, capturing however the 99.644% of the average energy. It is interesting, at this point, to provide the representation of the actual velocity field together with its recostruction obtained by POD. Since we are dealing with a two-dimensional velocity field u = (u,v), the natural description is provided by the stream function, namely a scalar field ip{x., t) such that u = dip/dy and v = —dtp/dx, which can be written in terms of real functions only as V>(x,t) = EtiMt)

fesin(k,

-x) +

fed

cos(k, -x)

^N

(25)

= Er=i*i(')Xi(x) with ci = (kitV mod 2). Recalling (19), we have the corresponding POD approximation M a

V>(x, t)^J2

n(t)Vn(x)

+

(26)

n=l

of the stream function tp (see Figs. 1-2) with N

Vn(x)=5>,nX«(x)

(n = l , 2 , . . . , M ) .

(27)

i=i

The above relations (26),(27) hold for any M


5. Characterization of periodic behaviour Both in experiments and numerical simulations a periodic behaviour is often observed, for which it is interesting to investigate existence when parameters are varied as well as its onset. In complicated experiments and heavy computer simulations, however, parametrical investigations are time expensive and delicate, expecially when a threshold has to be detected. Even if the model under study is a iV-dimensional dynamical system x = F(x,/x) obtained, for instance, from PDEs by Galerkin method, the existence and stability of a periodic solution, or equivalently a closed orbit 7, when parameters /j. are varied has to be discussed in terms of a Poincare map [12], namely a hyperplane £ transverse to 7 together with a first return map

46

Figure 1. Left: a sample snapshot at t = 0.22 of the periodic solution in (24) for R = 200 (contour plot of (25)). Right: the corresponding reconstruction obtained by the first four P O D terms (contour plot of (26) with M = 4).

Figure 2. T h e first 4 coherent structures. From the upper-left corner to the lower-right one: V1 (Ai = 28.671), V2 (A2 = 17.553), V3 (A3 = 0.706), V4 (A4 = 0.668).

The orbit 7 corresponds to a fixed point -P(xo) = xo G £ for the Poincare map and the stability of 7 is deduced from the eigenvalues of the jacobian matrix DP(XQ) at the fixed point Xo- To determine the eigenvalues of 7, the matrix W ( T ) is computed integrating the differential system /W^=£?=iift^,i

( M = 1,2,...,1V)

I W(0) = I along 7 for one period: the first JV —1 eigenvalues of W ( T ) are the eigenvalues of DP(xo) [11]. The inspection of the eigenvalues when the parameters H are varied leads to the description of the bifurcation that has originated

47

the orbit [12]. However, even in this case the computational problem is heavy and delicate when TV is large, as it often occurs in the cases of interest. A common way by which periodic behaviour can arise in a dynamical system is by supercritical Hopf bifurcation. A fixed point X 0 of the system x = F(x,/i) becomes unstable because a couple of complex conjugate eigenvalues of DF(Xo) crosses the imaginary axis at a certain fi = (ic- A stable closed orbit appears, coexisting with the unstable fixed point, with a period 27r/ |AO| where Ao is the imaginary part of the eigenvalues. In this case the periodic solution has an unstable equilibrium "related" to it. Suppose that a stable closed orbit is known for ji = /io in a dynamical system for which is a heavy effort to discuss existence, stability and bifurcations. If a "related" fixed point could be found and followed varying the parameters, then a Hopf bifurcation could be detected, the appearence of the closed orbit clarified and, at the same time, the stable equilibrium originating the orbit and the threshold value found. Clearly, to follow a fixed point for a dynamical system is a lighter computational effort when a starting point is known, in particular a fixed point for other parameter values. On the other hand the same problem is very hard, or indeed practically impossible, in large systems when a good initial guess is not available. For this reason an alternative technique, which reduces the problem to another for which computations can be made simpler, is strongly desired. A possible solution can be the POD. We assume that (15) admits a periodic solution X(£) = X(£ + T), i.e. a closed orbit 7. The algorithm to be used can be summarized as follows: (a) X(t) is used to build the (minimal) M-dimensional reduced system a = f(a,/io) that admits a closed orbit 7', &(t) = a(£ + T'), corresponding to 7 with T' = T. (b) The average point (a(t)) of the periodic orbit is taken as the initial guess for the Newton method, and the fixed point ao(/io)> "related" to the orbit, is found, unstable, for the reduced system. (c) By antitransforming ao we have X = Q M 3 O + (X). Starting from X, the fixed point Xo(/^o) of the full system is found, unstable, by Newton method. (d) When parameters are varied Xo(^) is followed by Newton method starting from Xo(/xo) and the threshold (j,c is found at which the fixed point becomes stable. (e) A new parameter value fix near fie is chosen, for which the periodic

48

orbit 71 exists for the full system and a new reduced model a = f(a, fii) is built. (f) The (unstable) fixed point ai (fi\) is found and its stability threshold fie is computed. (g) If \ic — \*c the algorithm stops, else \i
49 explicitally written as 7

DiR

^i,jaj

(i = l,2,3)

(28)

3 = 1 k=j

where B\i = 0 (i = 1, 2,3) and A1A = -.007 Ah2 = 17.421 AU3 = -5.996 A2,i = -12.695 ,42,2 = -014 A 2 , 3 = 7.741 A3,i = .002 A3,2 = --003 A3,3 = -9.696 B\a = -.013 Biz = .108 B\2 = -.086 5 ^ 3 = -.412 B\z = -.192 B ^ = .013 B\2 = .086 B\3 = -.682 £f' 3 = .121 B | 3 = -.120 B?' x = -.108 Bfi2 = 1.094 B{3 = .192 £ | 2 = -.121 £f' 3 = .120 Cj = -28.312 C 2 = 8.926 C 3 = 34.668 A = .320 £>2 = --101 D3 = -.392

Figure 3. Projection onto the plane (ai ,02) of the periodic orbit of the 3-mode reduced system (R = 88.5).

The reduced model (28) can be a substantial step toward center manifold and normal form calculations. At the bifurcation point P = (a?, a£, a^, Rc) = (.018, .018, .024, 87.96) , taken as the new origin, the reduced system (28) admits a 2-dimensional center subspace Ec = Ec {ui,U2} and a 1-dimensional stable subspace

50

Es = Es {U3}, therefore a 2-dimensional center manifold Wc, tangent to Ec, and a 1-dimensional stable manifold Ws, tangent to EsLet us suppose that a linear coordinate transformation has changed (0,1,0.2,0,3, R) into (ui,u2,U3,r) for system (28). Up to second order, Wc can be searched as the graph of a polynomial function u 3 = h(u\ ,u2,r) = G\u\ + G2u% + G3r2 + G4uiu2 + G5uir + G6u2r (see [12]). Substituting the above expression into the third of equations (28) we have a 6-equations linear system for the coefficients Gi, that gives Gi = -.036, G2 = .005, G 3 = .401 • 10" 4 , 4 G 4 = -.020, G 5 = -.937-10~ , G 6 = .872 • 10~ 3 . Once the center manifold has been found, performing the usual normal form calculation as in [12] we are led to x = [d/j, + a (x2 + y2)] x — [w + c/j, + b (x2 + y2)] y y = [u + cfi + b (x2 + y2)] x+ [d/j, + a (x2 + y2)] y which is just the normal form of the Hopf bifurcation, where u> = 14.870 is related with the period T = 2TT/UJ = 0.42 of the orbits when the new bifurcation parameter /J, oc (R — Re) tends to zero, and a = -.005, b = .005, c = .004, d = .010. We remark that the sign of a (coefficient of vague attractivity) determines the Hopf bifurcation type, in fact supercritical. 7. Concluding remarks In this paper the POD technique has been presented in details and used as a method able to reconstruct a periodic orbit by only a few eigenfunctions of the correlation operator. By means of a reduced model, the periodic behaviour consequent to a Hopf bifurcation has been also characterized. Expecially in the case of large dynamical systems, the localization by Newton method of the bifurcating equilibrium can become a very hard task when a suitable initial guess is not known. The POD approach is found very efficient in finding the fixed point of the originary system exploiting the low-dimensionality of the reduced model. At the same time the technique allows to clarify the origin of the periodic behaviour and the type of bifurcation when parameters are varied. In the considered truncated model of the Navier-Stokes equations a 3-dimensional system obtained by POD for R = 88.5 is able to describe the Hopf bifurcation at Re = 87.96. At least

51 in t h e examined case, t h e P O D represents a substantial step toward center manifold and normal form calculation t h a t avoids the analytical complications a n d t h e computational effort deriving from t h e full system when its dimension is high.

References 1. P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1996). 2. P. Holmes, J. L. Lumley, G. Berkooz J. C. Mattingly and R. W. Wittenberg, Phys. Rep. 287 (1997). 3. M. Ghil, Dynamic Meteorology: Data Assimilation Methods, Springer-Verlag, New York (1981). 4. L. Sirovich and J. D. Rodriguez, Coherent Structures and Chaos: a Model Problem, Phys. Lett. A120, no. 5 (1987). 5. N. Aubry, P. Holmes, J. L. Lumley and E. Stone, J. Fluid Mech. 192, 115 (1988). 6. N. Aubry, P. Holmes, J. L. Lumley and E. Stone, Physica D 37, 1 (1989). 7. S. Glavaski, J. E. Marsden and R. M. Murray, Model Reduction Via Centering and Karhunen-Loeve Expansion, in Procs. Conference on Decision and Control (1998). 8. C. Lopez and E. Hernandez-Garcia, Low-Dimensional Dynamical System for Observed Coherent Structures in Ocean Satellite Data (2003). 9. M. Basurto and C. Tebaldi, Coherent Structures and Scalar Fields Decomposition, in Procs. 10th W.A.S.CO.M., World Scientific Publishing (2001). 10. M. Basurto and C. Tebaldi, Riconoscimento ed Elaborazione d'Immagini con Applicazioni in Medicina ed Industria, Progetto M. U.R.S. T., Cluster C15/P.3, Workpackage 2, Technical Report (2002). 11. V. Franceschini and C. Tebaldi, Truncations to 12, 14 and 18 Modes of the Navier-Stokes Equations on a Two-Dimensional Torus, in Meccanica 20 (1985). 12. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo (1986). 13. V. Franceschini, C. Tebaldi and F. Zironi, Fixed Point Limit Behaviour of N-mode Truncated Navier-Stokes Equations as N Increases, J. Stat. Phys. 35 (1984). 14. L. Sirovich, Turbulence and the Dynamics of Coherent Structures, Pts. I—III, Quart. Appl. Math. 45 (1987).

I N V E R S E P R O B L E M S IN P H O T O N T R A N S P O R T PART I: D E T E R M I N A T I O N OF P H Y S I C A L A N D GEOMETRICAL FEATURES OF A N I N T E R S T E L L A R CLOUD

A. B E L L E N I - M O R A N T E Dipartimento

di Ingegneria

Civile,

Universita

di Firenze,

Italy

R. M O N A C O , R. R I G A N T I Dipartimento

di Matematica,

Politecnico

di Torino,

Italy

F . SALVARANI Dipartimento

di Matematica,

Universita

di Pavia,

Italy

We consider three inverse problems concerning the physical structure of interstellar clouds. In the problems the unknowns are, alternatively, t h e dimension of the cloud, its total cross section or the photon source. Here we solve numerically, with a similar procedure, the first two problems, since the third requires an additional analysis, reported in Part II (see next paper by Belleni-Morante and Riganti). To determine the unknowns one needs additional data, namely measured values of the photon density outside the cloud. Numerical results are provided by using a suitable iterative scheme.

1. Introduction Interstellar clouds are objects that occupy large regions of the galactic space and are composed of a low density mixture of gases and dust grains (mainly hydrogen molecules with some 1-2% of silicon grains) 1 , a . Since interactions between C/V-photons and such a mixture play a crucial role in the evolution of interstellar clouds, problems concerning such clouds, in general, are studied by using the photon transport equation (PTE) 3 . This equation has been recently applied to various problems considering that the width of the cloud is a time-dependent variable 4'5>6'7. Even in such conditions it has been shown that solution to the PTE can be approximated very well by the solution of the stationary version of the PTE, called the quasi-static equation (QSE) of photon transport, where the time

52

53

U = cos 01 + sinQ k (i = cosQ

Figure 1.

Geometry of t h e problem

is considered as a parameter. Under the simplified assumptions of plane simmetry and isotropic scattering, the QSE has the form

d 0 =-fi—n(x,fi)-a(x)n(x,/j.)

1 +-crs(x)

f+1 n(x,fj,')dfi' + q(x),

(1)

where n(x, /x) is the number density of photons which, in the position x, have velocity in the direction u such that u • i = fi, i being the unit vector in the positive x direction (see Fig.l). Moreover, a(x) and as(x) are, respectively, the total and the scattering cross-section (which are nonnegative functions of a;), and q(x) represents the UV-photon source. In the present paper, we shall consider the QSE defined in the slab [—L,L] (see again Fig.l), surrounded by two regions x < —L and x > L of vacuum. For such an equation we shall formulate and solve two inverse problems: in the first one the unknown is the half-width L of the cloud, in the second one the total cross-section a. For both problems we will assume to know the source q inside the slab and the solution n(x,fi), in an assigned direction fi, of the QSE at a point x » L, that is "far-away" from the cloud. A third problem, where the unknowns are the intensity and the position, within the slab, of the mean value of the source distribution q(x), will be treated in Part II (see next paper of this book). The theoretical background, concerning existence and uniqueness of the solutions to inverse problems, similar to the afore mentioned ones, has been provided in two recent papers 8 ' 9 .

54

2. Mathematical formulation of the problems With reference to Fig.l, consider a slab of width 2L with the half-length L of the slab such that 0 < Lm < L < LM, SO that the maximum length of the slab can be equal to 2LM- The slab "represents" an interstellar cloud where photons undergo absorption, scattering processes and can be produced by the source g, according to Eq.(l). To this equation we join non re-entry boundary conditions, i.e. n(—L,fi) = 0 n{+L,n) = 0

V/iG(0,l], V/xe[-i,0).

(2)

Hereinafter, we shall assume that the source q and the cross sections a and as are positive constants, with ers < a. We now re-write system (l)-(2) in its integral form /tx>0:

IfA \fA

n(x,y)

exp

M<0:

n(x,fi) = -

M

M

exp

V)

-(x-y)

-os

j

n(y,(j,')dn' + q > (3)

r+1

ri -crs /

n(y,n')d(i'

+q

(4)

where — L < x < L. Taking into account (2) and treating the problem in the interval [—LM,LM], to Eqs.(3) and (4) we add the following relations n(x,fj.) = 0, n(x,fj) = n(-L,fi), n(x,n) =n(L,fi), n(x, /j,) = 0,

x G [-LM,-L], x G \-LM,-L),

M> 0 V < 0

x G (L,LM],

A* > °

x G [L,LM],

A* < °-

(5)

Eqs.(5) take into account that, out of the interval (—L,L), no absorption, scattering and photon source are present. In paper 9 it has been shown that problem (3)-(5) can be put in the abstract form n(x,fi)=K(x,n),

xe[-LM,LM],

fi e [-1,1], n ^ 0,

(6)

admitting a unique solution in a suitable Banach space. Let us observe that the operator K depends on the two parameters L G [Lm,I/M] and a G [crm,CTM]-

After these preliminaries we are ready to formulate the two inverse problems discussed in the Introduction.

55

2.1. Problem

1

The aim of this problem consists in determining, in the range [Lm,Ljvf], the value of the unknown parameter L, under the assumption that the cross sections and the source are given. Thus let us re-write Eq.(6) in the following way nL{x,n)

= KL(x,n).

(7)

where KL(X,[I) is computed for a specific value of L and n/,(s,/i) is the corresponding photon density. Let us observe that the nonlinear operator KL links to every L S [ L „ I , L M ] the corresponding photon density n^. Consider now a point x » LM, "far away" from the slab and measured along the particular direction jl > 0. Prom (7) we have nL(x,fi)=nL(L,fi)=KL(L,fi)=:K(L).

(8)

The first equality in (8) is justified by the fact that out of the slab there are no sources and absorption or scattering processes. In paper 9 the following lemma was proven: Lemma 1. K(L) is a real, continuous and strictly increasing function of L e [Lm,L,M]It is now possible to formulate Problem 1: let us assume to know an experimental value of the photon density in (x,/t), say ft. The parameter L must then satisfy the equation h = K(L)

(9)

and consequently, thanks to Lemma 1 L = K-1(h).

(10)

The solution to Eq.(10) allows to evaluate the unknown value L = L corresponding to n, that is to the given "far-field" value of the photon density. 2.2. Problem

2

Similarly to Problem 1, Problem 2 consists in determining the unknown parameter a in the range [am,aM], with am > as, under the assumption that L, os and the source q are known, and a suitable "far-field" value n of the photon density is measured. Then Eq.(7) can be replaced, in this case, by na(x,fi)

=

Ka(x,n).

(11)

56

Here the nonlinear operator Ka links to every a e [o~m,o-u] its corresponding photon density na. In paper 9 again, it was proven that the above Lemma 1 can now be replaced by Lemma 2. Considering again a point x » LM, "far away" from the slab, and a particular value jl > 0, we have n,a(x,/}) = na{L,y)

= Ka(L,p,)

=: K(a),

(12)

where K{o~) is a real, continuous and strictly decreasing function of a £ [o~m,o-M\Thus Problem 2 can be formulated in the following way: let us assume to know an experimental value of the photon density in (£,/t), say n again. Therefore the parameter a must satisfy n = K(a)

(13)

and consequently o- = K-\n).

(14)

The solution to Eq.(14) allows to determine the searched value of the crosssection a = a, corresponding to h. 3. The numerical method We propose here a method to solve numerically equations (9) and (13), which is at the same time simple and "efficient", as it will be seen in next section. Consider Eq.(l) for a fixed value of /i > 0: d ji—n{x,jx)

1 = -an{x,pL) + -aaj

f+1 n(x, y!)dy' + q.

(15)

Integrate the last equation over x between [-L, L], taking into account the boundary conditions (2) and the definition of n, i.e. n = n(x, fi) — n(L, ft). We obtain L

/

-j

pL

p+l

n{x,ji)dx + -(Ja I j n(x,fi')dxdfi' + 2Lq. (16) Eq.(16) can be now used to express in a fixed point form both Eq.(10) and (14). After simple manipulations, one gets L> pLi p-\-l eh -,-1 rL /•+! 1 =•• G(L), ¥q / f ni(xJft)dx — -v8 I I ni(x,^)dxdfj. (17)

57

IqL - fih+ \
-j-^-

\

na(x, )i')dxd^' =

=: H(a),

(18)

na(x,jl)dx / -L which enable us to solve Problem 1 and 2, respectively, using the following iterative scheme Lk = G(Lk.{),

ak=H(ak-i),

k = l,2,....

(19)

Of course, formulae (19) converge provided that G(L) and H(a) are strictly contractive. Setting, for instance, L < L', the proof of such a property needs a careful evaluation of upper bounds of TIL and of TIL> • Such a boundness requires, in any case, the assumption as/<J < 1 made at the beginning of Sec. 2. 4. Numerical results Numerical simulations have been performed by assuming for Problem 1 a source with a constant intensity q = 0.8 inside the slab, a scattering cross section as = 0.2 and a total cross section a = 1. Moreover, in solving Problem 2, we have fixed q = 0.1, as = 0.1 and L/LM = 0.75. The numerical procedure to calculate the operators (17), (18) requires the knowledge of the solution to the direct quasi-static problem (2) in the slab [—L, L}. It was obtained for a set of discrete values of L,a and /i by means of a standard Runge-Kutta routine, after discretization of the spatial and angular variables x, /x and application of the well-known discreteordinate method 10 . The numerical evaluation of the integrals over the angular variable /u has been performed by using a 16-nodes Gauss-Legendre formula, in the same fashion as reported in 6 , and quadratures with respect to the spatial variable a; have been made through the Simpson's formula on a set of 9 • 102 nodes interior to the slab. The iterative schemes (19) have been applied by assuming separately four positive values of the velocity direction /} and various given experimental photon densities h. In Problem 1 for the slab half-width L, the convergence was attained, with a tolerance of 10~ 6 , after 10 -=- 20 iterations, with the exception of the cases with low angular component /t = 0.1834 and experimental densities n higher than 0.8, for which up to 60 -=- 80 iterations were necessary to obtain the convergence. The iterative scheme for Problem 2, to evaluate a, was applied with the same order of tolerance and the convergence was attained more easily, e.g. after a maximum of 30 iterations.

58

Figure 2.

Solution of the inverse problem 1

0.3

\

L/LM = 0.75 os=0.1 q =0.1

£ = 0.1834

0.2

A = 0.5255 ^ \ _ 0.1

A = 0.7966

"/ H = 0.9603 1.0

Figure 3.

1.2

1.4

Solution of t h e inverse problem 2

Figures 2 and 3 show the obtained results (solid line) and the comparisons with the ones given by the numerical solution of the direct problem (denoted by circles). The relative errors EL = | (L — L)/L\ and Ea = | (a — cr)/a\ have been also computed in each numerical simulation. The errors turned out to be of the order of 10~ 3 in the whole range of investigation of the parameters.

59

Acknowledgments This work has been partially supported by INDAM-GNFM (Gruppo Nazionale Fisica Matematica) and by PRIN 2003 "Problemi Matematici Non-Lineari di Propagazione e Stabilita nei Modelli del Continuo" (Prof. T. Ruggeri, co-ordinator). References 1. D.E. Osterbrock, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. University Science Book USA, 1989. 2. L. Spitzer, Physical Processes in the Interstellar Medium. Wiley, Toronto, 1978. 3. G.C. Pomraning, The Equations of Radiation Hydrodynamics. Pergamon Press, Oxford, 1973. 4. A. Belleni-Morante, R. Monaco and F. Salvarani, Approximated solutions of photon transport in a time-dependent region. Transp. Theory Stat. Phys. 30: 421-438 (2001). 5. R. Monaco, F. Salvarani and A. Belleni-Morante, Numerical simulations of a photon transport problem in a time-dependent domain, in Waves and Stability in Continuous Media (V. Ciancio et al., Eds.), World Scientific, Singapore, 2001, p. 314. 6. R. Riganti and F. Salvarani, A singularly perturbed problem of photon transport in a time-dependent region. Math. Meth. in Appl. Sciences 25: 749-769 (2002). 7. M. Lisi and S. Totaro, Quasi-static approximations of photon transport in an interstellar cloud, in Waves and Stability in Continuous Media (R. Monaco et a l , Eds.), World Scientific, Singapore, 2002, p. 271. 8. A. Belleni-Morante and F. Mugelli, Identification of the boundary surface of an interstellar cloud from a measurement of the photon far-field. Math. Meth. Appl. Sci., in press (2003). 9. A. Belleni-Morante, R. Monaco, R. Riganti and F. Salvarani, A numerical approach to inverse problems in photon transport inside an interstellar cloud. Appl. Math, and Comput., in press (2003). 10. S. Jin and C D . Levermore, The discrete-ordinate method in diffusive regimes. Transp. Theory Stat. Phys. 22: 739-791 (1991).

I N V E R S E P R O B L E M S IN P H O T O N T R A N S P O R T PART II: F E A T U R E S OF A S O U R C E INSIDE A N I N T E R S T E L L A R C L O U D

A. BELLENI-MORANTE Dipartimento di Ingegneria Civile, Universitd di Firenze Via S.Marta 3, 50139 Firenze, Italy R. RIGANTI Dipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 34, 10129 Torino, Italy A model for the spatial distribution of a UV—photon source inside a slab representing an interstellar cloud is proposed. Inverse problems for the related quasi static photon transport equation are studied, in order to determine the intensity and the location of the source by using measured data of the photon densities at points very far from the slab. Numerical simulations of the obtained results are also presented.

1. Introduction In Part I of this research 1 two inverse problems related to the interactions between f/V-photons and the medium in interstellar clouds have been studied by considering, in a one-dimensional model, a slab of width 2L, such that 0 < Lm < L < LM, representing the cloud where photons are produced by a source q(x) and undergo absorption and scattering processes. Both the source and the absorption and scattering cross sections were taken as positive constants q, a and as with as < a, and the inverse problems leading to the determination of the half-lenght L or of the total cross section a were considered. The aim of the present paper is to complete the above treatment, by determining the most relevant physical properties of the photon source q(x), under the assumption that the slab half-lenght is given, that the cross sections are known functions of the spatial variable x, and that a suitable number of experimental values of the photon densities with a prescribed velocity component are measured at a point x » LM far from the slab.

60

61

The mathematical formulation of these inverse problems is given in Sec. 2, and the analysis is completed by numerical simulations, whose results are reported and discussed in Sec. 3. 2. Formulation of inverse problems for the source An appropriate model of the source, which takes into account both its intensity and its spatial distribution inside the cloud, may be set in the form q{x) = 2Lqf{x;x,v)

(1)

with q defined in a prescribed interval [qm, QM] of R + , and where f(x; x, v) is a density function with mean value x € [xm,XM] C [—L,L], variance v, and satisfying f(x;x,i/)

= 0,

x € [-LM,-L)U(L,LM];

/

f{x;x,v)dx

= 1. (2)

It may represent a cluster of emitting stars, located in some region inside the cloud, that are centered at x and are spatially dispersed at a rate of order A/ZA If we consider that interstellar clouds occupy regions of the galactic space which are very large if compared with the size of the clusters we want to identify, it is reasonable to assume that in the above model the most relevant physical properties of the source are represented by the constant q, e.g. the intensity of the emitted t/l^-photons, and by the mean value x of its spatial distribution. Consequently, in the above model we will suppose that the density law / is given with a prescribed standard deviation ^/v « 2L, and that its mean value x must be determined, together with the photon source intensity q. To this aim, let us consider the quasi-static transport equation (QSE) of Part I. Eq.(l) of Part I is then re-written for a given velocity component fi = fli as p,i—n(x,ij,i;x,u)

+

a{x)n(x,(ii]x,v)
(3)

where n(x, /i; x, v) is the number density at point x of photons with velocity in the direction u such that u-T= /x, where MS the unit vector in the positive x direction.

62

Remark 1. An equation quite similar to (3) holds also if model (1) is slightly modified, by assuming that the source intensity is a random variable q(uj) defined in the abstract probability space (Q,B,P) of events w, with realizations in [qm,qM} and expected value E{q}. In fact, in this case the photon density will become a stochastic process n(u,x;iJ,, x, v) whose expected value E{n}(x; ^, x, v), owing to the linearity of the transport equation, must satisfy (3) with n and q replaced, respectively, by E{n} and E{q). • If both members of Eq. (3) with jli = jx\ are integrated over x € [—L, L] by taking into account the properties (2) of / and the non re-entry boundary conditions for n, we obtain — I fiifii + / - I

dx as(x)

a(x)n(x,fii;x,u)dxn(x,fi';x,u)dii'

I =: Fi(g,x,i/).

Likewise, multiplying by x both members of (3) with jli = fe and integrating over x € [—L,L], we have x= —

[fi.2Ln2 + J i ( s , i / ) - Jz(x,v)} =:F2(q,x,v),

Ji(x, u) =: I

\xcr{x) —

fi2}n(x,fli;x,u)dx

(5) (6)

J2(x,u) =: -2 / dx xas(x) I n(x,fi';x,i/)diJ,' (7) J-L L J-i Equations (4) and (5) supply an implicit solution for the inverse problem consisting in the evaluation of the two unknown "parameters" q and x. In fact, the evaluation of the right hand sides of Eqs.(4) and (5) requires the knowledge of the two data ni,fi2 and also of the solution of the QSE (3) which is itself depending on q, x and on the assumed form of the density / . Therefore, the solution of the above system (4)-(5) may be sought numerically by means of the iterative scheme lk»)=F(lk), * = 0,1,2,...; F = ( ^ S J V \ ) (8) Xk+iJ \XkJ \F2{q,x,^)J where the operator F : IR^. —> H + must be determined at each step k by making use of the solution of the direct quasi-static problem for the photon density, which is obtained from (3) by setting q — q^ and

63

The convergence of the above iterative scheme to a solution of the general inverse problem, consisting in the simultaneous determination of the two parameters q and x, obviously requires the strict contractivity of the operator F defined in (8). The proof of such a property may be obtained, under suitable simplifying assumptions on the form of the source density / and on the cross sections cr(x),as(x). 3. Numerical simulations and results In this section we present the results of some numerical simulations made for determining the solutions of the inverse problems considered above. The numerical treatment of the transport equation (3) was made by following the procedure presented in paper * (i.e. by using the discrete ordinate method 3 ). Namely, suitable discretizations of the spatial and angular variables x, /x were defined in order to determine, at each step of the iterations, the grid values of n{x,jli\x,v) interior to the slab, that were obtained by solving (3) with a prescribed distribution law f(x;x,v). Then, GaussLegendre and Simpson's formulas were used to perform the quadratures in equations (4), (5) and determine the operators F,. In these simulations, the total and scattering cross sections were assumed to be known and continuous functions of the space variable x G [-L,L] satisfying: 0 < as(x)/a(x) < x < 1; °{±L) = 0, as(±L) = 0. The total cross section was defined as ' 0, -LM <x<-L a0{l-[x2 + (2x + L- h)(L - h)]/h2}, a(x) = < cr0, -L + h<x < L- h 2 a0{l-{x -{2x -L + h){L - h)]/h2}, ,0, L < x < LM

-L<x<-L

+h (9)

L~h<x
where h is the amplitude of the uniform mesh used to discretize the slab [—L, L] and UQ is a known, strictly positive constant. The scattering cross section crs{x) was defined analoguosly, with <7so < (Jo- In particular, all the results reported in this section were obtained by assuming <JO — 1, aso = 0.2 and LM = 13.1. Uniform

distribution

The simplest assumption we can made is that the source is uniformly distributed within the slab, e.g. f(x,x,i/) = const. = 1/2L. In this hypothesis, that was the one assumed in Part I, the parameters x = 0 and

64

0.5

0

Figure 1.

0.2

0.4

0.6 0.8 Source intensity q

Solution of the inverse problem for q, with f(x;x,

1

v) = 1/2L.

v = L 2 / 3 are known and the inverse problem is reduced to the determination in [qm,qM] of the source intensity q by using the iterative scheme gfc+i^ifak.O.tf/S),

fc

= 0,l,2,...

(10)

with Fi defined by (4). Since the iterations (10) are made on a single scalar equation, to calculate this operator a unique experimental value of the photon density hi = n(x,/z,0, L2/3) with velocity component /} is needed at the point x » LM, and this value was used in Eq.(4). For given experimental values h\, the convergence to a solution q G [0,1] was attained within a maximum of k = 45 iterations, with a tolerance of 10~ 6 . The results obtained by setting L = 0.5, oo = 1,
distribution

Let us now assume that the photon source with intensity q is located at a point x G (—L,L) inside the slab. Then the law / is the impulsive Delta

65 1.0 f(x) = S(x- x) x = - 0.28 tx = 0.5255

CI0=1

J

oso= 0.2 L =0.5

0.5

®

5

^ = 0.1834

= 0.7966 r ^

0.2

Figure 2.

(1=0.9603

0.4

0.6 0.8 Source intensity q

Solution of the inverse problem for q, with f(x; x, 0) = <5(x — x).

function f(x; x, v) = 5{x — x) with variance v = 0'. Assuming that two experimental data hi = n(x,jii;x,0) and hi = n(x, U2',x,0) of the densities with different velocity components h\, fe have been measured at a point x, then the solution of the inverse problem consisting in the simultaneous determination of q and x may be obtained by using the bidimensional iterative scheme (11) xk+i

-

F2(qk,Xk,0),

k = 0,1,2,.

(12)

where at each step k the operators Fi and F% are given respectively by (4) and (5) with v = 0. The treatment of Eq. (3) in order to calculate the densities n(x, hi;xk, 0) for given values of q^, needs the numerical approximation of the Delta function. We have approximated it by the piecewise linear function: ' 0, (x — x)/h? + 1/h, A (a; — x) = < (x — x)/h2 + 1/h,

0,

x <x —h x —h<x < x x <x <x +h

(13)

x > x+h

where h is the constant amplitude of the mesh used to discretize the slab [L,L]. It was proved in a previous numerical application 6 that A(x — x) is an approximation of order two of the Delta function S(-). Of course, if either q or x are given, Eqs. (11) and (12) become independent and may be treated by using again a unique experimental datum, measured far from the slab.

66 0.50 f(x) = 5(x-x)

5255 0.25

H = 0.7966 \x = 0.9603

Figure 3.

Solution of the inverse problem for x, with f(x; x, 0) = S(x — x).

Two examples of the results obtained in this case are shown in Figs. 2 and 3. In the numerical simulation shown by Fig. 2, we assumed to know the location x = —0.28 of the source with unknown intensity q S (0,1), interior to a slab [—L,L] with L = 0.5, and applied the iterative scheme (11) to determine the source intensity q for given experimental values of h. Conversely, Fig. 3 shows the results obtained by assuming to know the intensity q = 0.1, and by applying the scheme (12) to determine the location x of the source interior to a slab with L = 0.75. 3.3. Gaussian

distribution

Let us now consider a Gaussian model of the spatial distribution of the source within the slab:

f{x;x,v)

: exp

(x — J) 2 " 1 2u

-L < x < L.

(14)

Simple computations show that all the distributions (14) with means such that \x\ < | i and variance v < 10~ 3 L 2 satisfy conditions (2) with a tolerance of order 10~ 8 . Therefore, these distributions may correctly simulate the presence in the slab of a cluster of emitting sources, centered somewhere in the region (—4L/5 < x < 4L/5). Moreover, conditions (2) are also satisfied with the same tolerance by all the Gaussian distributions with mean x = 0 and variance v < 2.5 • 10~ 2 L 2 , which therefore may correctly simulate a more dispersed cluster of emitting objects, placed in a central region of the interstellar cloud. In order to determine the " parameters" q and x by the iterative scheme (8), operators Fi, F2 were numerically found after calculation of the photon

67 0.4

f(x) = Gaussian x = -0.S x= 0.4 v =10- 3 L 2

L = 0.75 q = o.i

n

M = 0.1834

\\

uniform distribution,

0.5255]

\

(1 = 0.1834

0.7966^f^^.

0.9603jhp~zn 1'

o Figure 4.

.-' fflr

•••'' //

I-'' ^Jl\ -0.5

i i

, - ,

it

X~~

-__

0.5

Densities produced by a Gaussian source with intensity q = 0.1

densities satisfying the transport equation (3) with f(x;x,v) given by (14) and v suitably chosen in the ranges defined above, according to the size of the physical objects that are investigated. As an example, we report in Fig. 4 some results on the photon densities with different values of the velocity component p,, obtained by assuming that the source has a Gaussian distribution located at x = —0.5 or x = 0.4, with variance v = L 2 /1000 and intensity q = 0.1. The dotted line shows, for sake of comparison, the density with the smallest velocity component ji = 0.1834, which is obtained when a uniformly distributed source with the same intensity q is assumed. Note that at some fixed mean values x the densities n(L, jj.\x,u) are not monotonically decreasing functions of the velocity component jl. This is due to the different amount of the absorption effects represented by the term J\(x,v) defined in (6), and occurs when the source is far from the border x = L of the slab (see the results at x = —0.5 in the same figure). If we assume to know both the mean and the variance of the Gaussian distribution (in addition to one experimental data n of the density far from the slab), then the inverse problem consisting in the calculation of the source intensity q may be treated again by using the iterative scheme qk+i = Fi(qk',x, v), k = 0 , 1 , . . . The results are reported in Fig. 5, where a fixed variance v = 1 0 _ 3 i 2 and two values of the mean: x = 0 and 0.25 of the Gaussian distribution have been assumed. Again, they are affected by a maximum relative error of order 1 0 - 3 . If the results obtained with x = 0 are compared with the ones shown in Fig. 1, no appreciable difference with the case of a uniform distribution can be found at high velocity components, [i = 0.7966 and 0.9603. On the contrary, if lower values of jl are assumed, at a given experimental data n the resulting intensity of the source with a Gaussian distribution is remarkably higher than the one related to a

68 1.0 f(x) = Gaussian with variance v = 10"3L2 mean x=0 x = 0.25 L = 0.5

/ •

A ft = 0.1834 *

0.5

\ LI = 0.5255 ^f^

,*zs^ ^

\ * ^ ^ ^ '

A^p^^L ^ ^ ^

= S £ =

0.2

Figure 5.

% = 0.9603 0.4

0.6 0.8 source intensity q

1

Solution of the inverse problem for the intensity q of a Gaussian source

-1 Figure 6.

^

-0.5

0

0.5 —

1

Solution of the inverse problem for the mean value x of a Gaussian source

uniform law. Conversely, to obtain the results shown in Fig. 6, where the density at points x far from the slab are plotted versus x, we applied the iterative scheme Xk+i = ^{I'^k^v) by assuming to know fixed values q = 0.1 and 0.2 of the source intensity. The same figure also shows the existence of solutions to the inverse problem consisting in the simultaneous determination of q and x in the assumption that two experimental data satisfy a particular condition, and illustrates how these solutions may be found. Namely, let us suppose that a unique value n = n(x, fli; x, v) = n(x, p,i\ x, u) of the photon densities with two different velocity components jx\ and fa has been measured at the point

69 x. T h e n t h e solution (q, x), if it exists, is given by the intersection of t h e plots of n(x,fii;x,u) and n(x,(i2;x,v) versus x, obtained as solutions of the QSE with q as a parameter. In t h e numerical simulation presented here, six possible solutions, deduced by as many measured values of h and denoted by circles, may be singled out. Given q — 0.1, the plot of t h e photon densities with fi = 0.1834 intersects the ones with other velocity components at h = 0.0589; 0.06766 and 0.0901 respectively. Therefore, these measured d a t a identify sources with intensity q = 0.1 and spatial distribution with means placed respectively at x = - 0 . 0 3 5 2 ; 0.0213 and 0.1381. Moreover, by setting q = 0.2, intersections occur at equal values of x and correspond to values of h t h a t , owing t o t h e linearity of Eq. (3) with respect t o t h e parameter q, are exactly twice t h e previous ones. It follows t h a t t h e three remaining measured d a t a : h — 0.1178; 0.13532 and 0.1802 identify sources with a double intensity q = 0.2 and the same spatial distribution.

Acknowledgments This work has been partially supported by I N D A M - G N F M ( G r u p p o Nazionale Fisica Matematica) and by P R I N 2003, "Problemi Matematici Non-Lineari di Propagazione e Stabilita nei Modelli del Continuo" (Prof. T . Ruggeri, co-ordinator).

References 1. Belleni-Morante A., Monaco R., Riganti R., Salvarani F., Inverse problems of photon transport, Part I: Determination of physical and geometrical features of an interstellar cloud (these Proceedings). 2. G.C. Pomraning, The Equations of Radiation Hydrodynamics. Pergamon Press, Oxford, 1973. 3. S. Jin and C D . Levermore, The discrete-ordinate method in diffusive regimes. Transp. Theory Stat. Phys. 22: 739-791 (1991). 4. A. Belleni-Morante, R. Monaco and F. Salvarani, Approximated solutions of photon transport in a time-dependent region. Transp. Theory Stat. Phys. 30: 421-438 (2001). 5. A. Belleni-Morante and F. Mugelli, Identification of the boundary surface of an interstellar cloud from a measurement of the photon far-field. Math. Meth. Appl. Sci., in press (2003). 6. Riganti R., Salvarani F., A singularly perturbed problem of photon transport in a time-dependent region, Math. Meth. Appl. Sci 25: 749-769 (2002).

R I G O R O U S DERIVATION OF K A N E MODELS FOR INTERBAND TUNNELING

STEFANO BIONDINI & GIOVANNI BORGIOLI Dipartimento di Elettronica e Telecomunicazioni Universitd di Firenze - Via S.Marta 3 1-50139 Firenze, Italy Mathematical models describing interband tunneling in semiconductor heterostructure devices (superlattices) are discussed and compared to a model here rigorously derived from the Schrodinger equation. The continuity equation for the probability density is also derived.

1. I n t r o d u c t i o n Recent developments in nanometric semiconductor devices have compelled mathematical modelling to take in full account quantum aspects. We face here the problem of modelling interband tunneling, which is the main mechanism of working for some heterostructure devices (superlattices) like RITDs (Resonant Interband Tunnel Diodes). In these superlattices the contact of heterogeneous semiconductor materials enables to obtain potential barriers that interface the conduction and the valence band. The band diagram structure of the device is characterized by a band alignement so that the valence band at the positive side of the semiconductor device lies above the conduction band at the opposite side. The behaviour of such devices exhibits an I — V (current-voltage) characteristic with regions of negative differential resistance, which can be explained only taking into account quantum phenomena like interband tunneling. It is obvious that a rigorous derivation for the charge current I would be welcome. In literature a simple model was proposed in 1988 by M. Sweeny and J. Xu and successively, again, in a paper of 1991, by the same authors together with R.Q. Yang and D. Day 1,2 (in the following we shall refer to this model like to SX).

70

71

One of the authors of the present paper has used SX model for a formulation of interband tunneling in terms of the Wigner function formalism3, and inquired himself about its rigorous derivation from the Schrodinger equation. In the meanwhile new authors have proposed different, but essentially equivalent, models. We cite here J. Kefi4, L. Barletti and L. Demeio 5 . We shall label these models as K and BD, in the following. All of them can generally be denoted as Kane models since their background can be found in the papers of E. O. Kane 6 ' 7 ' 8 , where a multi-band approximated Hamiltonian is formulated for different semiconductors. The aim of the present paper is to propose a rigorous procedure for deducing a model for interband tunneling from the Schrodinger equation and compare it to SX, K and BD models.

2. Kane models The physical assumptions considered in SX, K and BD models are identical: electromagnetic and spin effects are disregarded, just like the field generated by the charge carriers themselves. Dissipative phenomena like electronphonon collisions are not taken into account, too. The dynamics of charge carriers is considered as confined in the two highest energy bands of the semiconductor, i.e. the conduction and the (non-degenerate) valence band, around the point k — 0, where k is the wave vector, The point k — 0 is assumed to be a minimum for the conduction band and a maximum for the valence band. The parabolic approximation is considered as valid. As a consequence of the assumed physical environment, it is proposed a very simple system of two Schrodinger-like equations, where focus is only on the interband tunneling mechanism. The BD model, as appears in 5 , is given by ih9*c

dt

=

& d2*c 2m dx2

t £

A

°

ftp m

g*« dx

cv

(BD) d$v n2 d2*v 2 + Ev^v ift-^rTT^T dt = —z2m dx

h d*c vc P vc m dx

where i is the imaginary unit, h is the Planck constant scaled by 2n, m is the bare mass of the carriers, Ec and Ev are the minimum of the conduction band energy and maximum of the valence band energy respectively, depending, through the a;—coordinate, on the layer composition, and Pcv, Pvc are the coupling coefficients between the two bands.

72

K model was presented in 4 and consists in the following system, very similar to BD formulation dt

2m dxz

ox (K)

..d$v h2 dH ^.hp9*c 2 in——= —-5+ E w + inP- dx v v dt 2m dx where P is the coupling coefficient between the two bands. Finally we show the system proposed in 1 : dt

2m dx2

i

d*v

n2 d2^

n o
dx

(SX)

2 in-—— —— + EVWV + i -P- dx dt = +-2m dx

It is important to remark that the second equation of (SX) exhibits a positive sign in the kinetic energy term. The authors present system (SX) as an evolution equations system for the wave functions of the charge carriers, with no further specification. We claim that all the models above shown do not concern full wave functions, but envelope functions (this fact is clearly asserted only in 5 ) , since the associated Hamiltonian is an approximated one (in the sense of a k • P Hamiltonian 8 ' 9 . In particular, as regards SX model, the second equation represents an evolution equation for a hole (positive charge carrier) envelope function. SX is not rigorously derived but can be obtained heuristically (but reasonably) following the derivation thay we propose in the sequel. 3. Derivation of Kane models We consider the Hamiltonian in the single electron Schrodinger equation d ih

jL

= Hj,

(1)

as composed by the kinetic energy term and by an electrostatic potential energy term which is the sum of a potential Ue{x), the "crystal" potential, represented by a function which has the periodicity of the one-dimensional lattice (say £) and by an "external" potential U(x), which takes in account impurities, barriers and the external applied voltage (bias). Let H = Ht + U(x) = - ^ - ~

+ Ut{x) + U(x) .

(2)

73

The Hamiltonian He admits as eigenfunctios the Bloch functions10 bn(k,x) = un(k,x)elkxx G R , n = 0 , 1 , . . . , i.e. Hebn(k,x)

= En(k)bn(k,x)

,

(3)

where En(k) are the energy bands and un(k,x) is a £—periodic function in x. Bloch functions constitute a complete generalized orthonormal basis in R for L2—functions defined on the whole one-dimensional lattice. The Bloch function u^(x) := w n (0,x) at the bottom or top of the bands, which are named the Luttinger-Kohn (LK) eigefunctions11 , form a complete orthonormal basis for any ^-periodic function in L 2 (R) In particular the ^-periodic part of the Bloch functions, un(k,x), expanded in the LK basis:

can be

oo

un(k,x)

= ^

Cnm(k)u°m(x)

(4)

7Tt=l

Any wave function ip, satisfyng the Schrodinger equation with the Hamiltonian H given by (2) can be expanded in the orthonormal basis of the Bloch functions, and the Bloch functions can be expanded in the LK functions basis: oo i>{x,t)=Y,^rn{x,t)u°m{x), m=l

(5)

where B is the first Brillouin zone and the functions »

*m(x,t)=

oo

J

/ 'VCn(k,t)Cnm(k)eikxdk,

m = l,2,...,

(6)

Bn=1

which are called LK envelope functions, do not depend on k. Consider a two-band dynamics, then n,m = c,v where c stands for "conduction band" and v stands for "valence band" and let u°(x) and u°(x) be the LK eigefunctions and $>c(x,t) and $v(x, t) the LK envelope functions. Expansion (5) becomes iP(x,t) = Vc(x,t)u°(x)

+ iS>v(x,t)u0v(x)

(7)

which is the envelope function expansion of the wave function ip- We introduce (7) in the Schrodinger equation (1) and take into account that, since

74

u° (x) and u°(x) are the Bloch eigenfunctions at k = 0 (LK eigenfunctions), we have h2 d2u° ~2m^~dx^

+ U

<W<«

= ^ . « ( ° X » := ^c%<„

(8)

Moreover it is consistent with the physical evidence assuming that only u°(x) and u°(x) vary significantly on the lenght scale of the unit cell. In terms of calculation this assumption leads to the following approximation: /

uJ*(£)*i(£K(Od£

- *<(*)*« , i,3 =c,v,

(9)

Ju — cell(x)

where * denotes complex conjugate and Sij represents Kronecker delta symbol. The meaning of assumption (9) is that we consider as negligible the difference between $i(x) and its average evaluated in a unit cell centered in x, u — cell(x). This approximation is extended till to the first (in t and x) and the second (in x) derivatives of the envelope functions involved in the integrals on the unit cell domain. Through (7) (8) and (9), integrating over the unit cell domain, we split the Schrodinger equation (1) the following two equations system: .5*c h2 dHc ^ T h2d*v f ih—^ = - — ^-f + £ c * c - — —?- / z at 2m ox m ox j u _ ce ii(x)

M/

,9<

u°*(x)^Ldx

°x (10)

. m v at

2

2

h d ^v 2m axA

^

2

T

h d^c r OT/ m ox Ju_ceii(a;)

where we have put Ec = E°c + U and EV=E° + U. Let us now define f du° Pi:j:=h u°*(x)~Ldx i,j =c,v,

,9«° ax

i^ j

(11)

°x

Ju-cell

which is independent on the choice of the center of the unit cell because of the periodicity of the functions involved. Applying definitions (11) to (10), we obtain 2m dx2

dt

m

dx (12)

ih9*v

dt

=

_¥_&3bL 2 + Ev*v 2m dx

- - P m

^ dx

u c

75

Note that, in general, Pcv = —P*c, as it is easily verified by direct integration by parts (or imposing to H to be hermitian). It is manifest that the choice of the LK eigenfunctions basis determines the numerical nature of parameters Pcv and Pvc, i.e. it is possible that the parameters are real, purely imaginary or complex. We now derive the continuity equation from (12) in a general form, i.e. without any further assumption on the choice of the LK basis. From the definition of probability density p := ipip* where ip is the wave function satisfying the Schrodinger equation, the probability density in terms of envelope functions reads as

p = |* c | 2 + |tf„|2 = * c * * + * „ * ; .

(13)

Now, following the well-known procedure used to derive continuity equation from the Schrodinger equation, we obtain the classical form dp_ dJ_ dt dx h Vlf* £ c lim dx h lim . " dx

(14)

0

dx

i

dx

)

Jvc

-

p l

r

vc

im p

Jcv

=

1

K^c

cv

im

It is manifest that our conclusions concide with system (BD). (K), moreover, can be found simply putting

-imP , P e

(15)

Concerning SX model we propose the following considerations. If we put "1" as the electron charge, the Hamiltonian of the electron, to be considered in the Schrodinger equation, reads as He — —^-§^1 + U(x). Hence the Hamiltonian of a positive carrier of mass m (a "hole") reads as Hh = .fii a2 U(x) and the related Schrodinger equation is in,-— = —

dt

h2 d2^ - U{x)ip . 2m dx2

(16)

If we take the conjugate equation to the above one, we get

ih

dtp* dt

h2 d2ip* + 2m dx2

U{x)ip*

(17)

76

Now, let iph = ip* (the choice for the definition for the wave function is "conjugation" insensitive!). We can describe the dynamics of holes as particles moving under the same potential energy of electrons, but as carriers of a negative mass —TO. As a consequence, the Schrodinger equation for the hole wave function (now displayed simply as ip) can be written as & d2*P

.dip ih dt

TT,

,,

(18)

The same interpretation arises if we apply the Ehrenfest theorem to a particle satisfying (18). Let us go back to system (12) and try to obtain (SX) from it. If we consider $!v as an envelope function of the hole (and not of the electron in the valence band) we replaceTOwith —m in the second equation of (12) and have dt

h2 d2^>c 2m dx2

d^v at

h2 d2^v 2m ox**

ihd^c

_

h

^ °

| £

^

pcvd*y dx

cv

TO T

(19)

h „ <9*c m ox

where, as usual, Ec := E° + U and Ev := E° + U. Now put Pcv

Tfl

:=

i

P,

wherePeR.

(20)

TTt

As a consequence Pvc = + — P and SX model is derived. i

Imposing to the hole a negative mass — m produces as a consequence that one assignes to the hole the same charge of the electron, i.e., in our case, 9 = 1. Hence, the charge density is still correctly given, as in the previous models, by p = |^cl + I^Sl a n d the derivation of the continuity equation follows the same procedure. The final result is a continuity equation like (14), with a different definition of the single terms: ;SX

Jvv

h 2im

*

h 2im

*.

;

dx dx

-*.

*„

d-$: dx

isx = - P W c Jvc

•sx dx

Jcv

p***, — *• * c^v

77

References 1. M. Sweeney and J. M. Xu, Resonant interband tunnel diodes, Appl. Phys. Lett. 54 (6), 546-548 (1989). 2. R.Q. Yang, M. Sweeny, D. Day and J.M. Xu, Interband tunneling in heterostructure tunnel diodes, IEEE Transactions on Electron Devices, 38(3) 442-446 (1991). 3. G. Borgioli, G. Frosali and P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transp. Teor. Stat. Phys., 32(3 & 4), 347-366 (2003). 4. J. Kefi, Analysis of multiband quantum transport model, Euroconference on Asymptotic Methods and applications in kinetic and quantum kinetic theory, Granada-Spain (2001). 5. L. Barletti, L. Demeio, Wigner-function approach to multi-band transport in semiconducotr devices, Proc. VI Congresso Nazionale SIMAI, Chia Laguna (Cagliari,Italy) May 27-31 (2002). 6. E. O. Kane, Energy band structure in p-type Germanium and Silicon, J. Phys. Chem. Solids 1, 82-89 (1956). 7. E. O. Kane, Zener tunneling in semiconductors, J. Phys. Chem. Solids 12, 181-188 (1959). 8. E. O. Kane, The k • p method, in Semiconductors and Semimetals, edited by R.K. Willardson and A.C. Bear (Academic, New York, 1966), Vol. 1, pp. 75-100. 9. M.G. Burt, The justification for applying the effective-mass approximation to microstructure, J. Phys: Condens. Matter 4, 6651-6690 (1992). 10. W.T. Wenckebach, Essentials of Semiconductor Physics, J.Wiley & Sons, Chichester (1999). 11. J. M. Luttinger, W. Kohn, Motion of electrons and holes in perturbed periodic fields, Phys. Rev., 97, 869-883 (1955).

Q U A N T U M H Y D R O D Y N A M I C EQUATIONS FOR A T W O - B A N D W I G N E R - K A N E MODEL

STEFANO BIONDINI, GIOVANNI FROSALI AND FRANCESCO MUGELLI Dipartimento di Elettronica e Telecomunicazioni and Dipartimento di Matematica Applicata "G.Sansone" Universita di Firenze

1. Introduction The numerical simulations of semiconductor devices are usually based on hydrodynamic models formulated in terms of macroscopic "quantities such as charge density, current, energy, and so on. In the classical frame, hydrodynamic models are derived from the hierarchy of the moments of the Boltzmann equation. Starting from the pionieristic paper by B^tekjasr 1 , in the 70's, the literature on hydrodynamic modeling, both theoretical and numerical, is very extensive. The reader interested in hydrodynamic modeling can refer to the papers quoted in the review by Anile and Romano 2 . As it is well known, the fluid dynamical description begins to fail when the carriers involved (i.e. electrons and holes) are few and the quantum aspects become not negligible or even predominant. Nevertheless it is worthwhile to recall that keeping an hydrodynamic formulation would still present some relevant advantages (also in the previously mentioned situation), since fluid dynamical equations are preferable on the computational side. Moreover, it easier to face the boundary conditions problem when macroscopic quantities are involved rather than the Wigner distribution function or the wave function. On the other hand, in practical applications, approaches based on microscopic models are not completely satisfactory, and it is useful to formulate semi-classical models in terms of macroscopic variables. Such models are generally built from a hierarchy of coupled moment equations and referred to as quantum hydrodynamic models. In this context some very interesting results are present in literature,

78

79

where a quantum hydrodynamic set of equations capable to describe the behaviour of nanometric devices like resonant tunneling diodes is introduced. We recall here the "smooth" quantum hydrodynamic model proposed by Gardner and Ringhofer3, who derive the fluid dynamical formulation from moment expansion of a Wigner-Boltzmann equation, and the approach by Jiingel 4 , whose starting point is the ansatz of a peculiar type of solutions to the Schrodinger equation. Published results are generally devoted to single-band problems. Quantum mechanical phenomena are essential in nanometer scale semiconductor devices, as for the Resonant Interband Tunneling Diode (RITD), whose properties differ from those of the Resonant Tunneling Diode (RTD) because of the role played by the valence band electrons in the control of the current flow5. For the new family of heterojunction resonant interband tunneling diodes, which make use of resonant interband tunneling through potential barriers, we haye to consider the multi-band structure in the transport computation of the current. In this context, a simple model introduced by E.O.Kane 6 in the early 60's describes the electron behaviour in a system equipped with two allowed energy bands separated by a forbidden region. The Kane model is the simplest framework capaple of including one conduction band and one valence band in each material of a heterogeneous device and it is formulated as the coupling of two Schrodinger-like equations for the conduction and the valence band wave (envelope) functions 7 . The typical band diagram structure of a tunneling diode is characterized by a band alignment such that the valence band of the positive side of the semiconductor device lies above the conduction band of the negative one. In this paper, we face the problem of a fluid dynamical formulation for a semiconductor device characterized by a two-band (i.e. conduction and valence band) structure where the carriers dynamics is driven by interband tunneling. We start from the formulation of the Kane model in terms of Wigner functions 7 , and derive formally a system for the zeroth, first and second velocity moments. Finally, a short discussion in given on such model and on related problems, such as closure and numerical implementation.

2.

Wigner formulation of the Kane model

Let ipc(x,t) be the conduction band electron wave (envelope) function and ipv(x,i) be the valence band electron wave (envelope) function.

80

In the one-dimensional case, the Kane model reads as follows6,7

r .M

2m 9a;2

lh

-bT =

m

h2 jP_

•

ft2p^c

,

•

ox

2

Ipv + —P-Zm scaled ox by 2ir, m is the where i is the imaginary unit, H is the Planck constant bare mass of the carriers, Vc and Vv are the minimum of the conduction band energy and maximum of the valence band energy respectively. Moreover P is the coupling coefficient given by 2m 9a;

r=mJm~E h\

9

2mm*

which is obtained through the energy dispersion relation and where m* is the effective electron mass depending, through the a;—coordinate, on the layer composition, but otherwise isotropic, and Eg = Vc — Vv is the ^-dependent gap energy. We define the density matrix pij(r,s,t) = rpi(r,t)t/jj(s,t), i,j = c,v. Taking formally the derivatives with respect to time t of the density matrix element and using Kane eqs., a set of four coupled evolution equations is derived for the density matrix elements pij. The Wigner function is defined by the inverse Fourier transformation x ^.-1 ( ft ft witj{x,v,t) = T V»,i [x + ^V, x - —77 Then we obtain the following system, which is the Wigner function formulation of the Kane model dwc ' .Jfflcc .,_ dlVcc ^ H2P dwvc - ihPv [wvc + wcv] in—-— = -ihv—— + Qccwcc + -— dx dx at ox 2m ih

dw. dt

ih

dwvv dwvv = —ihv—^ dt dx

= -inv—T ox

h Qvcwcv + —— 2m

dwvv dx

h2P 2m

dwc dx

•h ~vvw~vvvw " vv

+

dwc dx dwvc dx

ihPv \wv

+ iHPv [wcv + wvc

where wcc and IDy y 3X6 real, wcv — wvc and +00+00

^ijlVij

-hi I

— OO — OO

with i,j = CjV.

V

^

X

- ^

t

) -

V i

(

X +

^

t

Wij{x,v'^e^-^^dridv'

81

Let now define the particle density, the current density, the energy density and the third moment, limiting to the 1-dimensional case. If the quantum mechanical probability densities n^ are defined by +00 -t-00

iij(x,t)

= / Wij(x,v,t)dv

,

i, j

=c,v,

then ncc and the probability densities for the positions of electrons in band of conduction and valence, respectively. The interband terms ncv and complex functions such that ncv = nvc. It is worthwhile to remark that only ncc and real functions, corresponding to probability densities, but also we remark that the Wigner functions arise from the envelope function, not from the wave function and so the physical meaning of this quantities has to be understood in the sense of the envelope theory 8 ' 9 . It is well known that the first order moment of the Wigner function with respect to the velocity and multiplied by the charge — q is the quantum current density. Similarly, in the frame of a two-band system, we define +00

Jij — ~Q / vwij (x, v, t)dv ,

i,j = c,v .

—00

Also in this case we can recover the classical meaning of the conduction (valence) current density Jcc (Jvv) for electrons in conduction (valence) band. For i, j = c,v, the set of the higher order moments is +00

£ij{x,t) = —• j

v2Wij(x,v,t)dv:

—00 +00

M(ip{x,t)=

v3Wij{x,v,t)dv.

The equations for the moments of the Wigner functions can be derived by multiplying the equations of Wigner system by 1, v and v2 and by integrating over the velocity space. The first set of equations for the evolutions of the position number densities reads as follows

82

( dnc dt dncv Ot

ldJcc q dx - ~

.ftP dncv 2m dx +

dnvc dx

l

j.[Vc(x,t)

1 d Jvv q dx

\y cv "t~ Jvc\

-Vv(x,t)]nc

_ .hP_ l 2m dnvx ~8t

"T

07lcc

OTlvv

dx

dx

. HP dncv _ dnvc 2m dx dx

P

[Jcc

Jvv]

lu cv

+ Jvc q In the Kane model, in which only interactions among the conduction and the valence bands are allowed and all the others are neglected, the wave function ip is expressed by ipc{x)uc0(x) + tpv(x)uo(x) where UQ and UQ are Bloch functions 8,9,7 . Then, for the two-band Schrodinger-like Kane model, the total density is ntot(x,t)

= \ip\2 = \tpc\2 + \ipv\2 =ncc(x,t)

+nvv(x,t)

and the quantum continuity equation takes the form d (ncc + nvv) 1 — = -Aw J totdt q Using the expressions for the wave function in terms of Bloch functions, in the spirit of the envelope theory, the total current density for the two-band system is Jtot(x,t) = - x — W ' W ' - V»VV>) = Jcc + Jw lmncv. 2m m The second set of equations in the quantum hydrodynamic model for the two-band Kane system is given by the following first order moments equations : 2qP dJ, 2q d£cc q • HP d Jvc) ~r \&cv i &vc) = + V t)n +l [Jc -dT m^xm ^ - 2m-d-x m dJcv 2qd£cv q , , -W=m^x-+2m-[VAx>t)+VAx>t)K +

i^[Vc(x,t)-Vv(x,t)]Jc.

2~

^

2m dx

\*Jcc ' Jvv)

'

\&cc

m

&W)

83

dJ^ _2qd£v1 q_ < .hP_ d_ _ _ 2qP_ ~ ^ + Vv[X t)nvv+l { cv vc) [ cv + vc) 9* ~ m dx m ' 2mdx m ' The equations for the current densities (first order moments) contain the second order moments £ij(x,t), which can be interpreted as energy density terms. A simple quantum hydrodynamic model can be obtained directly from the first two moments, by manipulating the energy terms appearing under the divergence symbol and taking into account that 1

m d

d£jj

2

dx

Am

2q dx \ riij

l3

dx

d\ Ui dx2

/Tli

+

W_d_, 2mdx{Tlii

ij)

with i,j

= c,v, and applying the isothermal closure conditions on the 1 d2Jn-j temperatures T^. The quantum correction term /rTjJ dx2 can be interpreted as an internal self-potential, the so-called Bohm potential. The temperatures T^ are defined by T- — i

< P+P-Wij < Wij

- i

i

>

< P+Wij

>< P-Wij

< Wij

> i T-.

1 d

>

,

>

2

.mv

where < • > is the mean value and P± = - — ± 2 dx

i^^. n

For completeness we report the second order moment equations dSc, dt OCQV

dt

mP 2

dx

m 2

dM& dx

-iw.-vje UOyy

dt

m

dM$ dx

q +

2m dx

^ ' - ^ " ihP d \£"VV 2m dx -gKJvv

+

^ mP

I &C

- f £ | ^

+

[M&+MW]

^ (M$-M)

- em) - ^f[M^

+ jif W].

3. Concluding remarks A quantum hydrodynamic model is obtained directly from the set of zeroth, first and second order velocity moments of the Wigner equations for a twoband Kane model. In this contribution, we present some preliminary results for a two-band 1-D structure; the future research is oriented towards the

84

closure of the moment equations and towards the numerical validation of the model. The closure of the previous equations system can be done with different methods. The simpler one consists in assuming that the temperature is constant (isothermal assumption), for the zeroth and first moment equations. Otherwise, a closed set of equations can be obtained by means of the thermal equilibrium Wigner functions for the two-band semiconductor model, in the case of the Kane Hamiltonian 10 , or using the maximum entropy principle 11 . Acknowledgements This work was performed under the auspices of the National Group for Mathematical Physics INdAM and was partly supported by the Italian Ministery of University MURST (National Project "Mathematical Problems of Kinetic Theories", Cofin2000) and by the Italian Research National Council CNR (Strategic Project "Modem Matematici per Semiconduttori"). References 1. K.Bl0tekjaer, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices ED-17, 38 (1970). 2. A. M. Anile, G. Mascali and V. Romano, Recent development in hydrodynamical modeling of semiconductors, pagg. 1-54 in Mathematical problems in semiconductor physics, A.M. Anile editor, Lecture Notes in Mathematics, Springer 2003. 3. C.L. Gardner and C. Ringhofer, Smooth quantum potential for the hydrodynamic model, Physical Review E 53, 157 (1996). 4. A. Jiingel, Quasi-hydrodynamic Semiconductor Equations, Birkhauser, Basel, 2001. 5. N. C. Kluksdahl, A. M. Kriman, D. K. Ferry and C. Ringhofer, Self-consistent study of the resonant-tunneling diode, Phys. Rev B 39 (11), 7720 (1989). 6. E. O. Kane, Energy band structure in p-type Germanium and Silicon, J. Phys. Chem. Solids 1, 82 (1956). 7. G. Borgioli, G. Frosali, P.F. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transport Theory Statist. Phys. 32(3&4), 347 (2003). 8. M.G. Burt, The justification for applying the effective-mass approximation to microstructure, J. Phys: Condens. Matter 4, 6651 (1992). 9. T. Wenckebach, Essentials of Semiconductor Physics, Wiley, Chichester 1999. 10. L. Barletti, On the thermal equilibrium of a quantum system described by a two-band Kane Hamiltonian, Preprint 2003, (submitted). 11. V. Romano, Non parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices, Math. Methods Appl. Sci. 24, 439 (2001).

H Y D R O D Y N A M I C LIMIT FOR A GAS W I T H CHEMICAL REACTIONS

M . BISI Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano, Italy, E-mail: [email protected] G. S P I G A Dipartimento di Matematica, Universita di Parma, Via D'Azeglio 85, 43100 Parma, Italy, E-mail: [email protected]

A first approach to the hydrodynamic limit for a gas mixture undergoing moderately fast bimolecular reactions is presented by a Chapman-Enskog asymptotic procedure applied to Grad's 13-moment equations.

1. Introduction Kinetic formulation of chemical reactions in rarefied gas mixtures is a cumbersome but essential task which is being given some attention in the literature. 1 ' 2 One of the main outputs expected from the kinetic approach is derivation of macroscopic equations for the main observable fields. Typical tools used for such a purpose include asymptotic methods (ChapmanEnskog algorithm) and moment equations (suitable closure of a series expansion). Analytical manipulations are quite awkward for a mixture even in the absence of chemical reactions. However, it has been possible to determine the general structure of the Newtonian constitutive equations and of the resulting hydrodynamics (Navier-Stokes equations) via asymptotic analysis versus the small parameter constituted by the dimensionless molecular mean free path (Knudsen number). 3 On the other hand, the Grad 13-moment approach 4 can be extended to deal with gas mixtures, 5 though explicit expressions for coefficients are available, to our knowledge, only for Maxwellian molecules.6 To such a case we shall confine for simplicity also in the present paper, which is dealing with the kinetic model for a bi-

85

86

molecular chemical reaction proposed in Ref. 7. After scaling the relevant Boltzmann-type integrodifferential equations, two Knudsen numbers spontaneously arise, representing the ratio of a typical elastic and, respectively, chemical relaxation time to the chosen macroscopic time scale. Most of the pertinent literature is devoted to the case of slow chemical reaction, in which the elastic relaxation time is much shorter than the chemical one, and the elastic collision operator plays thus the dominant role in the evolution process. This is the scenario in which a simple closure at the Euler level provides macroscopic equations describing the chemical reaction on its proper (slow) scale, after the initial fast transient in which mechanical equilibrium has been reached (see for instance Ref. 8). Here we aim at considering physical situations in which the chemical reaction also becomes fast with respect to the macroscopic scale, but yet the process is mainly driven by elastic scattering. This can be modeled without loss of generality, by taking elastic and chemical relaxation times to be 0(e 2 ) and O(e), respectively, with respect to the typical macroscopic time, where e is a suitable small parameter. In this preliminary approach we are interested only in the first order corrections (namely O(e)) to the reactive Euler equations. Next order (0(e 2 )) corrections will hopefully be considered in a future paper. Our starting point is the set of 52 Grad's equations as worked out recently in Ref. 9. It is known in kinetic theory that the Chapman-Enskog procedure is much easier to apply to the Grad equations rather than to the kinetic equations, and yields the same result, at least for Maxwellian molecules.6 It is reasonable to expect the same in the present chemical frame, though the investigation of the cumbersome algorithm leading from the Boltzmann to the hydrodynamic level is also left as future work. Formulation of the problem and governing equations are presented in Sec. 2. The asymptotic limit for e —> 0 is then performed in Sec. 3 according to Chapman and Enskog after discussing the proper hydrodynamic variables to be used. 2. Chemical reaction model We consider a four-gas mixture whose molecules As, s = 1 , . . . , 4 interact by binary elastic collisions and by the bimolecular reaction A1 + A2 ^ A3 + A4

(1)

where, if Es is the internal energy of chemical link, we may always assume A£ = - E ! = i A s £ s > 0, where A1 = A2 = - A 3 - - A 4 = 1. The interested reader is referred to Ref. 9 for all details that will have

87 to be skipped here for brevity. Unknowns for Grad's equations to be studied are number densities ns, mean velocities w|, temperatures Ts, viscous stresses pf and heat fluxes qf. Global quantities for the mixture are labeled by the same symbols without any superscript. Masses are denoted by ms, with M = m1 + m2 = m 3 -f m4, and /xsr for the reduced mass of the (s, r) pair. For the physical situation described in the Introduction, scaled Grad's equations read as9 dn> dt du?

Jd{KTs) \ dt

3 2

s

sd{KT

d(n°uj) _ 1 dxi e

(2a)

^ ( n - K T - ) ^ dxj dxi

3 + u* dxj

dt

{

du

)\

ST^s

t

1

du

.

dxi

j dxi

l]

dxi

,

K

+

1

d

(2b) 1

l! dx;

. (2c)

dpfj dt +Pi

du*

d +

{UkPi )+n

dxk

K T

J

[dxj

+

dxt

3^dxk)

^r-^^P«^- + T ( ^ - + dxk 5 \dxj dxi dxk 3

f c

dqt

9 .

s

s.

7

s

ax.

at l

dul

2 , du]

dXj

dxi

s

s s

2 ° dxj \ m J

is^j dxj\rn n J

+Pik

3 2

l3

e2

dxk)

dus

ux dxkk e

l]

' (2d)

„ _ . 8 (KT 9ii V ms 5 •" dxj ' 2 a

Ptj dPkj msns dxk

5

tJ

s

1 2 ' '• e^W?

+ -W°, (2e)

where the elastic "collision terms" are given by 4

(3a) r=l

2"SE< r=l

V?.=2][>f

M' ms + mr M°

r=l 1 ^•(ur-u8^

^K{Tr

-Ts)

+ ^mr(ur

nsp^, - nrpf, + nsnrmr

Er=l

~us)2

(3b)

« - t O K - ««)

2<m s n s p^ 'TO5 + TOr)

-I^u'-u*) 2 J

+mrnrVsij (3c)

w =

5 n KTS < - l^ E "?vp»rM - o - - ^ E ^r^n-K -«;) 2 m in { 1 (/?f+ 3/?| )n #T E ^ ( m + m ) I 2 (<-«?) 4

r r

r

s

-5

T r

r

s

s

3

/?JP n r i f T r + m*n* /3f (u r -

us)2

mn

(3d) Under our assumptions and for our purposes, it is still consistent to use, for the chemical "collision terms", the close-to-mechanical-equilibrium expressions derived in Ref. 9, namely m1m?\

3„4

Q* = XaQ1 = X

2

AE

QKT

n n

—n^n2

,34 12

3 AE\ • r *, 0 r * \2' KTj ' (4a)

R^-Xsms{usi-ui)Q1,

S' =

(4b)

X'Q^n(KT)^-\(l-X')/SE^--l(KT') +

3 AEV~V --KT) 2' (4c)

I^-u) 2 + ( ^^r(^) Vg = As m* Q 1

W? = Xs Ql 5

- «

(AT*) «

-

(uf - «*)(«* -

Ui)

Ui)

Uj)

V[

- - Sij(us - u) 2

(KT) — + - (1 - A s ) « -

+ ^

1 («$ - «,-) - - m s « -

n

r

3

Ui)

(4d)

AE

U j )(u

s

M

- u

AE\

(i-i)K rlcr)

(4e)

89 The coefficients v[r, j / | r , (3%.r and u\^ denote suitable angle integrated microscopic collision frequencies for elastic (s,r) scattering or for the direct (endothermic) reaction, which turn out to be independent of molecular velocities in our hypothesis. 9 T stands finally for incomplete gamma function. Other symbols to be used later are the pressure tensor Pij — nKT 5ij + pij and the total (thermal plus chemical) energy density The model (2)-(4) shares several crucial properties with the actual kinetic equations. In particular, the same seven macroscopic conservation equations (corresponding to the seven collision invariants), which read as I - (n s + nr) + ^ - • (nsus + nrur) at ox d

d

dt

9x

0

(s,r) = (l,3), (1,4), (2,4)

iu + P ) = 0

d_

=l>"+") + £- (Xpu + di 2

f ^ u + P - u + q + ^ ] Esns(us

- u)

(5) Of course, they do not keep trace of collision terms (thus, are eindependent), and are exact but not closed. The Euler closure is achieved, as usual, by replacing undesired moments by their equilibrium values. Now thermodynamic equilibrium is characterized (see Ref. 9) by all equal mean velocities and temperatures (uf = Ui,Ts =T), all vanishing viscous stresses and heat fluxes i Pij = 0, qf = 0), C and densities related by the mass action law7 12 n1 n2 AE QKT (6) 3 4 34 n n Choosing to play with the 8 macroscopic fields n3, m, T, reactive Euler equations are then made up by the 7 partial differential equations following from (5) 9

I

s

n

d

dtX

(ns + nr) u = 0

( S ,r) = (l,3), (1,4), (2,4)

d d_ d_ (pu) + — • ( / ) u « u ) + nKT) = 0 dx di\2pU

U) +

d_ <9x"

1

2pU

2

U+nKT

u (7)

90 to be coupled to the algebraic equation (6). This differs significantly from the case of slow chemical reaction, where the above fields actually constitute the 8 elastic collision invariants, and, correspondingly, reactive Euler equations are 8 partial differential equations including chemical contributions. 8 The latter fact has bearing also on the Chapman-Enskog procedure to be developed in the next Section. There are indeed only 7 hydrodynamic variables, to be used as unknown fields and to be kept unexpanded. They are given by n 1 + n 3 , n 1 + n 4 , n 2 + n 4 , u, and U.9 This will put some constraints on the first order asymptotic expansions {0) {1) ns=ns(0)+ens(D u° = u° +eu: Ts=Ts(o) + eTs(i) s(0)

Pi)

,

s(l)

s(0)

s

+ePij

Qt =Qi

.

s(l)

+eQi

(8) to be introduced into the Grad equations. 3. Hydrodynamic limit Proceeding step by step, the 0(e~ 2 ) terms from (2) yield

nf0)=o,

ss<°) = o,

v# 0) = o,

w* (0) = o.

(9)

The first is, for fixed i, a linear homogeneous algebraic system for u* with matrix elements

,

4 sr

s 0)

r

sr

$"• = ^ > n ( n ( ° ) - 5 Y^ i/f'/i''n a ( 0 ) n' ( 0 ) -

(10)

Due to the properties of the matrix $ s r established in Ref. 10, there exist exactly oo1 solutions, namely u^ ' = u^ Vs,r. Since Uj must remain 6 unexpanded, we get then u^ — Ui. The second equation in (9) is again a linear homogeneous algebraic system for Ts^°\ with matrix elements $sr = 3K va{ _JL_ 1 ms +mr

ns(o)nr(0)

_Ssr

m

y v,i ^l n»V) nm. j - - 1 i ms +ml

( n )

Matrix ^sr has the same properties of $ s r , 1 0 and then there are oo1 solutions, namely Ts<°) = Tr^ Vs,r, from which Ts^ = T™. The third of (9) is, for fixed (i,j), a linear homogeneous algebraic system for p^ , where the matrix of coefficients Tsr (omitted here for brevity) has non-zero determinant, 10 so that only the trivial solution p^j = 0 Vs is left. The same occurs to the fourth of (9), for fixed i, versus the q^ , with a regular matrix of coefficients Esr,w and then with unique solution <jzs( = 0 Vs.

91

Until now, the situation is essentially the same as for slow chemical reaction. 10 The difference arises in the 0(e~1) terms, which yield

= 0, (0)

1)

^• + vJ -o,

S°W + s»W := o,

W?(0) + W-{1) = 0.

(12)

The first provides at once „K0) n 2(0) n 3(0) n 4(0)

/V 2 V

^

(13)

which is just the zero-th order approximation of the mass action law (6). From (4), it follows then immediately i?* (0) = <SS(°) = Vtf0) = W^ (0) = 0, so that the other four equations in (12) involve merely the mechanical terms and turn out to be the same as (9), only with different unknowns . Bearing s( 1)

in mind previous results, we may deduce then that all drift velocities u{ must take, for fixed i, the same common value, which is necessarily zero. For temperatures we get analogously Ts^ = T 1 Vs, but, contrary to the slow reaction case, T is not an hydrodynamic variable, and T 1 must coincide with the first order correction T^ to the gas temperature T = T^ +eT^\ As for p*j ' and q* , they must all vanish. Finally, invoking the fact that the quantities n1 +n3, n1 +n4, n2 + n 4 , U must be kept unexpanded, it is readily obtained n'l^A'n'W,

^I^TW. (14) v 2 AE Then, to first order in our scaling, neither Onsager relations nor Newtonian constitutive equations appear. Differences in the drift velocities, viscosity coefficient, thermal conductivity would arise as second order effects. In addition to u, one may choose as unknown fields the quantities ns(-°\ s = 1 , . . . , 4 (which determine also T^ by virtue of (13)), since they are in a one-to-one relationship with the hydrodynamic variables via nr+n'=

U=\nKT^

nr^

+ ns^

+ j^E*ns^.

n i(D

(s,r) = (1,3), (1,4), (2,4) (15)

The sought asymptotic closure of (5) is achieved thus by simply expressing T^1 in terms of the previous variables, which makes the tensor P explicit as nK(T^ + e T ^ ) times the identity tensor, with q = 0 and u s = u. The only correction to the Euler equations is an additional scalar pressure, as pointed out already on phenomenological grounds. 11

92

We pass then to the next step, namely to 0{l) terms, and it suffices to consider the equations for number densities and temperatures, i.e. Q -(I)

-

<9ns(°) dt

d(ns^Ui) dxi

S-<») + 5-(D ="-„•(«) ( ^ T ^ + u,l alKTl0)>) 2 ' \ dt dxi

(16a)

J

+ n^KTM ""

** dxi (16b)

It is easy to check that S s ( 2 ' = 0, and QS(I) =

n3(°)+n

y

I(1)Q=

_ \s i(i) 34 J_

T(3

AE \

-•i^G^M)''**"*"

S»W = \> n1™ Sa = \'nW

Q

T

,2(0)

(17a)

-^>-I(I-A*)AE -l

\2' KTWJ [ \2' KT(°)J (17b) Summing up Eqs. (16b) with respect to s, a little algebra yields

^ . m

»* S ^

+»

3 £ ^ +Bmm»!s.

(18)

2 dt 2 dxi dxi Equations (18) and (16a), whose unknowns are the function n 1 ^ and the operator -£, must be compatible. The relation between the zero-th order time derivatives of T^ and of the ns(°) is provided by (13) {KT^)2 AE

do(KT^) dt

9 0 n s <°) Zs

«s((0)

dt

(19)

In this way the ^ operator may be eliminated, yielding a linear algebraic equation for n1^1', whose unique solution provides the sought unknown T^\ \2 3 {KTW\ 1 + 2-n AE '

3 nK 3 2

2 4

{KT^KT?-*

- nuil [ —--=-

\ AE

E

dn8^ n (°) ftci XS

s

3 +

2

n U i

4

1

s =1

9 /AT<°A 9 x A AE J

+ n

i i T ' 0 ' <9u,; ' AE 0 ^ (20)

93 T h e 0 ( e ) correction terms to the pressure are t h e n affected by the spatial gradients of densities and t e m p e r a t u r e , and by the divergence of the mass velocity. Explicitly, our hydrodynamic limit reads as ( > ) + n r ( 0 >) + ~

|

[ ( n ' W + r f W ) u] = 0

( p u ) + — • ( p u ® u ) + — [nK (T<0> + 9

(l

2

TT\

9

(]- pu2 + U + nKT{0)

(s,r) eT^J

= (1,3), (1,4), (2,4) 0

+enKT{1n

u

0

(21) with U, r ( ° ) , and T™ given respectively by (15), (13), and (20). Euler equations are recovered for e = 0. Second order space derivatives do appear in (21) because of the constitutive equation (20) for t h e temperature correction T^ 1 ); such t e r m s are all affected by the multiplicative small parameter e.

Acknowledgments This work was performed in the frame of the activities sponsored by MIUR, by G N F M , and by t h e University of P a r m a (Italy), and by the European T M R Network "HYKE".

References 1. V. Giovangigli, Multicomponent Flow Modeling, Birkhauser, Boston 1999. 2. C. Cercignani, Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations, Cambridge University Press, Cambridge (U.K.) 2000. 3. J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam 1972. 4. H. Grad, Comm. Pure Appl. Math. 2, 331 (1949). 5. T. I. Gombosi, Gaskinetic theory, Cambridge University Press, Cambridge (U.K.) 1994. 6. M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York 1969. 7. A. Rossani and G. Spiga, Physica A 272, 563 (1999). 8. M. Groppi and G. Spiga, Trans. Th. Stat. Phys. 30, 305 (2001). 9. M. Bisi, M. Groppi and G. Spiga, Contin. Mech. Thermodyn. 14, 207 (2002). 10. M. Bisi, M. Groppi and G Spiga, preprint No. 322, Dept. Math., Univ. Parma (2003). 11. I. Samohyl, Collection Czechoslov. Chem. Commun. 40, 3421 (1975).

A N E X A C T M A C R O S C O P I C E X T E N D E D MODEL W I T H M A N Y M O M E N T S FOR ULTRARELATIVISTIC GASES

F. BORGHERO, F. DEMONTIS AND S. PENNISI Dipartimento di Matematica, via ospedale 72, 09124 Cagliari, Italy. E-mail address: [email protected], [email protected] Relativistic Extended Thermodynamics is a very interesting theory which is widely appreciated. Here its methods are applied to ultrarelativistic gases, and an arbitrary, but fixed, number of moments is considered. The kinetic approach has already been developed in literature; then, the macroscopic approach is here considered and the constitutive functions appearing in the balance equations are determined up to whatever order with respect to thermodynamical equilibrium. The results of the kinetic approach are a particular case of the present ones. 1. Introduction Relativistic Extended Thermodynamics is a well established and appreciated physical theory (see refs. [1,2] regarding the first pioneering paper on this subject and an exhaustive description of the results which has been subsequently found). More recent results regarding the kinetic approach are described in refs. [3-6] and many interesting properties are there obtained and exposed. The macroscopic approach has been also investigated, but the exact solution of the conditions which are present in Relativistic Extended Thermodynamics, for the case of an ultrarelativistic gas with many moments, is still lacking. This gap is here filled, and it is shown that the general solution can be obtained with little, but meaningful modifications of the kinetic approach. The balance equations of Extended Thermodynamics for an ultrarelativistic gas with many moments are QaA«h-f)n

=/ft-/3n

ft,™ = (),••• ,N.

(1)

The tensors Aa^'"^n and l^'P™ are symmetric and trace-less. In particular, Aa is also indicated with Va and denotes the particle number-particle

94

95 flux vector, while A01^1 is also indicated with Taf3\ i.e., the stress-energymomentum tensor. The entropy principle for the balance equations (1), thanks to Liu' s Theorem [7], amounts to assuming the existence of the Lagrange multipliers T,px...pn, symmetric and trace-less, and of a 4-vectorial function h'a (related to the entropy - entropy flux) such that dh'a

f;E ft ... A ./ ft - A '>o > n=2

where the Lagrange multipliers have been taken as independent variables and Pfa.'.'.'g^ is the constant tensor such that, for every tensor B^1'"^, the new tensor B^1'"^nPg\\\'S'n is symmetric, trace-less and is equal to the sum of Bai'"an and of a linear combination of its traces through constant tensorial coefficients. Its expression can be found for example in ref. [8] and reads [n/2] Pl-Bn

~~ 2—1

S

9

9

Qfos + l

9Bn

yPts&2s-\

90201 '

s=0

with P " defined by the following recurrence formula n + 1

~ 4 ^ n-s s+1 ' ° " Now, the eq. (2) show that Aa)3l'"^n are known in terms of h'a and their symmetry impose conditions on the representation of h'a. There are now three ways in which to proceed: (1) Approach at a macroscopic level: It uses the representation theorems for isotropic functions to write down the most general expressions for h'a, j£i""A*; after that, the symmetry conditions and the zero trace conditions for (2)i impose restrictions on h'a, while (2) 2 does the same for iPf'P™. This approach is here used in an expansion around thermodynamical equilibrium and up to whatever order. (2) Approach at a kinetic level: it proposes h'a, except for an arbitrary single-variable function / ( # ) , as

h'a = JF( SO + E A p* + • • • + Eft...^?"1 • • -P0N )PadP, (3)

96 where pa is the 4-momentum of the particle so that we have papa = 0 and dP = y/—g p 3, p is the invariant element of momentum space. It is easy to see that the expression (3) satisfies the symmetry and the trace conditions for (2)i. This expression has been obtained by Boillat and Ruggeri [5], who have studied also interesting mathematical properties of the subsequent model. (3) A generalized kinetic approach : It is here proposed starting from an arbitrary function / of N + 1 variables f(Xo,Xi, • • • ,-Xjv) and defining a

h" = y>(E,E / J l p'V.. .Eft.-.^p* •••/" )p dP.

(4)

It is easy to see that also this expression satisfies the symmetry and the trace conditions for (2)i. It is also proved that the first and third approach are equivalent, so that we can refer to the third approach also as the macroscopic approach. We note that, in the kinetic approach, the distribution function at equilibrium is determined in terms of F ( E o + E^p^ 1 ) so that the distribution function outside equilibrium can be simply obtained by substituting E 0 + E ^ 1 + • • • + E/3l.../3Np'31 •••pP" to So + S ^ 1 . Instead of this, the macroscopic approach (3) has i 7 '(Eo,E ; g 1 p' 3l ,0, • • • ,0) for the determination of the equilibrium distribution function and its knowledge doesn't determine F outside equilibrium. This is the reason of the one parameter family of single variable functions, arising from integration, which appear in the macroscopic approach. We remind also that the counterpart of this model with only 14 moments has already been treated in [9]. A similar result for the non ultra-relativistic case would be desirable, but it hasn't until been proved. In the next section the generalized kinetic approach will be exploited ; the first approach will be considered in section 3 and proved that it is equivalent to the third approach. 2. The generalized kinetic approach A particular general exact solution can be found as follows: A particular case of eq. (4) is that with F an homogeneous function of

97

degree hi in the variable Xi for i = 2, • • • , N, i.e., F = Fh2,...,hN{Xo,Xi)

(X2)

2

N

---(XN)

,

in which case eq. (4) becomes h,a = jFh2,..,hN(z,x0y>)

(EJiy2p^P^

•••

f Fhit... , h j v (E, E f t p " 1 ^ 1 • • •pA2h' • • -pMl • • -pA*hi • •

' E A 2 I • • ' ^A2h2

' ' " ^ A i i • • ' ^Aih.

• • • E , 4 m • • • ^ANhN

• ••P5-)hNpadP

(ZSI...SNP6*

• • •PANhNPadP

-PANI

,

(5) Air

where Air is a symbol standing for j3\r • • • (5ir and p p/?lr

stands for

...p0ir_

Now, if two functions h'a are such that the corresponding Aaai'"an are symmetric, also their sum satisfies the same condition, so that the solution (5) can be generalized into 00

M-h3

^=E

hN

M—hi+i

•••

E

M=0

hN

E

h2=0

M-hN

M

••• E E (6)

hi=0

hN^!=0

hN=0

j-,A21--A2h2---Aii---Aihi---ANi---ANhNOL h2,--- ih-N

•EA2I

• • • ^A2h2

•' ' E ^ j

•••^A

i h i

• • • E/IJV! " ' " ^ A

N h N

,

which is expressed as sum of homogeneous terms of degree M, with respect to equilibrium, and with j-,A2i---A2h2--Aa--Aihi---ANi---ANhN°' h2,-,hN

U

f Fh2i...ihN{i:,^pP0)pA^

/~\ \<)

-

•- - p

A

^ • • -pA^

• • -pAihi

• • -pA^

••

-pANhNpadP.

This last tensor can be calculated by integrating; because it is of the type B«i-ap

=

J Fh^

hN (E>

E ^ y * 1 ...p°*dP,

(8)

we will refer to this simplier expression; by integrating as in ref. [9], we find fp/2]

/

\

=

4

£«,«,...«„ = ^ ( _ i ) P ( ^ J _J_ 7 -(p+2) Gha ... ihjv (E) •

(9)

•

98 where

7 = (-E^)1/2

Ua = ( - E " E M ) - 1 / 2 £ a ,

,

(10)

/*oo

= 5 « ^ + UaUf>, Gh2t... >hN (E) = / F h2 ,... ,hjv (E, a K + 1 d a , a = E a P Q Jo We note that the function Gh2,--- ,hN(^) in eq. (10)4 is arbitrary; in fact, if it is given, we can define ^

Fh2,..

thN(S,

a) = _ L ^ G h 2 i . . .

jhN

(E)e"CT ,

(11)

which yields again eq. (10)4. In the next section we will prove uniqueness of the solution so far obtained. 3. The general solution of the macroscopic approach. Let us impose the symmetry conditions and the zero trace conditions for ec l- (2)i; regarding the second one of these conditions we see that (1) when n > 1, Aaai'"angaiaj = 0 is an identity for the presence of the tensor Pg^.'.'.'g" and similarly we find zero, if we contract the index a of Aaai'"an with another of its indices, because Aaai-ai-aj-a„gaai = A a i a i - a i - a - a „ g a a i = 0 ) where the symmetry of _4Qai"-Qr> has been used and aj is any other index different from c*j. (2) when n = 0, there is no trace condition for eq. (2), (3) when n = 1, the trace condition is -^fla/3l=0.

(12)

In other words, the zero trace conditions reduce to eq. (12). Let us now consider the polynomial expansion of h'a in the variables T,A2, • • • , T*AN a n d call h'fi ... h the homogeneous part of h'a of degree ht in E ^ for i = 2, • • • N; let us see how it is restricted by the symmetry conditions

Kt3:n = o,

(13)

which arises from eq. (2). Obviously, h'fi hN is restricted by (13) only when we impose this condition at the order hi in Ey^ for i ^ n and at the

99 order /ij_j for i = n. Moreover, this restrictions don't link h'fi ... h to the other orders of h'a. Similarly, the equilibrium value of h'a, i.e. h^... ,0' ^ restricted by (13) only for n = 1, and doesn't link it to terms of the other orders. Then, h'a is a sum of functions of the type h'g h , which are restricted only by eqs.(12) and (13). Therefore, we can restrict ourselves to consider such restrictions only on h'fi ... h . Now this tensor, being a polynomial, coincides with its Taylor's expansion around the values E ^ = 0, • • • , T,AN = 0: in this way we obtain the equation (6) with B

h2-hN

= 7T-T7 • • • 77—TT

(h2y.

Qh2-\

(hNy.

(14)

\-hN L/Q

9 E f t i • • • <9£/32h2 • • • 9 E f t l • • • dY,0ihi • • • dT,pNl • • • dT,pNhN PA1X . . . pA2h2 . . . pAtl •B2I Bzh2 Bn

pAihi Biht

pAN1 BJVI

pA2NhN BwhN '

where the tensors P'" take into accounts the fact that Yip1...pN is traceless. Now we see that in the tensor B, 21 h 2h2 l l 'h{ N1 Nh" defined by (14) we can exchange a with any other index, as consequence of eq.(13) or of a suitable of its derivatives. It follows that it is a symmetric tensor; in fact for any couple of indices /? and 7, we can exchange a and /?, then j3 and 7, after that 7 and a, i.e.: p>-f3--y---a

_ p--a--f--/3

_ p---a--0---y

_

p-~j-@--a

Moreover, 5.'.'.' is traceless because in B'"^""y'"gp1 we may use the symmetry to carry /3 and 7 in indices of the same tensor P.'.' and, after that the trace is zero. At the end we find the eq. (6) with Bh2,-,hN A*1'"A2h2'"Ail"'Aihi'"AN1"'ANhna symmetric and traceless; moreover, from eq.(13) with n — \ and eq.(12) we have that also its derivative with respect to E 7 is symmetric and traceless. Vice versa, if these conditions are satisfied, then eq.(12) holds and also (13). In order to simplify the notation and work with less indices, let us characterize a tensor Baia2'"ap which is symmetric with its derivative with respect to E 7 , and with zero trace. The symmetry of p,aia'2'"ap shows that it has* the form b/2] ,

.

100

In the Appendix A of ref. [10] it has been proved that the tensor as°ioxcp + 1 is symmetric iff gPt2S satisfies the following recurrence formula _ 9P,2S-2 -

-1 ^dg^ ^ p Y

+ {p_2S

+

1)gp2s

(16)

This equation determines all the functions gPt2S except for that with the greatest value of 5, i.e.,
can be expressed as 9p,2s+2 ~ Qp,2s = 0 .

(17)

This equation, together with eq. (16), reduces to the ordinary differential equation dgp,2S+2

7—^

. ,

. r,\

r,

+ {P + 2)5p,2S+2 = 0 ,

which can be integrated and becomes gP,2S+2 = (-l) p 4 7 rG h 2 ,..., h j v (E) 7 - ( p + 2 ) , with (—l)p47rGh2,.-.,/iN(S) a constant arising from integration. After that, the other relations in eqs. (16) and (17) are satisfied. In this way the eq. (9) of the generalized kinetic approach has been obtained with G^2i... thN an arbitray function of S. It remains to impose the condition ^hNgai

= 0.

(18)

But, when at least one of h2,--- ,/ijv isn't zero, the tensor -B£Ql.'ft has at least another index which we have called on; moreover, 7 and a\ can be exchanged because we have already imposed the symmetry condition. Therefore, eq.(18) becomes —g£ "hN gai — 0 which is an identity, because BlC'h ls traceless. Then, eq.(18) has to be imposed only for h2 = • • • = /ijv = 0 and becomes dh'a ^ ^

= 0,

(19)

101 where h'^q denotes t h e value of h'a at equilibrium. Now, for the representation theorems, we have h'ea = / i 0 ( £ , 7)C/ a , so t h a t eq. (19) becomes

7 — ^ + 3/i 0 = 0 , 07

i.e., ho = 7 _ 3 i 7 o ( S ) , i.e., h'^q has still t h e form (9) with p = 1 and Go(S) = -4^-. This completes the proof t h a t the macroscopic and the generalized kinetic approach give t h e same result.

References 1. Liu,I-S., MullerJ., Ruggeri,T. Ann.of Phys. 169 191 (1986). 2. I. Miiller, T. Ruggeri. Rational Extended Thermodynamics, second Edition Springer-Verlag, New York, Berlin Heidelberg (1998). 3. Boillat,G.,Ruggeri,T. Continuum Mech. Thermodyn. 11 107 (1999). 4. Boillat,G.,Ruggeri,T. J.Math.Phys. 40 6399 (1999). 5. Boillat, G.,Ruggeri, T. Continuum Mech. Thermodyn. 9 205 (1997). 6. Brini, F..Ruggeri, T. Continuum Mech. Thermodyn. 11 331 (1999). 7. Liu, I-S. Arch.Rational Mech.Anal. 46 131 (1972). 8. Pennisi, S. Meccanica 37 491 (2002). 9. Borghero F., Pennisi S. The non-linear Macroscopic Model of Relativistic Extended Thermodynamics for an Ultra-relativistic Gas. Submitted to Rend. Acad. Naz. dei Lincei, Roma. (2002). 10. Pennisi, S. Continuum Mech. Thermodyn. 14 377 (2002).

THE RIEMANN PROBLEM FOR A BINARY NON-REACTING M I X T U R E OP EULER FLUIDS

FRANCESCA BRINI AND T O M M A S O RUGGERI Department (CIRAM)

of Mathematics and Research Center of Applied Mathematics University of Bologna, Via Saragozza 8, 40123 Bologna, Italy E-mail: [email protected] ruggeriQciram.unibo.it

The Riemann problem for a binary mixture of Euler fluids without chemical reaction is numerically studied. This system is a particular case of hyperbolic systems with relaxation and the interest of such analysis could be found in the fact that, since now, there is no systematic theory about Riemann problem for this kind of equations. For relaxation terms not too big, we found that the solutions of this Riemann problem are a composition of shocks, constant states and "deformed" rarefaction waves. It was already shown t h a t t h e mixture model is simplified under the strong assumption that the constituent fluids have equal masses. The validity of this approximation is also studied for different binary mixtures.

1. T h e m o d e l In order to describe a binary non-reacting mixture of Euler fluids (that is to say fluids neither viscous nor heat conducting), we have considered the balance law system in the form proposed by Ruggeri 1. If we take into account a mixture without chemical reaction in a one-dimensional space, equations read 2 dp

9 ,

dpv

,

d (

dpc

2

ot

8i{

d(\pv2 + pe)

Ot

, ,o7_ +2jv +

d \(\ +

T.

J2

:

d(pcv + J) , d (_ + pcv

d .

d-x{{-2

pV

2

J2 yc+p')

= -^

\ f ) {PW^C)

+P£+P V+

102

Jv

1\ +

1 n «)J\=0>

103

where p, c, p, v and J denotes respectively the total mass density, the concentration of the first constituent, the total pressure, the center mass velocity and the diffusion flux vector. Moreover, if m-i and 7; represent the atomic mass and the ratio of specific heats of the i-constituents, it holds 1 c (1 — c) m{c) mi m2 ' 1 _ pm(c) ( 7l

1 _ cm(c) (1 —c)m(c) 7(c) — 1 (71 — l ) m i (72 —l)"i2 72 A , J2{I-2c) +

r. 2 „ 2 / l

„\2

p \ m i ( 7 i - 1) "12(72 — 1 ) / 2/9 2 c 2 (l-c) 2 p J m(c) rpi = p cpe = — r r — + — 7 r 7(c)-1 2pc(l-c) ^ mi ' a

VZJ

In (1), (3 is a positive quantity, that plays the role of the inverse of a relaxation time and is supposed to be constant for the sake of simplicity. Concerning all the properties of the system, we refer to 2>3'4 and the references therein. For such a system Ruggeri and Simic have already studied analytically shock waves, obtaining a complete explicit description of them under two hypothesis: a) the unperturbed diffusion flux vector is equal to zero and b) the atomic mass of the two constituents are equal. Unfortunately this is not the most general case and also if only one of the previous hypothesis fails, it is no more possible to get explicit analytic expression for shock waves. In the following sections we will study numerically the Riemann problem related to the model. To this aim, equations can be rewritten in dimensionless variables

„ _ x_ X

f-~

l^

'- —

X V po

po J

J

VP0P0 where X, po and po represents respectively a space scale and suitable reference values of total mass-density and total pressure. Through this transformation the previous system remains the same except for the" symbol. Hence, for the sake of simplicity, from now on we will always use the dimensionless field variables, denoting them without 'hat'. 2. The Riemann Problem The Riemann problem for a system of conservation laws in a onedimensional space du dt

d f (u) dx

0

(3)

104 consists in determining a generally weak solution u(x,t) (—00, 00) x [0,00], starting from initial condition of the type u(x, 0) = ui if x < 0

and

G Rn

u(a:, 0) = u r if x > 0,

on (4)

5

with ui and u r given constant states. It is well-known that it is possible to construct the solution u(x,t) as a combination of constant states, rarefaction and shock waves. There are methods aimed to the analytic selection of the admissible solutions 5 , while numerically the solution can be selected, for example, thanks to the presence of artificial viscosity due to approximation errors. It is still an open question the complete description of the Riemann problem for hyperbolic systems with relaxation terms du

ir

+

d f(u) . , i = g(u)

.. (5)

-to

for g(u) of relaxation type. In this case one has to take into account the competition between dissipation and hyperbolicity. Comparing the solutions of the systems with and without relaxation, surely the propagation velocity of shock waves decreases in presence of relaxation, since also the amplitude of shocks decays. In Section 4, we will present some numerical results related to model (1), in which it is possible to observe such phenomena. The results confirm the possible presence of shocks in the solution together with constant states and "deformed" rarefaction waves. 3. Numerical Methods The following numerical results were obtained using the Uniformly Implicit Central Scheme of order 2, proposed by Liotta, Romano and Russo in 6 . The algorithm, which belongs to the group of the finite-volume methods, is based on the discretization of space-time with staggered cells. Starting from an equation system as (5), the scheme leads to U

n+l j + l/2

\ (»? + u?+i) + \ K - «5+i) - A (f (u^ 172 ) - f ( u ; ^ 2 ) ) +Ai Q g « + 1 / 3 + | g ( u ^ 3 ) + \g(u^

+ 1/2) ,

(6)

where, Ax, At and A are respectively the space and the time grid steps and their ratio. Moreover, if xm and tk are the generic nodes of the space and time grids and if u^ 1 denotes u(x m ,tfc), we have u n+i/*

= u? + ^

( g ( u ; + 1 / f c ) - %A

for k = 2, 3

105 1 0.95 a

I

0.9

0.85

^

0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

,

0.8

1

X

1 0.95

1

\

a. 0.9

0.85 0.8

"

1 ]

,

Figure 1. Two examples of p behavior for different mixtures, for the same interval of time, the same initial condition and the same value of /3: t = 0.5, /3 = 10, pi = pi = 1, Pr =Pr = 0 . 8 , vi =vr = Ji = Jr = 0, ci =Cr = 0.4. On the top: the mixture of O (mi = 16) and O2 (1712 = 32) is simulated with the real masses (continuous line) or supposing t h a t the constituents have equal masses (dotted line). On the bottom: the mixture of Ar (mi =39.9) and CO2 (m,2 =44.0) is simulated with the real masses (continuous line) or supposing that the masses are equal (dotted line). The approximation with equal masses works very well when the mass ratio is not so far from 1. Ax = 1/800.

where u^/Ax and fj/Ax represent a suitable approximation for the space derivatives at Xj of the field and of the flux. For more detail see 6 .

4. Numerical Results We considered system (1) with initial condition of the type (4) and used pi and pi as reference values (po and po), in order to obtain dimensionless variables. The first question we try to answer is about the validity of the approximations considered in 2 ' 3 . The assumption that the constituent fluids have the same atomic masses could appear too restrictive to be realistic. In Figure 1 we consider two different mixtures and compare for each one the solutions of equations (1) with the solutions of the approximated system, obtained from (1) assuming that the masses of the constituents are equal. It turns out that, as expected, the approximation by Ruggeri and Simic works as better as closer to one is the ratio of the masses of the two constituents and as smaller is the relaxation time compared to the simulation

106

time. In spite of the relaxation, for the model with unequal masses it is possible to distinguish at t = 0.5 the second sound and the sonic shocks on the right and the second sound and the sonic rarefaction waves on the left (although very damped). In Figures 1 and 3 we present only the behavior of the p variable since the one of the other fields is completely similar.

1

^ 0.41

0.95 0.9 0.85

1 '

'• p

"

0.4

0.39

0.8

l

u -0.5

0.5

0.01

0.005

Figure 2. An example of solution for system (1) when /? = 1, t = 0.5 and the initial condition is the same as in Figure 1. On the top left pressure (dotted line) and massdensity (continuous line) are presented in the same picture. T h e constituents of the mixture in this example are O and O2; Ax = 1/800.

Without relaxation, the solutions for equal or unequal atomic masses remain different for all the time, since they are self-similar with respect to the ratio space/time. Nevertheless it is easy to verify that the characteristic behavior of the shocks described in 2 is not so far from the one observed numerically if the relaxation time is not too small (compared with simulation time) and the constituent masses not too different. Indeed, in Figure 2 it is possible to distinguish essentially three types of waves: the one that behaves as the characteristic one (with propagation velocity close to zero and negligible jumps in velocity, concentration, pressure and diffusion flux

107 vector), the one that behaves as diffusive wave (with the intermediate propagation velocity and jumps in all the variables) and the one that recalls the sonic wave (with the highest propagation velocity, but, unlike the case 2 , with jumps in all the variables). Hence the qualitative behavior predicted in 2 could be useful also if the assumption mi = m^ is no more valid. As

1\

0.96

o- 0.9

0.65

\ "l I

08

-1

-0.8

-0.6

-0.4

-02

0

0.2

0.4

0.6

0.8

1

Figure 3. The behavior of p in a mixture of O and O2, at t = 0.5, for the same initial condition used in Figure 1, but for different values of /3: (3 = 0 (dotted line), /3 = 1 (continuous line), /3 = 10 (dotted-dashed line), Ax = 1/800.

already anticipated, Figure 3 shows a possible behavior of the solutions for hyperbolic systems with dissipation. Looking at the figure with attention, we can see that shocks moves slower as greater is (3. In this last figure it is possible to notice that diffusive waves decade quite fast and the rate of this decay depends, obviously, on relaxation time. Acknowledgments This paper was supported by fondi Cofin 2003 MIUR Progetto di interesse Nazionale Problemi Matematici Non Lineari di Propagazione e Stabilita nei Modelli del Continuo Coordinatore T. Ruggeri and by GNFM-INDAM. References 1. T. Ruggeri, The binary mixtures of Euler fluids: A unified theory of second sound phenomena in Continuum Mechanics and Applications in Geophysics and the Environment, Eds. B. Straughan, R. Greve, H. Ehrentraut and Y. Wang, 79, Springer-Verlag, Berlin (2001).

108 2. 3. 4. 5.

T. Ruggeri & S. Simic, Contin. Mech. Thermodyn. 15, No.7, (2003). T. Ruggeri & S. Simic, contribution to the present Proceedings (2004). T. Ruggeri, contribution to the present Proceedings (2004). C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer Verlag, Berlin (2000). 6. S. F. Liotta & V. Romano & G. Russo, SIAM J. Numer Anal. 38, No.4, 1337-1356 (2000).

O N T H E ONSET OF C O N V E C T I O N FOR A D O U B L E DIFFUSIVE MIXTURE IN A POROUS MEDIUM U N D E R PERIODIC IN SPACE BOUNDARY DATA

F. CAPONE AND S. RIONERO University of Naples Federico II Department of Mathematics and Applications l'R. Caccioppoli" Complesso Universitario Monte S. Angelo Via Cinzia, 80126 Naples - ITALY E-mail: [email protected]; [email protected]

The nonlinear stability analysis of the motionless state for a binary fluid mixture in a porous layer, under horizontal periodic temperature and concentration gradients, is performed.

1. I n t r o d u c t i o n Let Oxyz be a cartesian frame of reference with the z-axis vertically upwards and S = M2 x [0,d\ a horizontal (infinite) porous layer filled by a binary fluid mixture. In t h e past as nowadays double-diffusive convection in porous media, when S is strictly uniformly heated and salted from below, has been widely studied since it has geophysical and technical applications J" 3 - 6 . u - 1 5 . Recently an analysis of this problem has been carried out by many authors, on assuming a linear spatial variation of temperature and concentration along t h e boundaries 8 - 1 0 > 1 6 . This assumption is more realistic t h a n t h e strictly uniform heating and salting. B u t , because implies infinite temperature and concentration on the horizontal planes at large spatial distances, does not appear completely acceptable. In order t o overcome this difficulty, we consider a model in which diffusion-convection is driven by horizontally periodic t e m p e r a t u r e and concentration gradients. We assume t h a t : i)both the horizontal bounding planes are kept to time-independent spatially periodic temperatures and concentrations in the horizontal direction; iijboth the vertical differences in temperature and concentration between the horizontal planes are not constant. T h e plan of the paper is t h e following.

109

Section 2 is devoted t o t h e

110

mathematical statement of convection problem for a double - diffusive fluid mixture modelled by the Darcy-Oberbeck-Boussinesq (D.O.B.) equations. The existence of a motionless state is shown. In Section 3 we study the nonlinear stability of this state by means of the Liapunov direct method and obtain conditions ensuring global nonlinear exponential stability. 2. Basic equations and steady state solution As it is well known the Darcy-Oberbeck-Boussinesq (D.O.B.) equations governing the motion of a binary porous fluid mixture are 3 ~ 6 f Vp = -(ji/k) v + po[l - 7 T ( T - T 0 ) + V-v = 0 ATt + v • VT = kTAT l
1C{C

- C 0 )]g v ;

with: 7c and yr, respectively, the solute and thermal expansion coefficients; tp the porosity of the medium; To and Co, respectively, the reference temperature and reference concentration; v the seepage velocity; p the pressure; C the concentration; // the viscosity; T the temperature; kx and kc the thermal and salt diffusivities, respectively; c the specific heat of the solid; cp the specific heat of the fluid at constant pressure; A = (poc)m/ (pocp) fa. To (1) we append the boundary conditions TL(x) = a\ sin(x/d) + Tu Tu(x) = {a$/e) sm(x/d),

CL{x) = a* sm{x/d) +2CX onz = 0 Cv\x) = (a*/e) sin(x/d) + Cx on z = d

. > ^'

in which TL > 0, C\ > 0 and a*, a\ are two positive constants having, respectively, the dimension of concentration and temperature such that a*=a*^(>0).

(3)

IT

On assuming 0
ps(z) = -pogiAt z +

T,(x, z) = a\e-z/dsm{x/d) a

(TTTI

- lcCl)/{2d)

(4)

z2}

+ Ti(l - z/d)

T h e subscript m and / refer to the porous medium and to the fluid, respectively.

(5) (6)

111

Cs(x,z)

= a*e-z'dsm{x/d)+C1{2

- z/d)

(7)

where A\ = 1 + 7c(2Ci — C 0 ) — 7r(Ti — To) is a constant. The solution (5)-(7), because of (4) corresponds to the case of a porous fluid mixture layer heated and salted from below b . Let u = (u,v,w), 9, r , ir be the perturbations to the (seepage) velocity, temperature, concentration and (reduced) pressure fields, respectively, and introduce the non-dimensional quantities 5 ~ 6

x = dx*,

i=

ff,

u = ^ u *,

kT RZT

Ti-yT

d

k

S,C

kc

*> =A' *, f = , f e , e =y y 91rkd' R\ = S2. =

;

* = &.ir\e = fp,

c l ^ g^ kk d' c

c

(> 0)

vertical thermal Rayleigh number

(> 0)

vertical solutal Rayleigh number.

[ikc

Dropping all asterisks, it turns out that (V(a;, y, z) £ M2 x [0,1]):

Vir = -u +

Rzek-Szrk

V-u = 0 01 + u • V6> = - a i Rz e • u + # z w + A9 ,Leip0Ttt + L e u - V r = - a L e S z e • u + Szw + AT

(8)

where e = [erz cosx, 0, — e - 2 sinx). To (8) we append the boundary conditions Ou w= — =9 = T = 0on z = 0,.z = l . (9) In the sequel we shall assume that the perturbation fields are periodic functions of x and y of periods 2ir/ax, 2n/ay c and we shall denote by fi = [0,2-7r/ax] x [0,2n/ay] x [0,1] the periodicity cell. Finally to ensure that the steady state (5) - (7) is unique, we assume that < u > = < v > = 0. b Let us observe that temperature and concentration fields Ts(x, z), Cs(x,z) exhibit a bounded periodic behaviour in the unbounded x-direction. c When Q T^ 0, r = 27r can be the only admissible period in the x-direction 7 .

112

3. Global Nonlinear Stability Let us consider the Liapunov function

v(t)=1-\\8\f

+

^\\v\\\

(io)

Along the solutions of (8), on accounting for the boundary conditions (9), since Sz = \/XeN Rz and a\ = aLeN2, it follows that dV (11) dt <{RZ \r + ai(Fl+j:2)\-l}D where j :

2 < w 6 >

(12)

=

l

||U||2 + | | V ^ | | 2 '

D = IHI2 + Iiw|| 2 + Iivr|| 2 , < e~z cosxuS > +\/Le/N 1=

(is)

< e~z cosxuT

>

2

IHI + ||v^|p + ||vr||2 . _ < e~z sinx w9 > +V~Le/N < e~zsmxu>r > ||u||2 + ||V^||2 + ||Vr||2 '

Moreover, since

>

( ) ( }

3 n

'

-— = max^", 2-7T

(16)

A

we set

wrmAx:Fi'

w2=mFjr2'

(17)

where A = {u,9,T regular infi, periodic in x and y, satisfy the boundary conditions (9); V • u = 0 a n d D < oo}

(18)

is the class of the kinematically admissible perturbations, and underline that (17)i and (17)2 are new and characteristic of the problem at hand. In order to solve (17)i and (17)2 we introduce the transformation #i = — 8 cosrr,

8-2 = 9 sinx,

Y\ = — V cos a;,

I"^ = T smx.

Then it follows that Fi =

< e~z u8i > +VLe/N

< e~z uTx >

2

lulP + D l l v ^ + iivr^-H^-liri i=l

(19)

113

f2

z < e~ _ w62 > +y/Le/N

=

< e~z wT2 >

.

l|u||2 + E[ll V ^ll 2 + l l V r ^ 2 - | l ^ l | 2 - H r ^ 2 ] i=l

The boundary conditions for 0j, Ti (i = 1,2) are: 0! = 02 = Ti = T 2 = 0

on

z = 0,z = l.

(20)

Hence in the set of the most destabilizing perturbations (i.e. T% > 0, % — 1,2), because of the Poincare inequality, it follows that T\ < Fi,

Fi < ^2

(21)

with < e~z uTx >

< e~* u6x > +VU/N

1

iiuip + Hv^ip-ii^ip + iivriip-iirxip < e-z w92 > +VU/N ||u||2 + ||v0 2 || 2 -||0 2 || 2

2

[

>

< e~z wT2 >

+ ||vr 2 ||2-||r 2 ||2

w

and hence —— < maxf,* = -r—-, l Mi ~ A\ M{

—— < max?? = —— M2 ~ A*2 2 M2*

(24) '

v

where .4* = {u, 9i, Irregular in fi and periodic in x and y, satisfying the boundary conditions (9), (20), V • u = 0 and Di < oo} 2 2 2 2 d 2

(25)

with A = ||u|| + IIV^H - INI + iivr-H - H^ii , (i = i,2) . From (11), on accounting for (21) and (24), we obtain dV

f

Rz

2 7 T + Q l \M;

+

M2*

1}D.

(26)

The following (global) nonlinear stability theorem holds true. Theorem 1 - The condition Rz

< j ^ 1 +c*i

(27)

iih M

*

d

=

M* 4- M* ,!,. , , » 2

<""*

" l = 27rai M * ,

The maxima 1/Mj* and 1/Mj exist by virtue of the Rionero's theorem

(28)

114 ensures the global nonlinear asymptotic exponential stability of the motionless state (5) — (7), with respect to the V'-norm, according to the inequality

2TT

V{t) < V ( 0 ) e x p ^ 7 Rz

7 = (1 + di)7r min \ 1,

I

}•

1+dx

Vi>0 (29)

—V >0.

fo Le J

In Table 1 we collect t h e values of M*, (i = 1,2) obtained by solving the variational problems 1 - = m a x J7*, M. * A*

1,2

(30)

for some values of t h e ratio \/~Le/N. It is not difficult t h e n t o evaluate the values of d\ t o determine the energy stability threshold (27). Table 1. Solutions of the variational problems (30). VL^/N

M{2

M2*2

0.1

9.254

118.184

0.01

9.346

119.476

1

4.669

58.192

10

35.202

39.635

20

35.476

28.436

Acknowledgments This work has been performed under the auspices of t h e G.N.F.M. of I.N.D.A. M. and M.I.U.R. (COFIN2003): "Nonlinear m a t h e m a t i c a l problems of wave propagation and stability in models of continuous media".

References 1. G.I. Barenblatt, V.M. Entov, V.M. Ryzhik (1990), Theory of fluid flows through natural rocks. Kluwer Academic Publishers. 2. J. N. Flavin and S. Rionero (1996), Qualitative Estimates for Partial Differential Equations. CRC Press: Boca Raton, Florida.

115 3. D.A. Nield and A. Bejan (1992), Convection in Porous Media. Springer-Verlag: New York. 4. D.D. Joseph (1976), Stability of fluid motions I, II. Springer Verlag, New York. 5. B. Straughan (1992), The Energy Methods, Stability and Nonlinear Convection. Springer-Verlag: Berlin. 6. B. Straughan (2004), The energy method, stability, and nonlinear convection. Appl. Math. Sci. Ser. Vol. 91, Springer-Verlag, Second edition. 7. F. Capone and S. Rionero, Cont. Mech. Thermodyn. 15, 529 (2003) 8. P.N. Kaloni and Z.C. Qiao, Int. Journal Heat Mass Transfer 40, 1611 (1997). 9. P.N. Kaloni and Z.C. Qiao, Cont. Mech. Thermodyn. 12, 185 (2000). 10. J. Guo and P.N. Kaloni, ZAMP 46, 645 (1995). 11. S. Lombardo and G. Mulone and S. Rionero, Math. Meth. Appl. Sci. 23, 1447 (2000). 12. S. Lombardo and G. Mulone and S. Rionero, J. Math. Anal. Appl. 262, 191 (2001). 13. S. Lombardo and G. Mulone and B. Straughan, Math. Meth. Appl. Sci. 24, 1229 (2001). 14. G. Mulone, Cont. Mech. Thermodyn. 6, 161 (1994). 15. G. Mulone and S. Rionero, Rend. Mat. Ace. Lincei Series 9 9, 221 (1998). 16. D.A. Nield and D.M. Manole and J.L. Lage, J. Fluid Mech. 257, 559 (1993). 17. S. Rionero, Ann. Mat. Pura Appl. 78, 339 (1968)

A VARIATIONAL P R O B L E M W I T H NON-LOCAL CONSTRAINTS

S. C A R I L L O Dipartimento

di Metodi e Modelli Matematici per le Scienze Universita di Roma "La Sapienza", Via A. Scarpa 16, 1-00161 Roma, Italy E-mail: [email protected]

Applicate,

M. C H I P O T Institute of Mathematics, University of Zurich, Winterthurerstr. 190, 8057 Zurich, Switzerland E-mail: chipot@amath. unizh. ch G. V E R G A R A C A F F A R E L L I Dipartimento

di Metodi e Modelli Matematici per le Scienze Universita di Roma "La Sapienza", Via A. Scarpa 16, 1-00161 Roma, Italy E-mail: [email protected]

Applicate,

In this note, we summarize some recent results concerning a study a non-local variational inequality. The non-locality appears in the coefficients of the operator through some integral representing some elastic energy. It appears also in the constraints which have been called soft in the literature 7 .

1. I n t r o d u c t i o n This Section is concerned about the introduction of the problem and, in particular, those notions needed to achieve the results which are comprised in the subsequent Section 2. All the proofs, here not included, are given in The N-Membrane Problem with Soft Constraints2 where also a discussion concerning possible applications of results here presented is comprised. Let fi C W1, n > 1 be a bounded open set, with boundary dQ.. If 4>{x)eH\tt),

0
then the convex set

116

ondQ.

117

K^ = {veH^{Q) is not empty and for / G H^1(fl),

, v>4> a.e. x G Q}

,

(1.1)

there exists a unique solution to

u G /C<£ f (X-2) / Vu • V (i> - u) dx > (f,v — u) Vi> G K$ . Ja u is the solution of the so called obstacle problem. (See Lions and Stampacchia 6 , Brezis 1 , Kinderlehrer and Stampacchia 5 . The reader is referred to Kinderlehrer and Stampacchia 5 for an introduction to variational inequalities and for the notation on Sobolev spaces.) Physically (1.2) models the position of an elastic membrane to which a vertical force / is applied and which is constrained to stay above an obstacle represented by the function . In an other framework, u can be thought to represent the stabilized temperature of a plate figured by 0, and which is supposed to be that such a at each point above some level given by and 0 at the boundary. Then, / is the density of heat provided to the system. In this situation, one can "soften" the constraint by allowing only the average temperature to be above or below some level. Introducing for a G R IveHliSl)

K-n

K.a = iv e i ^ ( f i )

,

J v dx>a\

,

J v

,

(1.3)

(1.4)

dx
one is led to study the problem u € K.a

(resp. /Ca)

/ Vu-V(w-u) Jn in

dx>(f,v-

u)

Vu G Ka

{resp.

Ka)

(1.5)

It is easy to see that (1.5) admits a unique solution. Moreover, as for the obstacle problem (see Brezis 1 ), some regularity theory can be developped (cf. Chipot and Kis 3 ). Note that in the definition of Ka (resp. K,a) we did not consider the average but the integral which, up to a scaling factor, is perfectly equivalent. Instead of a fixed obstacle, one can consider the action of a second membrane lying below the first one. This, has been introduced by Vergara Caffarelli8. More generally, one can consider the

118

case of N membranes (see Kinderlehrer and Stampacchia 5 ) lying above of each other. To describe the problem, consider N functions ^ i ? 1 ^ ) , i = l...N,

(1.6)

such that 0i > • • • >

,

4>N

(1.7) 2

(the above sequence of inequalities is taken for instance in the L (dQ.) sense). If we denote by / ; the forces applied on the different membranes and set /C := {v = (vi, v2, • • •, vN) G (H1^))1*, i = 1 . . . N, vi > V2 > • • • >

UJV

| Vi=4>i , on dCl ,

, a.e. x £ Q} ,

(1.8)

we are led to the problem of finding u solution to u G /C , N

\]

N

/ Vw, • V (vt - u{) dx > S~] (/j, Vi — Ui) Vv G K

(1.9)

f~f

~1 Jn

As above the /» are supposed to satisfy fiEH-1^)

Vi = l,...,N,

(1.10)

( , ) denotes the duality bracket between i J - 1 ( f i ) and HQ(Q,). Note that if u, v G /C, then, for any i, Vi — Ui G HQ(Q). If JC is not empty it results from the usual theory of variational inequalities that (1.9) admits a unique solution. A i72(f2)-regularity theory can be also developped and we refer the reader to Chipot and Vergara Caffarelli4 for details. In this note we would like to "soften" the constraints appearing in (1.8) and at the same time introduce some non-local nonlinearity in the operator and show that a regularity theory still remains available. So, assuming now only (1.6) we replace K. by the closed convex set, that for simplicity we still denote by /C, given by

vl = 4>i , on dfl,

Vi = 1,...,N

,

(1-11)

/ v\ dx > / i>2 dx > • • • > / VN dx> . JQ. Jrt ' It is easy to see that K. is not empty. Indeed, consider a function such that JQ

cp£ V{n)

,

J
.

(1.12)

119

Setting Vl = l , V2=<^2+A 2 (A,

.-.,

VN = (j)N + \N(f> , .

(1-13)

it is clear that A2 can be choosen large enough so that / vi(x) dx > v2(x) dx . Jn Jn Then, repeating the process, the choice of A3, . . . , A;v such that / v\(x)

dx >

Ja

V2{x) dx > • • • > / VN(X) dx ,

Jn

(1-14)

(1-15)

Jn

will provide us with an element v, given by (1.13), which is in /C. We denote by a,i(s), i = 1 , . . . , N positive functions such that di(s)

is continuous

0<m
V i = 1 , . . . , ./V ,

(1-16)

Vi = l , . . . , J V ,

(1.17)

for some positive constants m and M. Then, for fi G H~1(Q.), we would like to consider the problem of finding u solution to u£ tC , y2

/ Oi I / \Vui\2dx)

Vu, • V(t>j - Ui)dx > V" (/»,u» - u») Vv G/C . (1.18)

Remark 1.1. We could choose a more general operator than the one we have here, i.e. we choosed -atA

Vi = l , . . . , J V .

As we will see also, the non-locality inside the operator is playing no role. We choose the Dirichlet intergal for simplicity and also for its meaning in energetical terms. 2. Existence and regularity results This Section is devoted to a brief survey of the main results here presented; for a complete proof of them all we refer to 2 . We will set for convenience f:=(A,/2,...,/jv)

, *:=(0i,02,...,&v)

Then we proved the following existence theorem

•

(2.1)

120 Theorem 2.1. Suppose that (1.6),(1.10),(1.16),(1.17) hold then exists a solution to (1.18). Subsequently, the following regularity result, 2 , can be recalled.

there

T h e o r e m 2 . 2 . Let u represent a solution stants Ai, A 2 , . . . , AJV such that u satisfies

con-

to (1.18).

There exist N

2

|Vui|

. Am = fi + Xi, 2,ny on 9fi ,

. . (2.2)

in a weak sense. T h e existence result stated in t h e first theorem follows from a fixed point argument according to t h e deatiled proof comprised in Carillo, Chipot and Vergara Caffarelli 2 . T h e proof of t h e second theorem, which concerns t h e regularity of solutions, again see 2 for details, is based on a classical lemma for test functions with zero mean value. 2

is a constant. T h e n 2,n, the regularity of M, solution to (2.2) is the regularity of the corresponding Dirichlet problem. For instance for fi e LP(Q), fc = 0 V i = 1 , . . . , N, Q, smooth then

R e m a r k 2 . 1 . One should r e m a r k t h a t a.

|V«i

A c k n o w l e d g e m e n t s : T h e second author would like t o acknowledge t h e kind hospitality at Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universita di R o m a "La Sapienza", Rome, Italy and the support of t h e Swiss National Science foundation under t h e contract # 2067618.02. T h e support of G.N.F.M.-I.N.D.A.M. and of M.I.U.R., P r o g e t t o Cofmanziato 2002 Modelli Matematici per la Scienza dei Materiali, are also gratefully acknowledged.

References 1. H. Brezis "Problem Unilateraux" J. Maths. Pures Appl. (1972) 2. S. Carillo, M. Chipot and G. Vergara Caffarelli The N-Membrane Problem with Soft constraints" J. Math. Anal, and Appl., submitted to, (2003) 3. M. Chipot and L. Kis "On Some Class of Variational Inequalities and their Regularity Properties" Coram. Appl. Anal. 6, 27-48 (2002) 4. M. Chipot and G. Vergara Caffarelli "The N-Membrane Problem" Appl. Math, and Opt. 13, 231-249 (1985)

121 5. D.Kinderlehrer, G.Stampacchia "An introduction to Variational Inequalities and their Applications" Academic Press, New York (1980) 6. J.L. Lions, G.Stampacchia " Variational Inequalities" C.P.A.M. 20, 493-519 (1967) 7. O. V. Titov "Minimal Hypersurfaces over Soft Obstacles" izn.Akad.Nauka S.S,S.R. Ser. Mat. 38, 374-417 (1974) 8. G. Vergara Caffarelli "Regolarita di un problema di disequazioni variazionali relativo a due membrane " Rend. Acad. Lincei 50, 659-662 (1971)

ON THE RIEMANN PROBLEM FOR A R E A C T I N G M I X T U R E OF GASES*

F . C O N F O R T O + A N D A. J A N N E L L I Dipartimento Contrada

di Matematica, Universita di Messina, Papardo, 31, 98166 Messina, Italia R. M O N A C O

Dipartimento

di Matematica,

Corso Duca degli Abruzzi,

Politecnico

di

24, 10129 Torino,

Torino, Italia

T. R U G G E R I C.I.R.A.M.

e Dipartimento di Matematica, Universita Via Saragozza, 8, 40123 Bologna, Italia

di

Bologna,

This paper deals with the Riemann problem for a gas mixture undergoing irreversible bimolecular reactions governed by a suitable closure at Euler level of the Boltzmann equation. The aim of the paper is to propose a 'simplified' solution procedure for the Riemann problem in absence of chemical reactions (conservation laws), providing its validation by means of numerical simulations, and to investigate the influence of chemical reactions, as well as of the initial data, on the space-time evolution of the initial jump.

1. Introduction In two recent papers 1 ' 2 the propagation of steady detonation waves driven by a bimolecular reaction of type 1 + 2 ^ 3 + 4 has been studied. In order to solve that problem, the authors started from the full Boltzmann equation, extended to a gas mixture undergoing the above reaction, deriving the reactive hydrodynamic equations at the Euler level and determining an exact expression of the reaction rate. By using these equations the steady propa*This work is supported by Intergroups Project 2003 of INdAM-GNFM,GNCS (coordinator A. Quarteroni). t\Vork partially supported by a Junior Project 2002 of INdAM-GNFM (coordinator M. Groppi).

122

123 gation problem of a detonation was formulated and solved, by means of an appropriate numerical scheme. A natural continuation of investigation on detonation waves, in authors' opinion, consists in the study of the onset of the detonation itself. This kind of study may be approached by formulating a Riemann problem for the afore-mentioned reactive Euler equations in the sense that, at the initial time, in the space domain of the gas mixture a jump discontinuity occurs and its consequent relevant motion is then driven by the chemical processes governing the hydrodynamic equations. Moreover, it seems natural to compare the time-space evolution of the Riemann problem for the reactive hydrodynamic equations, with that obtained by using the Euler equations for the same gas mixture in the non-reactive case. From a mathematical point of view this corresponds to compare the timespace evolution of the same hyperbolic system which, in the reactive case, presents balance laws for densities and total energy, while, in the chemical inert one, is a set of conservation equations. As it is well-known 3-6 , the Riemann problem for a system of conservation laws, under the assumption of not too large initial jumps, admits an unique solution consisting in a superposition of shocks, contact discontinuities, rarefaction waves and constant states. Conversely, the same problem in presence of source terms cannot be analytically solved since rarefaction waves and constant states are not in general admitted by inhomogeneous systems. What one should expect in this case is that the solution is, in some sense, a 'deformation' of the solution in the inert case. Therefore, the aim of this paper consists in solving the Riemann problem in the non-reactive case, using a numerical method for the reactive one in order to compare their evolution. The admissible characteristic velocities of the system are the classical ones of Euler equations. Thus, for what concerns the non-reactive case, the solution is formed by four constant states (two are the ones on both sides of the initial jump) connected by a contact discontinuity and different combinations of shocks and rarefactions. For the reactive case, an ad hoc method is proposed here with the aim of providing the 'deformed' shocks or rarefactions. On the numerical ground, the simulations have been performed by using the Strang splitting method 8 which consists in separating, in the evolution equations, the hydrodynamic part from that accounting for chemical processes. For the hydrodynamic part, the Nessyahu and Tadmor scheme9 has been used because, as it does not need the solution of the Riemann problem, it allows, when the initial jumps are not too large, to validate the results obtained by the proposed solution procedure. Let us finally observe that the aim of the numerical simulations consists in underlining the influ-

124

ence, besides that of the initial datum, of the type of chemical reaction on the time-space evolution of the initial jump. At this end the simulations have been carried on by considering both exothermic and endothermic irreversible reactions.

2. The reactive Euler equations In a recent paper 1 a mixture of four gases, undergoing a chemical reaction 1 + 2 ^ 3 + 4, has been treated starting from the Boltzmann equation, and the hydrodynamic equations have been derived. In one dimension, these equations can be re-written in the following form drii d •g7+o-M=

S

i.

(1)

i = 1,2,3,

i + sC)-"-

(2) (3)

d_ 1 PV dt 2

2

. 3 2)

+ P

d +

di\2PV

+ VP

2

= ES,

(4)

where rij, i = 1,2,3,4, are the number densities of the four species, v is the mean velocity, p = nkT is the pressure and T the temperature. Moreover, n = J2i=i ni a n d P = S i = i mini are the total number and mass densities, respectively, being m, the molecular masses, with mi + m-2 = ms + m^, k the Boltzmann constant and E > 0 the activation energy of the reaction. Note that Eqs.(2-4) are the reactive Euler equations. In Eqs.(l), Si = <JiS, <Ji = 1 if i = 1,2, <7j = — 1 if i = 3, are the reaction rates and S is given by a i n 3 n 4 - a2nin2

exp

/m3m4\3/2 \ __ I exp mim2

V

M34X

2kT

VM34

kTJ

l-erf(W||x

.(5)

Here, j3 is the Arrhenius parameter, x the exothermic threshold velocity and ^34 = ™ar% • In (5), the three choices ot\ = a2 = 1, a i — 1 «2 = 0, and a i = 0 a2 = 1 correspond to reversible, exothermic irreversible and endothermic irreversible reactions, respectively.

125

3. The Riemann problem For what concerns the Riemann problem for the hydrodynamic equations (1-4) in the non-reactive case, that is by setting 5 = 0, the procedure is the standard o n e 3 - 6 . Accordingly, if one indicates with U = (ni,n2,ri3, p,v,p)T the vector of the field variables, the initial data for such a problem are given by

{

Ur

for x > 0

Vt

for x < 0

(6)

where XJr and U^ are constants. The eigenvalues of system (1-4) are Ai = v - c;

A2,3,4,5 =v;

A6 = v + c;

c= ^5p/(3p)

(7)

meaning that the initial jump gives rise to three propagation modes: two related to the characteristic velocities Ai and A6, respectively, which can be either shocks or rarefactions, and one, related to A2,3,4,5, being a contact discontinuity. The gas is then divided into four regions: besides the state U r of the gas ahead the fastest wave and the state U^ of the gas behind the slowest one, there are the unknown states between the fastest wave and the contact discontinuity and between the contact and the slowest wave denoted, respectively, by U i and U2. The two intermediate states can be found using the standard procedure 6 . Instead, in the reactive case, according to the discussion reported in the Introduction, it is impossible to obtain, in analytical form, the 'deformed' rarefaction waves as a solution to the Riemann problem (1-4) and (5). Therefore, the only possible approach is a numerical procedure. Nevertheless, in order to have an estimate of the intermediate states (at least in the neighborhood of t = 0) our procedure consists in supposing, as a first step, that the solution exhibits two shocks, propagating with velocities s+ and s-, s+ > s_. By imposing the Rankine-Hugoniot conditions 6 across the two shocks, we find that the state U i is related to U r by the following expressions: Tin = n f r 3

4M? + M 2 .

* = 1.2,3,

Pi = f t - 3

4M? + M 2 '

(8)

126

while the state U2 is related to U^ by 3 + M2_

7121 = nu

3 + Ml

p2 = PI

^MT'

(10)

~^MT'

where M+ =

S

-±^,

M_ = £ ^ * .

Cr

(12)

C2

Eqs. (8-11), together with the relations across the contact shock, na — na

arbitrary,

S — V2-V1,

p? — pi

arbitrary

P2=Pl,

(13) (14)

provide the solution, s being obviously the velocity of propagation of the contact discontinuity. In the sequel, the common values of velocity and pressure on both sides of the contact jump will be denoted by v and p, respectively. In fact, (8-14) can be seen as a system of twelve independent equations in the twelve unknowns riji, n»2 {i = 1,2,3), p\, P2, v, p, s+, s_. By introducing n = pe/pr and r = pe/pr, after standard calculations, one gets an equation for the quantity M+ Ml-1

|

M

+

(5M?-1-4TT)

_A(ve-vr)

^5r(5M*-l+7r)

3c

-

Note that the knowledge of M+, at least numerically, allows to recover all the unknowns, provided that the relation between M+ and M__ is explicitly given by:

«—^K'+SSFO-

(16)

Once this kind of solution is achieved, we need some conditions which assure its admissibility. Naturally, Lax conditions 7 allow us to distinguish between stable shocks and rarefactions. Prom s+ > s_, it is clear that s+ represents the propagation velocity of a 3-shock and must satisfy the constraints: S+ > Vr + Cr,

V\ < S+ < V\ + C\

(17)

which, in terms of the Mach number M+, read as M + >1,

V

-±-^<M+
(18)

127

Conversely, s_ is the propagation velocity of a 1-shock and, consequently, must satisfy the other constraints: (19)


V2 — C 2 < S - < 1>2,

which, for the Mach number M_, read as - 1 < M_ < 0,

M_ <

Vj-

Cg-

V2

(20)

If one of the Lax conditions (17), (19) is not satisfied, then U i or U2 (or both) will not belong anymore to a shock curve, but will lay on the corresponding rarefaction curve and the solution must be performed following the well-known procedure 3 - 6 . Even if this procedure can be considered a bit rough, the numerical results of Sec.5 will validate such a method. In view of the numerical simulations presented in Sec.5, we select three different cases, called in the sequel Test 1, 2, 3. In particular, we set, once for all, the following values for some of the macroscopic variables of the states U r and U^: nir = 0.06, nir = 0.05, nzr = 0.15, pr = 2.36 • MT 3 , vr = 0, Vr = 5 • 102; nu = 0.03, n2e = 0.02, nu = 0.1, pt = 3.6 i-10 - 3 . the remaining ones, vg and pg, are varied as reported in the second and third columns of Table 1. The solution in the non-reactive case has been carried out in each of the three cases and the two intermediate states U i and U2, as well as the velocity of propagation of shocks, if they occur, are listed in the last six columns. Observe that in Tests 1 and 2, the solution presents one shock wave propagating to the right and a rarefaction wave to the left; conversely, Test 3 exhibits two shock waves both propagating to the right. Table 1. Test

Data and solutions of the Riemann problem for three different tests.

ve • 1 0 3

n • io3

M+

M_

Pi • I O " 3

p2 • IO"3

v- IO3

p - IO 3

1

0

2.69

1.56

2.62

0.40

1.38

0

5.99

2.11

-

4.21

2

5.64

2.41

0.73

2.63

3

3

2.69

4.41

-1.55

8.18

8.61

1.86

11.93

128

4. The numerical m e t h o d In this section, we present the numerical method used in order to solve the hydrodynamic equations, considered in Sec.2, with initial data (6). By using a fractional step approach, one alternates between solving the homogeneous hyperbolic equations, ignoring the source term, and solving the time-dependent ordinary differential equation with sources. In this context, for the homogeneous nonlinear hyperbolic equations, we use the non oscillatory central scheme of Nessyahu and Tadmor 9 (NT) that is second-order accurate in regions of smooth flow and first-order accurate near discontinuities. The equations with source terms are handled by a second order Runge-Kutta scheme. In the last years, high-resolution non-oscillatory central schemes have become popular in solving nonlinear hyperbolic conservation law because of the simplicity in their construction. In fact, by the NT scheme, we obtain second order accuracy in smooth regions without knowledge of the characteristic structure of equations. No Riemann solvers and no field-by-field decomposition are needed in the computation of the smooth numerical fluxes along the mid-cells. The results obtained by analytical study of the Riemann problem on the homogeneous hyperbolic equations are in accordance with the numerical results obtained implementing this method without requiring complicate Riemann solvers. Regarding the fractional step method, we use the Strang splitting approach which consists in advancing first by a half time step on PDE equation, then by a full time step on the ODE equation, and ends with a half time step on the first equation again. As far as the accuracy question is concerned, the Strang splitting 8 yields second order accuracy in time if each sub-problem is solved by a second order accurate method. For what concerns the NT method, let us observe that the central scheme is implemented as a predictor-corrector method W

J

=

w

""2

w £ j = \W

^ '

+ w?+i] + \W

- w;-+1] - A[f(w£*) - f ( w ^ ) ] ,

where w? represents the approximate cell-average at the discrete time level n+tn and A = At/Ax. f(w- 2 ) denotes the numerical flux evaluated at the pointwise values at the half-time steps. w^/Aa; and i'JAx are approximations of the space derivatives of the field and of the flux, such that w'j ~ Ax • wx(Xj,tn)

+ 0{Ax)2

,

fj ~ Ax • fx(Xj,tn)

+

O(Ax)2.

129 For the stability of this scheme it is necessary to require the following CFL (Courant-Friedrichs-Lewy) condition 10 A max /9(A(wn)) < - , where p(-) represents the spectral radius of the Jacobian matrix A(-) = dfi/dwk- In order to ensure that this scheme is non-oscillatory, the numerical derivatives should satisfy •w'j = MinMod(Awj+i,Awj_A

,

fj = MinMod (MJ+ I , Af,-_ i ) , where A w , + i = w J + i — MVJ and MinMod MinMod(a,b)

stands for the slope limiter 11

= - [ sgn(a) + sgn(6) ] • min(| a \,\ b |).

5. Numerical results In the present section we present the simulations concerning the integration of the reactive hydrodynamic equation with the aim of showing the influence of chemical source terms on the shape of the profiles. For the simulation performed, we set the following values of the parameters occurring in the governing system: mi = 0.018, m 2 = 0.001, m 3 = 0.017, m 4 = 0.002, E = 63300, \ = 3927. The numerical results are obtained at final time t m a x = 100, on a computational domain of dimension [—200,200] with Aa; = 0.2 and variable time step At, chosen in order to ensure a Courant number 10 equal to 0.4. We always apply the MinMod slope limiter function for each wave; moreover, Neumann boundary conditions are used. In Fig.l we plot the total mass density p versus space at different instants of time with a step of 20 units. In the first picture, using initial data of Test 1, we solve the non-reactive Riemann problem. If we compare the values of p\ and pi in Table 1 for Test 1 (where a 3-shock and a 1-rarefaction are present) with the profiles of p in the first picture of Fig.l, we can note that the procedure adopted here to solve the Riemann problem is validated by the numerical simulation. In the second picture of Fig.l, for the same initial data, we present the solution in the case of an exothermic irreversible reaction driven by (3 = 0.5. In the third one, again for data of Test 1, we show the profiles for an endothermic irreversible reaction driven by (3 = 105. Let us firstly comment that the

130 No Reaction (Test 1)

wM

: -100

-50

0

:

50

100

150

Exothermic Reaction (Test 1) 1

i

^ 1

-200

\ 1

V-

JH '

1

y\

8f

I

•

-150

Endothermic Reaction (Test 1)

x10

'. -200

\

WW Itfrm -150

200 Exothermic Reaction (Test 2) 1

-

. —v —s— \ — r —i r

^ 4 -

^^Q.XV-^. 1

-200

1

1

1

1

1

-150

Figure 1.

Density profiles at different times.

1

131

profiles in the reactive eases can be interpreted as 'perturbations' of the corresponding non-reactive ones. In particular, the rarefaction wave in the inert case results to be 'deformed' by chemical reactions, but still maintains its shape. Furthermore, the presence of exothermic or endothermic reactions increases in any case the speed of all the wave fronts. Moreover, the exothermic reaction exhibits a rapid growth in the amplitude of the shock front which then slowly decreases for longer times of observation. On the other hand, such a behavior is not present in the case of an endothermic reaction, where, conversely, the amplitude of the shock quickly becomes lower then the one in the non-reactive case. Finally, the last picture of Fig.l (which should be compared with the second one), obtained for the initial data of Test 2 in the case of exothermic reaction, shows that when a greater initial jump for the pressure is imposed, both the speeds of propagation and the amplitude of the shock are greater. In addition, it should be noticed that in this case also the amplitude of the contact discontinuity is increased. The case presented in the second picture of Fig.l is reported also

Figure 2. Wave configuration for Test 1 (left) and for Test 3 (right) in the case of an exothermic reaction.

in Fig.2 (left) where a view from above is shown in the plane x, t in order to outline the evolution of the fronts, which can be easily identified. This picture must be compared with the analogous one of Fig.2 (right) where Test 3 for an exothermic reaction is performed. Here, we want to stress that a sufficiently great initial jump of the mean velocity changes the structure of the solution to the Riemann problem, in the sense that, instead of having a shock propagating to the right and a 'deformed rarefaction' propagating to the left, Test 3 exhibits two shocks both moving to the right. As a last

132 comment, let us note t h a t t h e higher is the initial velocity j u m p the higher are t h e propagation speeds of t h e three fronts.

References 1. F. Conforto, R. Monaco, F. Schiirrer and I. Ziegler, J. Phys. A: Math. Gen. 36, 5381 (2003). 2. F. Conforto, R. Monaco, F. Schiirrer and I. Ziegler, Proceedings XI International Conference on Waves and Stability in Continuous Media, World Scientific, Singapore, 161 (2002). 3. J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer-Verlag, Berlin, 1983. 4. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, SpringerVerlag, Berlin, 2000. 5. A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, Oxford, 2000. 6. R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser, Basel, 1992. 7. P.D. Lax, Comm. Pure Appl. Math. 10, 537 (1957). 8. G. Strang, SIAM J. Num. Anal. 5, 506 (1968). 9. H. Nessyahu and E. Tadmor, J. Comput. Phys. 87, 408 (1990). 10. R. Courant, K.O. Friedrichs, H. Lewy, Math. Ann. 100, 32 (1928). 11. P.K. Sweby, SIAM J. Num. Anal. 2 1 , 995 (1984).

P L A N A R D V M S M I X T U R E S MODELS W I T H T H E S A M E GEOMETRICAL S T R U C T U R E

HENRI CORNILLE Service de Physique Theorique,DSM/Saclay

91191 Gif-sur-Yvette,

FT.

For planar (x,z coordinates) discrete velocity models (DVMs), we present binary mixtures (masses m = 1, M > 1 for light and heavy species) for a gas in a halfspace (x, 2 ^ 0 ) , filling with the same geometrical structure, all integer coordinates of the half-plane. The interface (without velocities) is located at z = 0 and only a z spatial dependence of the gas (densities with (±x,z) are equal). We prove that the models are "physical", with only physical invariants: masses, energy and momentum along the z-axis. First we obtain the same geometrical structure in the plane (|x| + \z\ odd or even) for an unbounded of heavy mass values: M even, odd divided by 2q. Second, we present M = (2p + l)/5 models with the heavy species in dodecagons, while the light fills all empty sites

1.

INTRODUCTION

The recent [1] planar binary mixtures M = 2,3,3/2,4 half-space models with |a;| + \z\ either odd or even for the heavy or light species, are extended for M even, odd divided by 2q. For M = (2p + l ) / 5 , p > 4, we present models with the heavy in dodecagons and the light filling the empty sites. The first work, [2], for mixtures models filling all the integers coordinates and different geometrical structures, was for M = 2,3,4,5 without boundary conditions (not half-plane), with also velocities along the x-axis. Tools: We add new momenta to a preliminary "physical" simple model 1) With 3 known momenta of a collision of the same species in rectangles or squares [2], we add the last. 2) For a 9 momenta square (one species) and the center and 4 tops along the diagonals (or medians), we can add the tops of the medians (or diagonals). 3) For a collision mixing heavy and light species and 3 momenta belonging to a physical model, we can add the last. 4) In a physical model, we can add the densities symmetric to the bisectors z = ±x: if exist other previous densities with the same z. 5) We add new densities if adding a linear combination of their evolution equations, we only get the correct new conservation laws. 6) If, [1] for one

133

134

species \x\ + \z\ = 1,3 (or 0,2,4) are physical, we extend to all odd (or even). 2.

M EVEN,

ODD/21,

filling

all |JC| + \z\ odd, even, fig3

2.1 M = 2p > 4 E V E N figla,p = p. We start with a 9 heavy F physical, 6 ind. F, and 3 invariants: mass Mi, energy IE, momentum J. ^ ( 0 , 1 + 2p),F4(0,-l + 2p),F6(0,-3 + 2 p ) , ^ ( ± l , 2 p ) , F 5 ( ± l , - 2 + 2p), F 3 (±2, - 1 + 2p). Collisions (3 known), add F (2.1) p = p : F 3 F 6 - F 4 F(2,2p - 3), F(2,2p - 3)F 6 - F 5 F(1,2p - 4), F(2,2p 3)2-F4F(0,2p-5), F(2,2p-3)F(0,2p-5)-F6F(2,2p-5).. (2.2) F ( l , 2 p - 6 ) , F ( 0 , 2 p - 7 ) , F ( 2 , 2 p - 7 ) . . The z values of F(x,z) decrease, we stop after F(0,3),F(1,2),F(0,1),F(2,1) with \x\ + \z\ = 1,3 -> all odd. For a mixture with light densities f(x,z) we fill all |a;| + \z\ even. To the old heavy (2.1), we add 2 light: / o ( 0 , 0 ) , / i ( l , 1), write L2,Li,l0,l\ (evolution equations) and an energy exchange collision T, deduce the light mass Mi, add A/i to the old conservations and verify the correct A values. r = / ! ( l , l ) F 4 ( 0 , 2 p - 1) - / 0 F 2 ( l , 2 p ) -» h = -T,l0 = 2 r , L 4 = 2T,L2 r -»• Mi = / 0 + 2ii = 0, 2 F = ..2(4p2 + 1)L2 + (2p - 1) 2 L 4 + Xh r(2(4p 2 + l ) - 2 ( 2 p - l ) 2 - A ) = 0-> A = 8p, J = ..(2p-l)LA + lpL2 + \h T(-2(2p - 1) + 4p - A) = 0 -> A = 2, Collisions (3 known) -> / / ! ( 1 , 1 ) 2 _ - /o/(0,2),/ 0 F(3,6p) - /(3,3)F(0,3(2p - 1)), / 0 F(10p 7,4) - _ / ( l , 3 ) F ( 2 ( 6 p - 4),1), / ( 1 , 3 ) 2 - / ( 0 , 2 ) / ( 0 , 4 ) , /(1,3)/(1,1) / ( 0 , 2 ) / ( 2 , 2 ) , / ( 3 , 3 ) / ( l , 1) - / ( l , 3)/(3,1), / : |x| + \z\ = 0,2,4 -» even

= = = -

2.2 M = 2p + 1 ODD, flgla, p = p + 1 in 2.1 For the heavy, we start with eqs(2.1-2), p = p+ 1 for a physical single-gas. We still deduce for the heavy |a;| -I- \z\ = 1,3 and extend to all odd. For a mixture with equal light f(±x,z) we show that we can fill all |a;| + \z\ even. We add 2 light: /o(0,0), /i(0,2), write L,, li for the evolution equations and an energy exchange collision T. We deduce the light mass Mi and for the old conservation laws, add A/i and verify the correct A values. T = / 0 F 2 ( l , 2 p + 2) - / ! ( 0 , 2 ) F 5 ( l , 2 p ) -> h = 2 r , / 0 = -2T,L2 = -T,L5 = r -> Mi = l0 + h, 2E = ..2(4(p + l ) 2 + 1)L 2 + 2(1 + 4p 2 )L 5 + A/i = T(2A - 8(1 + 2p)) = 0 -> A = 4(2p + 1 ) , J = „4(p + 1)L2_+ 4pL 5 + A/j = T ( - 4 + 2A) = 0 -*• A = 2. Collisions (3 known) -» add / / 0 F ( l , 4 ( p + 1)) - 7(0,4)F(l,4p), / ( 0 , 2 ) / ( 0 , 4 ) - _ / ( l , 3 ) 2 , / ( 0 , 4 ) / 0 /(2,2)2, / ( 0 , 2 ) / 0 - / ( 1 , 1 ) 2 , /(0,4)/(2,2)-/(0,2)/(2,4), / ( 2 , 2 ) / ( 2 , 4 ) /(l,3)/(3,3), / ( 3 , 3 ) / ( l , l ) - / ( l , 3 ) / ( 3 , l ) (2.3)

135

The light f(x,z) filling \x\ + \z\ = 0 , 2 , 4 can be extended to all even. 2.3 M = (2p + l ) / a , a = 2, figlb. We start with a 9 F heavy physical single-gas, 6 ind. F(±x,z) written (x, z), two (x,2p — l),x = 0,4 and 3 invariants: mass ML, energy 2 F and momentum J: From squares and center (cent.) known, we add the tops of medians, deduce the 9 F (2.1) with p = p + 1 and extend to all |a;| + \z\ odd. 9 F : (0,2p + 3), (±x,2p - 1), ( ± 2 , 2 p + 1), ( ± 2 , 2 p - 3 ) , ( 0 , 2 p - 5 ) - » sq:(±2,2p + 1), (±2, 2p - 3) cent. (0, 2p - 1) -> (0,2p + 1) sq:(0,2p + 3 ) , ( ± 2 , 2 p + l ) , ( 0 , 2 p - l ) cent. (0,2p+ 1) -> (±l,2p+ 2), (±l,2p) (2.4) For a mixture figs2a-b, with energy exchange collisions, squares (3 known), tops of medians, we add new / , fill |x| + \z\ = 0,2,4 ->• all even. r = 7 o F 3 ( 2 , 2 p + l ) - 7 ( 2 , 2 ) F 6 ( 0 , 2 p - l ) , /0F(0,2p + 3 ) - 7 ( 0 , 4 ) F ( 0 , 2 p 1), / o _ F ( 1 0 p - 6 , 7 ) - / ( 2 , 6 ) F ( 1 0 p - 8 , l ) sq: (±2,2), (±2,6) cent (0,4) add /(0,2),/(±_2,4) sq: (0,0), (0,4) cent (0,2) add / ( ± 1 , 1 ) , / ( ± 1 , 3 ) / ( 2 , 2 ) / ( 2 , 4 ) - / ( l , 3 ) / ( 3 , 3 ) , / ( l , l)/(3,3) - / ( l , 3 ) / ( 3 , 1 ) (2.5) 2.4 M = (2p + l ) / a , a = 2 i > 2 figslc-2a-b: l)zo = 2p + 3.We start with a 9 heavy square with a > 2, Ai = 0 , 2,4, A2 = 1,3 and from squares and centers known, we add the tops of medians: (0,z o -Aia), (±a,zo-\2a), (±2a,z0-2a), -> sq:(±5, z0-a), (±a,zQ-3a), cent (0, z0 - 2a) ->• (0, z0 - a) figlc, sq:(0, z0), (0, z0 — 2a), (±5, z0 - a), cent (0, z0 - a) ->• (±5/2, z 0 - a/2), (±5/2, 20 - 35/2) (2.6a) We stop here, like in 2.3 for a = 2 and go on, figlc, for a = 2\i = 4,8.... sq:(±/i,z 0 -M),(±A t ^o-3/x) , cent (0,zo-a) -> (0,zo-n) sq:(0,z 0 ), (0,zoa), (±/z, z0 - /x),cent (0, z0 - fi) -> (±/i/2, z0 - /Lt/2), (±/u/ 2 , z0 - 3/u/2) We stop for 5 = 4, go on figlc, for a = 8,16.... For all starting a we get the 9 heavy F(x, z), eq(2.1), p = p + 1, 2.1 and extend to all |x| + |z| odd. 2) z 0 = 2p + 1. For a mixture, figs2a-b, We add 2 light: / 0 (0,0), /1 (5, a), with an energy exchange collision V, write Z/j,/« (evolution equations), deduce the mass Mi, add A/i (old conservation laws) and verify the A.

r = /oF2(a,zo)-/i(a,5)F1(o,^-a) ->i1=r,i0 = -2i\Li =2r,L2 = - r -s- M, = /o + 2/i,2F = ..2[a,2 + z$\L2 + (zo-a)2Li+\h = T(\-4az0) -» A = 4az 0 , J = ..2z 0 L 2 + (20 - a)Li + A/i = T ( - 2 a + A) -> A = 2a (2.66) We add new /(3a, 3a), / ( a , 3a) (energy exchange) and other / (squares): / 0 F(35,3z 0 ) - 7(35,3a)F(0,3(z 0 - 5)), / 0 F(2J> 0 + 3p) - 45,35 + 1) /(o,3o)F(2(z 0 + 3p) - 55,1), / 2 ( 5 , 5 ) - / 0 / ( 0 , 2 5 ) , / ( 5 , a ) / ( 5 , 3 5 ) -

136

/(0,2a)f(2a, 2a), / 2 (2a, 2a) - / 0 / ( 0 , 4 a ) , / ( a , a)/(3a, 3a) - / ( a , 3a)/(3a, a) Start: /(x,-z) : (0,0), (a,a), (0,2a), (2a,2a), (0,4a), (a,3a), (3a,a)

(2.6c)

fig2b: With squares (3known) or tops of diagonals and center known, (i) sq.cent (±a, 3a), (±a, a), (0,2a) giving (a, 2a), (0,3). (ii) sq.cent (±a,a), (0,0), (0, 2a), (0,a) giving (a/2, a/2), (a/2,3a/2). (iii) / ( a , a)/(a, 2a) - / ( a / 2 , 3 a / 2 ) / ( 3 a / 2 , 3 o / 2 ) (iv) / ( a / 2 , a / 2 ) / ( 3 a / 2 , 3 a / 2 ) - / ( a / 2 , 3 a / 2 ) / ( 3 a / 2 , a / 2 ) . We have new / : a -» a/2 ->• f(x/2,z/2) which for a = 2 -» |x| + |z| = 2,4. Similarly a/2 ->• a/4 -> / ( x / 4 , z/4) -»• the 2,4 sums for a = 4 and a/8 -» f{x/8,z/8).. For any a = 2", we fill all |x| + \z\ even.

3.

Half-Space M = ( 2 p + l ) / 5 , Heavy Dodecagons,figs4

3.1 LIGHT physical double-Octagons fa,i = 0,1,..16, figs4a-b. L e m m a l a The double-Octagons: 17 Vi (velocities), f(—x,z) = f(x,z), 9 ind fi,i = 0, odd < 15 is physical. We write 7 collisions A*, the evolution equations / j , a sum X = YlaiU (ao = 1> a i arbitrary) and the Afc —> 0. (0,0),iTi(l,2) = -v3,v5(2,l) = -v7, v9(l,3) = -t?„,t/ 1 3 (3,1) = -v15 (3.1) c = 0,2 : A 2 + c / 2 = -/0/9+c + fb+cf\+c, A 4 + c / 2 = /0/13+c - / i + c / 7 - 0 Ai = /7/5 - /1/3, A6 = / 9 / n - / i 3 / i 5 , A 7 = /5/15 - /3/5 def: x~ 6 = x a Xb,-X'o.t.c,. = x a + x;, + x c .J 0 = 2(A2,3 - A4>5),/1 = Ai )4 - A 2 , k = Ai, 5 A3, ..0 = Ai(a 13 - a 5 , 7 ) = A2(2 + a 9 - ai, 5 ) = A3(2 -I- an - a 3 , 7 ) = 0.. (3.2) 3 invariants (a^,fc= 0,1,5) are equivalent to Mi, Ei, Jf. X = Zakh = a0[I] + ai[II] + a5[III] [I] = Jo - 2/ 9ill ,i3,i5, [II] = h,9 - hz - 2/ 7 - 3/3,15 - 5/n, [III] = k,9 + 3/7,13 + 4/3 + 5/is + 7/n Mi = l0 + 2ZioddJ = I + 2(11 + III), Et/5 = Ji,3,6,7 + 2/9,11,13,15 = II + III, J,/2°= 2 / - 5 + l~7 + H~n + ff3)15 = 21 + III (3.3) L e m m a l b If the 5 /», i = 0,1,9 (3 ind.) (3.1) belong to a physical model, we can add 5 new / ( 0 , j), j = 1,2,3 and / ( ± 1 , 1 ) (4 ind). Ai = / ( 0 , 3 ) / ( 0 , l ) - / 2 , A 2 = / ( 0 , l ) / 9 - / ( 0 , 3 ) / ( l , l ) , A 3 =_f(0,2)f0 /(1,1) 2 ,A 4 = /(O^fo-JJiO^Jo = -A 3 ,«i_= A ^ . J Q = - A 2 i l 1(0,3) = -Ax + 2A2,4, /(0,2) = -A3 - 2A 4 ,1(0,1) = - 2 A 2 - Ai, 1(1,1) = A2,3 (3.4) X = ^2a(x,z)l(x,z), linear combination of the 4 new l(x,z) For each old conservation law we add X and A, -» 0, we verify the correct a(i,j): X = -Aj (o(0,3) + o(0,1)) + A 2 (2(a(0,3) - a(0,1)) + a(l, 1)) + A 3 (a(l, 1) -

137

a(0,2)) + 2A 4 (a(0,3)-a(0,2)) M, = l0 + 2llfi + X = 2Aj - A 3 - 2 A 2 + X -»• o.(0,j) = 1,J = l , 2 , 3 , a ( l , l ) = 2, 2 £ = lWi + 2O/9 +X = 10Ai - 20A2 I O A 4 + X ->• a(0,2) = 4 = a ( l , l ) , a ( 0 , 3 ) = 9, J = 4k + 6lg + X = 4Aj - 6A2 - 2A4 + X a(0,2) = 2 = a ( l , 1), a(0,1) = 1, o(0,3) = 3. fi -> fi+2, i = 1,9 -•, we can add /(O, - j ) , j = 1,2,3 and / ( ± 1 , - 1 ) . (3.5) L e m m a l c With collisions (3 known), we add f(±l,z),f(0,z),

z ^ 0.

(i)M ? 9/5, j arbitrary:A 5 = f(l,j)f(-l,j) - f(0,j - l ) / ( 0 , j + 1), A6 = f(0,j + l ) / ( ± l , i ) - f(0,j)f(±l,j + 1) -> / ( 0 , i + 1), / ( ± l , i ) , fig4c (3.6) starting with j = 3, we can add all / ( 0 , j + 1), / ( ± l , j + 1) (also z < 0). (ii) M = 9/5,p = 4, with 2 heavy at (±1,4), (±1,5). To lemmasla-b (with collisions,3 known) we add new densities and apply lemmalc with j = 6 /92-/(0,2)/(0,4), / 9 / ( - 3 , l ) - / 0 / ( - 2 , 4 ) , / 5 / ( - l , 4 ) - / 0 / ( 0 , 5 ) , / ( - 2 , 4 ) / ( 2 , 4 ) - / ( 0 , 2 ) / ( 0 , 6 ) , / ( 0 , 6 ) / ( ± l , 3) - / ( 0 , 3 ) / ( ± l , 6), fig4b (3.7) 3.2 HEAVY: 4 Physical Dodecagons, D o d ,

figs4a-b-c.

To the light Lemmasla-b, we add one heavy DodI, 12 V,, 12 Fi(x,z), p>4, and 3 other Fi(—x,—z) —> Fi+2, Fi+\(—x,z) = Fi(—x,z) . DodI Vrip - 2,p - 2),Vh(p,p - 3), V9(p - 3,p), V13(p - 3,p + 1), V17(p 2,p + 3),V21(p + l,p-3), V25(p + 3,p-2),V29(p + 3,p + 3), V33(p+l,p + 4),V37(p,p + 4),V41(p + 4,p+l),V45(p + 4,p), Vi+2 = -Vi,i = 1.5..45 (4) Lemma2a: add 4 Fi, Li,i = 1,3,9,11, X = Yl AtLi, Ai arbitrary, c = 0,1 Tl+c = /o^9+2c

ft-2cF9+2c --^1,9, £9 =

—

/l+2c-Fl+2c,r3+c = /5+2c-^9+2c — fs+2cFl+2c,^h+c

- fn+2cF1+2c,T7 _

= F1f(0,-p+

=

2) - F 3 / ( 0 , p - 2) = L 3 ,n =

r i , 3 , 5 , i l l = _ r 2 , 4 , 6 , X = AltgTit3^

+ A3nT2t4te

+

A31T7

Heavy mass: Mi = 2 ^ L j = 0,X = 0—> Ai = 2. Energy, momentum: add X to old k,2Ei,J[ —> . one parameter A\, Mi = 0 —> heavy 2Ei, Ji:

k = ri,z 3 = r2,/5 = — r3j6,/7 = —r4,5,/9 = r 3 ,?i 3 = rs,/n = v^,^ = T 6 , / ( 0 , p - 2 ) = - / ( 0 , 2 - p ) = 2 r 7 , 2E« = 2(2p+l)[ii,3,5,7 + 2/9,ii,i3,i5 + (p - 2) 2 (J(0,p - 2) + 1(0, -p + 2))/10] = 2(2p + 1)^,2,3,4,5,6 # 0 , 2E = 2Ei + X = 0 -» X = AXML/2 + 2(2p + l)L 9 ,n -> 2 ^ with ML = 0. J,/2 = /5-7 + /r 3 , 15 + 2 ^ 3 + 3 / 9 - n + (p - 2)(/(0,p - 2) - 1(0, -p+ 2))/2 ^ 0, J / 2 = Ji/2 + X = 0 -» X = A 1 M L / 2 - 2 ( p - 2 ) L 3 + 2L 9 + L 1 1 (2-2p) = J L / 2 + (Ai - p + 2)ML/2 - • J L / 2 with M L = 0 -> physical model (3.8) Lemma2b For a physical mixture model we add all other (4) heavy F,: F(±p,±(p — 3)) (z = x symmetric to Fg,Fu same z as / ( 0 , ±(p — 3))). -^13,^17 (energy exchange), F 2 1 ( p , p - 3 ) , F 2 5 ( p + 3 , p - 2 ) (z = x symmetry)

138 and with squares (3known) ^29,^33,^37,^41,^45 for x > 0,z > 0 / ( - l , 3 ) F i ( p - 2,p - 2) - f0F13(p - 3,p + 1), Fi/(0,5) - / 0 F i 7 ( p - 2,p + 3),Fi 7 F 2 5 - F x F 2 9 (p + 3,p + 3),FiF 2 9 - F5F33(p + l , p + 4),F 3 3 F 5 F21 F 3 7 (p,p + 4), F33F5 - F 9 F 4 i(p + 4,p + 1), F41F9 - F13F45(p + 4,p) With the a:, z symmetries for the other quadrants, we have 4 heavy dodecagons (16, not presented) for mixtures models, fig4a with M = (2p + l ) / 5 . With Lemmasla-b-c, (collisions,3 known), we fill the empty sites with new fu (figs4b-c M = 9/5,11/5). References [1] Cornille H. Ann.Inst.H.Poincarr'e, Int.Conf.Theo.Phys.4,741-54(2003) 2 Cornille H, Cercignani C. JPA.Math.Gen. 34 2985-98 (2001)

139 f i g l b : M = ( 2 P + l ) / 2 : 9 Heavy D-*add x tz 2p+c figla: p=p-+M=2P + l o d d (l,2p+2) p->p-l-»M=2P even: 9 Heavy • physical-* a d d x /' (4,2?-1) 2 2P + 3JB (l,2p+2)V | T2p^l|V(2,2p+l)x^2p-3

'

CH2p) 2 N

'jk' IH

4^

2 P :^

jk{i.i2p-2) |I *(2 x ( 2 2, 2pp - 3^)

figlc:M=(2p+l)/a,a=2,4,8..Heavy:D,W a / 2 -»a/4. o=2p+3

V(l,2p-4)

~X

- -0 /(2,2p-3)

r jX,( a /l: z o- a /4) _ (a/2,Zo-a/2) ta/4Xi-3a/4) z0-a| (a/2,.z0-^(a,z0-a)
(0,-l> -*(2,-l) 9 Heavy • physical->addx ->lx| 3->odd v a l u e s 1 1 +1 |z[=l, 1 x tz X p

Q -

*

X 0 ' .X S X ~S x' Cf X y' x' U^ X

.<4j3a)

fe2iT)

y

K t\ X

3atz - ^v

X


3

V

tx'x^

(3a,3a)

/•

. N .(3a/2,3a/2)

-i 3 a / 2 ' a / 2 )

'x a X"B X V I ^ X pa I \P a/2p pa/2 3^, fig2a:o light with a - > a / 2 - > a / 4 . . . 5 • - - 3 - - -B-l-X- - - 2 4 x

• x •

X

Q X D X 0

X •

'xt3sx"nxnr'X£Jx' ..

•

X

, \

BX X \j

/

B X X cf

.

..

0 X x'

' ^^ / - ' X

E X

fig3: |x| + |z| o d d ( h e a v y ) even(light)

•

X

z

..(2,4) 4& A A 3 X ,^f'' ? 1 ^ (3,3) x' 2 < - ( - - * '(£,2)

VnXl)^

(3 1}

'

-2-r*.-ti •& Hl

(i^T)TS.-D

fig2b:Light o:squares->x->f(x,z) ^|x| + | z | = 0 , 2 , 4 ^ e v e n (light)

140

'(\o^r-

1

3

' '.^pJop-3

p p + l p + 3 p+4

/

p-4 F35^F39 fig.4a o l i g h t : 2 O c t a g o n s , D h e a v y : d o d e c a g o n s , DodI

WW ll

V o V

-t O

D

I*

H O

filftlj aifii III o v

q o M^ - 2 4 o\o

lA

81 S181 ov 6\*P o P ^ I

fie.4b M = 9 / 5 , f n , 4 D o d e c a g o n s , D h e a v y H old l i g h t , o . n e w light:I,n,III,IV,V,VI fig.4c fi. M=ll/5,f0,4 Dodecagons a old l i g h t , o n e w light:!,II,III..n h e a v y

SHOCK-LIKE TRAVELLING WAVE SOLUTIONS FOR A H Y P E R B O L I C T U M O U R G R O W T H MODEL*

C. C U R R O A N D D . F U S C O Dipartimento di Matematica, Universita di Messina Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy

For a hyperbolic model describing tumour growth an approach is developed to construct travelling wave-like solutions which are discontinuous across a strong shock line in the phase plane. Connection of the resulting equilibria is also investigated.

1. Introduction In view of modelling tumour growth and encapsulation within the theoretical framework of wave propagation at finite speed, in a recent paper 1 there has been derived the following hyperbolic system of PDEs dm

dJ

...

dJ dt

d~f (m) dx

=

j (m) 9{m,c)

l

'

l{Hc)j)=0

dt + dx being x and t the space and time coordinates, while m (x, t) and c (x, i) represent the concentrations of the tumour cells and of the connective tissue respectively; J (x, t) accounts for the effects of connective tissue on the overall flux of the tumour cells; / (m) governs the local proliferation of the benign tumour cells and k (c) takes into account the rate of the matrix movement or the strenght of connective tissue convection per cell, 7' (m) = 7oexp(am) with r = (I/7') being the relaxation time. Moreover 8(m,c) *This work is partially supported by Intergroups Project 2003 of INdAM-GNFM,GNCS (coordinator A. Quarteroni) and by PRA 2001 of University of Messina (coordinator D. Fusco).

141

142

is a response function to be determined. By suitable rescaling of the cell density, it is assumed m 6 [0,1], m = 1 being the equilibrium level within the tumour so that / (0) = / (1) = 0. System (1) includes in the limit case 70 —> 00 the parabolic model proposed in 6 . In order to outline the prominent role of the hyperbolic structure of the governing system (1), in 1 the analysis was carried on along the main lines of investigation considered in 6 . Since the early growth of a tumour has the form of a travelling wave moving outwards from an initial site of disease into the sorrounding normal tissue, attention was first focused on travelling wave solutions of (1). As well known, travelling wave solutions are thoroughly studied in the range of reaction diffusion systems 2 . For a hyperbolic system however such solutions are not necessarily smooth and may contain discontinuities. Recently, Marchant et al. 3 ' 4 ' 5 , by combining Rankine-Hugoniot shock conditions and the behaviour in the phase plane, found a novel family of solutions called 'discontinuous travelling waves'. These solutions possess singular behaviour near several points in the phase plane. As far as the hyperbolic set of balance equations (1) is concerned, the analysis worked out in1 evidentiate the existence in the phase plane of a line of a strong discontinuity (singular barrier) for the travelling wave solutions under interest. The resulting problem of connecting phase trajectories crossing the shock line is the main goal of this paper. In Sect. 2 there are outlined the results obtained in1 on the trajectories behaviour in the phase plane associated to the travelling wave solutions of the governing system (1) and the role of singular barrier as strong discontinuity line for the solutions under interest is evidentiated. Furthermore, in Sect. 3 an approach is developed to construct discontinuous travelling wave solutions which are also valid across the singular barrier. As it is usual for hyperbolic equations the approach in point is based upon the R-H conditions and the Lax conditions to be obeyed by the wave speeds. 2. General remarks and outlines In order to make a comparison with the corresponding results obtained in 6 , in 1 , there was assumed 6 = 0(c)

k{c) = kc

9 (c) > 0, fc

= -f

9' (c) < 0,

9 (1) > 0

(2)

(3)

143

After the above assumptions, we search for travelling wave solutions of the system (1) m = M(z),

J = G(z),

c = C(z)

(4)

being z = x — at, which obey the conditions: lim M = 0 2—»-j-00

lim G = 0 2—J-±00

lim M = 1 Z—>• — CO

(5)

lim G = 1 2—»±00

Insertion of relations (5) into system (1) yields the set of ODEs ( y _ a 2) iM

= a M ( 1

_

M )

_

iG

h' - «2) f = t M (! - M ) - f G) i

(6)

GW = f ( l - ^ ) The equilibrium states of (4) are defined by o M ( l - M ) - 1G = 0 (7) M(l-M)-fG = 0 whereupon we have Af = 0,

G = 0,

C = 1

(8)

M = 1,

G = 0,

C = 1

(9)

The searched travelling wave solutions correspond to connection between two equilibria. In 1 in order to investigate such a connection, linear stability analysis has been performed. Bearing in mind (7), a direct inspection shows that the system of ODEs becomes singular along the straightline M = i l n ^ . (10) k a As noticed elsewhere 3,5 , the phase plane trajectories have no meaning on the locus (10) and cannot cross it. In the following we refer to (10) as singular barrier. In 1 attention was focused on wave speed a satisfying the following limitation ^ ® < a < v ^ e x p ( - f c ) , (11) 7o + 0(1) whereupon the singular barrier (10) does not interfere with the phase plane trajectories. According 1 , eteroclinic connection between the two equilibria exists and the resulting phase plane trajectories terminate at (0,0,1).

144

The main scope of the present analysis is to outline the behaviour of the travelling wave solutions in point when the phase plane trajectories cross the singular barrier (10) whence a must obey the condition V ^ e x p (-k)
v ^

(12)

As far as the nature of the two equilibria is concerned we have 1 • (0,0,1) is a stable node if 70 > #(1), an unstable node otherwise; • (1,0,1) is an unstable node. The singular barrier (10) is to be considered as a shock line so that the connection between the steady states given by the equilibria can be completed by using appropriate Rankine-Hugoniot conditions accounting for discontinuous (or shock-like) travelling wave solutions of the hyperbolic governing model. Such a kind of approach was also considered in 3,5 for studying singular systems of ODEs associated with travelling wave solutions of the system in point. Intimately connected to the shock relations are the Lax conditions 7 which select appropriate wave speeds for propagating shocks. Within the latter theoretical framework we will show later that there exist shocks propagating with wave speed a satisfying the condition (12). These shocks, in fact, incorporate the existence of the singular barrier in point so that the searched discontinuous travelling wave solutions are fully determined.

3. R-H and Lax conditions The R-H conditions permit to define weak solutions of hyperbolic systems of balance laws 8 . The system (1) is in conservative form so that the associated R-H are - s [m] + [J] = 0,

-s [J] + [7] = 0,

-s [c] + [kcJ] = 0

(13)

where s denotes the velocity of the shock and [ • ] = (-)L — (-)R the jump between the limit value of a generic quantity ( • ) on the left and on the right of the shock front in the x, t plane. A direct inspection of the set of relations (13) shows that the shock speed must satisfy the condition 2 _ 70 exp {-2kmL) - exp (-2kmR) 2k rriR — TUL

.

145

As well known the physical relevant shocks are those propagating with speeds fulfilling the Lax entropy condition. In the present case the characteristic wave speeds, are given by 1 Ai = -y/joexp

(-km),

A2 = k J,

A3 = ^Toexp (-km)

(15)

so that the admisssible shock velocities for system (1) must obey the following Lax conditions

A£}

< - < A£ < * < 4+1) < - < >$

1 s \L v. Aj? A^ < v. ... ... < v A< /\L \, A£U ... < AJT '
(16)

- nl i" =

3

Q

whereupon, for a fixed value of i, we have A^<s
i = 1,2,3.

(17)

Trajectories of travelling wave solutions of (6) which in the phase plane meet the singular barrier (10) are associated with the wave speed a subject to the condition (12) which in terms of the eigenvalue (15) writes A3 (1) < a < A3 (0).

(18)

The relation (17) for i = 3 specializes to A3 (mR) < a < A3 (mL)

=$•

mR > mL

(19)

The scope of our later analysis is to investigate which of the two equilibria (0,0,1), (1,0,1) can be connected through the singular barrier by means of appropriate Rankine Hugoniot conditions to another state representing in the phase plane the end point, or alternatively, the initial point of a smooth travelling wave trajectory. I) First we assume (m,L,JL,CL) = (0,0,1) so that the R-H conditions evaluated at s = a yield l-exp(-2kmR) a2 1 C 7T, = —> R = -, ; , 2kmR 70 1 - kmR Since mR € [0,1], Eq. (20) requires 1 - exp (-2A;) a2
2T-^
JR

= amR

(20)

,

,

(21)

As far as the Lax conditions (16) are concerned, a direct inspection shows that (16)i is identically satisfied while (16)2 leads to the further restriction

146

Therefore we are in position to define a discontinuous travelling wave solution which connects the equilibria (8), (9). Following1 we introduce now the phase plane (M, W) being W = -

kG 0(C)

so that the solution in point contains a shock which jumps from the point (MR,WR) , WR = - gfp'Z)' t o *^ e s t e a d y stable state (0,0) as well as a smooth orbit which starts from (1,0) and terminates in (MR,WR). A phase plane plot of a discontinuous travelling wave is shown in Fig.l.

0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 -1.75

•i

0

k —=, 1 0 . 5

0.2

0.4

0.6

i

0.8

]

M Figure 1. Phase plane plot of discontinuous travelling wave solutions 6(c) = \ (3 - tanh ( § - l ) j , a = 0.85, MR = 0.033, WR = -0.363 and 70 = 10 1 0 , k - 10.5. The unstable manifold (solid line) from (1,0) connects to the point (MR,WR) (marked by •) and then the jump take the profile to the steady state (0,0). The vertical line represents the singular barrier.

II) Next we assume (THR, JR, CR) = (1,0,1) so that the R-H conditions evaluated at s = a yield exp (—2kmi,) — exp (—2k) 2k(l-mL) C

L

= ., , ,.,, ——, l + fc(l-mL)

JL

a2 7o'

(23)

=a (mL - 1) .

Since m t £ [0,1], the equation (23) requires exp(-2fc) < — < 7o

1 - exp (-2k) 2k

(24)

147 As far as t h e Lax conditions (16) are concerned, a direct inspection shows t h a t (16)i is identically satisfied while (16)2 leads to the further restriction 2a 2 In (a + ^a2 -

/ v

.

+ exp (-2fc)) +

a + v / a 2 + exp (-2k))

J

2ka2+

+ exp(-2fc)<0,

'

Equation (6)1 evaluated at (ML,GL,CL)

a

a = ——y/lo

(25)

yields

(£),>"• whereupon the resulting point (ML,WL) in t h e phase plane cannot represent the starting point of a smooth travelling wave solution. References 1. C. Curro, D. Pusco, G. Valenti, A Hyperbolic Model for describing growth and encapsulation of tumour, submitted. 2. J.D. Murray, Mathematical Biology, Springer, Berlin, 1989. 3. B.P. Marchant, J. Norbury, A.J. Perumpanani, SIAM J. Appl. Math. 60 2, 463 (2000). 4. B.P. Marchant, J. Norbury, J.A. Sherratt, Nonlinearity 14, 1653 (2001). 5. B.P. Marchant, J. Norbury, IMA J. Appl. Math. 67, 201 (2002). 6. J.A. Sherratt, SIAM J. Appl. Math. 60 2, 392 (1999). 7. P.D. Lax, Comm. Pure Appl. Math. 10, 537 (1957). 8. A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, London, 1976.

SOLITON-LIKE SOLUTIONS FOR A CAPILLARY FLUID

S. D E M A R T I N O Dipartimento di Fisica dell'Universita di 1-84081 Baronissi (SA), Italy E-mail: [email protected]

Salerno

G. L A U R O Facoltd di Architettura, Seconda Universitd di 1-81031 Aversa (CE), Italy E-mail: giuliana. lauro @unina2. it

Napoli

The hydrodynamical equations governing the irrotational flow of an isothermal and inviscid fluid whose pressure tensor depends on the density and its first and second space derivatives are shown to describe a capillary fluid with a suitable form of t h e free energy. Moreover, such a system is equivalent to a nonlinear Schrodinger-like equation, with logarithmic nonlinearity, t h a t admits soliton-like solutions.

1. Introduction In the literature concerning the description of fluids, the notion of diffuse interface 3 and the use of a capillary stress tensor 4 have been employed successfully in modelling a wide class of phenomena involving fluid-fluid systems. Recently 2 , it has been shown that the presence of capillarity 3 in classical hydrodynamic systems allows the reduction of the governing dynamic equations to a nonlinear Schrodinger-like equation. In particular, in 2 the authors have proved the reduction to the integrable cubic Schrodinger equation in (1+1) dimensions showing, hence, a connection between fluid dynamics and soliton theory as investigated, among others, in 5 . The central postulate in the model proposed in 3 states that the fluid entropy depends additionally on the gradient of the density. In the present paper we consider an isothermal fluid with the effect of viscosity neglected and assume that the pressure tensor depends, in a suitable form, on the density and its first and second space-derivatives 9 . Such a constitutive law allows, at equilibrium, the hydrostatic pressure to

148

149 be compatible with the law of ideal gases as it happens for a Boltzmann gas 8 . As shown in 9 , the conservation equations of mass and momentum for such a fluid are equivalent to a nonlinear Schrodinger-type equation with logarithmic non linearity. We recall that the hydrodynamic form of Schrodinger equation has its origin in a work of Madelung 1 in 1926. It is worthwhile to stress that such an equation is integrable and furnishes soliton-like solutions in (3+1) dimensions, the so called gaussons, as proved in 7 . Here, once again, we find a link between soliton theory and fluid dynamics. Moreover, we point out that the form of our pressure tensor is of the Korteweg-type 4 , who first introduced in 1901 the so called "capillary tensor" that takes into account the capillary forces associated to the interfacial region of a two-fluid system. In this paper we show, in fact, that, by a suitable choice of the free energy, the system of equations governing the flow of the fluid described by our model corresponds to the system given in 3 for a capillary fluid. 2. H y d r o dynamical e q u a t i o n s The classical conservation equations of mass and momentum for an isothermal and inviscid fluid, in absence of external forces, are: dtp = - V • pu

(1)

dtpu = - V • (puu + P)

(2)

where p(x,t) is the density, u the velocity, P the pressure tensor, x € R3 the position and t € R+ the time variables. In order to close system (1),(2) we assign the following constitutive relation to the pressure tensor P 9 : Pij = (ap)5ij + P^

(3)

where

p[3 = -L[pXidx.p-

l

-dXiPdXjp)

(4)

and where a and (3 are positive dimensional parameters and i,j — 1,2,3. Namely, we are assuming Py functions of the density p and of the immediate environment which is modelled by first and second space derivatives of p. We remark that in kinetic theory of gases eqns.(l),(2) are a direct mathematical consequence of the Boltzmann equation 8 . In particular, as it is

150

well known, the absolute equilibrium state of a Boltzmann gas is characterized by the constitutive equations of a perfect (Euler) fluid,i.e.,for the hydrostatic pressure we have: *eq

=

~X^ii ~ Peq^B^-eq

\p)

where fcs is the Boltzmann constant and T the temperature. Note that here and in the sequel we used the convention of summing over repeated subscripts from 1 to 3. Hence, (3) can be interpreted as the the deviation from the ideal gas behavior due to the fluctuations in the density of the environment. Further, let us recall that the "capillary tensor" first introduced by Korteweg in 1901 4 has the following expression:

Tij = adXidXjp + bdXipdXjp + (cAp + d(Vp)2J / that corresponds to our P^ with a = — 4~> b — + §-, c = d = 0. In Section 4 we will show, indeed, that the choice (3),(4) yields a conservation equation of momentum valid for a capillary fluid as proved in 3 . 3. Soliton-like solutions By taking in account (1), (3) and (4) we can write (2) as follows 9 : dtu + [u - V ] u = - a V l o g

P +

^-V[^]

(6)

Now if we assume the fluid be irrotational, u — VS, S real function, then (6) furnishes: dtS+±(VS)2

+ a\ogp-^^/l=0

(7)

It is easy to check that (1),(7) are the hydrodynamic form of a nonlinear Schrodinger-like equation with logarithmic non-linearity, where the fundamental unit of action H has been replaced by a suitable dimensional constant (3. Namely: ij3dt$ + ^ - A * + alog(| * | ) 2 * = 0

(8)

with \I/ given by *(x,t)

= y/pexp[^S{x,t)]

(9)

151

We remark that equation (8) is integrable and furnishes soliton-like solutions, namely, uniformly moving Gaussians of the form (7) tf(x, t) =

(ZTT1^)-! e x p

[jC?-)t

+ i v • x] exp -(x-r,-

wtf /2l2

(10)

where 77 is the initial position, v is the mean velocity corresponding to given values of a and /? and I2 = 01 /(2a). Hence, the proposed fluid model admits a density profile of gaussian shape. 4. Capillary fluid Let us recall that in 3 it has been shown that the conservation equation of momentum for an isothermal, inviscid, capillary fluid takes the following form: St«+[u.V]« = - V [ ^ ]

(11)

where the free energy / is such that / = / (p, | | Vp | 2 ) and where (12)

Tp is the variational derivative 2 , i.e.: <50

dG

„

,d®„

,

v

TP^irP- -^ti

,

x

as)

with7=-|V^|2. In order to show the correspondence of our model with a capillarity model, let us assume for the free energy,/, the following relation : f = alogp-^-L 2 4 p

(14)

It is easy to check [see the righthandside of (6) in Section 3] that

v[^]

= av*,-£v[^]

«15,

Hence, by means of assumption (14) on the free energy, equations (6) and (11) are equivalent, that is, our model behaves as a capillarity model. Finally, let us note that the capillary pressure for such a model is given by p = p j - =ap-

—pAlogp

(16)

152 In terms of the pressure tensor given by (3) and (4), relation (17) becomes:

P=\Pii + \pL-

(17)

We observe t h a t , under a suitable choice of the parameter a, at equilibrium such a pressure reduces to (5), valid for a perfect (Euler) fluid.

5.

Conclusions

T h e central assumption on t h e pressure tensor, (3), (4) states t h a t it depends not only on the s t a t e of the fluid, which can be described solely by density (being the fluid isothermal), b u t also on the s t a t e of its local environment which is here modelled by first and second space derivatives of t h e density. Such a choice corresponds t o a suitable "capillary tensor" 4 and it is in agreement with a capillarity model, proposed in 3 , for which we have determined the density profile (gaussian shape), the free energy and the capillary pressure and fits the t h e equilibrium state of a Boltzmann gas. We show t h a t t h e t h e hydrodynamical equations associated t o this model of fluid reduce t o a nonlinear Schrodinger-like equation with logarithmic nonlinearity whose solutions are expressed by soliton-like waves, the so called gaussons, as proven in 7 . Such a result shows another link between fluid dynamic a n d soliton theory involving model laws and it is in agreement with t h e conclusion stated in 2 on the central role played by capillarity in the exact reduction t o a nonlinear Schrodinger-like equation. References 1. E. Madelung, Z. Physik 40, 332 (1926) 2. L.K. Antanovskii, C. Rogers, W.K. Schief, J. Phys. A Math. Gen. 30, L555 (1997) 3. L.K. Antanovskii, Phys. Rev. E,54, 6, (1996) 4. D. Korteweg Arch. Neerl. Sciences Exactes et Naturelles, Series II, 6, 1-24 (1901) 5. B. Konopelchenko, C. Rogers, J. Math. Phys. 34, (1993) 6. P. Resibois, M. De Leener Classical Kinetic Theory of Fluids, (John Wiley and Sons, N.Y. 1974) 7. I. Bialynicki-Birula, J. Mycielski, Ann. of Phys. 100 62, (1976) 8. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, (John Wiley and Sons, N.Y. 1975) 9. S. DeMartino, S. DeSiena, F. Illuminati, G. Lauro, Proc. WASCOM XI (World Scientific, Singapore) (2002)

RATE OF C O N V E R G E N C E T O W A R D T H E EQUILIBRIUM IN D E G E N E R A T E SETTINGS

L. D E S V I L L E T T E S CMLA, ENS Cachan, 61 avenue du President Wilson, 94235 Cachan, FRANCE E-mail: [email protected] C. V I L L A N I UMPA, ENS Lyon, 46 allee d'ltalie, 69364 Lyon Cedex 07, FRANCE E-mail: cvillaniQumpa. ens-lyon.fr

We describe here some of the aspects of a general method enabling to yield quantitative estimates about the rate of convergence toward the equilibrium. We are interested in solutions of P D E s in which dissipative effects can be easily obtained only with respect to part of the variables. Then, one has to use in a subtle way the interplay between the different variables in order to conclude. The example of spatially inhomogeneous kinetic equations (in a confining potential, or in a box with boundary conditions) is detailed.

1. Introduction : The entropy/entropy dissipation method We first describe the traditional strategy (sometimes called entropy/entropy dissipation method in the context of kinetic equations) used to get explicit estimates of convergence toward equilibrium when dissipative effects are predominant. We consider an abstract equation dtf = Af,

(1)

where A is an operator which can be linear or nonlinear, and can involve derivatives or integrals (examples are developed later). We suppose that there exists a (bounded below) Lyapounov functional H = H(f) (usually called entropy (or opposite of the entropy)) and a

153

154

functional D = D(f) (usually called entropy dissipation) such that (what is called in kinetic theory the first and second part of Boltzmann's H-theorem) holds :

D(f) = 0

dtH(f)

= -£>(/) < 0,

«=*

Af = 0

<=>

(2) f = feq,

(3)

where feq is a given function. Then, one can often prove that the decreasing function t i-4 H(f(t)) converges toward its minimum, that t >-» D(f(t)) converges toward 0, and that as a consequence lim f(t) = feq

(4)

t—^+oo

in a convenient topology (a precise theorem could for example be written down, in the case when H is coercive). The situation described above is however often slightly more intricate, because one has to take into account quantities which are conserved in the evolution of eq. (1). In particular, in eq. (3), feq is in general not a given function, but a set of functions depending on a number of parameters equal to the number of conserved quantities. In order to obtain an explicit rate of convergence toward equilibrium, one looks for functional inequalities of the form D(f)>$(H(f)-H(feq)),

(5)

where $ : 1+ -> M+ is a function such that $(x) = 0 •*=>• x = 0. One tries to find a function $ which increases as much as possible near 0. In the case when there are conserved quantities in the evolution of eq. (1), it is enough to prove estimate (5) when the corresponding quantities are fixed. Assuming that estimate (5) holds, we get thanks to estimate (2) the differential inequality 8t(H(f)

- H(feq))

< -*(H(f)

- H{feq)),

(6)

and GronwalPs lemma ensures that H(f) - H(feq)

< R(t),

(7)

where R is the reciprocal of a primitive of —1/$. Then, if H has good properties of coercivity, we obtain

ll/-/ e ,ll<5(t),

(8)

155

where S is related to R, and || || is some norm which depends on the problem. When one can take $(a;) = Ctex, one gets R(t) < e~ctet. Sometimes, It is however only possible to take $(x) = Ctee x1+e for some (or all) e > 0, and consequently R(t) = Ctee t~l/£. Note also that generically (that is, if H behaves quadratically in a neighborhood of f e q ) , one can take R(t) = CteS(t)2. 2. Two examples : Fokker-Planck and Boltzmann kernels We explain here how the previous method can be applied in two specific situations.

2.1. The Fokker-Planck

operator

let us first consider / = f(t, v) > 0, with v € M.N, and Af(v)

= W-{Vf

+ vf).

(9)

In the context of kinetic theory, A is called the Fokker-Planck operator. Then, one can show that one quantity (the total mass) is conserved :

dt [

f(t,v)dv = 0.

We restrict ourselves in the sequel to the case when JVGKN f{t,v)dv One defines the (relative) entropy (or free energy) by : H{f)=f

= 1.

f{v)log-fPrdv, N JveR veR1*

M v

(>

with M(v) = ^ j -

2

.

(10)

Differentiating H(f) along the flow (1) with A given by (9), we see that (2) holds, with

D(f) = I f

/

2

dv. (11) M R That is, the dissipation of the (relative) entropy is exactly the so-called relative Fisher information. Then, (3) holds with feq = M (and H(feq) = 0). V l 0 S

156

In this situation (and still assuming that fveRN f(t, v) dv = 1), estimate (5) can be obtained with $(3;) = 2x thanks to the logarithmic Sobolev inequality of Gross 13 , that is : D(f)>2H(f).

(12)

As a consequence, we can take R(t) = e~2t in eq. (7). Using now the Csiszar-Kullback-Pinsker inequality (Cf.

5

and

H{f)>\\\f-feq\\h,

14

) :

(13)

we see that we can take S(t) = 2e~* in (8). This result is sharp since — 1 is the first nonpositive eigenvalue of A. Of course, since eq. (9) can be solved explicitly, the entropy/entropy dissipation method is not the simplest way to study the long time behavior of its solutions. This is not so in next example. 2.2. The Boltzmann

operator

We now wish to consider the (spatially homogeneous) Boltzmann equation of rarefied gases. One still takes / = f(t,v) > 0, with v € E ^ (f(t,v) is the number density of molecules of a gas which at time t have velocity v), but now the operator A is a quadratic and integral operator modeling the binary collisions between the molecules. It is defined by : Af(v) = Q(f)(v)

= [ f

{/(„') / K ) - f(v) f{v.)\

(14)

x B\ \v — vJ, -.— *, • a } dadv* \v-vJ with v

,

v + v„ = —7;

. v + v* v. = —

\v — vJ 1

s — a1

\v-vJ

a.

and B > 0 (a.e.) is a cross section depending on the interaction between the molecules. For more details about this kernel, we refer to 4 . The conserved quantities for this model are the mass J f dv, the momentum J fvdv

and the energy / / ^ - dv : One defines the (relative) entropy

157

by

H{f)=f

f(v)\og^-dv, JveRN

(15)

Mf{v)

where

M v) = p

*

(16)

'Wrtfm

is the Maxwellian function with same mass, momentum and energy as / :

H

Pf

x

-

'

1

Differentiating (1) along the flow (with A given by (14)), we get (2) with

D

W = \ Iff

x log (

«

«

)

{/(«')/(<)-/(«)/« * ( | « - „.|, 1 ^ 1 . . a)

dadVtdv.

(17)

Then (still assuming that B > 0 a.e.), (3) holds with feq = Mf (and H(feq) = 0). Another possible presentation of this result is that Vv€KN, <S=>

Af(v) = Q

«=•

D(f)=0

/ ( v ) = e x p ( a + b • v — c\v\2)

for some a > 0, c > 0, and b £ M.N. In this situation, eq. (6) can be obtained (for all physically relevant cross sections B) with $ ( x ) = Ctes x1+£ for all e > 0, provided t h a t / belongs t o a set S of functions which satisfy smoothness conditions which are detailed in 21 (together with the precise assumptions on B). Since for many initial data and cross sections B (including the physically relevant cases), it is possible to prove that the solution t H-> /(£) of eq. (1), (14) remains in 5, one can take R(t) = Ctee t~xle and, still using Csiszar-Kullback-Pinsker's inequality (13), S(t) = Cteet-lK2e\ We refer t o 3 , 1T, 18 and 21 for a complete picture of the state of the art concerning the use of the entropy/entropy dissipation method for the

158

spatially homogeneous Boltzmann equation. Those results should be compared with what is obtained thanks to spectral theory and linearization, Cf. for example 1 . Note finally that the entropy/entropy dissipation method can be used in other fields than kinetic theory. We refer for example to 10 for an application in the context of nonlinear elliptic PDEs. 3. Degenerate setting We now consider a situation in which we still have the abstract equation (1), and a couple H,D of entropy and entropy dissipation which satisfies (2). However, we do not suppose anymore that eq. (3) holds, but assume instead that D(f) = 0 *=> feM,

(18)

where M is an infinite-dimensional set of functions, and that feM

and

Af = 0

=•

/ = /„,,

(19)

where feq is a given function (once again, a finite number n of quantities can be conserved along (1), so that feg may in fact depend upon n parameters). Assumption (18) basically means that the dissipation phenomena in eq. (1) concern only some part of the variables of/, and assumption (19) means that the corresponding so-called local equilibria (that is, the elements of M) are unstable by the flow along eq. (1), except when they are global (that is, when / = f e q ) . Under assumptions (18) and (19), it is usually possible to prove that t t-» H(f(t)) still converges to its minimal value, that t i->- D(f(t)) still converges to 0, and that (4) holds in a convenient topology. We refer for example to 6 for a proof in the case of the spatially inhomogeneous Boltzmann equation in a box with specular reflexion. In order to obtain explicit rates of convergence in (4), one cannot hope to use an estimate like (6), since it is not compatible with assumption (18). We suggest to look instead for an intermediate functional K(f) > 0 for which

D(f) > *(#(/))

(20)

159 with K(f) = 0

<=>

fsM,

(21)

and which moreover, when differentiated twice along the flow of eq. (1), yields a differential inequality of the form :K(f) > * i ( F ( / ) - H(feq)) - * 2 ( i f (/)), (22) dt2' where * i , * 2 : 1K+ -> K+ are functions such that the property ^i(x) = 0 «=>• x = 0 holds. Usually, (22) can be proven only under certain assumptions (typically, of smoothness) on / . One then has to prove that those assumptions are compatible with the properties of the solutions of eq. (1). Before explaining how to use formula (22), let us say that looking for an estimate on the second derivative in time of K{f) is reasonable since if at some time to > 0, one has /(to) £ -M, then K(f)(to) = 0 because of (21), and jiK(f)(t0) = 0 because K is nonnegative. Suppose now that estimates (20) and (22) hold. Using eq. (2), we can rewrite them as a system of two differential inequalities : --(H{f)-H(feg))>*{K(f)), dt"

(23)

—K(f) > * i ( H ( / ) - H(feq)) - 92{K{f)). (24) dt2' When $, $ ! and *2 behave nicely (that is, when they grow fast enough near 0), this system will imply (thanks to a lemma which replaces Gronwall's lemma in this context) that H(f(t))-H(feq) c for use (24) under the form ^K(f)

+ 92(K(f))

> c.

is the following : as that thanks to (23), if at some point to, some c > 0, and we

160

Then, if $2 behaves nicely, it is possible to show that K(f) cannot remain close to 0 (except for a small interval of time), so that f(t) will not stay close to M for long, and we can again use (23). A complete proof in the case when $(x) = Ctex, ^i(x) — Ctex, $ 2 (a;) = Ctee xl~e (for all e > 0) can be found in 8 . In this particular case, one can take R(t) = Cte£ i 1 / , £ _ 1 . One of the advantages of the strategy described above is that it is quite robust. We now detail two examples issued from kinetic theory. In the first one, the strategy can be directly applied, while in the second one, many details have to be modified. 4. The Fokker-Planck equation in a confining potential We consider here the operator . Af = - v Vxf

+ VXV - V „ / + V , • (V„/ + v / ) ,

(25)

N

where x,v £ M. , f = f(t,x,v) > 0, and V = V(x) is a confining potential satisfying the normalization property JRN e _ v ^ x ' dx = 1. One quantity is conserved along the flow of eq. (1), (25), namely the total mass : — / at JRN

/

JRN

f(t,x,v)dvdx

= 0.

As a consequence, we suppose in the sequel that fRN JRN f(t, x, v) dvdx = 1. Then, one introduces the (relative) entropy (or free energy)

ff(/)

/< )los


=tL " (^&^))* '

where M is given by formula (10). Eq. (2) is satisfied with the entropy dissipation D defined by 2

M

dv.

(26)

18) is verified with It is easy to see that (18) M = {(x,v) H^ p(x) M(v),

p>0,

pdx = 1}.

That is, the set M of local equilibria is formed by the functions which have a (centered, normalized) Maxwellian profile with respect to the v variable, and are arbitrary with respect to the x variable.

161

Then, (19) holds with feq(x,v) = e~v^ M(v) since f eMtmdAf =0 implies that vVx{p{x) M{v))-VXV •Vv(p{x)M(v)) = 0, so that Vx(p + e~v(xl PV) • {vM{v)) = 0, and finally p(x) = The intermediate functional K(f) that we introduce here is the relative entropy of / with respect to the local Maxwellian pf{x) M(v) (where pf(x) = f f(x,w)dw), that is K(f)=

f

[

f(x,v)\og(

{(f J .

.)dxdv.

It is clear that (21) holds. Estimate (20) with $(x) = 2x is then a simple consequence of Gross's Sobolev logarithmic inequality. Estimate (22) holds with ^i(x) = Ctex and ^ ( z ) = Cte£ xl~e as soon as / 6 S, where 5 is the set consisting of functions / having bounded Hk derivatives (for all k e N) and such that afeq 0. Noticing that 5 is stable by the flow of eq. (1), (25) when V has a behavior at infinity close to that of the quadratic potential, we obtain

H{f){t)-H{feq)
Theorem 1 : Let f0 = fo(x,v) > 0 be an initial datum of mass 1 for eq. (1), (25), such that a feq < /o < b feq for some a, b > 0, and V be a potential of the form V{x) = ^f + ${x) + V0 with V0 6 K, / e~v^ dx = 1 and $ € H°°{RN). Then, for all e > 0, there exists Ce > 0 explicitly computable such that \\f(t)-feq\\L^Mn
One can compare this result with the theorems obtained in We also refer to 2 and n when A is a linear integral operator.

15

We now turn to the case of the Boltzmann equation in a box.

and

162 5. The Boltzmann equation in a bounded box with specular reflexion We consider the spatially inhomogeneous Boltzmann operator Af = -v-Vxf

+ Q(f),

(27)

with Q defined by (14) and we suppose that x varies in a bounded box Q which is supposed to have no axis of symmetry. We supplement (27) with the boundary condition ,x,v) = f(t,x,v

— 2 (v • n(x)) n(x)),

(28)

4

where n(x) is the exterior normal vector to dft. We refer to for details on the underlying modeling. The conserved quantites for eq. (1), (27), (28) are the total mass and energy : -77 / / f{t,x,v) dvdx = 0. M2 Ut JKN JRN y J-J- J We shall suppose in the sequel that / is normalized in such a way that its total mass and energy are respectively equal to 1 and N/2. We define the (relative) entropy H by H(f)=

f f JRN JRN

f(x,v)log(^l)dvdx. \ M(v) J

Then, 4 JRN JRN Jy,N JSN-I

x log

—7

r—;

(

)

) B [\v - v* , -.

r•a

dadv*dvdx.

According to 4 for example, we see that eq. (18) holds with M =

{{X'V) ^ (2 7 rT(x))^ e " b i i ^' f

,

f, \u\2

P{X)

~ °Mx)

Nm ,

N)

J pdx = l,J(pi-±- + -T)dx =

£ RjV r

' W > °'

-j.

Note that M. can also be defined by M = < (x,v) H-> exp (a(x)+b(x)-v-c(x)

\v\2),

a(x) G R,b(x) £ RN,c(x)

> 0,

163

LLf{^)dvdx=il)}Then, property (19) is a consequence of the following computation of Grad (in dimension 3) : v • Vz(exp (a{x) + b(x) • v - c(x) \v\2)) - 0

=>•

a(x) = a0,

b(x) = b0 + b\ x x,

c(x) = c 0 ,

for some constants a0 G R, bo,bi € R 3 , Co S R. Using the boundary condition (and the fact that Q, has no axis of symmetry), we get bo = b\ = 0, so that feq{x,v)

= exp(a 0 - c0 \v\2).

Integrating with respect to x estimate (5) (with D, $ as defined in subsection 2.2), it is easy to see that (20) holds with $(x) = Cte£ xl+E, and K defined as the relative entropy of / and M/ :

K(f) = f [

f(x,v) log (-£%£T)

dvdx.

In this case, unfortunately, eq. (22) has to be replaced by a variant for many reasons. First, one needs to consider K'(f) = \\f — Mf||| 2 instead of K(f) in order to avoid the assumption that / decreases when \v\ —> +oo like a Gaussian (such a decay is in general not propagated by the flow of eq. (1), (27), (28)). Then, one has to replace V2(K(f)) (or rather * 2 ( # ' ( / ) ) ) by ty2(K'(f)/S) + SH(f) (for any 6 > 0). Those two changes do not lead to conceptual difficulties. Finally, the most important change with respect to the model differential inequality (22) is related to the fact that it is not possible to reconstitute the term * i ( f f ( / ) - H(feq)) in (22) by using directly K(f) or K'(f). One in fact needs to introduce other intermediate functionals, so that one ends up with a differential system of more than 2 inequalities (actually : 4). When all those modifications have been performed, one can prove the following result (Cf. 9 ) : Theorem 2 : Let f be a (strong) solution of the Boltzmann equation (1), (27) (with a cross section B satisfying assumptions detailed in 21) on a

164

bounded regular open set 0. with no axis of symmetry and with the specular reflexion boundary condition (28). We assume that for all k,l € N, sup/

\Vkxvf{t,x,v)\2{l

/

+ \v\2)1 dvdx < +oo,

and that there exists C\,C 0, q > 2, such that f(t,x,v) d exp(-C 2 \v\"). Then for all e > 0, f(t,x,v)

-

M(v)

where Ce is an explicitly computable

>


Before concluding, we notice that the assumptions of this theorem are not empty, since solutions close to equilibrium satisfying them can be builded. We refer to 16 and 19 for a comparison with theorems obtained by spectral theory and linearization. 6. Conclusion We now briefly discuss the advantages and drawbacks of the entropy/entropy dissipation method, in particular with respect to methods based on spectral theory. Note first that the entropy/entropy dissipation method is robust because it is mainly based on functional inequalities. For this same reason, some monotonicity with respect to the parameters of the model can usually be exploited. Finally, one obtains at the end explictly computable constants. The main drawback of the method is that one often loses some sharpness in the estimate : typically, a polynomial decay is obtained when an exponential decay is expected. Moreover, in order to get a polynomial decay of high degree, one needs to be able to control the propagation of a large number of derivatives of the solution of the equation. References 1.

ARKERYD, L. Stability in L for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 2 (1988), 151-167. 2. CACERES, M., CARRILLO, J., GOUDON, T. Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Comm. Partial Differential Equations 28, 5-6 (2003), 969-989.

165 3. CARLEN, E., CARVALHO, M. Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74, 3-4 (1994), 743-782. 4. CERCIGNANI, C. The Boltzmann Equation and its Applications. Springer, 1988. 5. CSISZAR, I. Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 2, (1967), 299-318. 6. DESVILLETTES, L. Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rational Mech. Anal. 110, 1 (1990), 73-91. 7. DESVILLETTES, L., VILLANI, C. On the spatially homogeneous Landau equation for hard potentials. Part II : .ff-theorem and applications. Comm. Partial Differential Equations 25, 1-2 (2000), 261-298. 8. DESVILLETTES, L., VILLANI, C. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54, 1 (2001), 1-42. 9. DESVILLETTES, L., VILLANI, C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Preprint. 10. D E L PINO, M., DOLBEAULT, J. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 9 (2002), 847-875. 11. FELLNER, K., NEUMANN, L., SCHMEISER, C. Convergence to global equi-

librium for spatially inhomogeneous kinetic models of non-micro-reversible processes. To appear in Monatsh. Math. 12. GRAD, H. On Boltzmann's H-theorem. J. Soc. Indust. Appl. Math. 13, 1 (1965), 259-277. 13. GROSS, L. Logarithmic Sobolev inequalities and contractivity properties of semigroups. In Dirichlet Forms (1992), F. et a]., Ed., vol. 1563, Lect. Notes in Math., Varenna, Springer-Verlag, 54-88. 14. KULLBACK, S. A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. The. 4, (1967), 126-127. 15. HERAU, F . , NlER, F . Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential. Preprint 16. ASANO, K., SHIZUTA, Y. Global solutions of the Boltzmann equation in a bounded domain. Proc. Jap. Acad. 53, (1977), 3-5. 17. TOSCANI, G., VILLANI, C. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203, 3 (1999), 667-706. 18. TOSCANI, G., VILLANI, C. On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98, 5-6 (2000), 1279-1309. 19. UKA'I, S. On the existence of global solutions of mixed problems for nonlinear Boltzmann equation. Proc. Jap. Acad. 50, (1974), 179-184. 20. VILLANI, C. Personal Communication. 21. VILLANI, C. Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234, 3 (2003), 455-490.

A M A T H E M A T I C A L MODEL FOR A SOLE LARVAL P O P U L A T I O N W I T H T W O T I M E SCALES

SILVIA DONATI AND SILVIA TOTARO Dipartimento

di Scienze

Matematiche ed Informatiche Via del Capitano 15, 53100 Siena, Italy E-mail: [email protected]

"Roberto

Magari",

M E R I LISI Dipartimento

di Matematica "Ulisse Viale Morgagni 67a, 50134 Firenze, Italy E-mail: [email protected]

Dini",

In this paper we propose a model of a sole larval population. We distinguish two time scales: at the fast time scale we have the migration dynamics and at the slow time scale the demographic process. At a first step, we study the so called "aggregated" system by means of the semigroups theory. Then, we study the model by using the Chapman-Enskog procedure. We show that the solution of the aggregated system is a good approximation to the solution of the exact model.

1. Introduction Lot of research concern vertical migrations related to the movement of larvae offish or early juvenile fish: this is because many fish, especially flatfish, spawn offshore but early juveniles develop inshore. These migrations towards the sea bed are due to light, while the lack of light is followed by migrations towards the surface. Thus, the migration process is performed daily, which is a fast time scale compared with the demographics process. In the case of the common sole, Solea solea, active vertical migrations are involved during the larval stage, [1]. This study is interesting also because european catches for common sole are the greatest one. In particular, we study a model, which takes account of the main characteristics of the dynamics of the sole larval population of the Bay of Biscay, a region of the North Atlantic, along the West Coast of France, extending

166

167

from the Brittany peninsula in the north, to Spain in the south. The life of the sole of Bay of Biscay has been intensively studied in the field as well as in laboratory (see, for istance [1], [4] and the references therein). The life cycle is divided into four stages: egg, larva, juvenile and adult. The model proposed by Arino et al. in [1] takes as the state variables the densities of the four stages as functions of time, age and distance from the coast. Nevertheless, it seems to be important to consider also the daily ascending migrations of larvae after sunset and the descending ones before sunrise. This is because of the different predations rates depending on the depth and because of the tidal stream transport at the surface. Thus, we include the influence of depth in the demography of larvae, studying a model of an age-structured population divided into N spatial patches (with different mortality rates) that distinguishes the two time scales involved. We can describe the evolution of the population in a standard way, [1], [8]: drii = -fii (a) rii + -

J2f=i

k

ij (a) ni ~ D j L i kji (a) ni\ >

ni(0,t) = f+°°(3i(a)ni{a,t)da, rii(a,0) —
t > 0, a > 0,

W

with rij = rii (a,i) the population density in patch i (i = 1,...,N), a > 0, t > 0, drii = drii/dt + drii/da and where e > 0 is a small constant which describes the fact that the migration process evolves at a faster time scale compared to that of the demographic one. Moreover, k^ (a) represents the migration rate from patch j to patch i, with i ^ j , whereas /i., (a) and /% (a) are the mortality rate and the fertility rate in patch i, respectively. We consider k^, ^ and /% not depending on time t. By putting n(a,t) = (n1(a,t),...,nN(a,t))T, M(a) = diag {/zj (a),..., fiN (a)}, B ( a ) = diag {Pi (a), ...,(3N (a)}, * (a) = (4>i (a),..., (/>N (a)) T and K (a) = (k^ (a))i system (1) becomes: dn = - M ( o ) + - K ( a ) n (0, t) = / 0 + o ° B (a) n (a, t) da, t > 0, n (a, 0) = * (a), a > 0,

(2)

where n = n (a, t), a > 0, t > 0, dn = dn/dt + dn/da and 0 < e < < 1. 2. The aggregated model We want to build up an approximated model for the evolution of the total population n (a, t) = J2i=i ni (a> 0 •

168

The exact model for the total population n (a, t) can be obtained adding up i = 1 to N in system (1): dn{a, t) = - J2i=i M» (a) ni (a> *)' +o

a> 0,t> 0,

n (0, t) = J 0 ° [ E i l i ft (a) "i (a, *)J da, n{a,0) = 3>{a) = ^=1i{a),

t > 0,

(3)

a>0,

where 9n(a, t) = <9n(a, t)/dt + dn{a, t)/da. In order to simplify the system, without loosing of generality, we assume that the N x N matrix K ( a ) is age-independent and irreducible, [1], [4]. Hence, the left eigenvector associated to its dominant eigenvalue 0 is 1 = (1,..., 1) , whereas the right eigenvector is a vector v that we choose having positive entries and such that lTv = v\ + ... + v^ — 1. Assume that rii(a,t)/n(a,t) ~ Vi, and put X)i=i niM« ( a ) vi = M* ( a )j This yields the following approximation of (3), [1]: On (a, t) dn(a.t) ^, . K ga ' =-ii*(a)n(a,t), dt ' + n (0, t) = J 0 +o ° (3* (a) n (a, t) da, n (a, 0) = $ (a),

o>0,t>0, t > 0, a > 0.

(4)

This model is called the aggregated system. To study system (4) by using the theory of semigroups of operators ([2], [5]), we introduce the space X = L1 (R + ), with norm ||/|| = J °° \f {a)\ da and we define the operator A{f) = —/' — /J,* (a) / , with D (A) = {/ G X : / ' e X,f (0) = J 0 +o ° /?* (a) / (a) da). The following assumptions on /J,* and /3* are made: h • M* (a) > 0, /3* (a) > 0, a.e. in R+; i2 : H*, [3* e L°° (R+), i.e., 3 / i , / 3 e R + such that fi* (a) < p,, [3* (a) < f3; is : 3rj > 0 such that /J.* (a) > r\ > 0, a.e. in M + ; it : 3 £ > —77 such that l i m a _ + 0 0 e?a/?* (a) < +00; *5 : Io°°P* (a)e-«°-/o>*< s ) d ^a > 1. By using the properties of the operator Ao (/) = —f — p* (a) / , with D {Ao) = {/ e X : / ' 6 X, / (0) = 0} and by applying Theorem 1 of [3], we can prove that D {A) = X. By using the Hille-Yosida Theorem, the following theorem can be proved. T h e o r e m 2 . 1 . The operator A generates a strongly continuous semigroup {Z{t) = exp {tA), t > 0} and the type of the semigroup w {A) is equal to A, where A satisfies J0+°° S3* (a) e~~Xa-$S »"(s)dsda = l.D

169 In order to have more informations about the essential type of the semigroup uess (A), [7], and the type of the semigroup u){A), [5], we introduce the operators Ti 6 S (L 1 (0, t)), t > 0 and T2 G B ^ f O . + o o ) ) , such that T a (/) = f*0*(t-a)f(a)e~Si~aii'(t)d'da and T2(f) = J 0 +o ° (3* (t + a) f(a)e~ti+"^^dsda. We can prove that T2 is compact and that Z (t) f = Zx (t) f + Z2 (t) / , where

(ZAt)f)(a) = {^f^>:^:

(5)

'0,

a>t,

(Z2 (t) f) (a) = I

(6)

( / - T i ) _ 1 T 2 / l (t-a)e-/o0"'W*>)

a

< t.

Because of the properties of T\ and T2, it follows that a(Z(t)) = a{Z\{t)) < \\Z\(i)\\ess — e _ r ? a i where a(-) is the measure of noncompactness. In fact Z2 (t) is compact and a (Z2 (£)) = 0. Thus, ojess (A) = -rj. It can also be proved that {Z(t), t > 0} has the property of asynchronous exponential growth, [9], i.e., there exists a constant C(i>) such that Km e-'xtn (a, t) = C (*) e-laSo

»" (s)ds,

t—*+oo

where A is defined in Theorem 2.1 and the limit is taken in the L1 norm. 3. T h e C h a p m a n - E n s k o g p r o c e d u r e In this section, we examine the asymptotic behaviour of the solution of the exact model (2), by applying the Chapman-Enskog procedure in the modified version introduced by [6], i.e., the compressed asymptotic approach. Let us consider the Banach space XN = Ll (M + ,R J V ), with the norm llflU = Eti Wfil w h e r e f (*) = (/i (x),...,fN(x)f, fi G L'(R+) and ||/i|| = / 0 + o o |/<(a:)|da:,i = l ) 2 ) „ , J V . Let us define the operator Af = —f — M (a) f = (Ai,..., An) , with D ( A ) = {f G XN : f (0) = | 0 + o o B (a)f (a) da} and Atft = - ^ - ^ (a) / , with I> (Ai) = [h&X-.fi (0) = / 0 +o ° ft (a) / 4 (a) d a } , for * = 1,2,.... iV. Thus, the abstract version of (2) reads as follows: - ^ = A n ( « ) + -Kn(t), n(0) = ,

* > 0,

(?)

170

where n (t) = n (•, t) is now a function from t to XN and = * (•) is a given element i n X w . Theorem 3.1. The operator A generates a strongly continuous semigroup jexp (tA) ,t >0> and exp [tAj — diag {exp(t J 4 1 ), ...,exp(L4jv)} • • Since - K is a bounded perturbation of A, from the previous theorem, e we can prove existence and uniqueness of the solution of (7) and if Z e (t) is the semigroup generated by ( A + ^ K J, we have that n (a, t) ~ Ze (t) $ . Now, we examine the asymptotic behaviour of the solution n, by using the compressed Chapman-Enskog procedure, [6]. Let us define the projection operators V, Q : RN —• KiV, such that P w = ( l r w ) u, Qw = (I — P)w = w — ( l T w ) u, for any w in RN, where u is the right eigenvector associated to the simple eigenvalue A = 0 of the matrix K. It can be proved that R (Q) = S = {w e 1 ^ : l T w = 0}. Hence / ^ ( R + . R * ) = (L1 (R+.R) ® v) © L1 (K+, 5) and RN = \v] © 5 , where [i/] is the subspace spanned by v. We can write n (a, £) = p (a, t) v + q (a, £), where pi/ is the hydrodynamic part of the solution and q is the kinetic one, [6]. Thus, system (2) can be written in terms of its projections on the subspaces [v] and S: dpu + dq = - [ l T M (a) + M S (a)] {pv + q) + i K s q , p (0, t) v + q (0, t) = / 0 +o ° [1 T B (a) + B S (a)] (pi/ + q) da, . p (a, 0) v + q (a, 0) = # (a) = p i r l (a) i/ + q i n (a),

(8)

where pv = p (a, i) i/, q = q (a, £), a > 0, i > 0, dpu = dpv/dt + dpu/da and 9q = dq/dt + dq/da. Moreover l T M ( a ) , 1 T B (a) represent the projections on [i/] and M s (a), Bg (a) the projections on S. Now, expanding the kinetic part q in a series in powers of e, and leaving the hydrodynamic part pu unexpanded, we have: dpu = - 1 T M (a) (pu + q 0 + eqi + e2q2 + •••) , edqo +
(9)

(a) (epu + eq 0 + e2q1 + ...) + Ks (q 0 + eqi + ...).

Comparing the terms of the same order, and by using the properties of K, we obtain: q 0 = q 0 (a,t) = 0,

ql = qx (a,t) = KjlMs

(a) up{a,t).

(10)

171 We note t h a t substituting (10) in (9) we obtain the following equation: ^

+ ^

= " 1 T M (a) p(a,t)

[1+eK^Ms

(*)] •

In particular, if we neglect the terms of order s, we have: dp(a.t) dp(a.t) .. . , + da Ot = ' ^ (fl)P(Q^)P(0,t) = j+°°f3*(a)p(a,t)da, , P (a, 0) = Pin {a),

a>0,t>0, t > 0, a> 0.

(U)

Hence p (a, t) = exp (tA) pin (a) . We can notice t h a t system (11) is the same as the aggregated system (4): thus, t h e first approximation of the hydrodynamic p a r t of the exact solution satisfies t h e aggregated model. A similar result is proved in [4] by using other tecniques and after lenghtly computations. In order to complete our study, we should also consider the corrections due to the initial and t h e b o u n d a r y layer effects. This will be done in a further work.

References 1. O. Arino, E. Sanchez, R. Bravo De La Parra, Math. Comput. Modelling, 27, 4, 137-150 (1998). 2. A. Belleni Morante, Applied semigroups and evolution equations (Clarendon Press, Oxford, 1979). 3. A. Belleni Morante, G. Busoni, Math. Comput. Modelling, 2 1 , 8, 13-19 (1995). 4. R. Bravo De La Parra, O. Arino, E. Sanchez, P. Auger, Math. Comput. Modelling, 3 1 , 17-26 (2000). 5. T.Kato, Perturbation theory for linear operators (Springer Verlag, New York, 1984). 6. J. R. Mika, J. Banasiak, Singularly perturbed evolution equations with applications to kinetic theory (World Scientific, Singapore, 1995). 7. M. Mokhtar Kharroubi, Mathematical topics in neutron transport theory (World Scientific, Singapore, 1997). 8. G. F. Webb, Theory of nonlinear age-dependent population dynamics (Marcell Dekker, New York, 1985). 9. G. F. Webb, Transactions of the American Mathematical Society, 303, 2 (1987).

Acknowledgments This work was partially supported by the Italian "Ministero dell'Universita e della Ricerca Scientifica e Tecnologica" research funds, as well as by G N F M of Italian CNR.

F R E E ENERGIES IN T H E MATERIALS W I T H FADING M E M O R Y A N D A P P L I C A T I O N S TO P D E S .

MAURO FABRIZIO Department of Mathematics. University of Bologna

1. Introduction In the fading memory materials, the state is given by a history. Here, we show as it is very natural to consider the state by the response of the stress submitted to a constant deformation. In this work (preview of the paper 5 ) , we want to put in evidence the meaning and importance of determining a natural norm on the space of processes and states for a material with fading memory. Moreover, we will show how the idea, considered by Gentili in 13 , of introducing a topology on the space of the processes through the continuity of the work provides a natural topology on the space of the states, which Gentili proves to be the dual one. This point of view seems particularly useful in the study of PDEs connected with such problems. For this reason this paper ends with an application to the integro-differential equation of viscoelasticity. For such problem, by means of this new space, we are able to prove the existence, uniqueness and asymptotic decay of the solutions for a wide family of kernels. 2. Fading memory and thermodynamics A viscoelastic material is defined by a constitutive equation which related the stress tensor T and the deformation gradient F by a functional of the type

T(x,t)=f(Ft(x)) where Fl(x, s) = F{x,t — s) is the history of the deformation gradient F. In the linear case /•OO

G'{x, s)Et(x, s) ds

T(x, t) = G0(x)E{x, t)+ Jo

172

173

where E = V"+(2V") j s the infinitesimal strain tensor. Moreover, G0(x) and —G'{x, s) are symmetric and positive defined tensors, as well as /•OO

G^x)

:= G 0 (x) + / G'{x,s)ds Jo Of course, this study can be applied to a non-linear constitutive equation of the type /»00

T(x,t)

= G0(x)$(F(x,t))

+ /

G'(a:, *)$(**(!, s))ds

(1)

Jo

where $ : IR3 —> IR3 is a non-linear, regular and odd function of F. A map P : [0, dp) —> - Lin piecewise continuous and defined as: P(t)=E(t),

te[0,dp)

is called kinetic process of duration dp € IR+. Definition 2.1. A simple material is given by the set (II, E, /3, T) : a- II is the set of kinetic processes defined as: II = {P : [0, dp) —• Lin piecewise continuous; if Pi, P2 € II, P\ * P2 6 II; where 'Pi(i), \ P2(dPl -t),

»/ if

iG[0,dpi) t G [d Pl , d Pl + d P2 )

ifPen,^1,ta)en)[ti>t2)c[0,dp)} 6- 52 i s the state space, their elements are denoted by a, c- the map p : 52 x n —> 52 is called evolution function, such that if 1 a is the initial state and P is a process: p (a1, P) — af , d- the map T : E x II —> F is the response function which, to any state a and process P, assigns the stress tensor process Tp over all the time interval [0,dp). Namely Tp(t)

= f(a,P)(t)

,

tc[0,dp)

Definition 2.2. The system is called causal, if the property d- can be replaced by the new condition d'- the map T : E x Lin —> .Sym is such that T(t) = f ( a ( t ) , P ( i ) ) Remark 2.1. In this paper we consider only causal systems. For these systems, because a(t) = p(a,Pt) we have that the function T is connected to T by f(p(o-,Pt),P(t))=f(o-,P)(t)

174

A viscoelastic material is a simple material where the state is given by the history a = {El) = (E (t), El) and the process by P{t) = E(t),

te[0,dp).

In the following we consider the work W on the process P, defined as W(a,P)

P

= [ Jo

f(a(t),P(t))-L(t)dt

Definition 2.3. (Noll, 1 6 ). Two strain histories E[ , E\ are said to be equivalent if for every process Ep : [0, dp) —> Sym , they satisfy f(E\,EpS)

=f(Ei,EpS)

or equivalently, as proved by Gentili, 13 , two histories are said be equivalent if for every process Ep : [0, dp) —> Sym we have

W(E\,EP}

=W(E%,EP}

Theorem 2.1. (Del Piero-Deseri, 6 , 7 ) . In the linear case, two histories E\ , E\ are equivalent if and only if E\ (t) = Ei (t) and / G'(s + T)E\{s)ds= I G'is + ^E^ds, VT > 0 (2) Jo Jo According to the definition of state a, two couples (Ei(t),E\(r)) and (E2(t), E\{T)) equivalent (in the sense of Noll) are represented by the same state cr(t) (see 1 4 ). In other words the state can be represented by the pair
7*(T) = - / G'(s + T)Et(s)ds Jo We use also the representation /•OO

/*(r) = - / Jo Definition 2.4. (Gentili, a finite work process if W(Q\EP)=

13

G'(s + r)(E*(s) -

E(t))ds

) A process Ep : [0, oo) —> Sym is said to be

( Jo

P

T(0\EPtT))-Ep(r)dr
175

Moreover, the work W{Q\EP) following form W(0\Ep) -|

+

/'CO

in the linear case can be written in the

= ^G^Eidp)

• E(dP)+

/-CO

2Jo

6{ T s P{ T

\ - ^ - )-ZP^dTds=

Jo

I

1

= -GxE(dP)

(3)

fOO

GC{U)EP{UJ)

• E(dP) + —J

• EP(u)du

where G = G(s) — G ^ , Gc and Ep the Fourier transforms of G and The set of the finite work processes is given by HG

= IEP;[

J

G(\T

• Ep{s)dTds

- S\)EP(T)

< ooj

Ep.

(4)

This set is an Hilbert space if the kernel G satisfies the II Law of the Thermodynamics (i.e. Gc(ui) > 0 )and the norm is given by 2

Ep

/*oo /•CO

/ Jo -

/-oo rOO

/ Jo /

- S\)EP(T)

G(\T

GC{LJ)EP(LU)

• Ep(s)dTds

• EP(to)dui

" " J — oo

The domain of definition of the admissible states is the set of all strain histories rendering the work well defined when the process belongs to TicW(P, Ep) = hd^Eidp) 1

+

2j0

/»CO

• E(dP)+

/»CO

L 6(\r-s\)zp(Tyzp^dTds+

(5)

/•CO

+ G " 1 / t ( 0 ) • E(dP) - / Jo

P{T) • Ep{s)dr

< oo

Therefore the set of admissible states P{T) is given by the dual H'G of HG, namely H'G = (lt(T);JCCIt(T)-Ep(s)dT

< oo,\/Ep

G HG\

Definition 2.5. A pair (
=

176

Second Law for isothermal processes. For every cycle (cr,P) £ ^2 xU the inequality

£

T (a ( r ) , P (r)) • L (r) dr > 0

(6)

/o

holds. For materials with fading memory, cycles are quite rare, because usually the material gets to a state which, although close to the initial state, is different from it. Za:={o-';3PeTL;p(o-,P)

= o-'}

Strong Form of the Second Law (for isothermal processes) 8 . The set of the works done in passing from a to any state a £ ^2a W(a):={W(a,P);

Pell}

is bounded below. There is a state <7o , called zero state, such that infW(cro) = 0 This last definition of the Second Law holds the Dissipation Principle given by Gurtin & Herrera in 15 . As show in ( 10 ) from the Second Law of the Thermodynamics we have />oo

I G'(s)smwsds < 0 , / o r o l / w £ R + + Jo while from Strong Form of the Second Law, we have G'S{UJ)=

Goo > 0 Definition 2.6.

(7)

(8)

A function V : 2?i/> —»• JR-+ is called free energy if

a the domain V^ c V is such that for every o\ e V^ and P e II , the state a = p (<7i, P) £ V^, b (T0 € V^ , and ip (a0) = 0 , c for any pair Q\ , o~2 € ^V anc ^ -P G n such that p {o\ ,P) = &2 w e have: 1>(
(9)

In the Linear Viscoelasticity, there are many free energies 2 . The family J- of the free energies is a convex set. J- , which has a minimum and a maximum element ipM, ^m •

177

Maximum free energy was considered in ^M{Et) 1

/>00

=; •* Jo

8

l

=

-G00E{t)-E{t)

+

/»00

/ G 1 2 ( S l - s2) (E* ( S l ) - E (t)) • (E* (s 2 ) - E (*)) ds2ds2 Jo

(10)

where Gi 2 (si - s 2 ) = 5 $ ^ < 3 ( l * i - s 2 |). There is an intermediate free energy, called Graffi-Volterra free energy given by 4G(Et)

=

±G00E(t)-E(t)-

G'(s)(Et(s)-E(t))-(Et(s)-E(t))ds

--J

(11)

where G' (S) < 0, G" (s) > 0. Finally, the minimum free energy was found by Bruer and Onat in 1964 1 in the following form

^m(Em)

= \GooE{t) • E(t) +

+o / / G{\s1-s2\)Em{s1)-Em(s2)ds1ds2 (12) * Jo Jo where Em is the optimal process which yields the maximum recoverable work, but it is not a functional of the history Et (s), or of the state / ' = (E(t)Jt(T)). Golden 12 was able to give a representation of the minimum free energy in term of /* (r) as * m ( / ' ) = \GooEit) -t

-Jo

POO

• E(t) + /*00

Jo

H{T,T')l\T).l\T)dTdT>

(13)

where H(T,T') is a suitable kernel depending by G. Finally, let me introduce a new free energy written as functions of /*(• (see 5 ) 1

* 2 (/ 4 ) = ^Eit)

1 f°°

• E(t) + -J

G~\T)PT(T)

• PT{r)dr

178

where G(s) < 0, G(s) > 0. In order to prove the Clausius-Duhem inequality, consider the time derivative of ^(-f*) r°° . d * 2 ( J 4 ) = Goo£?(t) • £ ( t ) - J G - 1 ^ ) - / ^ ^ ) • 7*(r)dr because d r°° - / * ( r ) = - G ( r ) £ ( * ) -J G(r +

s)Et(s)ds

rOO

= -G(r)E(t)

-

G(T + *)(£'(«) -

E(t))ds

Jo = -G(r)E(t)

+

lUr)

we have * 2 ( / ' ) = GooE(t) • £ ( i ) - j( = G^Eit) + f

G-Hr)^/*^) • £(r)dr

• E(t) G~\T)

(G(r)E(t)

= T(t) • E(t) + ±G-\0)It(0)

+ [^ G {r + s)(£'(s) - E(t))ds)

• PT{r)dT

• 11(0)

1 r°° . .. ~2j0 G-2(r)G(r)^(r).^(r)dr

(14)

Finally, when there is a constant a G M + + such that 0 < -aG(x,s)

< G(x,s) < oo, for all (x, s) £ £1 x IR +

then the equality (14) assume the form * 2 ( / * ) < T(t) • E(t) + I G - ^ O J / ^ O ) • /<(0) +

1 r°° -aJQ G-^T^-IUTW

(15)

3. Weak formulation and applications to linear viscoelasticity Because the constitutive equation of linear viscoelasticity can be written in the form T(t) = G(i)Vu(0) + f G(tJo

T)Vu{r)dT + !°(t),

(16)

179

then the equation of motion becomes pu(t)

= V• /

G(t - T)VU{T)(LT

+ b(t)

(17)

Jo where b(t) = f(t) + y-[I°(t)

+ G(t)Vu(o)}

is a given function in Q x IR + . Letting v — u we can write the differential e q u a t i o n in t h e form pv{t)=V • [ G{t-T)Vv(T)dT + b(t)

onftx]R + .

(18)

Jo This is a differential equation in the unknown function v, together with the initial and boundary conditions v(x,0) = v0(x),

v(x,t)\m

=0

(19)

in the following we suppose, vo{x) = 0 and b(x, 0) = 0. a Let w be a smooth vector function on Q. x 1R+, vanishing on the boundary dQ, and such that w(x,T) = 0 . Inner multiply by w(t) and integrate on ft x [0, T] to obtain Jo In Pv(t) • ^)dx

dt

-So Sn^W-^Wxdt-tf

= Jo In So G ( * " r ) V u ( T ) • Vu>(t)dT dx dt

fQf(t).w(t)dxdt.

which corresponds to the Virtual Work Principle and is able to provide a new definition of weak solution. So that, in order to obtain a rigorous definition of solution according to the equality (20) , we need tofixthe domain of the functions v(x,t) and w(x, t). As we will see, the free energy in the form of Brueur and Onat l is able to characterize the set of solutions. So that, we introduce the spaces nG(^+;m(Q)) = {v€ L? oc (IR+;#i(fi)) ; Jo°° Jo°° Sa G(x> \T ~ r'\)Vv{x, r') • Vv(x, ^dxdr'dr

< oo}

a So that, if v is a solution of (18) with initial data v(x, 0) = VQ(X) and source b(x, t) such that fc(x,0) = bo(x), then the function z = v + w, with w e C°°(IR+, H^ (Q)) and such that « J ( X , 0 ) = — VQ(X) , iii(x,0) = — b(x,0) and w(x,-) = 0 for every t > to, satisfies a problem formally equal to (18) -(19) with source

b(x, t) = b(x, t) + w - V • / G(t- s)w(x,

Jo

s)ds

180

T{Q) = Hi(lR+;L2(n)) while the states I°(x,t)

nnG{M+;H^))

will be elements of the space

W£(]R + ;L 2 (fi)) = {1° G L2oc{TR+;L2{Sl)) + Jo

00

/ J " / „ G*{x, \T - T'\)I°{X,

; JQGoo{x)Vv{x.0)

T') • I°{x, T)dxdT'dT

-Vv{x.0)dx

< Oo}

where G*{x, \T — T'\) is defined by / G*(X,T - T')G(x,T')dT Jo The spaces H.G,J~{Q) /•OO

(vi,v2)nG= ./o

/>T

= IS(T)

are Hilbert spaces respect to the inner products />

/ / (GOOCXJ + ^ . T - T ' ) ) Jo JQ

[VVI(X,T')

• VV2{X,T)

+ VVI(X,T)

and {yi,V2)r = -1 / / (vi(x,t)-v2{x,t)+v2(x,t) Jo Jo. = / vi(x,0) •v2(x,0)dx Jn

+

• Vv2{x,T')]dxdT'dT

•vi(x,t))dxdt

+ (vi,v2)Ha

(v1,v2)nG

Definition 3.1. A function v G J~{Q) is called a virtual work solution of the problem (18), (19) with data v0 G Hi {SI), f G H'i{M+;L2{n)) U Hh{JR+;H'\n)), 1° G W£(IR + ;L 2 (fi)), if it satisfies the identity ( 2 0 ; for any w G J~{Q)Now we are in a position to give the following T h e o r e m 3 . 1 . Assume that the kernel G{x,-) G L1 (IR + , L°°(fi)) satisfies the thermodynamic condition Gc{x,u>) > 0, /or any (x,a;) G Jl x IR. TTien £/iere exists a unique virtual work solution v G 3~{Q) of the problem (18 ) , (19) for any / , / ' G HG{TR+; H'1 {Q)) andl°,l0' G H*G{M+; L2{Q)). Proof. Let us consider the Fourier transform of the system (18), (19) iLov+{x,u)-V-G+{X,LO)VV+{X,U>)

= f+{x,Lu) v

+(x,u)\dn = °

+ v0{x)-V-I°+{X,UJ)

(21) (22)

181

For any fixed w £ IR, the sequilinear form a{v+{w),w+(u>)) = /

(23)

iu>v+(x,Lv) • w+(x,co) + G + (a;,w)Vv + (i,w) • V w + ( i , uJ)

Jn L

dx(24)

is a bounded and coercive in HQ(Q). It is easy to verifies that is bounded. In order to obtain the coercivity, we have to prove that the inequality for any fixed w € IR \a(v+{u),v+(u))\

> k(u) \\v+(u>)\\Hi

holds for all v+ £ HQ(Q.) , while k(w) is a positive constant. So that, by means of the definition (23) we have for any UJ £ IR |a(u+(o;),v + (w))| > / Gc{x,u)Vv+(x,u)-Vv+(x,uj)dx Jn

>

k(w)\\v+(w)\\Hi

where fc(w) = inf {GC(X,LJ) , i e S l } , This prove that for any fixed w £ IR, the problem (21 ), (22) admits a solution v+ £ HQ(£1), if the supply F+(u) =

f+(w)+vo-V-I°+(uj)

is an element of H~ (fi). Now, in order to study the behavior of v+ when w —» oo, we consider the Parseval theorem applied to (20) IZO In {v+(x>w) • {iuw+{x,u)

~ w0(x))

—G+(x,u>)Vi>+(x,u;) • Vw+(i,w) > dxdui = — 2n J„vo(x) • wo(x)dx - IZO In \f+(x'UJ)

• w+(x'w)

+ If+(x^)

(25)

• Vw+(x, w ) | dxdoj

It is known that for any vector function / £ H.Q(M.+ ;H~ (fi)) there exists at least a tensor function A £ HQ(M.+ ; L 2 (fi)) such that for all

v e HQ) /»O0

/ Jo

r

/»00

/ f(x,t)-v(x,t)dx= Jn

/ Jo

/>

/ A(x,t) • \7v(x,i)dx Jn

Hence, the equation (25) can be written for w(x,t) = I^oc In \~V+(X'LJ) A

xw

' iuv+(x,u)) + GC(X,UI)VV+(X,OJ) 7

x w

= IZV In( + i ' ) + + ( > )) •

Vw

•

v(x,t) VV+(X,OJ)

\ dxdus

x w

+ (> ) ^ w (26)

182 from which /-oo fnGc(x,u)Vv+(x,uj)

•

^ IZo In G'1(X,U)(A+{X,UJ)

Vv+(x,u))dxdu> + Il(x,u>)

• (A+(x,w)

+ 7°

(x,w))dx)dxdu) < i Z o I n G + i ^ ^ + ^ ^ + t l M )

(A+(x,uj)+I°+(x,cu))dxdw (28) we have from (25)

Finally, when we put w+ = isigncov+

•

IZCIQW V+(X,OJ) • V+(X,UJ) +signujGs(x,uj)Vv+(x,ui) • Vv+(x,u>)dxdu; = /_

Jnisignw(A+(x,ui)

if G'(0) / 0 , we have for w » \GS{UJ)\VV+(X,W)

-VV+(X,UJ)

+ I\(x,u)))

•

(29) Vv+(x,ui)dxduj

1 < (1 + \CJ\)GC(LJ)VV+(X,U)

• VU+(X,W)

then if we p u t in (25) w+(u>) = \u>\ V+(UJ) we obtain X!^, $a\u\Gc(x,w)Vv+{x,u)) • \/v+{x,uj)dxduj -oo Jn A x / ;r w poo = /-oo In, M ( +( >") + + ( . ) ) • Vv+(:r, u;)dxdiv

(30)

from which J_00

JQ \UJ\GC(X,LU)'VV+(X,W)

< J , " ! Ig Gj(x,w)

•

Vv+(x,ui)dxdcJ

M ( A ^ s . ^ ) + 1° (s.u;))

(31)

• |w| (A+(x,w) +/°(a:,a;))c!xdw T h e n by means of the inequalities (27-31) we have

/-oo In M v+ix^) • M 3 ^ ) +Gc(x,w)V«+(i,w) •

Vv+(x,uj)dxdio

1

= IZO Ini + M K ? + ( ^ ) 0M*,") + Il(x,u)) •(A+(x,ui)

+

(32)

I9(x,ij))dxduj

Hence, if the supplies / , / ' G HGQR+; H'1 (fi)) and J 0 , / 0 ' € W£,(IR + ;Z, 2 (ft)) , then we see t h a t the function v belongs to T{Q) and it is a virtual work solution of the problem (18), (19) in the sense of Definition 1.1.

183 4. S e m i - g r o u p t h e o r y In t h e study of t h e differential system (17) by means of t h e semi-group theory, we can rewrite the problem in the form ii(x, t) = v(x,t)

(33)

v(x, t) = V • (GooVu(x, t)+

G'(x, s)Wud{x,

s)ds)

(34)

Jo Ed{x,s)=~Ed(x,s)-E(x,t)

(35)

where Ed(x,s) = Et(x,s) — E(x,t). Now t h e state is given by x = (u,v,Ed) S K,. In t h e works 4 , 9 t h e space K, is a subset of the domain S^G a n d t h e stability theorems was proved with initial conditions in t h e space K . W h e n we use the function I1{T), the system (33-35) is equivalent t o t h e following problem u(x,t)

=

v(x,t)

v(x, t) = V • (GooVw(x, t) + 7*(0) I\X,T)

= -^P{X,T)

-

[G(T)E(x,t)Y

where t h e s t a t e x = {u->v: If) G HI 3 xIR 3 xQ. Now we can use the free energy ip2 a s a n o r m on the space state, then it is easy t o prove t h a t the state space is coincident with the domain S^2 = V D S^,G. T h e o r e m 4 . 1 . Under the hypotheses 1. G'S(OJ) = J0°° G'{s) simosds < 0, w > 0 2. 3 a € IR++; 0 < -aG(x, s) < G(x, s), s € IR+ any solution (\\u(t)\\L2

{u,v,P),

with initial conditions

+ * 2 ( V u ( i ) ) ) < Me-^(\\u(0)\\L2

where M and /x are suitable

xo € Sfat *s such that + *2(Vu(0J)

constants.

References 1. S. Breuer & E.T. Onat, On recoverable work in linear viscoelasticity. Z. Angew. Math. Phys. 15, 13-21 (1964). 2. B.D. Coleman & D.R. Owen, A mathematical foundation for thermodynamics. Arch. Rational Mech. Anal. 54, 1-104 (1974). 3. W.A. Day, The thermodynamics of materials with memory. Materials with Memory, D. Gram ed., Liguori, Napoli, 1979.

184 4. C. M.Dafermos, Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297-308 (1970). 5. L. Deseri & M. Fabrizio & M. Golden, Applications of a new concept of state in linear viscoelasticity, to appear. 6. G. Del Piero & L. Deseri, On the concepts of state and free energy in linear viscoelasticity. Arch. Rational Mech. Anal. 138, 1-35 (1997). 7. G. Del Piero & L. Deseri, On the analytic expression of the free energy in linear viscoelasticity. J. Elasticity, 4 3 , 247-278 (1996). 8. M. Fabrizio, C. Giorgi & A. Morro, Free energies and dissipation properties for systems with memory. Arch. Rational Mech. Anal. 125, 341-373 (1994). 9. M. Fabrizio & B. Lazzari, The domain of dependence inequality and asymptotic stability for a viscoelastic solid, Non linear oscillations, 1, 117-133 (1998). 10. M. Fabrizio & A. Morro, Viscoelastic relaxation functions compatible with thermodynamics. J. Elasticity, 19, 63-75 (1988). 11. M. Fabrizio & A. Morro, Mathematical Problems in Linear Viscoelasticity. SI AM, Philadelphia, 1992. 12. J. M. Golden, Free energies in the frequency domain: the scalar case. Quart. Appl. Math. 58, 127-150 (2000). 13. G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity. Quart. Appl. Math, 60, 153-182 (2002). 14. D. Gram & M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate", Atti Ace. LinceiRend. fis, (8), 83, 201-208 (1989). 15. M. Gurtin & Herrera, On dissipation inequalities and linear viscoelasticity. Quart. Appl. Math. 23, 235-245 (1965). 16. W. Noll, A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48, 1-50 (1972).

C O M P U T A T I O N OF T H E P H A S E F R A C T I O N IN A DISCRETE MODEL OF A P S E U D O E L A S T I C MATERIAL

MARCO FAVRETTI Dipartimento

di Matematica Pura ed Applicata Universita di Padova via Belzoni, 7 - 35131 Padova Italy E-mail: favrettiQmath.unipd.it

We consider a discrete model of a pseudoelastic material formed by a chain of bistable elements (snaps springs). This model, considered in e.g. 1 , 2 , 3 exhibits non monotone stress—strain relation due to the non convexity of the potential energy and it can mimic the hysteretic behaviour observed under cyclic loading in a hard device. We compute all the possible stress states compatible with a given total strain and we apply the Maximum Entropy Principle. Given an arbitrary hysteresis cycle, we are able to infer the evolution of the phase fraction and the (information) entropy during the cycle.

1. Computation of the system equilibria We consider a discrete system formed by a chain of n identical elements (bi-stable springs) whose potential energy is described by the bi-parabolic function w(u) — — (u — sgn(u)a)2

for u ^ 0,

w(0) = —a2

where a > 0, and h > 0 are fixed constants, u is the elongation of the single element (spring), and sgn(u) = u/\u\. See Fig.l a. We start by computing the equilibrium configurations of the system under prescribed total elongation y = ]^Mi. We are therefore lead to the following constrained extremization problem: find the stationary points of n

n

J{uu... ,un) = Y2w(ui)-°C}2ui i=l

-y)-

i=l

By applying the necessary condition for extremization we get r\ T

— = w'(ui) - a = 0 i = l,...,n.

185

(1)

186

Figure 1.

a ) potential energy and /3) applied force, h = a = 1

In the above formula, the Lagrange multiplier a has the physical meaning of applied force (stress) on the clamped ends of the chain. From Fig.l f3 it is clear that for | ha, all the springs have the same elongation, while for — ha < a < ha there are two possible values for u at equilibrium: u+ and u~ := u+ — 2a, with 0 < u+ < 2a,

and

- 2a < u~~ < 0.

(2)

Let I € {0,... ,n} be the number of springs in the chain with u > 0 in a equilibrium configuration of the system. The number z = l/n has the physical meaning of phase fraction of the " + " phase. All the equilibrium solutions (1) can therefore be labeled by / and they are found by imposing the constraint that lu+ + (n-

l)u~ = y.

(3)

Hence, the above argument yields that i) for \y\ > 2an we have the uniform strain solution Ui(y) := - i - 1 , . . . ,n n with I = n for y > 2an and I = 0 for y < —2an; ii) for —2an < y < 2an we have I g {0,... , n} and the solution is, up to a re-ordering of the chain Ui =uf{y),

i = 1 , . . . ,/ and Uj = uj{y),

j = 1 + 1,...

,n

where uf{y):=-+2a( ), u~(y := ^ - 2a-. ' n n n n The set E(y) C { 0 , . . . ,n} of the possible equilibrium solutions is determined again by applying (2)i to uf{y) for fixed y. We have E(y) = {I e{0,...

,n} such that •%- < I < ^- + n}.

187

We denote with N(y) = | E ( j / ) | = n - [ j p ] , where - 2an < y < 2an, (4) la the cardinality of the set S(y) and we extend the domain of N(y) setting N(y) = 1 for \y\ > 2an. 1.0.1. The space of states and observables In a tensile experiment with a specimen constrained to a given total displacement y, the applied force 4> o n the clamped end of the specimen is usually the result of the measurement. The values of cf> for the possible values of I can be computed through the formula
= -y + ha(?——),

I = 0 , . . . n.

(5)

In Fig.2 we have plotted the functions ; and their domain; they describe the multivalued relation between total displacement (elongation) and applied force. For a given y £ D = (—2an,2an), the values 4>i{y), I € E(y)

Figure 2.

Multivalued stress-strain relation n = 10, h = a = 1

represents the possible values of the applied force in the equilibrium configuration ui — (u\,.. . U p t i j ] , ; , . . . ,u~) which are local minimizers of the potential energy. Note that, on the one hand, each possible value for I € E(y) represent a possible value of the phase fraction; on the other hand, the knowledge of 4>i does not give any information on the possible dispositions of u+ and u~ in w;, that is it does not allow to infer the distribution of interfaces between + and — phases. As a first problem, one is interested into resolving the multivocity of the stress-strain relation inside the multivocitiy domain D.

188

Motivated by the classical work of Jaynes 4 , in this paper we propose an approach similar to the standard approach of statistical mechanics. The total elongation y € D is thought as a parameter, the I € S(y), are the possible states in which the system can be found, and the i(y) = w'(uf(y)) are the possible values of the observable . Note that the number of states N(y) also depends on the parameter y. Our task now is to assign a probability pi(y) to each state I given the value F = ((f)) resulting from the measurement, which is to be understood as the average value of <j>. In this way the expected value of the phase fraction, see formula (12) below, is recovered from the experimentally measured hysteresis cycle JP = F(y). 2. Maximum entropy method The problem outlined above has a well known solution (see Jaynes 4 ): the maximum entropy probability distribution is the one maximizing, for fixed y, the entropy (uncertainty) S(p,y) = -

£

PilnPi

(6)

among all the probability distributions satisfying the constraints

£

PI = i,

J€2(!/)

£

PIMV)

= <<£> = F.

(7)

*e£(y)

The (unique) solution to (6), (7) is Pl

M= TaTT-

z

&>y)= £

e

My)

(8)

where Z(X, y) is the so-called partition function. The unknown Lagrange multiplier is to be determined from the constraint (7)2

The partition function for our problem becomes

Z{\ y) = £ e~Mi(v) = e~Xkv) E ^ i=0

a:

=

eA5

> S= ^

i=0

and the associated maximum entropy probability distribution is eWi(y) ai Pi(y) =

(10)

189

where the unknown a is to be determined trough the constraint equation (7)2- This latter is a nonlinear equation hence an explicit solution cannot be found, however a number of features of the solution are known (see Kapur 5 ): N 6) 0 < a < 1 if F > cf>{y) - 5—,

a) a = 0 if F = <£(y), N c) cr = l if F = cj)(y)-S—,

d)l
N F<(j>{y)-8—

Hence pi decreases with i if case b) holds and it increases with i if case d) holds. Moreover pi = \/N if case c) holds and po = l,Pi = 0, i > 1 if case a) holds. 3. Computation of the phase fraction Remember that / € £(?/) is the number of springs in the + phase. Therefore, the maximum entropy probability distribution allows the theoretical computation of the expected phase fraction through the formula n We find {z) F)

^

= l

+

i ^ - ^

-2an
(12)

This formula has a simple graphical interpretation: suppose that F = F(y) is given. Then d{z)/dy = 0 if and only if dF/dy = 2ha/2an, that is the tilted lines in Fig. 2 are lines of constant phase fraction. The same formula (12) is obtained in Huo and Muller6 using the principles of Thermodynamics. 4. Computation of the entropy of the system The entropy of the system in the state (y, F) is given by (6) evaluated at the maximum entropy distribution. It is given by the formula (see Jaynes 4 ) S(F,y) = \nZ(X,y) + XF The formula for the variation of S(y, F) for a variation in the variables y (strain) and F (stress) is well known (Jaynes4) d dS = XdF + —\nZdy dy

d l / , n --f-dy)+p