proceedings

Sergiù Klainerman

Damiano Foschi

On bilinear estimates for wave equations

Alexander Shnirelman

On the ${L}^2$-instability and ${L}^2$-controllability of steady flows of an ideal incompressible fluid

Richard Péreyrol

Une inégalité isopérimétrique en courbure négative d'après Christopher B. Croke

Jacques Lafontaine

Luc Rozoy

Courbure scalaire et trous noirs

Philippe G. Ciarlet

Daniel Coutand

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
PHILIPPE G. CIARLET
DANIEL COUTAND
Un théorème d’existence en théorie non linéaire des coques minces
Journées Équations aux dérivées partielles (1999), p. 1-4
<http://www.numdam.org/item?id=JEDP_1999____A1_0>
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Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
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Journees Equations aux derivees partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
A minimization problem arising in nonlinear thin
shell theory
Philippe G. CIARLET Daniel COUTAND
Resume
Les equations bidimensionnelles (Tune coque non lineairement elastique
"en flexion" ont ete recemment justifiees par V. Lods et B. Miara par la
methode des developpements asymptotiques formels appliquee aux equations
de Pelasticite non lineaire tridimensionnelle. Ces equations se mettent sous la
forme d'un probleme de point critique d'une fonctionnelle dont Pintegrande
est une expression quadratique en termes de la difference exacte entre les
tenseurs de courbure des surfaces deformee et non deformee, sur un ensemble
de deformations admissibles qui preservent Ie tenseur metrique de la surface
moyenne et satisfont des conditions aux limites ad hoc.
Nous montrons ici comment Pexistence d'un minimiseur peut etre etablie
1. The two-dimensional equations of a nonlinearly elastic
^flexurar shell.
Greek indices and exponents take their values in the set {1,2}, Latin indices
and exponents in the set {1,2,3}, and the summation convention with respect to
repeated indices and exponents is used. The scalar product, the exterior product,
and the Euclidean norm of a, b € R3
 are denoted a" b, a A b, and | a |.
Let (jj be a bounded open connected subset of R2
, with a Lipschitz-continuous
boundary 7, the set uj being locally on one side of 7. We denote by y = {y^) a generic
point of uj and we let 9a == 9/Qya and 9a^ = (f'lQyoQy^-
Let 6 6 C^cc^R3
) be an injective mapping such that the two vectors a^ == 9^6
are linearly independent at each point ofuJ. We define a normal vector at each point
of the surface 5=0(cD) by
ai A as :
as = i——-——, . | ai A a2 |
The vectors a^ and the vectors a1
 defined by a^ • a1
 = ^ respectively constitute the
covariant and the contravariant basis at each point of the surface S=0{Cj).
We denote by a^ = a^ • QLQ and a^ = a° • a^ rhe covariant and contravariant
components of the metric tensor of 5, and we let a = det(a^). The covariant
components of the curvature tensor of S are defined by ba6 = a3
 • c^a^.
1-1
Using a formal asymptotic analysis of the three-dimensional equations of nonli￾near elasticity with the thickness as the "small" parameter, Lods & Miara [4] have
identified the two-dimensional equations of a nonlinearly elastic ^flexural" shell with
•middle surface S and thickness 2e. constituted by a Saint Venant- Kirchhoff mate￾rial with Lame constants A > 0 and IJL > 0, and clamped over a portion of its lateral
surface.
These two-dimensional equations take the form of variational equations, whose
solutions are the critical points of a functional, the two-dimensional energy of the
shell, over a manifold of admissible displacements, defined in the next theorem.
The unknown is the deformation (p : uj —> R3
 of the middle surface of the shell:
This means that, for each y € uj, (p{y) is the position taken by the point 0(y) of the
undeformed middle surface under the action of the applied forces.
2. The existence theorem.
We show here how the existence of at least one minimizer of the energy of the
shell can be established; for a detailed proof, see Ciarlet & Coutand [3]. Given
any field ^ G H^ci^R3
), the vector fields a^(^) == Qa^ and the functions a^z^) =
8iaW ' ^W are defined almost everywhere in u}. If the two vectors a^(^) are linearly
independent almost everywhere in GJ, the vector field
ai(i/QAa2(^)
^W =
^
/ i „ /,^ . „ /.^ i [ai^Aa^WI
the vector fields a^
 (^?) defined by the relations a^ (^?) • a1
 (^?) = 51
 and the functions
baoW = ^zW ' 9/3^-aW ^re also defined almost everywhere in uj.
Theorem 1 Let 0 € 1^(0;; R3
) (p > 2) be such that the two vectors a^e) == 9a0
are linearly independent at each point of uj. Let there also be given a portion 70
of 7 with length 70 > 0 and a mapping ^o '' 7o ~^ R3
 such that the manifold of
inextensional admissible deformations^ defined by :
$p(^) = {^ € H2
^; R3
); OaftW = CLa0 in UJ\ ^ = ^o OH 7o}
is not empty.
Then, if ^ G $^0^)? the vectors a^(^) = Qa^ are linearly independent almost
everywhere in uj and the functions ba/?(^) belong to L2
^).
Let there be given a continuous linear form L on ^(o^R3
) and let the energy
IF '' ^p{^) —> R &e defined by
IFW =- [ a^^W - b^){b^W - b^^aduj - L^)
o Juj
for all ^ € 4>^(o;)^ where
^ar ^ ^__^^r ^ ^aa^r ^ ^ar ^
denote the contravariant components of the two-dimensional elasticity tensor of the
shell.
1-2
Then there exists at least one (p (E ^p^) such that
IF(^) = mf (^)/F(t/Q.
t/?e^F
3. Sketch of the proof of the existence theorem.
The proof is "naturally55 broken into several steps, of which only the statements
are given (we refer to Ciarlet &: Coutand [3] for complete proofs) . The first four
steps deal with properties of isometric surfaces, while the remaining two deal with
properties of the functional I p over the manifold ^p(uj}.
In the following, the usual norm ofL2
^) is denoted |[ . ||o,^ and the usual norm
of the Sobolev space W^o^R3
) (m > 0, p > 0) is denoted || . \\m^ '
Step 1. For each ^ € ^p^), the vectors (^(i/O are linearly independent a.e. in uj.
Consequently, the functions b^^) are well defined and belong to the space L2
^)
and thus, the functional I p is well defined over the manifold ^p^)'
Step 2. There exists a constant C\ > 0 such that:
I I ^ lkoo^< C^ for all ^ € $p(a;) .
In other words, the manifold ^p((jj) is included in a bounded subset of W^^; R3
).
Step 3. There exists a constant 62 such that:
EllM^llo2
. ^ I I ^\&. + C^ foralliAe^o;).
Q,/?
This means that over ^p(u}) the L^o^-norm of the curvature tensor (which is
well defined by step 1) "uniformly controls" the H^c^R^-norms of the fields in
the manifold $^(0;).
The next step can be easily derived from the compactness of the Sobolev imbed￾ding of H^c^R3
) into H^o^R3
) and the compactness of the trace operator:
Step 4. The manifold ^p(u) is sequentially weakly closed in H^o^R3
) (we let —"
denote weak convergence):
^ G ^F(^), I > 1, and ^ -^ -0 in H2
^; R3
) =^ ^ € $^(0;).
The next steps establish that, while the functional I p is not defined in general
"outside57 the manifold ^p(^}^ it nevertheless possesses the usual properties required
to apply the fundamental theorem of the calculus of variations.
Step 5. The functional I p is sequentially weakly lower semi-continuous over the
manifold ^p^):
^ € ^pM^ ^ 1, and ^ -^ -0 in H^R3
) => IpW < liminfJp^).
I—>00
Step 6. There exists constants €3 > 0 and €4 such that:
IFW > €3 || zp ||j^ +^4, tor all ^ € ^p(^).
The proof is then concluded by combining steps 4, 5 and 6.
1-3
4. Commentary.
(1) While Lods & Miara [4] assumed 6 e C^c^R3
) in order to carry out the
asymptotic analysis that leads to the two-dimensional equations of a "flexuraF shell,
the assumption 0 6 W2
^^; R3
) for some p > 2 is enough to ensure the existence of
a minimizer of the energy over the manifold of admissible deformations.
(2) The existence result immediately extends to the case of a flexural shell sub￾mitted to the boundary conditions "of clamping" tp = <&o and 9^(p = 9y^o along
7o-
(3) The '"interesting" cases covered by this existence theorem are those where
^p(^} does not reduce to a discrete set, i.e. ^p(uj} is a "genuine" manifold. This
is for instance the case if the middle surface of the shell is a portion of a cone
(excluding its vertex), or a portion of a cylinder clamped along a portion of one of
its generatrices (if S is a non-flat cylinder, it may be clamped along portions of two
generatrices).
(4) A comprehensive presentation and analysis of the different plate and shell
problems is given in Ciarlet [1, 2].
References
[1] Ciarlet P.G., 1997. Mathematical Elasticity, Volume II : Theory of Plates, North
Holland, Amsterdam.
[2] Ciarlet P.G., 2000. Mathematical Elasticity, Volume III : Theory of Shells, North
Holland, Amsterdam.
[3] Ciarlet P.G., Coutand D, 1998. An existence theorem for nonlinearly elastic
"flexuraF shells, J . Elasticity, 50, p. 261-277.
[4] Lods V., Miara B., 1998. Nonlinearly elastic shell models: A formal asymptotic
approach. II. The flexural model, Arch. Rational Mech. Anal., 142, p. 355-374.
LABORATOIRE D'ANALYSE NUMERIQUE, UNIVERSITE DE PARIS 6, 4 PLACE JUS￾SIEU, 75005 PARIS
pgcQann.jussieu.fr, coutand@ann.jussieu.fr
1-4

Un théorème d'existence en théorie non linéaire des coques minces

Georgi Popov

Georgi Vodev

Resonances for transparent obstacles

Philippe Martin

Pierre Rouchon

Systèmes plats :  planification et suivi de trajectoires

Nicolas Bergeron

Sur l'homologie et le spectre des variétés hyperboliques

Gilles Carron

Déterminant relatif et la fonction Xi

Mario Pulvirenti

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
MARIO PULVIRENTI
Kinetic and hydrodynamic equations for granular media
Journées Équations aux dérivées partielles (1999), p. 1-8
<http://www.numdam.org/item?id=JEDP_1999____A11_0>
© Journées Équations aux dérivées partielles, 1999, tous droits réservés.
L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www.
math.sciences.univ-nantes.fr/edpa/) implique l’accord avec les conditions générales d’utili￾sation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression sys￾tématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi￾chier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Journees Equations aux derivees partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
Kinetic and Hydrodynamical equations for
one-dimensional granular media
M.Pulvirenti
Abstract
In this lecture I present some open mathematical problems concerning
some PDE arising in the study of one-dimensional models for granular media.
1. The microscopic model.
Roughly speaking a granular media is a collection of particles (grains, seeds,...)
interacting via inelastic collisions. Although each individual can have a complex
internal structure, we shall consider it as the basic microscopic object. The main
purpose in the study of such systems is to outline its collective behavior, attempt￾ing more fruitful descriptions that those arising by the mere consideration of the
equation of the motion for the microscopic system. In other words we try to intro￾duce and investigate reduced descriptions similar to that used in kinetic theory to
derive macroscopic (thermo-dynamical or hydro-dynamical) behaviors of the usual
non-dissipative mechanical systems.
It is surprising how a small inelasticity on the system can change so drastically
its macroscopic behavior. As a consequence we face aspects of the theory which are
absolutely new and challenging.
In the present lecture we review some of the very few rigorous results we are
aware and discuss some open problems.
The simplest model of granular media (see e.g. [?, ?, DLK]...) is constituted
by a system of point particles on the line, interacting inelastically.
More precisely, we consider a system of N particles in R. Let Xi, vi € R be the po￾sition and the velocity of the i-th particle and Z^ = (XN
, V^) == (a:i, 1:1, .. . XN , VN )
be a state of the system. The dynamics is the following. The particles move freely
up to the first instant in which two of them are in the same point. Then they collide
according to the following rule:
v' = v — e(v — v\)
v[ =Vi+£(v-v^), (1)
where i1
', v[ and 2;, z^ are the outgoing and incoming velocity respectively, and
e € [0,1/2] is a parameter measuring the degree of inelasticity of the collision.
XI-1
Note that the collision preserves the total momentum and dissipates the kinetic
energy. Moreover for e == 0 we have the free particle system, while for 5=1/2 we
have the so called sticky particle model in which the particle pair remains attached
after the collision.
The particles are assumed to be identical so that it is completely arbitrary how to
call the particles after a collision. The collision law is useful for many purpose being
a small perturbation of the free motion, however a more intuitive representation
could be:
v
1
 =?;i +e(v -v^)
v[ = v — e(v — i;i),
which is a slight modification of a perfectly elastic collision.
The most relevant qualitative feature of the systems is the possibility of delivering
collapses in a finite time for a suitable values of the main parameters e and N, namely
the particles may suffer infinitely many collisions in a finite time in which some of
them reach the same position with the same velocity. After a collapse the dynamics
is not prolongable anymore.
The first example of collapse has been shown for three particles (see Ref. [CGM]).
They can collapse if and only if e > 2\/3 — 3. Then (see Ref. [BC]) it was proved
the existence of collapses for suitable values of the main parameters.
Since we are mostly interested in the collective behavior of the system rather
than to specific solutions of the dynamical equations, it seems natural to apply the
methods of the kinetic theory to understand the general behavior of the system
under suitable scaling limits.
According to such a prescription we rescale suitably the degree of inelasticity,
as well as the total number of particles (which is assumed to diverge), to obtain
a kinetic equation for the one-particle probability density. Such a (purely formal)
drivation can be done (See [BCP] and [BCPe]). The result is the following kinetic
equation:
{9t+v9^f^v)=-WFf)^ (3)
where:
F(^)- [dv(t>(v-y)f(v.t). (4)
2. The mean-field equation.
Eq.3-4 of the previous section is sometimes called mean-field equation because
it is obtained in the so called quasielastic limit: N —^ oo e —^ 0 Ne —^ \ which is a
sort of a mean field limit.
Such an equation possesses a system of characteristics which allows a rigorous
study of the behavior of the solutions. The existence problem is not an easy task.
It is not hard to show that homogeneous solutions exist globally in time for a fairly
general class of initial conditions. For a general (non-homogeneous) initial datum /o
however, we cannot exclude the appearance of singularities in a finite time. These
(if any) should correspond to the presence of collapses for the particle system in the
regime in which A = NE is sufficiently large. As matter of fact we are not able to
answer (even at heuristic level) to the following questions:
XI-2
i) Do weak solutions exist globally in time for the non-homogeneous equation?
ii) Can occur singularities in a finite time and if it is so, can the solution con￾centrate (that is produce a component of 8 type in the (rc,z;)-space)?
iii) If there are many weak solutions, do exists a criterion which select the physical
relevant solution ?
Notice that the only extra a-priori information we have on the solution, is a
reverse H -theorem, which describes the obvious tendency of the system to concen￾trate. Namely, defining H{f) = f dxdvh{f(x,v,t)), with h convex function, then:
H(f) ^ 0. (5)
Unfortunately inequality 5 does not seem to exclude or guarantee the concentra￾tion feature in a finite time.
The only regularity result we have, (beyond the local existence thorem for clas￾sical solutions which is, of course, rather easy to obtain) concerns the case in which
A is sufficiently small compared with the size of the initial condition /o (m terms of
the Loo norm of /o and its gradient), because the dispersivity of the free flow allow
us to prevent singularities (see Ref. [BCP]).
The features of the dynamics of inelastic particles has been widely investigated in
numerical simulations. Let us briefly describe what happen for a large number N of
particles in a box, or with periodic spatial boundary conditions, initially distributed
homogeneously in x and v. For A == eN sufficiently large the system exhibits an
inelastic collapse in a finite time. If A is small, as the time goes on, the particle
distribution remains spatially homogeneous, while the velocity distribution concen￾trates toward two peaks, at distance of order ]
.. After this, for a larger number
of collisions, the space homogeneity breaks down and correlation between particles
becomes important.
The two-peaks feature can indeed be explained in terms of the behavior of the
homogeneous solution via a careful rigorous asymptotic analysis when t —^ oo (see
[BCP]).
This universal behavior suggests that it can be interesting to consider the evo￾lution of initial data for the kinetic equation concentrated into two hydrodynamic
profiles. This can be easily done at a formal level. Let p\^{x^t} and u\^(x^t) with
HI < ^2 be solutions of
9tpi +^(pi^i) = 0,
QtP2+Qx{p2^2) = 0,
9t(piui) + 9x(piui) = >PiP2^ - ^i)2
,
9t{p2U2) + 9x(p2^) = -Api/92(^2 - ^l)2
.
(6)
Then the following measure in the phase space
f(x,v,t)dxdv == (p^{x,t)6{v - Ui(x,t)) + p^{x,t)8(v - u^(x,t))) dxdv, (7)
is a weak solution of the mean field equation, as it follows by a direct inspection.
Eq.s 6 describe the evolution of two profiles of density and velocity, interacting
via a friction term. Obviously Eq.s 6 can be generalized by considering k profiles of
density and velocity p^ u^ interacting via the friction terms \pipj(ui — Uj}2
.
XI-3
The conservative part of system 6 is not symmetrizable and the reaction term
\p\p<
2.{^2~~^i
\)2
 c^i favorite the occurrence of shocks in presence of concentrations of
the densities. Nevertheless an existence theorem for regular solutions, locally in time,
can be obtained by considering a particular change of variables which transforms
eq. 6 in a linear wave equation. (Work in progress).
3. One-dimensional Boltzmann equation.
Standard arguments of kinetic theory will lead us to consider the following equa￾tion for the unknown / = /(re, v^ t) that is the probability density of a single particle:
9tf{x, v, t) + v9^f(x, J \^1 ^ v,5
 vt) f =
Ifdv , \v - v,\ (^ff^'^ - f{x,v,t)f(x^t)) ,
- ii-,} ?;* == ??i — ——f'?? — v.}^ are the nre-collisional v
(8)
where v* == v + —r(v — z;i), v^ = v\ — —-r(^ — v\)^ are the pre-collisional velocity
and I > 0 is the mean free time inverse of the theory.
How to justify the introduction of this equation on the basis of logically well
founded arguments? One can just say that Eq. 8 is a simplified model of the more
difficult two and three dimensional Boltzmann equation for rarefied gas of inelastic
balls in the so called Boltzmann-Grad limit .
Let us now pass to the mathematical analysis of Eq. 8. The main difference
between Eq. 8 and analogous one-dimensional kinetic equations relative to systems
without energy dissipation is that, here, we do not have an H-theorem as a con￾sequence of the tendency of the system to clusterize. As a technical consequence
we do not know how to construct global solutions for large data. However we can
construct global solutions under suitable smallness assumptions (A small compared
with the Li norm of the initial datum /o- Note the difference with the smallness
assumption for the mean-field equation !) exploiting the one-dimensionality of the
system.
The proof (see Ref. [BCP2]) is similar to that presented in Ref. [A] for conser￾vative systems. In this last case there is a standard way, based on the H-theorem,
which allows us to extend theorems for data with small mass to general situations.
The lack of an H-Theorem however, prevent us to apply these arguments to the
present situation.
It is not worthless to mention that also the Bony's approach to one-dimensional
kinetic models (see Ref. [B]) which do not make use the entropy functional, does
not prevent, in our case, a blow up in a finite time.
Therefore we cannot exclude, for a general initial datum, the possibility of a blow￾up of the solutions in a finite time. We believe that this cannot happen because we
can exclude at least a total concentration of the solution in a finite time, namely
that f(t) -> 6(x,v) weakly when t —> to for some to. See [BCP2].
There is an interesting connection between the one-dimensional Boltzmann equa￾tion and the mean-field equation introduced in Sect. 4.
If we consider the limit I —)• oo, e —> 0, Ie = A, we find that he collision operator
of the Boltzmann equation we are considering in this section, converge to
-X9^f{x,v^)F{x^^t))^
XI-4
where:
F(x, v, t) == dv(v - v)\v - v\f(x, v, t).
It is not difficult to show that this limit holds for solutions of the homogeneous
Boltzmann equation.
4. Heating the system.
The usual hydrodynamical description for rarefied gases does not apply to gran￾ular media. The major point is that, due to the energy dissipation, the system has
no thermal equilibria described by Gibbs states. Moreover it is also well known that
it is not sufficient to pump energy to the system locally, for instance by means of
an hot wall, to thermalize the system (see Ref.[DLK]).
As a consequence we cannot consider local equilibria and a hydrodynamical
picture.
A way to overcome this difficulty is to put the system in a thermal bath at
a constant temperature. As usual the action of the reservoir on the system will
be described by a Fokker-Planck term. The system itself will be considered in the
kinetic picture arising by the quasielastic limit formalized in the mean-field equation.
Therefore the equation which we are interested on reads:
(9t + v9^f = -\9,(Ff) + ^vf + a9^f. (9)
The physical motivation of such a study is to detect the response of the system to
a strong thermalization action. Moreover we want also to investigate the equilibria
of the system in view of a possible definition of a local equilibrium concept which will
be fundamental for establishing hydrodynamical equations. To this purpose we first
consider the homogeneous version of Eq. 9 because locally, in the fast thermalization
scale, the homogeneous regime is dominant.
We shall see that any solution of the homogeneous version of Eq. 9 converges,
as t —> oo, to an equilibrium which is not Maxwellian. Such stationary solution is
described implicitly by an equation of mean-field type, and, for large v, behaves as
exp—C|^|3
. In other words, the inelastic interaction makes the equilibrium distri￾bution more picked around zero with respect to the usual Maxwellian distribution.
We do not know whether this solution is a stable stationary solution for the
general non homogeneous case. Also we do not know whether the thermal bath is
sufficient to prevent the solution to have a collapse in a finite time.
A trivial calculation shows that the (formal) equilibria of Eq. 9 , satisfy the
following equation:
7^) = Le-^^n-^w-^ (ID)
^j
where
Z^ f dve-^^S^^^.
It is possible to show (see Ref. [BCCP]) that there exists a solution to Eq. 10
which minimize the free-energy functional :
ri{f)= ff(v)\ogf(v)dv+-^ It
\v-v\3
f(v)f(v)dvdv+^- [dvv2
/^).
XI-5
A remarkable property of the free energy functional is that it decreases along
the solutions as follows by a direct computation and an integration by parts. As a
consequence one can prove that any solution to Eq. 9 converges in L^(x,v) to the
unique minimum / of the free-energy functional (which satisfies Eq. 10) See Ref
[BCCP] for details.
5. A hydrodynamical picture.
A natural problem arising in the study of a granular medium is that of the
possibility of describing the system in terms of a hydrodynamical picture.
It is well known in kinetic theory that a large dense system is often conveniently
described by hydrodynamic (macroscopic) variables rather than by kinetic (micro￾scopic or mezoscopic) variables. The transition is nothing else than a change of
variables motivated by the scaling properties of the system. If we apply the same
procedure to the granular systems, we face at once a difficulty when considering
the mean-field or the Boltzmann description. The mass and the momentum are
preserved by the collision mechanism so that we hope to write down the continuity
equation, expressing the mass conservation and an equation for the momentum field.
However we do not have a good notion of a local equilibrium: the only steady states
are the delta functions around the mean velocity, so that we get the pressureless
gas. Nevertheless if we pump energy to the system by a thermal bath, we can hope
to extract a reacher hydrodynamical behavior. Thus let us consider the mean-field
equation with the additional Fokker-Planck term:
9if + vQ.f + Q^\F - f3v)f] = aQ^f (11)
Let us remark, preliminary, that if /3 > 0, the momentum is not conserved
anylonger. To have a non-trivial hydrodynamics we assume /3 = 0 (the infinite
temperature case) and study:
9tf + v9^f + [(WFf) - aQ^f] = 0. (12)
The usual space-time scaling and the (formal) Hilbert expansion techniques, lead
us to the following hydrodynamics:
9tp(x^t)+9^pu){x^t)=Q^
9t{pu){x^t)+9^fdvv2
f(x,v^)=0^
that is
Qt{pu){x, t) + 9^pu2
){x^ t) = -9, f dv[v - u(x^ ^p/Cr, ^ t).
The system is not in a closed form because we have not yet used the local
equilibrium hypothesis. In doing this we can compute the pressure term and find
the following apparently familiar system:
9tp(x,t)+9^(pu)(x,t)=0
9t{pu)(x^)+9^pu2
+p)(x^)=0
XI-6
(13)
with p = p7
^ = 1/3.
Eq.s 13 are exactly the equations of the gas dynamics of one-dimensional isen￾tropic gases. The only important difference with the gas dynamic case is that, in
general, 7 ^ 1 while in the granular case 7 = 1/3 < 1.
A mathematical study of Eq.s 13 is not standard. In fact the qualitative prop￾erties of the isentropic gas dynamics change radically if 7 < 1 (see Ref. [BCGP]).
Now the sound speed diverges if the density vanishes, and it is not straightforward
to establish even a local existence theorem for the case of finite mass in the vacuum.
Moreover it is possible to find solutions which collapse in finite time, i.e. the density
concentrates in a delta function.
Finally let us notice that the derivation of the system 13 is formal, and it is not
easy to prove the validity of the Hilbert expansion even for short times.
A parabolic scaling is also possible to get a nonlinear diffusion equation for the
density (0 ^ O):
9tp(x, t) ~
 1^ t dvv^f^x, v, t), (14)
where /o is the local equilibrium. Thanks to the structure of the local equilibrium
solution, it is clear that the above integral is of the form:
dvv2
fo(x,v,t)=^{p(x,t))
for some function zp.
To obtain Eq. 14 we use an Hilbert expansion which makes a rigorous sense at
least term by term (work in progress).
References
[A] L.Arkeryd Existence theorems for certain kinetic equations Arch Rat Mech and
Anal 103 pp 139-149 , (1988)
[B] M.Bony Solutions globales bornees pour Ie modeles discrets de Inequation de
Boltzmann en dimension 1 d'espace Act. Jour. E.D.P. St. Jean de Monts (1987)
[BC] D. Benedetto, E.Caglioti Collapse/or one'dimensional granular media preprint
Roma (1998)
[BCCP] D. Benedetto, E. Caglioti, J.A. Carrillo, M. Pulvirenti A non Maxwellian
equilibrium distribution for one-dimensional granular media, Jour. Stat. Phys.
91 pp 979-990 (1998)
[BCGP] D. Benedetto, E. Caglioti, F. Golse, M. Pulvirenti A Hydrodynamic Model
Arising in the Context of Granular Media, preprint Roma ( 1998)
[BCP] D. Benedetto, E.Caglioti and M. Pulvirenti A Kinetic Equation for Granular
Media, Math. Mod. and Num. An. 31 n. 5, pp 615-641 (1997)
XI-7
[BCPe] D. Benedetto, E.Caglioti and M. Pulvirenti A Kinetic Equation for Granular
Media .'erratum, (to appear)
[BCP2] D. Benedetto, E. Caglioti, M. Pulvirenti A one dimensional Boltzmann
equation with inelastic collisions, (in press on Rend. Sem. Mat. Fis. Milano
1998)
[BM] B. Bernu and R. Mazighi One-dimensional bounce of inelastically colliding
marbles on a wall, Jour. of Phys. A: Math. Gen. 23 pp. 5745-5754 (1990)
[C] S. Campbell Rapid Granular Flows, Ann. Rev. of Fluid Mech. 22 pp. 57-92
(1990)
[CGM] P. Constantin, E. Grossman, M. Mungan Inelastic collisions of three parti￾cles on a line as a two-dimensional billiard, Physica D 83 pp. 409-420 (1995)
[DLK] Y. Du, H. Li, L. P. Kadanoff, Breakdown of Hydrodynamics in One￾Dimensional System of Inelastic Particles, Phys. Rev. Lett. 74 (8), pp. 1268-
1271,(1995)
DIPARTIMENTO DI MATEMATICA, UNIVERSITA DI ROMA LA SAPIENZA, PlAZ￾ZALE A. MORO 2, 1-00082 ROMA, ITALY
pulvirentiQsci.uniromaO.it
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Kinetic and hydrodynamic equations for granular media

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On bilinear estimates for wave equations

On the ${L}^2$ -instability and ${L}^2$ -controllability of steady flows of an ideal incompressible fluid