WASSERSTEIN METRIC AND LARGE–TIME ASYMPTOTICS OF
NONLINEAR DIFFUSION EQUATIONS
J.A. CARRILLO
Departament de Matemàtiques - ICREA, Universitat Autònoma de Barcelona,
E-08193 - Bellaterra Spain
E-mail: [email protected]
G. TOSCANI
Dipartimento di Matematica, Universitá di Pavia, 27100 Pavia Italy
E-mail: [email protected]
We review here various recent applications of Wassertein–type metrics to both
nonlinear partial differential equations and integro–differential equations. Among
others, we can describe the asymptotic behavior of nonlinear friction equations
arising in the kinetic modelling of granular flows, and the growth of the support
in nonlinear diffusion equations of porous medium type. Further examples include
the approximation of nonlinear friction equations by adding viscosity, and the
asymptotic behavior of degenerate convection–diffusion equations.
1
Introduction
In recent years, due to its increasing importance both in the treatment of mass
transportation problems and gradient flows 14,16 , Wasserstein metric became popular fifty years after its introduction into probability theory 13 . While applications
in this area are widely known 17 , further possibilities of application are presently
not well established. The connection between probability theory and nonlinear evolution equations comes through two main properties of solutions, namely positivity
and mass preservation. Suppose we are considering the initial value problem for
the evolution equation
∂f (x, t)
= Q(f )(x, t),
∂t
(x ∈ R, t > 0),
f (x, t = 0) = f0 (x) ≥ 0,
(1)
(x ∈ R)
where Q denotes here an operator acting on f which preserves positivity and mass,
Z
Z
f (x, t) dx =
f0 (x) dx.
(2)
R
R
Then, given a initial datum which is a probability density (nonnegative and with
unit mass), the solution remains a probability density at any subsequent time. Let
F (x) denote the probability distribution induced by the density f (x),
Z x
F (x) =
f (y) dy.
(3)
−∞
Since F is not decreasing, we can define its pseudo inverse function by setting,
for ρ ∈ (0, 1), F −1 (ρ) = inf{x : F (x) > ρ}. Among the metrics which can be
Wasserstein: May 5, 2003
1
defined on the space of probability measures, which metrize the weak convergence
of measures25 , one can consider the Lp -distance of the pseudo inverse functions
µZ
1
dp (F, G) =
[F
−1
−1
(ρ) − G
¶1/p
(ρ)] dρ
,
p
1 ≤ p < ∞.
(4)
0
As we shall see later on, d2 (F, G) is nothing but the Wasserstein metric 20 . Metrics
(4) can be fruitfully used to obtain results on uniqueness and large–time asymptotics
of the solution every time equation (1) for f (x, t) takes a simple form if written in
terms of its pseudo inverse F −1 (x, t),
∂F −1 (ρ, t)
= Q∗ (F −1 )(ρ, t),
∂t
(ρ ∈ (0, 1), t > 0).
(5)
This strategy has been recently applied to nonlinear friction equations arising in the
modelling of granular gases 9,15 , to nonlinear diffusion equations of porous medium
type 6 , and to degenerate convection–diffusion equations 5 .
2
Extremal distributions and Wasserstein–type metrics
Denote by M0 the space of all probability measures in R and by
½
¾
Z
p
Mp = F ∈ M0 :
|x| dF (x) < +∞, p ≥ 0 ,
(6)
R
the space of all Borel probability measures of finite momentum of order p, equipped
with the topology of the weak convergence of the measures. On Mp one can consider
several types of metrics 25 . Among them, an important class is given by the socalled minimal metrics.
Let Π = Π(F, G) be the set of all joint probability distribution functions H on
R2 having F and G as marginals, where F and G have finite positive variances.
Within Π there are joint probability distribution functions H ∗ and H∗ discovered
by Fréchet 8 and Hoeffding 10 which have maximum and minimum correlation. Let
x+ = max{0, x} and x ∧ y = min{x, y}. Then, in Π(F, G) for all (x, y) ∈ R2 ,
H ∗ (x, y) = F (x) ∧ G(y) and
H∗ (x, y) = [F (x) + G(y) − 1]+ .
The extremal distributions can also be characterized in another way, based on certain familiar properties of uniform distributions. If X is a real–valued random
variable with distribution function F , and U is a random variable uniformly distributed on [0, 1], it follows that F −1 (U ) has distribution function F , and, for any
F, G with finite positive variances the pair [F −1 (U ), G−1 (U )] has joint distribution
function H ∗ 24 . Let
Z
|x − y|p dH(x, y).
(7)
Tp (F, G) =
inf
H∈Π(F,G)
1/p
Then Tp metrizes the weak-* topology T W∗ on Mp . For a detailed discussion,
and application of these distances to statistics and information theory, see Vajda 19 .
Wasserstein: May 5, 2003
2
1/2
We remark that T2 is known as the Kantorovich-Wasserstein distance of F and
G 13,20 . In this case
Z
Z
d2 (F, G)2 = T2 (F, G) =
inf
|x − y|2 dH(x, y) = |x − y|2 dH ∗ (x, y). (8)
H∈Π(F,G)
In fact, if the random vector (X, Y ) has joint distribution function H with marginals
F and G, and E(·) denotes mathematical expectation,
Z
£
¤
|x − y|2 dH(x, y) = E (X − Y )2 = E(X 2 ) + E(Y 2 ) − 2E(XY ).
(9)
Since X and Y have marginals F and G respectively, the quantity E(X 2 ) + E(Y 2 )
remains constant for H ∈ Π(F, G). On the other hand, thanks to a result by
Hoeffding 10
Z
E(XY ) − E(X)E(Y ) = [H(x, y) − F (x)G(y)]dxdy ≤
Z
≤
[H ∗ (x, y) − F (x)G(y)]dxdy.
Subtracting the constant quantity 2E(X)E(Y ) on both sides of (9), we obtain (8).
Recalling now that [F −1 (U ), G−1 (U )] has joint distribution function H ∗ 24 , we
conclude that the Wasserstein distance between F and G can be rewritten as the
L2 -distance of the pseudo inverse functions
µZ
d2 (F, G) =
1
[F
−1
−1
(ρ) − G
¶1/2
(ρ)] dρ
.
2
(10)
0
The previous result can be generalized to any convex cost instead of the quadratic
cost as pointed out in 23 . Therefore, dp (F, G)p = Tp (F, G) for any 1 ≤ p < ∞.
1/p
Moreover, Wassertein distances Tp form an increasing sequence in p by Hölder
inequality and thus we can always define the Wasserstein distance for p = ∞ as
d∞ (F, G) = lim Tp1/p = lim dp (F, G) = kF −1 − G−1 kL∞ (0,1) .
p→∞
p→∞
(11)
Let us remark that this ∞-Wasserstein distance has not been used up to our
knowledge to study supports of the probability densities, despite of the clear relation
to them. Let us make several remarks to this respect:
• Assume f (x) is compactly supported and g(x) verifies that d∞ (F, G) < ∞,
then g(x) is necessarily compactly supported.
• Assume f (x) verifies that its ∞-Wasserstein distance with respect to the Delta
Dirac centered at 0 is finite, then f (x) is necessarily compactly supported.
We will make use of these properties in the next section.
Wasserstein: May 5, 2003
3
3
Porous medium equations
Let us consider the Cauchy problem for the porous medium equation
ft = (f m )xx
f (x, 0) = f0 (x) ≥ 0
1
∞
∈ L (R) ∩ L (R)
(12)
(13)
with kf0 kL1 (R) = 1. It is well-known that this Cauchy problem has a unique solution
in a strong sense uniformly bounded in time and space 21,22 . Mass conservation then
implies that the solution f (·, t) is a time–dependent probability density on R. The
discovery of similarity solutions to equation (12), the so-called Barenblatt-Pattle
solutions, and its finite speed of propagation produced a great interest in studying
the time evolution of the support of solutions. By means of heavy analytical tools,
it was possible to show that the long time asymptotics and the velocity of expansion
of the support was given by the similarity solution 21,11,12 . Here we will address
the finite speed of propagation property from a completely different perspective.
First, we will show formally that the flow for this equation is a contraction for all
Wasserstein distances of order 2n. Second, we will deduce from this result the speed
of propagation of the support.
By an induction argument one can easily show that moments of the solution at
any order are bounded if so are for the initial data. Precisely, given
Z
m2n (t) =
x2n f (x, t) dx
R
we have that
d
m2n (t) = 2n(2n − 1) m2n−2 (t),
dt
for any n ≥ 0, and thus, m2n (t) = O(tn ). Now, one can compute the equation
satisfied by the pseudo-inverse distribution F −1 (ρ, t) function obtaining
·µ −1 ¶−m ¸
∂F −1 (ρ, t)
∂
∂F
=−
.
(14)
∂t
∂ρ
∂ρ
It is quite natural now to wonder how the Wassertein distances between two solutions f, g of the equation evolve in time. It is straightforward to derive by formal
integration by parts that
Z
d 1 −1
(F − G−1 )2n dρ =
dt 0
Z 1
h¡
¢−m ¡ −1 ¢−m i
(F −1 − G−1 )2n−2 (Fρ−1 − G−1
2n(2n − 1)
Fρ−1
− Gρ
dρ ≤ 0
ρ )
0
for any n ≥ 1 since the function x−m , m ≥ 1, is decreasing. Note that the boundary
term vanishes due to the compact support of the solutions, which implies
¡
¢−1
¡
¢−1
lim+ Fρ−1
= lim− Fρ−1
= 0.
ρ→0
ρ→1
Therefore, we deduce that
d2n (F (t), G(t)) ≤ d2n (F0 , G0 )
Wasserstein: May 5, 2003
4
for any t ≥ 0 and n ≤ 1 and thus,
d∞ (F (t), G(t)) ≤ d∞ (F0 , G0 )
for any t ≥ 0. This fact has been done rigourously and exploited further for bounded
supported initial data in 6 to show an estimate of the growth of the support of the
solutions.
Theorem 3.1 Given two postive initial densities f0 , g0 for the porous medium
equation in L1 (R) ∩ L∞ (R), with the same mass and compactly supported, then
the solutions f (t), g(t) remain compactly supported and their supports verify the
following estimates:
| inf{supp{f (t)}} − inf{supp{g(t)}}| ≤ d∞ (F0 , G0 )
| sup{supp{f (t)}} − sup{supp{g(t)}}| ≤ d∞ (F0 , G0 )
for any t > 0.
The main consequence of the previous theorem is that we can use as g(t) the
explicit self–similar solution of Barenblatt-Pattle translated in time (see 22,4 ), to
recover that the support of any solution f (t) grows in time as the support of the
Barenblatt, which is known to be of order t1/(m+1) .
There are two simple extensions of the previous results:
1. General nonlinear diffusion equations:
ft = (Φ(f ))xx
f (x, 0) = f0 (x) ≥ 0
∈ L1 (R) ∩ L∞ (R)
with kf0 kL1 (R) = 1. Here, the diffusion function Φ : [0, ∞) −→ [0, ∞) is nondecreasing with Φ(0) = 0. Computing formally the equation satisfied for the
pseudo-inverse distribution function, one finds
· ·µ −1 ¶−1 ¸¸
∂F −1 (ρ, t)
∂
∂F
=−
Φ
,
∂t
∂ρ
∂ρ
and thus, Wassertein distances of exponent 2n between any two solutions of
these equations are non increasing in time.
2. Nonlinear Fokker-Planck equations:
µ
¶
dV
ft =
(t, x)f
+ (Φ(f ))xx
dx
x
f (x, 0) = f0 (x) ≥ 0
∈ L1 (R) ∩ L∞ (R)
where V (t, x) is a smooth strictly uniform (in x and in t) convex potential
2
( ddxV2 (t, x) ≥ λ > 0) and Φ as above. In this case the equation for the pseudoinverse distribution function reads
· ·µ −1 ¶−1 ¸¸
∂F −1 (ρ, t)
dV
∂
∂F
−1
=−
(t, F ) −
Φ
,
∂t
dx
∂ρ
∂ρ
Wasserstein: May 5, 2003
5
and thus, Wassertein distances of exponent 2n between any two solutions of
these equations are decaying exponentially fast as t → ∞ at a rate e−λt .
Moreover, in this case also we have that for any two solutions
d∞ (F (t), G(t)) ≤ d∞ (F0 , G0 ) e−λt
and therefore, supports of solutions are converging to the support of the stationary solution. This argument gives a very simple proof of the finite speed of
propagation and convergence of the support of the solutions when the unique
steady state is compactly supported.
4
Nonlinear friction equations
In this section we illustrate shortly a further application of the Wasserstein metric
to the study of uniqueness and large–time behavior of the equation
·
¸
Z
∂
∂f (v, t)
=
f (v, t) |v − w|γ (v − w)f (w, t) dw ,
(15)
∂t
∂v
R
where the unknown f (·, t) is a time–dependent probability density on R, and γ >
−1. This equation, called nonlinear friction equation, arises in the study of granular
flows, and has been introduced in 18 , in connection with the quasi-elastic limit
of a model Boltzmann equation for rigid spheres with dissipative collisions and
variable coefficient of restitution (see 1,2 for the γ = 1 case). The variable v ∈ R
represents the velocity of particles. The nonlinear friction equations exhibit the
main properties of any kinetic model with dissipative collisions, like conservation
of mass and mean velocity and decay of the temperature. Since the mean velocity
is conserved in time, without loss of generality one can assume as initial values
only probability measures with expectation equal to zero. The equilibrium state is
given by a Dirac mass located at the mean velocity of particles. In addition, these
equations exhibit similarity solutions, which are in general of noticeable importance
to understand the cooling process of the granular flow. The passage to the pseudo
inverse function shows a remarkable simplification. A direct computation shows
that, if the probability density f (v, t) satisfies (15), F −1 (ρ, t) solves
Z 1
∂F −1 (ρ, t)
=−
|F −1 (ρ, t) − F −1 (p, t)|γ (F −1 (ρ, t) − F −1 (p, t)) dp .
(16)
∂t
0
In view of (16), it becomes natural to look for the time evolution of the square of
the Wasserstein metric. This evolution is easily found to satisfy
Z
d 1 −1
[F (ρ) − G−1 (ρ)]2 dρ
dt 0
µ
¶
Z 1
∂ −1
∂ −1
−1
−1
=2
dρ[F (ρ) − G (ρ)]
F (ρ) − G (ρ)
∂t
∂t
0
Z 1
Z 1
=−2
dp
dρ[F −1 (ρ) − G−1 (ρ)] (|H(ρ, p)|γ H(ρ, p) − |K(ρ, p)|γ K(ρ, p)) ,
0
0
Wasserstein: May 5, 2003
(17)
6
where
H(ρ, p) =: F −1 (ρ) − F −1 (p);
K(ρ, p) =: G−1 (ρ) − G−1 (p).
(18)
The functions H and K are deeply linked to Wasserstein metric any time we are
considering distribution functions with the same momentum. In this case in fact
one can rewrite Wasserstein metric in the following form
Z 1
Z
1 1
d2 (F, G)2 =
dρ
dp [|H(ρ, p)| − |K(ρ, p)|]2 .
2 0
0
Owing to this definition, one can easily handle the right–hand side of (17) to
obtain the evolution equation for the Wasserstein metric 9 , which reads
Z
Z 1
d 1
dρ
dp (|H(ρ, p)| − |K(ρ, p)|)2
dt 0
0
Z 1 Z 1
= −2
dρ
dp (|H(ρ, p)| − |K(ρ, p)|)(|H(ρ, p)|1+γ − |K(ρ, p)|1+γ ).
0
0
Equation (19) can be studied in details, to obtain both uniqueness and asymptotic behavior of the solution to (15). While uniqueness is a direct consequence of
the fact that the Wasserstein metric is non increasing, the study of the large–time
behavior requires a deeper analysis. This has been done in 9 , and it can be resumed
as follows
Theorem 4.1 Let F (v, t), G(v, t) ∈ C 1 (R+
t , M2 ) be two solutions to the initial value problem for equation (15), corresponding to the initial distributions
F0 (v), G0 (v) ∈ M2 , respectively. Then, if 0 < γ < 2, the Wasserstein distance
of F (v, t) and G(v, t) is monotonically decreasing with time, and the following decay holds
d
1
d2 (F (t), G(t)) ≤ − γ−1 d2 (F (t), G(t))1+γ .
dt
2
(19)
Moreover, if −1 < γ < 0 and the initial density f0 (v) has bounded support,
Supp(f0 ) = L < ∞, the support of the solution decays to zero in finite time, and
the following bound holds
·
Supp(f (v, t)) ≤ Supp(f0 )|γ| −
1
¸ |γ|
|γ|
2|γ|−1
t
.
(20)
+
Furthermore, if both initial densities f0 (v), g0 (v), have bounded supports, the
Wasserstein distance of F (v, t) and G(v, t) decays to zero at finite time, and the
following time-decay holds
·
d2 (F (t), G(t)) ≤ d2 (F0 , G0 ) 1 −
2|γ|
t
(2L)|γ|
1
¸ 21 ( |γ|
−1)
,
(21)
+
where L denotes the maximum of the supports.
Wasserstein: May 5, 2003
7
5
Nonlinear friction equations with viscosity
An interesting application of Wasserstein metric comes out from the problem of
the numerical approximation of nonlinear friction equations 15 . As explained in the
previous section, the solution to this equation converges to a Dirac mass located
at the mean velocity of particles. The use of spectral methods in this situation is
not allowed. To overcome the problem of the approximation of a Dirac mass, it is
convenient to add a small diffusion. The new equation to be considered is
¸
·
Z
∂f (v, t)
∂2f
∂
=
f (v, t) |v − w|γ (v − w)f (w, t) dw + ² 2 ,
(22)
∂t
∂v
∂v
R
where ² > 0 denotes a small parameter. The solution to (22) has an equilibrium
which is a smooth function 3 , and the new problem can be well approximated by
spectral methods 15 . The question now is to understand if the correction due to
the small diffusion added can be controlled in time. Denote by f² the solution to
equation (22). Coupling the computations of Sections 3 and 4, we obtain that the
pseudo inverse function F²−1 (ρ) satisfies
Z 1
∂F²−1 (ρ, t)
=−
|F²−1 (ρ) − F²−1 (p)|γ ·
∂t
0
"µ
¶−1 #
∂
∂F²−1 (ρ)
−1
−1
·(F² (ρ) − F² (p))dp − ²
.
∂ρ
∂ρ
Let f (t) be the solution to the nonlinear friction (15). Then, owing to Theorem
4.1 we can bound the time evolution of the Wasserstein distance between f and f² ,
obtaining
Z
d
d 1 −1
1
d2 (F² , F )2 =
[F² (ρ) − F −1 (ρ)]2 dρ ≤ − γ−1 d2 (F² (t), F (t))2+γ
dt
dt 0
2
"µ
¶−1 #
Z 1
−1
∂
∂F² (ρ)
−2²
[F²−1 (ρ) − F −1 (ρ)]
dρ.
(23)
∂ρ
∂ρ
0
Integrating by parts the last integral we get
"µ
¶−1 #
Z 1
∂
∂F²−1 (ρ)
−1
−1
−
[F² (ρ) − F (ρ)]
dρ
∂ρ
∂ρ
0
Z
0
1
¢
∂ ¡ −1
[F² (ρ) − F −1 (ρ)]
∂ρ
µ
∂F²−1 (ρ)
∂ρ
¶−1
Z
dρ =
0

1
1 −
∂F −1 (ρ)
∂ρ
∂F²−1 (ρ)
∂ρ
Indeed, the boundary term is equal to zero, due to the fact that
µ −1 ¶−1
∂F² (ρ)
= f² (F²−1 (ρ)) → 0
as ρ → 0+ , 1− .
∂ρ

 dρ ≤ 1.
(24)
Moreover we deduce from (24) the non negativity of the derivatives of F −1 (ρ)
and F²−1 (ρ). Thus we showed that the distance d2 (F² , F ) satisfies the differential
Wasserstein: May 5, 2003
8
inequality
d
1
d2 (F² (t), F (t))2 ≤ − γ−1 d2 (F² (t), G(t))2+γ + 2².
dt
2
(25)
Let D solve
−
1
D2+γ + 2ε = 0
2γ−1
Then, if at some time t = t0 it holds d2 (F² (t0 ), F (t0 )) ≤ D, we conclude from
(25) that the maximum value for d2 (F² (t), F (t)) at any subsequent time t can not
overcome D. We proved 15
Theorem 5.1 Let f (v, t), f² (v, t) ∈ C 1 (R+
t , M2 ) be the solutions to the initial
value problems for equations (15), (22) respectively, corresponding to the same initial distribution f0 (v) ∈ M2 . Then, if 0 < γ < 2, the Wasserstein distance of f (t)
and f² (t) is uniformly bounded in time, and the following bound holds
2/(2+γ)
d2 (f² (t), f (t)) ≤ (2γ ²)
6
(26)
Degenerate convection–diffusion equations
Our final application of these Wasserstein metrics will be done in the case of equations which are perturbation of nonlinear diffusions. For instance, some degenerate
convection-diffusion equations. Let us consider the initial value problem for
us + (uq )y − (um )yy = 0
u(s = 0) = f0 ≥ 0 ∈ L1 (R) ∩ L∞ (R)
(27)
with m ≥ 1, q > m + 1. We set ku0 kL1 (R) = 1. Note that mass conservation holds.
We refer to 7 for the discussion of the well-posedness and asymptotic behavior for
this problem.
In order to study the asymptotic behavior of this equation, it is convenient to
rescale the variables and the solution
y = xR(s),
t=
1
ln (1 + λs) ,
λ
f (x, t) = R(s)u,
where R(s) is defined as
1
R(s) := (λs + 1) λ ,
R(t) = et
and λ = m + 1. We emphasize that this rescaling leaves the initial data f0 =
u0 unchanged and preserves the L1 -norm kf (s)k1 = ku(s)k1 for all s ≥ 0. The
corresponding Cauchy problem for f (x, t) becomes
ft + R(t)−δ (f q )x − (f m )xx − (xf )x = 0
f (t = 0) = f0 ≥ 0
(28)
∈ L1 (R) ∩ L∞ (R)
with δ = q − m − 1.
Wasserstein: May 5, 2003
9
By mimicking the procedure in previous sections we derive the formal equation
verified by the pseudo-inverse distribution function that in this case reads as:
µ −1 ¶1−q
µµ −1 ¶−m ¶
∂F −1 (ρ, t)
∂F
∂
∂F
−1
−δ
= −F + R(t)
−
.
∂t
∂ρ
∂ρ
∂ρ
As shown in 7 , this problem can be considered in this range of exponents as
an asymptotic perturbation for large times of the corresponding porous medium
equation. In fact, a naive approach to previous equation implies that the convection
term should give an asymptotic negligible contribution.
In fact, this intuition can be made precise by using the L1 − L∞ smoothing
effect of these equations. It is indeed possible to prove that the new term coming
from the convection is negligible for large times and in fact, the Wasserstein d2
distance between any two solutions converges exponentially to zero as t → ∞ for
the rescaled equation. We refer for the details to 5 in which the following result is
proved:
Theorem 6.1 Given two initial data f0 , g0 for the convection-diffusion equation
for equation (28), in L1 (R) ∩ L∞ (R) with with the same mass and bounded second
moment, then the Wasserstein distance of the solutions f (t), g(t) verifies
lim d2 (F (t), G(t)) = 0,
t→∞
and in fact, it decays asymptotically as e−αt with α = min( 2δ , 1). This result implies that the the Wasserstein distance of the solutions to equation (27) decays as
s−α/(m+1) .
7
Conclusions
We briefly described here some recent applications of Wasserstein metrics to one–
dimensional nonlinear diffusion equations and to nonlinear equations arising in
kinetic theory of granular gases. The common features of these models is that, in
consequence of mass conservation and positivity of the solutions, they can be easily
reformulated in terms of the pseudo inverse function. These applications enlighten
the power of Wasserstein metrics in deriving uniqueness, asymptotic behavior and
approximations of solutions to nonlinear problems. Similar studies are in progress
to understand if these applications can be easily extended to higher dimensions,
where the representation of these metrics in terms of the pseudo inverse function
fails. Likewise, we believe that, even in one–dimension of space, the use of these
metrics allows for further applications to nonlinear Fokker–Planck type equations.
Acknowledgments
The authors acknowledge financial supports both from the project HYKE, “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis” financed by the European Union (IHP), Contract Number HPRN-CT-2002-00282, and from the bilateral project Azioni integrate Italia–Spagna, “Models of diffusion in partial differential equations for thin films, viscous fluids and semiconductors”. G.T. acknowledges
Wasserstein: May 5, 2003
10
support from the Italian MIUR project “Mathematical Problems of Kinetic Theories”. J.A.C. acknowledges the support from the Spanish DGI-MCYT/FEDER
project BFM2002-01710.
References
1. D. Benedetto, E. Caglioti, M. Pulvirenti, M2AN Math. Model. Numer. Anal.
31, 615 (1997).
2. D. Benedetto, E. Caglioti, J.A. Carrillo, M. Pulvirenti, J. Statist. Phys. 91,
979 (1998).
3. J.A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation
estimates, Revista Mat. Iberoamericana (in press).
4. J.A. Carrillo, G. Toscani, Indiana Univ. Math. J. 49, 113 (2000).
5. J.A. Carrillo, K. Fellner, Long–time asymptotics for degenerate convection–
diffusion equations via entropy methods, (work in preparation) (2003).
6. J.A. Carrillo, M.P. Gualdani, G. Toscani, Wasserstein metrics and the propagation of the support in nonlinear diffusion equations, (work in preparation)
(2003).
7. M. Escobedo, E. Zuazua, J. Funct. Anal. 100, 119 (1991).
8. M. Fréchet, Ann. Univ. Lyon Sect. A 14, 53 (1951).
9. Hailiang Li, G. Toscani, Long–time asymptotics of kinetic models of granular
flows, Arch. Ration. Mech. Anal. (in press) (2003)
10. W. Hoeffding, Schriften des Math. Inst. und des Inst. Angewandte Mathematik der Universität Berlin, 5, 179 (1940).
11. S. Kamin, Israel J. Math., 14, 76 (1973).
12. S. Kamin, Arch. Ration. Mech. Anal., 60, 171 (1976).
13. L. Kantorovich, Dokl. AN SSSR, 37 , 227 (1942).
14. R.J. McCann, Adv. Math. 128, 153 (1997).
15. G. Naldi, L. Pareschi, G. Toscani, M2AN Math. Model. Numer. Anal. 37, 73
(2003).
16. F. Otto, Comm. Partial Differential Equations 26 , 101 (2001).
17. S. Rachev, L. Ruschendorf, Mass transportation problems, Vol. II: Probability
and its applications, Springer Verlag, New York, 1998.
18. G. Toscani, M2AN Math. Model. Numer. Anal. 34 (2000), 1277-1292.
19. I. Vaida, Theory of statistical Inference and Information, Kluwer Academic
Publishers, Dordrecht 1989.
20. L.N. Vasershtein, Probl. Pered. Inform., 5, 64 (1969).
21. J.L. Vázquez, Trans. Amer. Math. Soc., 277, 507 (1983).
22. J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed
in the whole space, J. Evol. Equ. (in press) (2002).
23. C. Villani. Topics in optimal transportation, AMS, Providence, (in press) 2002.
24. W. Whitt, Ann. Statist., 6, 1280 (1976).
25. V.M. Zolotarev, Theory Prob. Appl., 28, 278 (1983).
Wasserstein: May 5, 2003
11