On a nonlinear, nonlocal parabolic problem with conservation of
mass, mean and variance
A. Tudorascu∗
Department of Mathematics
West Virginia University
Morgantown, WV 26506, USA
[email protected]
M. Wunsch†
Research Institute for Mathematical Sciences (RIMS)
Kyoto 606-8502, Japan
[email protected]
Abstract
In this paper we prove that the steepest descent of certain porous-medium type functionals
with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed)
manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to
the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen
and Gangbo on constrained optimization for steepest descent of the negative Boltzmann
entropy in the Wasserstein space is generalized to porous-medium type functionals. An
interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term
occurring at the same time as its nonlinearity.
1
Introduction
1.1
Overview
In this paper we prove that the steepest descent of the functional
{∫
log F dv
if m = 1
RN F
∫
Sm (F ) :=
1
m
dv if m > 1
m−1 RN F
with
to the quadratic Wasserstein distance over the constrained manifold of nonnegative
( respect
)
L1 RN functions of given mass, mean and variance gives rise to the nonlinear, nonlocal (if
m > 1) parabolic partial differential equation
[
(
)]
∂
v − u(t)
m
m
F (t, v) = θ(t) ∇v · ∇v (F (t, v)) + ∥F (t, ·)∥Lm (RN )
F (t, v) ,
(1)
∂t
θ(t)
∗
AT gratefully acknowledges the support provided by the Department of Mathematics at WVU.
MW gratefully acknowledges the support provided by RIMS. Key words: nonlinear, nonlocal parabolic PDE,
gradient flow, constrained optimization, optimal mass transport, Wasserstein metric. AMS code: 35G25, 46N10,
49J99.
†
1
where u(t), θ(t) denote the mean and the variance of F (t, v), respectively. An interesting feature
of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same
time as its nonlinearity. Nonlocal parabolic problems have been studied by several authors (cf.
[12, 13, 34] and the references therein). Such models find applications in mathematical biology
(e.g., population dynamics) and mathematical physics (e.g., material science), where coefficients
are often found to be nonlocal in nature. For instance, the population density and mobility of
a species might be affected by the crowdedness of a region in their habitat. Moreover, from an
experimentalist point of view, it certainly makes sense to introduce nonlocal quantities, since
measurements are often (local) averages.
We point out, however, that equation (1) differs from those models in various respects: First,
it is set in the whole space RN ; second, it involves a drift term not inherent in the investigations
of the aforementioned references. Third, the drift coefficient, and not the diffusion coefficient,
is nonlocal. Finally, we deal with a nonlinear diffusivity.
1.2
The Jordan-Kinderlehrer-Otto scheme
It is known that the Gibbs distribution
Fs (x) = Z
−1
∫
exp(−Ψ(x)),
Z=
exp(−Ψ(x)) dx
RN
is the unique stationary solution of the Fokker-Planck equation
∂F
= ∇ · (∇Ψ(x)F ) + ∆F,
∂t
F (x, 0) = F 0 (x),
x ∈ RN ,
(2)
as long as the potential Ψ grows sufficiently rapidly so that the partition function Z is finite.
If the Gibbs distribution exists, then it minimizes the free energy functional (potential energy
plus the negative Boltzmann entropy)
∫
L(F ) := S1 (F ) +
ΨF dx.
(3)
RN
over all probability density functions on RN . Even if there is no Gibbs distribution, a free energy
functional, along which solutions to (2) dissipate, can be defined. This is an analogue to the
famous H-Theorem [4] for the Boltzmann equation.
It was the merit of Jordan, Kinderlehrer, and Otto [20] to discover that one can regard
the Fokker-Planck dynamics as a gradient flow of its own free energy. Before illustrating this
observation, recall that the heat equation ((2) with Ψ ≡ 0) can be solved by the classical implicit
variational scheme:
{
Determine Fk minimizing
∫
1
h
2
2
2 ∥Fk−1 − F ∥L2 (RN ) + 2 RN |∇F | dv,
where h > 0 denotes∫ the time step size. Observe that we need at least H 1 (RN )-regularity for the
Dirichlet integral 21 |∇F |2 dx to make sense. On the other hand, the Jordan-Kinderlehrer-Otto
(JKO) time-discrete scheme is implemented as follows:
{
Determine Fk minimizing
(4)
1
h
2
2 W2 (Fk−1 , F ) + 2 S1 (F )
2
in the set of all probability density functions with finite second-order moments. Here, W2 (F, G)
denotes the quadratic Wasserstein distance, which can be thought of as the minimal expectation of |U − V |2 , with U , V being random variables with laws Φ and Γ (corresponding to the
probability densities F and G), respectively.
For the Fokker-Planck equation (2), replace S1 in (4) by L from (3). Under suitable conditions, it can be shown that there exists a unique minimizer of (4). Next, define the timedependent probability density F h by setting F h (t, ·) = Fk if kh ≤ t < (k + 1)h. Then, as the
time-step h tends to zero, it can be shown that Fh converges weakly in L1 (RN ) to the unique
solution F ∈ C ∞ ((0, ∞) × RN ) of (2) – given that the initial datum F0 has finite free energy [20].
What are the advantages of the JKO scheme? First, by the very definition of Wasserstein distances, minimizers must stay probability densities (non-negative, having unit mass) with finite
second moments. Second, less smoothness on the minimizer F (k) is required as L involves only
the function itself, and not any of its derivatives. Third, it establishes a new link between one of
the most essential equations (the Fokker-Planck equation) and a basic quantity (the Boltzmann
entropy) of statistical mechanics. Finally, it is perfectly sensible with respect to the second law
of thermodynamics: The decrease of the (mathematical) entropy leads to a smoothing out of differences in the density (i. e., diffusion), but this process is penalized by the cost of transporting
mass, given by the Wasserstein distance.
1.3
The Constrained JKO Scheme
While [20] studied purely dissipative equations, it was the idea of Carlen and Gangbo [6] to
apply and extend the JKO scheme to a more general set of equations: equations that not only
dissipate, but also satisfy some conservation laws. As a model for the (space-homogeneous)
Boltzmann equation, [6], [7] propose the nonlinear Fokker-Planck equation
[
(
)]
∂
2
2
F (t, v) = θ(t)∇ · e−|v−u(t)| /2θ(t) ∇ e|v−u(t)| /2θ(t) F (t, v) ,
(5)
∂t
where u is the first moment (momentum, bulk velocity), and θ is the dimensionally normalized
second moment (energy, temperature)
∫
∫
1
u :=
vF (v) dv
and
θ :=
|v − u|2 F (v) dv.
N RN
RN
It is, however, easy to see that once the initial datum F0 is a given probability density (thus
having unit mass) with bounded second moment, a smooth solution of (5) can be readily obtained
by solving the linear Fokker-Planck equation
∂
F (t, v) = ∇ · [(v − u0 )F (t, v)] + θ0 ∆F (t, v),
(6)
∂t
where u0 , θ0 are the momentum and temperature of F0 , respectively. This is based on the
observation that smooth solutions of (5) with probability density functions as initial data conserve mass, mean and variance. While the conservation of mass and the mean is natural for the
diffusion equation with initial data having unit mass, these conservative properties nevertheless
constitute the main appeal for studying the kinetic Fokker-Planck equation (5) as a model for
the Boltzmann equation. Existence of smooth solutions for (6) is by now classical (see, e.g.,
[20]).
Let P2ac denote the set of all Borel probability densities with finite second moments. For given
u ∈ RN and θ > 0, we define the constrained manifold
∫
∫
1
ac
2
Eu,θ := {F ∈ P2 :
|v − u| F (v) dv = θ &
vF (v) dv = u}.
N RN
RN
3
We remark
√ that Eu,θ is part of a sphere in the 2-Wasserstein metric centered at the Dirac δu and
of radius N θ. Next, we introduce the entropy
∫
S(F ) :=
β(F (v)) dv
(7)
RN
for a convex function β : R → R+ with super-linear growth [21, 10, 19]. Now we are in a position
to formulate the constrained JKO scheme: For a given time-step h > 0 and Fk−1 ∈ Eu,θ ,


Determine Fk minimizing
(8)
IFk−1 [F ] := 1θ W22 (Fk−1 , F ) + hS(F )


over all F ∈ Eu,θ .
Our aim is to find the optimizer Fk of this constrained minimization problem (8), to calculate
the corresponding Euler-Lagrange equation, and to describe the properties of the minimizer.
The strategy which we will use was first suggested in [6] and rigorously put into practice in
[28]: After proving that the unconstrained minimization problem for IFk−1 [F ] over P2ac increases
second moments (see Theorem 3.4), we
the minimization problem over P2ac for the
∫ consider
2
functional with Lagrange multiplier λ |v − u0 | F (v) dv,
∫
∫
2
IFk−1 ,λ [F ] = W2 (F, F0 ) + h
β(F (v)) dv + hλ
|v − u0 |2 F (v) dv, λ ≥ 0.
RN
RN
The existence and uniqueness of such an optimizer is by now classical (cf. [20]). In view of the
growth of second moments for the problem of minimizing IFk−1 [F ] over P2ac , one can show that
there exists a λk−1 > 0 so that the second moment of the optimizer F (λk−1 ) corresponding to
the unconstrained minimization problem for IFk−1 ,λk−1 agrees with that of Fk−1 :
∫
∫
|v − u0 |2 F (λk−1 ) (v) dv =
|v − u0 |2 Fk−1 (v) dv,
RN
RN
so that F (λk−1 ) is a solution to (8).
We will also demonstrate that the time-dependent densities tend to the solution of a nonlinearly
diffusive Fokker-Planck equation with nonlocal drift.
Let us stress that, even if some of the results (such as the first part of Proposition 3.3 and
Lemma 4.1) leading to the proof of our main theorem hold for general displacement convex
entropy functionals S [21], Theorem 4.3 (concerning the existence and uniqueness of minimizers
for the constrained variational scheme (8)) and Theorem 1.2 are proved solely for entropies
having integrands of the form β(z) = (m − 1)−1 z m , m > 1 or β(z) = z log z.
1.4
Nonlinear Fokker-Planck equations
An important link to the study of filtration equations (cf. [25, 17, 30, 11, 8] and the references
therein) should be mentioned. For porous media (slow diffusion) equations,
∂t u(t, x) = ∆um (t, x), m > 1
(9)
time-dependent intermediate states can be shown to exist: these are the Zel’dovich-BarenblattPattle profiles. Defining λ := N (m − 1) + 2, they are given by the formula
(
)1/(m−1)
u∞ (t, x) = t−N/λ C − k|xt−1/λ |2
,
+
4
k=
m−1
λ,
2m
where C > 0 is a constant normalizing the mass of u∞ to unity. By the time-dependent scaling
{
N
u(t, x) = R(t)− λ v(τ, y)
1
y = xR(t)− λ , τ = λ1 log R(t), R(t) = λt + 1,
the porous media equation (9) is transformed into the nonlinearly diffusive Fokker-Planck equation
∂τ U (τ, y) = ∇ · (∇U m + yU ).
(10)
By the same transformation, the Barenblatt-Pattle profiles become
(
) 1
U ∞ (y) = C − k|y|2 +m−1 ,
(11)
and thus are exactly the stationary states of the aforementioned Fokker-Planck equation (10).
Cast in a variational framework, the steady states (11) minimize the free energy given by
∫
1
Em (U ) = Sm (U ) +
|y|2 U dy,
(12)
2
along which solutions of (10) dissipate with respect to the 2-Wasserstein distance. Numerous
works have dealt with the convergence of the implicit Euler scheme for Em in the Wasserstein
space (e.g., [23], [27], [2]). It has been shown that the time interpolators of the minimizers over
P2ac arising from these schemes converge to the solution of (10).
In [2], the authors present a systematic way of dealing with gradient flows in the Wasserstein
space P2ac (Borel probabilities on RN with finite second moments). However, their methods
do not apply to the constrained submanifold Eu,θ which is not even weakly closed in P2ac (see
[6]). The natural thing to do here is to investigate the case where the underlying manifold is
Eu,θ . For any Gibbs-Boltzmann or power-law type entropy, we shall demonstrate existence and
uniqueness of an optimizer of (8). We shall prove that the scheme provides us with the solution
to the nonlinear, nonlocal Fokker-Planck equation (1). Just as in the case of (5) one can show
that, at least formally, the solutions preserve initial total mass, mean and variance. Thus, again
as in the case of (5), one can formally solve (1) for given initial F0 ∈ P2ac by solving
[
(
)]
v − u0
∂
m
m
F (t, v) = θ0 ∇v · ∇v (F (t, v)) + ∥F (t, ·)∥Lm (RN )
F (t, v) .
(13)
∂t
θ0
Note, however, that not only is this equation nonlinear but also nonlocal. This sets it apart
from (6) in both respects. Assuming temporarily that the second moment N θ(t) in (13) is
time-dependent, let us formally compute its ”evolution”:
∫
d
′
N θ (t) =
|v − u0 |2 F (t, v) dv
dt RN
)]
[
(
∫
v − u0
m
m
F (t, v)
dv
= −2θ(t) v · ∇v (F (t, v)) + ∥F (t, ·)∥Lm (RN )
θ(t)
[
]
∫
m
= 2∥F (t, ·)∥m N θ(t) − v · (v − u0 )F (t, v) dv
[
]
∫
∫
2
= 2∥F (t, ·)∥m
N
θ(t)
−
|v
−
u
|
F
(t,
v)
dv
+
u
·
(v
−
u
)F
(t,
v)
dv
0
0
0
m
[
]
∫
∫
= 2∥F (t, ·)∥m
vF (t, v) dv − u20 F (t, v) dv = 0.
m u0 ·
This shows that, at least formally, solutions to (13) conserve the second moment.
Let u ∈ RN and θ ∈ (0, ∞).
5
Definition 1.1. A weak solution for (13) with initial datum F (0, ·) = F0 ∈ Eu,θ is a function
F (t, v) satisfying
(
)
a. F ∈ L1 ∩Lm (0, T ) × RN ;
[
]
b. ∥F (t, ·)∥m
F (t, ·)(id − u) + ∇v F (t, ·)m ∈ L1 (RN ; RN ) for any t ∈ (0, T );
Lm (RN )
c. For every ζ ∈ Cc∞ ([0, T ) × RN ), we have
∫ T∫
{
[
(
)
F (t, v) ∂t ζ(t, v) − θ∇v F m (t, v)
0
RN
}
+∥F (t, ·)∥Lm (RN ) F (t, v)(v − u)] · ∇v ζ(t, v) dv dt = −
∫
RN
F0 (v)ζ(0, v) dv.
Our main result is:
( )
( )
Theorem 1.2. Let m ≥ 1 and F0 ∈ L1 RN ∩ L∞ RN ∩ P2 (RN ). Then there exists a nonnegative and essentially bounded weak solution in the sense of Definition 1.1 to Equation (13).
The proof follows from Proposition 4.6 and the compactness result in Section 5. The rest of this
paper is organized as follows: Section 2 assumes the existence of a minimizer and deals with its
regularity as well as the Euler-Lagrange equation it satisfies. A discrete comparison principle
for the unconstrained minimization problem is obtained in Section 3, along with a monotonicity
property for the second moments. These will be of help to establish existence of minimizers in
Section 4. Section 5 contains a proof of convergence of the scheme to a weak solution of (13).
Remark 1.3. In the study of intermediate asymptotic profiles for the porous media equation (9)
(cf. [26]), and, more generally, the filtration equation ut = ∆ϕ(u) (cf. [8, 11]), the scaling of the
unique generalized (not necessarily classical) solution by its own second moment
∫
(
)
1
−N/2
−1/2
u(t, x) = θ(t)
G θ(t)
x, τ , τ = log θ(t), θ(t) =
|x|2 u(t, x) dx
(14)
2
RN
has been employed. Notice that this scaling involves, via θ, the solution itself, and that it
corresponds to a projection of u(t, x) of (9) onto the constrained manifold E0,1 . The projection
G should then be a (weak) solution to the nonlocal and nonlinear equation
{
}
∂
G(τ, v)m
G(τ, v) = ∆
+ ∇ · (v G(τ, v)), v = θ(t)−1/2 x.
(15)
∂τ
N ∥G(τ, .)∥m
Lm (RN )
In contrast to (13), the nonlocality in (15) appears at the denominator of the diffusion term,
but we observe that this can be easily remedied by a further time-dependent scaling involving
N
1
∥G(τ, .)∥m
Lm . The authors of [8] are exclusively interested in the long-time asymptotics in L+ (R )
of solutions to the filtration equation and thus do not consider any approximative schemes for
weak solutions to (15) or associated regularity issues. What sets our study apart from [8] is
the scope: since the rescaled solutions in [8] are projections of the original solutions onto the
constrained manifold, it is natural to expect that they can be obtained directly by performing
steepest descent over the constrained manifold. Thus, we solve the problem of existence and
uniqueness for the constrained minimization problem (8) with power-law entropies and show
that the steepest descent over the constrained manifold is, indeed, implementable, furnishing at
once (without further rescaling) a weak solution to (13) (see Definition 1.1). It is interesting,
however, that we do not get the same equation as the one obtained by performing the rescaling
in [8], but further rescaling in terms of the Lm -norm of the solution is necessary for agreement.
6
2
Properties of the constrained minimizer
Here we assume that (8) admits a minimizer, and study its properties.
2.1
Euler-Lagrange Equation
In this section, we follow [6], and, in particular, Theorem 4.1 of [19] to derive the first variation
of the minimizing principle of (8). To this end, let us assume the existence of a minimizer F1
(uniqueness is immediate from the convexity of the constraint manifold and the strict convexity
of the functional to be minimized), and define α(z) := zβ ′ (z) − β(z).
Now consider a vector field ξ : RN → RN satisfying
∫
∫
ξ(v)F1 (v) dv = 0
and
(ξ(v) · v) F1 (v) dv = 0.
(16)
RN
RN
Next, define the flow Tt (v) = v + tξ(v) and the curve of densities G(t) = Tt #F1 , and let G̃(t) be
the projection of the curve G(t) onto the constrained manifold (see Theorem 3.1 of [6]). Note
that there is a flow T̃t corresponding to these projections, so that one can write G̃(t) = T̃t #F1
(see [6], Theorem 4.1). Let us check the entropy term first:
∫
S(G̃(t)) =
β(G̃(t, v)) dv
∫
=
β(G̃(t, T̃t (v))) det ∇T̃t (v) dv
)
∫ (
F1 (v)
=
β
det ∇T̃t (v) dv.
det ∇T̃t (v)
Therefore, we have that
)
(
∫
∫
d d F1 (v)
det ∇T̃t (v) dv
β(G̃(t, v)) dv = β
dt t=0 RN
dt t=0 RN
det ∇T̃t (v)
∫
=−
α(F1 (v))∇ · ξ(v) dv.
RN
We proceed with the analysis of the Wasserstein term. For the sake of completeness, we record
the relevant computations from [6] here.
Denote by ψ the convex function on RN such that ∇ψ is the optimal map from F1 to F0 such
that ∇ψ♯F1 = F0 ), whence, in view of T̃t ♯F1 = G̃(t), ∇ψ ◦ T̃t−1 ♯G̃(t) = F0 . Hence
∫
1
W22 (G̃(t), F0 ) ≤
|∇ψ ◦ T̃t−1 (v) − v|2 G̃(t, v) dv
2 RN
∫
1
=
|∇ψ − T̃t (v)|2 F1 (v) dv
2 RN
∫
2
≤ W2 (F1 , F0 ) − t
(∇ψ − v) · ξF1 (v) dv + o(t).
RN
As a consequence, one has that
W 2 (G̃(t), F0 ) − W22 (F1 , F0 )
lim sup 2
≤
t
t→0+
7
∫
RN
(v − ψ(v))F1 (v) · ξ(v) dv.
Since F1 is a minimizer of the functional I(F ), and because we may replace ξ by −ξ and obtain
the reverse inequality, we get
)
∫ (
∫
F1 (v)
· ξ(v) dv = +h
(∇ψ(v) − v)
α (F1 (v)) ∇ · ξ(v) dv.1
θ
N
N
R
R
Thus we can give the representation
F1
− h∇v α(F1 (v)) = (A + b(u − v)) F1
(17)
θ
for some vector A and some scalar b.
Integrating both sides of (17) in v, we see that A = 0. Next, if we multiply the equation
with u − v and again integrate in v, we see that
∫
W 2 (F1 , F0 )
N θb = + 2
− Nh
α(F1 (v)) dv
θ
RN
(∇ψ(v) − v)
because
∫
RN
(∇ψ(v) − v) · vF1 (v) dv = W22 (F1 , F0 ).
As a consequence of the foregoing deliberations, we have the following proposition.
Proposition 2.1. The Euler-Lagrange equation associated with the constrained minimization
problem (8) reads
[
]
∫
∇v α(F1 (v)) u − v W22 (F1 , F0 )
−
− hθ
α(F1 (w)) dw .
(18)
∇v ψ(v) = v + hθ
F1 (v)
θ
N
RN
Thus we recover the Euler-Lagrange equation of [6], in which case αB (z) = z:
)
[ 2
]
(
W2 (F1 , F0 )
F1
− (u − v)
,
∇ψ(v) = v + hθ∇v ln
M F1
Nθ
where MF1 denotes the isotropic Gaussian density with the same mean and variance as F1 , i.e.
(
)
|v − u|2
−N
2
exp −
.
MF1 (v) = (2πθ)
2θ
Now we shall formally compute the equation which the minimizing probability densities F h
satisfy in the time-step size limit h → 0. Observe that the velocity in each time step is given by
∂
Vh = v−∇ψ(v)
. From the continuity equation ∂t
F + ∇ · (F limh→0 Vh ) = 0 we recover
h
(
)
∂
v − ∇ψ(v)
F (t, v) = −∇ · (F lim Vh ) = −∇v · F (t, v) lim
h→0
h→0
∂t
h
= ∇v · {F (t, v)
∫
1
∇v α(F ) u − v
2
× lim [−hθ
+
(W2 (F0 , F ) − N hθ α(F ) dw)]}
h→0 h
F (t, v)
Nθ
by the Euler-Lagrange equation (18). If W22 (F1 , F0 ) = o(h) as h → 0, we obtain the nonlocal
and nonlinearly diffusive equation
{
}
∂
u−v
F (t, v) = θ∇v · ∇v [α(F (t, v))] − ∥α(F (t, ·))∥L1 (RN )
F (t, v) .
(19)
∂t
θ
Setting β(z) = z log z, we retrieve equation (5), while β(z) = z m /(m − 1) for m > 1 gives (13).
1
Observe that this corrects a wrong sign in the corresponding computation of [6].
8
Remark 2.2. We will justify this passage to the limit rigorously in the last section; more precisely,
we will show that the interpolators F h converge strongly to a weak solution of (13). However,
before demonstrating this convergence, we have to deal with several important properties of the
sequence of minimizers {Fk }k≥1 .
3
3.1
Properties of the unconstrained minimizer
A discrete comparison principle
The following proposition and its proof are inspired from the maximum principles in [1], [27]
(bounded domains), [23] (m = 2) and [28] (β(z) = z log z). As announced in the introduction, we
are only concerned with Boltzmann- or power-law entropies, β(z) = z log z or β(z) = z m /(m −
1), m > 1, or linear combinations thereof. This sort of discrete comparison principle for β(z) =
z m /(m − 1) in the whole space RN is studied here for the first time.
Proposition 3.1. If F0 ∈ P2ac ∩ L∞ (RN ), then the minimizer F1 of IF0 is also bounded in RN
and satisfies
∥F1 ∥∞ ≤ ∥F0 ∥∞
Proof. Take p ∈ Π(F0 , F1 ) to be the optimal transfer plan from F0 to F1 , and assume that
E = {v ∈ RN : F1 (v) > M }, where M = ∥F0 ∥∞ , has strictly positive measure. We will argue
by contradiction.
Observe that p((RN \E) × E) > 0. Otherwise,
∫
M |E| <
F1 (v) dv = p(RN × E) = p(E × E) ≤ p(E × RN )
∫E
=
F0 (v) dv ≤ M |E|,
E
a contradiction. Now let us define
∫
∫
u0 ξ dv :=
ξ(v) dp(v, w)
RN
(RN \E)×E
∫
∫
u1 ξ dv :=
ξ(w) dp(v, w)
RN
(RN \E)×E
for all test functions ξ ∈ C(RN ). Observe that, as a consequence of the definitions, u0 E = 0
and u1 RN \E = 0. One can also check that
0 ≤ u0 ≤ F0 ≤ M and 0 ≤ u1 ≤ F1 .
(20)
We can now define a transference plan ps ∈ Π(F0 , F1,s ), where F1,s = F1 − s(u1 − u0 ) and
0 < s ≪ 1, by
∫
∫
ξ(v, w) dps (v, w) =
ξ(v, w) dp(v, w)
R2N
R2N
∫
+s
(ξ(v, v) − ξ(v, w)) dp(v, w),
(RN \E)×E
the identity holding for all ξ ∈ C(RN × RN ). Taking test functions just depending on the first
variable, the right hand side becomes
∫
ξ(v) F0 (v) dv,
RN
9
showing that the first marginal is indeed F0 . As for the second marginal, we compute
∫
∫
∫
∫
ξ(w)dps (v, w) =
ξ(w) F1 (w) dw + s
u0 ξ dv − s
u1 ξ dw,
RN
RN
RN
RN
which shows that ps ∈ Π(F0 , F1,s ). It follows that
∫
1 2
W2 (F0 , F1,s ) +
β(F1,s (v)) dv
h
RN
∫
s
1
[β(F1,s (v)) − β(F1 (v))] dv −
≤ IF0 [F1 ] +
h
h
RN
∫
|v − w|2 dp(v, w) (21)
(RN \E)×E
the inequality holding since we defined ps to be just a transport plan, not necessarily the optimal
one, between F0 and F1,s .
For β(z) = z log z, one finishes the proof as in [28]. Let us now conclude for the porousmedium case. Due to the convexity of β,
∫
(m − 1)
RN
∫
m−1
β(F1,s (v)) − β(F1 (v)) dv ≤ m
(F1,s − F1 )F1,s
dv
N
R
∫
∫
= −s
u1 (F1 − su1 )m−1 dv + s
u0 (F1 + su0 )m−1 dv. (22)
RN \E
E
If 1 < m ≤ 2, we have
(F1 − su1 )m−1 + sm−1 um−1
≥ F1m−1 and F1m−1 + sm−1 um−1
≥ (F1 + su0 )m−1 ,
1
0
which gives that the right hand side of (22) is bounded from above by
∫
∫
( m
)
sm
u1 + u m
dv
−
s
(u1 − u0 )F1m−1 dv.
0
RN
RN
Due to (20), both u0 and u1 are in Lm (RN ). Also, (20) and the fact that u1 − u0 has zero
average imply
∫
RN
∫
(u1 −
u0 )F1m−1
dv
=
RN
(u1 − u0 )(F1m−1 − M m−1 )
dv
≥
0,
which clearly contradicts the optimality of F1 (the right hand side of (21) will drop below IF0 [F1 ]
for 0 < s ≪ 1), so that E = ∅, or ∥F1 ∥∞ ≤ M .
If m > 2 one reaches the same conclusion by noting that in this case the convexity of x → xm−1
implies
−u1 (F1 − su1 )m−1 ≤ −u1 F1m−1 + (m − 1)su21 F1m−2
≤ −u1 F1m−1 + (m − 1)sF1m
on E, and
u0 (F1 + su0 )m−1 ≤ u0 F1m−1 + (m − 1)su20 (F1 + su0 )m−2
≤ u0 F1m−1 + (m − 1)s(2M )m−2 M F0
on RN \E if 0 < s < 1. We then apply (22) and conclude as in the case 1 < m ≤ 2.
10
We obtained a maximum principle for the unconstrained problem of minimizing
∫
2
IF0 [F ] = W2 (F, F0 ) + h
β(F (v)) dv.
(23)
RN
under the assumption that F0 ∈ P2ac ∩ L∞ (RN ). However, what we will actually need later in
the proof of existence of minimizers is a comparison principle for
∫
∫
IF0 ,λ [F ] = W22 (F, F0 ) + h
β(F (v)) dv + hλ
|v|2 F (v) dv
(24)
RN
RN
for some λ > 0 and h > 0.
Corollary 3.2. Let F0 ∈ P2ac ∩ L∞ (RN ), λ > 0, h > 0 and denote by F (λ) the minimizer of
IF0 ,λ over P2ac . Then, F (λ) ∈ P2ac ∩ L∞ (RN ) and
∥F (λ) ∥∞ ≤ (1 + λh)N ∥F0 ∥∞ .
(25)
Proof. Denote by ε = (1 + λh)−1/2 . Let Gε (v) := ε−N F0 (v/ε), and for any density F , introduce
Fε (v) := εN F (ε v). Finally, set βε (s) := εN β(ε−N s). Then
∫
2
IF0 ,λ [F ] = W2 (Gε , Fε ) + h
βε (Fε (v)) dv + (1 − ε2 ) M2 (F0 ) =
RN
=: JGε ,λ [Fε ] + (1 − ε2 ) M2 (F0 ),
where M2 (F0 ) is the second moment of F0 . Since the functional JGε ,λ does not, as IF0 [F ] in
(23), contain the second moment of the minimizer, we conclude that there holds a maximum
principle for Fε . Rescaling back to the minimizer F (λ) of (24), we get (25).
3.2
Regularity of the unconstrained minimizer
It was one of the crucial observations of [28] that optimizers of particular variational problems
in the 2-Wasserstein space actually enjoy certain regularity properties. We motivate this finding
and its generalization below.
Consider the nonlinearly diffusive Fokker-Planck equation
Ft (t, v) = ∇v · [∇v α(F (t, v)) + 2λhF (t, v) v] .
Recall [16] that the enthalpy h = h(s) is defined as
∫ s ′
α (r)
dr.
h(s) =
r
1
(26)
(27)
It turns out that the unique equilibrium state of (26) minimizing the free energy of (26) given
by
∫
∫
β(F (v)) dv + λh
|v|2 F (v) dv
(28)
RN
can be written down as
RN
(
)
λh 2
Fλ (v) = g c −
|v| ,
2
(29)
where g is the generalized inverse of the enthalpy h (cf. [9]). The constant c is determined by
the mass constraint on F : namely, c is such that
(
)
∫
∫
λh 2
Fλ (v) dv =
g c−
|v|
dv = 1.
(30)
2
RN
RN
11
For example, in the case of linear diffusion we get g(σ) = exp(σ). Moreover, if β(z) =
1
1
m
m−1 z ,
m−1 (for porous media-type nonlinear diffusions). For later
m > 1, then g(σ) = ((1 + σ m−1
m )+ )
reference, we give the following regularity result. Note that for the local Lipschitz property we
require α(z) = z m (for some m > 1), F0 ∈ L∞ (RN ) and Φ(v) := λ|v|2 for some λ > 0. These
were not required in [28] where m = 1, but we have not been able to reproduce the proof in [28]
in the case m > 1.
Proposition 3.3. For every h > 0 and every F0 ∈ P2ac for which S(F0 ) < ∞, the minimizer
F ∗ over P2ac of
∫
2
IF0 ;Φ [F ] := W2 (F0 , F ) + h
Φ(v) F (v) dv + hS(F )
(31)
RN
for a (potential
Φ satisfying |∇Φ(v)| ≤ C(1 + Φ(v)) for any v ∈ RN is such that α ◦ F ∗ ∈
)
W 1,1 RN . Furthermore, if α(z) = z m (for some m > 1), F0 ∈ L∞ (RN ) and Φ(v) := λ|v|2 for
1,∞
some λ ≥ 0, then α ◦ F ∗ ∈ Wloc
(RN ).
Sketch of Proof. As before, let p ∈ Π(F0 , F ∗ ) be optimal in the Kantorovich problem [32]. According to [19] and [20],
∫
∫
(w − v) · ξ(w) dp(v, w) − h
α(F ∗ (v))[∇ · ξ(v) − ∇Φ(v) · ξ(v)] dv = 0
(32)
R2N
RN
for all test functions ξ ∈ C(RN ; RN ). Moreover,
∫
∫
φ(v, w) dp(v, w) =
R2N
F ∗ (w)φ(∇ψ(w), w) dw
RN
for all test functions φ ∈ C(RN × RN ) of at most quadratic growth. We apply this to φ(v, w) =
v · ξ(w) and get from equation (32)
∫
∗
α(F (v))∇ · ξ(v) dv
∫
=
RN
−
1
F ∗ (v){ [∇ψ(v) − v] − ∇Φ(v)} · ξ(v) dv
h
∫R
N
=:
RN
U (v) · ξ(v) dv
for all ξ ∈ Cc∞ (RN ; RN ). It follows from the growth condition on Φ and the minimizing property
of F ∗ that α ◦ F ∗ ∈ W 1,1 (RN ). Proving local Lipschitz regularity under the extra-assumptions
is somewhat trivial. Indeed, according to Corollary 3.2, F ∗ (and, thus, α ◦ F ∗ ) is essentially
bounded in RN . But ψ is convex and proper, which gives that U is locally essentially bounded
in RN .
3.3
Monotonicity of second-order moments
For the subsequent discussion, we assume, without loss of generality, that u = 0 and θ = 1.
We follow [28] (simply set β(z) = z log z and therefore α(z) = z for the special case treated
therein), generalizing the result for any α derived from the entropies introduced before. The
main theorem of this section reads as follows.
Theorem 3.4. Let α(z) = z m for some m ≥ 1. For every F0 ∈ P2ac ∩ L∞ (RN ) and every
time-step h > 0, the minimizer
F1 := argminF ∈P2ac IF0 [F ]
12
satisfies
∫
∫
|v| F1 (v) dv ≥ N h
∫
2
RN
α(F (v)) dv +
RN
RN
|v|2 F0 (v) dv
(33)
Before giving the proof of the theorem, we state a helpful lemma [2], [28].
1,∞
Lemma 3.5. Let Ψ : RN → R be convex and f ∈ L1 (RN ) ∩ Wloc
(RN ) be nonnegative (of
1
N
positive total mass). Also, suppose that |∇Ψ|f ∈ L (R ) and ∇Ψ · ∇f ∈ L1 (RN ). Then
∫
∇Ψ · ∇f dv ≤ 0.
(34)
RN
For a proof, the reader may consult [2], Lemma 10.4.5.
Proof of the Theorem. Let f := α ◦ F1 . We subtract the variances of F1 and F0 , use the pushforward property of ∇ψ♯ F1 = F0 , and employ the Euler-Lagrange equation corresponding to
the unconstrained minimization problem for IF0 [F ] over P2ac , cf. [22, 19]:
∫
∫
∫
[ 2
]
2
2
|v| F1 (v) dv −
|v| F0 (v) dv =
|v| − |∇ψ(v)|2 F0 (v) dv
∫
= − [∇ψ(v) + v] · [∇ψ(v) − v]F1 (v) dv
∫
∫
= −h ∇ψ(v) · ∇f dv − h v · ∇f dv
The last two integrals are legitimate. Indeed, we know that both F0 , F1 ∈ P2ac and that the
Euler-Lagrange equation reads h∇α(F1 ) = [∇ψ − id] F1 . It easily follows that id · ∇f and
∇ψ · ∇f both lie in L1 (RN ).
Since F0 is essentially bounded (with essup norm denoted by M ), we use Proposition 3.1 to
conclude 0 ≤ f ≤ M m−1 F1 . Thus, since F1 ∈ P2ac , we get f, |id|f ∈ L1 (RN ). Along with
id · ∇f ∈ L1 (RN ), this implies (exactly as in [28])
∫
∫
v · ∇f dv = −N f dv.
∫
The proof is complete if we show that ∇ψ · ∇f ≤ 0. Note that the second statement in
1,∞
Proposition 3.3 gives f ∈ Wloc
(RN ). We have already seen above that f, ∇ψ · ∇f ∈ L1 (RN ).
Thus, in order to conclude, it suffices to prove |∇ψ|f ∈ L1 (RN ). This follows immediately from
0 ≤ f ≤ M m−1 F1 and ∇ψ# F1 = F0 ∈ P2ac , F1 ∈ P2ac .
4
Existence of minimizers for the constrained problem
Before stating the main theorem, some preliminaries are necessary. We first give a lemma
resembling Lemmata 2 and 3 of [28].
( )
Lemma 4.1. Let F0 ∈ P2ac RN and h > 0 be given. For every λ ≥ 0, denote by F (λ) the
unique minimizer of
∫
∫
2
IF0 ,λ (F ) := W2 (F0 , F ) + h
β(F (v)) dv + λh
|v|2 F (v) dv
(35)
RN
RN
P2ac ,
over
where β is such that the corresponding entropy S is displacement convex. Then there
exists a λ1 > 0 such that
∫
∫
2 (λ1 )
|v| F
(v) dv ≤ |v|2 F0 (v) dv.
(36)
13
Proof. Let F (λ) be the minimizer of IF0 ,λ [F ]. We use the unconstrained minimizer F1 of IF0 [F ]
to write
[
]
IF0 F (λ) ≥ IF0 [F1 ] =: m ∀ λ > 0.
This inequality leads to
∫
h |v|2 F (λ) (v) dv =
≤
1
{IF0 ,λ − IF0 } (F (λ) )
λ
(
)
]
1[
IF0 ,λ F (λ) − m =: n(λ).
λ
We contend that lim supλ→∞ n(λ) = 0. Indeed, it is straightforward to see that
[
]
IF0 ,λ F (λ) ≤ IF0 ,λ [Fλ ].
(37)
Furthermore,
∫
+ hS(Fλ ) + λh |v|2 Fλ (v) dv
)
∫ (
λh 2
2
= W2 (F0 , Fλ ) + h β g(c −
|v| ) dv
2
(
)
∫
λh 2
2
+λh |v| g c −
|v| dv
2
∫
∫
( (
))
2
−N/2
≤ 2 |v| F0 (v) dv + hλ
β g c − h/2 |v|2 dv
∫
(
)
2 + hλ
+ (N +2)/2 |v|2 g c − h/2 |v|2 dv.
λ
IF0 ,λ [Fλ ] =
W22 (F0 , Fλ )
This proves that n(λ) tends to 0 as λ becomes infinitely large. And this fact, in turn, asserts
the statement of our lemma.
For the next lemma, the comparison principle of Section 3.1 is essential.
Lemma 4.2. Let F0 ∈ P2ac ∩ L∞ (RN ) and h > 0 be given. Then there is a λ0 > 0 such that
∫
∫
2 (λ0 )
|v| F
(v) dv =
|v|2 F0 (v) dv.
(38)
RN
RN
Proof. We define the function φ : [0, ∞) → R by
∫
∫
2 (λ)
φ(λ) :=
|v| F (v) dv −
RN
RN
|v|2 F0 (v) dv.
From Theorem 3.4 we know that φ(0) is strictly positive, and from the last Lemma we conclude
that there is a λ1 > 0 such that φ(λ1 ) is non-positive. Let us assume that φ(λ1 ) < 0 (otherwise
we are done, and λ1 = λ0 ). The proof thus reduces to demonstrating continuity of φ on the
interval (0, λ1 ); its zero will be assumed at λ0 .
Fix λ∗ ∈ [0, λ1 ]. The minimizing property of F (λ) implies
IF0 ,λ [F (λ) ] ≤ IF0 ;λ [F ] ∀F ∈ P2ac ∩ L∞ (RN ).
(39)
If we let λ → λ∗ , we deduce from the superlinearity of β that there exists a F ∗ ∈ P2ac such that
F (λ) ⇀ F ∗ weakly in L1 (RN ) as λ → λ∗
14
up to a subsequence. By a lower-semicontinuity argument [20],
W22 (F0 , F ∗ ) ≤ lim inf
W22 (F0 , F (λ) )
λ→λ∗
∫
∫
∗
β(F (v)) dv ≤ lim inf
β(F (λ) (v)) dv.
∗
λ→λ
RN
RN
From (39), we conclude that F ∗ minimizes IF0 ,λ∗ over P2ac . However, the unique minimizer is
∗
∗
∗
F (λ ) , so that F ∗ ≡ F (λ ) , and the weak convergence F (λ) ⇀ F (λ ) is true for the whole range
of parameters [λ → λ∗ .
]
Now let fs := (1 − s) idRN + s∇ψ (λ) ♯ F (λ) be displacement interpolant between f0 = F (λ) and
f1 = F0 , where ∇ψ (λ) is the Brenier map from F (λ) to F0 . It is well-known that
∫
s ∈ [0, 1] → M4 (fs ) =
|(1 − s)v + s∇ψ (λ) (v)|4 F (λ) (v) dv
(40)
RN
is convex. Thus
(
)
M4 (F0 ) − M4 F (λ)
≥
d
ds
∫
∫
RN
|v|4 fs (v) dv s=0
d
|(1 − s)v + s∇ψ (λ) (v)|4 F (λ) (v) dv
ds
s=0
∫
]
}
( 2 ) {[
|v| v · ∇ψ (λ) (v) − v F (λ) dv.
= 4
=
(41)
( )
We know that for the unconstrained problem with a potential Φ, the minimizer F1 over L1+ RN
satisfies the Euler-Lagrange equation
{
}
1
∇α(F1 ) =
[∇ψ(v) − v] − ∇Φ(v) F1 (v)
(42)
h
a. e. in RN [22, 19]. Recall from Proposition 3.3 that α ◦ F1 ∈ W 1,1 (RN ) and set Φ(v) = λ|v|2 .
Then substituting (42) into the moments’ estimate (41), we get
∫
(
)
}
(
)
( 2 ) {
(λ)
|v| v · ∇α F (λ) + 2λvF (λ) (v) dv
M4 F
− M4 (F0 ) ≤ −4h
∫
(
)
(
)
(λ)
= −8λhM4 F
+ 4(N + 2)h |v|2 α F (λ) dv.
Corollary 3.2 applies to F (λ) to yield
( (
))
M2 α F (λ)
≤ (1 + λ1 h)N (m−1) ∥F0 ∥m−1
M2 (F (λ) ).
∞
Thus the fourth-order moments M4 (F (λ) ) are uniformly bounded, whence
∫
C
|v|2 F (λ) (v) dv ≤ 2
R
|v|≥R
for some positive constant C independent of λ. So for any R > 0, one has
∫
∫
2 (λ)
2 ∗
(λ)
∗
|v| F (v) dv −
|v| F (v) dv |M2 (F ) − M2 (F )| ≤ |v|<R
|v|<R
∫
∫
+
|v|2 F (λ) (v) dv +
|v|2 F ∗ (v) dv.
|v|≥R
15
|v|≥R
(43)
As R ↗ ∞, the last integral on the right-hand side obviously vanishes. Together with (43) and
the weak L1 -convergence F (λ) ⇀ F ∗ , we obtain convergence of the second-order moments, hence
continuity of φ.
As we shall presently see from Theorem 4.3, the λ0 just calculated above gives exactly the
minimizer on the constrained manifold:
F (λ0 ) = arg min
IF0 ,λ0 [F ] = arg min IF0 [F ] = F1 .
ac
P2
E0,1
(44)
Theorem 4.3. Let the initial datum F0 ∈ Eu,θ ∩ L∞ (RN ). Then there exists a unique minimizer
F1 ∈ E ∩ L∞ (RN ) of the constrained minimization problem
F1 = argminF ∈Eu,θ I(F ).
(45)
Proof. Uniqueness is a consequence of the strict displacement convexity of the entropy S on
Eu,θ .
Without loss of generality, we prove existence for E = E0,1 only. Let us write down the
minimizing property of F (λ0 ) :
∫
) ∫
1 2(
W2 F0 , F (λ0 ) +
β(F (λ0 ) )(v) dv + λ0
|v|2 F (λ0 ) (v) dv
h
N
N
R
R
∫
∫
1 2
2
≤
W (F0 , F ) + β(F (v)) dv + λ0 |v| F (v) dv,
h 2
true for all F ∈ P2ac . In particular,
∫
∫
1 2
1 2
(λ0 )
(λ0 )
W (F0 , F
)+
β(F
(v)) dv ≤ W2 (F0 , F ) + β(F (v)) dv
h 2
h
RN
∫
∫
for all F ∈ P2ac such that |v|2 F dv = 1 = |v|2 F0 dv. Let us write F1 = F (λ0 ) . Since
α ◦ F1 ∈ W 1,1 (RN ) (see Proposition 3.3), it follows that
∫
∂(α ◦ F1 )
dv = 0, i = 1, ..., N.
∂vi
Integrating the Euler-Lagrange equation for the unconstrained problem with a potential λ0 |v|2
(cf. (42)) componentwise, the last identity implies
∫
∫
∂ψ
(2λh + 1) vi F1 (v) dv −
(v)F1 (v) dv = 0,
∂vi
which gives the desired result, as ∇ψ♯F1 = F0 :
∫
∫
∂ψ
F1 (v) dv = vi F0 (v) dv = 0.
∂vi
The next proposition shows that the number λ0 from (44) can be computed explicitly and that
the sequence (λk )k corresponding to the minimizing scheme is bounded uniformly.
16
Proposition 4.4. Consider the sequence {Fk }k consisting of probability density functions Fk
minimizing, for each k ∈ N,
∫
∫
h
IFk−1 [F ] = W22 (Fk−1 , F ) +
F m (v) dv + hλk−1 |v|2 F (v) dv,
m−1
∫ 2
∫
where λk−1 is chosen so that |v| Fk−1 (v)dv = |v|2 Fk (v)dv.2 Then
a. λk =
1
2
2N h W2 (Fk , Fk+1 )
b. λk ≤
1
2
(
1+
1
N (m−1)
)
+ 12 ∥Fk+1 ∥m
, and
Lm (RN )
∥F0 ∥m
.
Lm (RN )
Proof. Using the notation of (44), the Euler-Lagrange equation for the constrained optimization
problem of finding minE0,1 IFk [F ] reads
[
(∇ψ(v) − v)Fk+1 (v) − h∇[Fk+1 (v) ]
m
=
1 2
W (Fk , Fk+1 ) + h∥Fk+1 ∥m
m
N 2
]
vFk+1 (v),
while the Euler-Lagrange equation for the unconstrained minimization problem for IFk ,λk [F ]
over P2ac is
h∇[F (λk ) (v)m ] = [(∇ψ(v) − v) − 2hλk v] F (λk ) (v).
Our knowledge that F (λk ) ≡ Fk+1 allows us to plug the latter identity into the former, so that
we get
(∇ψ(v) − v)Fk+1 (v) − (∇ψ(v) − v)Fk+1 (v) + 2λk hvFk+1
[
]
1 2
m
=
W (Fk , Fk+1 ) + h∥Fk+1 ∥m vFk+1 (v),
N 2
equality holding for any v ∈ RN . Thus
[
]
1 2
m
2λk hFk+1 (v) =
W (Fk , Fk+1 ) + h∥Fk+1 ∥m Fk+1 (v).
N 2
With Fk+1 not vanishing identically, there exists a vector v0 ∈ RN such that Fk+1 (v0 ) ̸= 0, so
that we may infer
1
2λk h = W22 (Fk , Fk+1 ) + h∥Fk+1 ∥m
m,
N
which gives the first statement of the proposition. As for the second, remember that for the
constrained minimizer Fk+1
∫
∫
1 2
1
1
m
W (Fk , Fk+1 ) +
Fk+1 dv ≤
Fkm dv,
h 2
m−1
m−1
so that, in particular,
1
1
1
W22 (Fk , Fk+1 ) +
∥Fk+1 ∥m
∥Fk ∥m
m ≤
m.
2N h
2N (m − 1)
2N (m − 1)
As a consequence, we can bound λk as follows:
λk ≤
1
1
m
∥Fk ∥m
m + ∥Fk ∥m .
2N (m − 1)
2
This estimate, together with the decay of the free energy, entails the second statement.
2
This is possible due to Lemma 4.2.
17
Remark 4.5. As we have seen in the previous proposition, λk ≤ c∥F0 ∥m
=: c̃, where c
Lm (RN )
depends on m and N exclusively. Thus we have, for any k ≥ 0,
∥Fk+1 ∥L∞ (RN ) ≤ (1 + λk h)N ∥Fk ∥L∞ (RN )
≤ (1 + c̃h)N ∥Fk ∥L∞ (RN )
≤ (1 + c̃h)2N ∥Fk−1 ∥L∞ (RN )
≤ (1 + c̃h)(k+1)N ∥F0 ∥L∞ (RN ) .
Note that the time-step size is given by h =
T
n,
and that, of course, k + 1 ≤ n. Thus
T
∥Fk+1 ∥L∞ (RN ) ≤ (1 + c̃h) h N ∥F0 ∥L∞ (RN )
≤ ec̃T N ∥F0 ∥L∞ (RN ) .
This demonstrates the existence of an L∞ (RN )-bound (uniform in k) for any minimizer Fk ,
k = 1, ..., n. We will frequently refer to this result in the proof of convergence of the constrained
JKO scheme, where we shall use the notation
K := ec̃ T N .
4.1
(46)
The approximate Euler equation
Let α(z) = z m for some m ≥ 1. Furthermore, fix u ∈ RN and θ ∈ (0, ∞).
Proposition 4.6. Let F0 ∈ L∞ (RN )∩Eu,θ and let {Fk }k∈N be the sequence of minimizers for the
constrained JKO scheme. Then, for any h > 0 and all integers k ≥ 1, we have Fk ∈ L∞ (RN ),
α ◦ Fk ∈ W 1,1 (RN ), and
∫ {
} C 2
]
[
1
N h (Fk − Fk−1 )ζ + θ∇(α ◦ Fk ) + ∥α ◦ Fk ∥L1 (RN ) (u − v)Fk · ∇ζ dv ≤ h W2 (Fk−1 , Fk )
R
(47)
∞
N
for any ζ ∈ C (R ). Here C is a real constant depending only on N , θ and ∥∇ζ∥W 1,∞ (RN ;RN ) .
Proof. The proof of this result follows, mutatis mutandis, from [23, 27, 29], so we only point out
the differences while very succinctly mentioning the similarities. Just as in these references, the
inequality (47) is obtained from (18). One starts with the obvious inequality
1
ζ(ψ(v)) − ζ(v) − ∇ζ(v) · [∇ψ(v) − v] ≤ ∥∇2 ζ∥L∞ (RN ) |∇ψ(v) − v|2 ,
2
then multiplies both sides by Fk (v) and integrates over RN . Taking then the absolute value on
the left-hand side, this gives
∫
∫
1 2
≤
ζ(v)(F
−
F
)
+
∇ζ
·
[∇ψ(v)
−
v]F
(v)dv
∥∇
ζ∥
|∇ψ(v) − v|2 Fk (v) dv
∞
N
k
k−1
k
L (R )
N
2
N
R
R
1 2
= ∥∇ ζ∥L∞ (RN ) W22 (Fk , Fk−1 ).
2
Next, we insert the identity for ∇ψ(v) − v from (18) into this inequality and divide both sides
by h, which transforms the left hand side into
∫
1
ζ(v)(Fk (v) − Fk−1 (v)) +
h
N
R
[
] W22 (Fk , Fk−1 )
+∇ζ · θ∇(α ◦ Fk (v)) +
(v − u)Fk (v) + ∥α ◦ Fk (v)∥L1 (RN ) (u − v)Fk (v) dv .
hN θ
18
We discern here, in contrast with the case of unconstrained minimization, an additional nonlocal
term involving the Wasserstein distance, which we estimate using the Cauchy-Schwarz inequality
∫
W22 (Fk , Fk−1 )
∇ζ(v) · (v − u)Fk (v) dv
hN θ
RN
(∫
)1/2 (∫
)1/2
W22 (Fk , Fk−1 )
2
≤
∥∇ζ∥L∞ (RN )
|v − u| Fk (v) dv
Fk (v) dv
hN θ
RN
RN
W22 (Fk , Fk−1 )
√
∥∇ζ∥L∞ (RN ) .
=
h Nθ
In order to get (47), we use the triangle inequality, and obtain:
∫ {
} [
]
1
N h (Fk − Fk−1 )ζ + θ∇(α ◦ Fk ) + ∥α ◦ Fk ∥L1 (RN ) (u − v)Fk (v) · ∇ζ dv R
≤
C 2
W (Fk , Fk−1 ),
h 2
where C has the dependencies as described in the statement of the Proposition.
5
Convergence to weak solutions
Let F0 ∈ L1 (RN ) ∩ L∞ (RN ) be an initial datum and denote by Ωt the cylinder (0, t) × RN ,
t ∈ (0, T ].
We will apply the following lemma [5] in order to show strong convergence of the piece-wise
constant interpolators {F h }h .
Lemma 5.1 (Riesz-Fréchet-Kolmogorov). Let Ω ⊂ RN be open and let F be a bounded subset
of Lp (Ω), 1 ≤ p < ∞. Suppose that
a. ∀ ε > 0
∀ ω ⊂⊂ Ω
∃ δ > 0, δ < dist(ω, Ωc ) such that
∥f (· + h) − f ∥Lp (ω) < ε
b. ∀ ε > 0
∃ ω ⊂⊂ Ω
∀h ∈ RN
such that ∥f ∥Lp (Ω\ω) < ε
with |h| < δ and ∀ f ∈ F,
∀ f ∈ F.
Then F is relatively compact in Lp (Ω).
Let us write U h := (F h )m and Uk = (Fk )m , where m > 1. Also, recall that the time-interpolators
F h (t, v) are given by
F h (t, v) = Fk (v) if kh ≤ t < (k + 1)h.
While each optimizer Fk is, of course, implicitly dependent on the time-step size h > 0, we avoid
highlighting this by yet another index in order to keep the notation as succinct as possible.
We proceed by validating the hypotheses of the Riesz-Fréchet-Kolmogorov lemma (Lemma
5.1). Our approach resembles that of [29, 27].
Remark 5.2. In the first part of the proof the following lemma, we will actually demonstrate the
uniform integrability outside compact sets of the sequence {F h }h .
19
Lemma 5.3. (Precompactness in space) There exists a function l : RN → R for which
lim|e|↓0 l(e) = 0 and
∫ T∫
|F h (t, v + e) − F h (t, v)|m dv dt ≤ l(e)
RN
τ
for all τ , h.
Proof. Let us first verify condition (b) of the Riesz-Fréchet-Kolmogorov lemma. Recall the
ε
definition of K in (46). For given ε ∈ (0, 2KT ), choose 0 < δ < (2K)
m and ωT = [2δ, T − 2δ] ×
B(0, 1) ⊂⊂ (0, T ) × RN . Then (0, T ) × RN \ ωT ⊂ ((0, δ) ∪ (T − δ, T )) × RN , so that
∫
∥F h ∥m+1
Lm+1 ((0,T )×RN \ω
∫
( )m+1
Fh
dv dt +
≤
T)
T
T −δ
Ωδ
∫
(
Fh
)m+1
dv dt
RN
≤ 2δK m < ε.
We proceed with the proof of condition (a).
∫
|U h (t, v + e) − U h (t, v)| dv dt
(0,T )×RN
=h
n−1
∑∫
k=0 R
n−1
∑∫
N
≤ |e|h
= |e|h
|Uk (v + e) − Uk (v)| dv dt
∫
N
k=0 R
n−1
∑∫
k=0
RN
1
|∇Uk (v + se)| ds dv
0
|∇Uk (v)| dv.
We need a bound on the L1 (RN )-norm of the gradient of Uk . To this end, we employ the
Euler-Lagrange equation (18) and estimate
∫
∫
1
h
|∇Uk (v)| dv ≤
|ψ(v) − v|Fk (v) + |bk ||v|Fk (v) dv
θ
RN
[ 2
]
∫
√
1
W2 (Fk−1 , Fk ) h
U
dv
Nθ
≤ W2 (Fk−1 , Fk ) +
+
k
θ
N θ2
θ
[ 2
]
1
W2 (Fk−1 , Fk ) h m−1 √
≤ W2 (Fk−1 , Fk ) +
+ K
N θ,
θ
N θ2
θ
∫
,Fk )
W 2 (F
where we used the notation bk = − 2 Nk−1
− hθ Uk (v)dv and used that 0 ≤ Uk ≤ K m−1 Fk .
θ2
The
Wasserstein distance in the above expression is uniformly bounded in k because
∑n−1squared
2 (F , F
W
k
k+1 ) ≤ Ch, while the term W2 (Fk−1 , Fk ) can be estimated analogously by using
2
k=0
the generalized Young inequality
( n
)2
n
∑
∑
xk
≤n
x2k .
(48)
k=1
k=1
We have proven so far that
∫
(0,T )×RN
|U h (t, v + e) − U h (t, v)| dv dt ≤ C|e|.
20
The proof is completed by the noticing that, as F h ≥ 0 almost everywhere,
∫
∫
|F h (t, v + e) − F h (t, v)|m dv dt ≤
|U h (t, v + e) − U h (t, v)| dv dt.
(0,T )×RN
(0,T )×RN
Lemma 5.4. (Precompactness in Time) There exists a function q : R → R with limτ ↓0 q(τ ) = 0
and such that
∫
J :=
|F h (t + τ, v) − F h (t, v)|m+1 dv dt ≤ q(τ ) ∀ τ < T ∀ h.
ΩT −τ
Proof. Fix τ, h > 0; there exists a real number γ ∈ [0, h) such that τ = jh + γ for a given
integer j > 0, so that we can write
T − τ = nh − jh − γ = (n − j)h − γ ≤ (n − j)h.
This inequality implies
∫
J
≤
|F h (t + τ, v) − F h (t, v)|m+1 dv dt
Ω(n−j)h
=
n−j−1
∑ ∫
∫
RN
=
k=0
n−j−1
∑ ∫
|F h (t + τ, v) − F h (t, v)|m+1 dt dv
kh
∫
RN
k=0
(k+1)h
(k+1)h
|F h (t + τ, v) − Fk (v)|m+1 dt dv.
kh
We know that τ = jh + γ, so that t + τ = t + jh + γ. Thus
(k + j)h + γ ≤ t + τ < (k + j + 1)h + γ.
With γ lying in [0, h) and t in [0, (n − j)h], two possibilities now arise:
• Either (k + j)h + γ ≤ t + τ < (k + j + 1)h, so that
F h (t + τ, v) = Fk+j (v),
• or (k + j + 1)h ≤ t + τ < (k + j + 1)h + γ ≤ (k + j + 2)h + γ, in which case
F h (t + τ, v) = Fk+j+1 (v).
Hence we arrive at the estimate
∫
n−j−1
∑ [
J ≤
(h − γ)
RN
k=0
]
∫
|Fk+j − Fk |
m+1
|Fk+j+1 − Fk |
m+1
dv + γ
RN
dv
By the inequality |a − b|m+1 ≤ (a − b)(am − bm ) for a, b ≥ 0 and m ≥ 1, we get the bound
J
≤
n−j−1
∑
∫
[(h − γ)
k=0
∫
+γ
RN
RN
(Fk+j − Fk )(Uk+j − Uk ) dv
(Fk+j+1 − Fk )(Uk+j+1 − Uk ) dv]
21
We observe next that h − γ ≤ τ . Indeed, if this were not the case, we would have h − γ > jh + γ;
in other words, h > h − 2γ > jh, a contradiction (since j ∈ Z>0 ). Now we claim that
√
n−ι−1
∑ ∫
ι
,
(49)
(Fk+ι (v) − Fk (v))(Uk+ι (v) − Uk (v)) dv ≤ Cι + C
h
RN
k=0
where ι equals j or j + 1. If this claim is true, then
(
(
)
√ )
√
j
j+1
J ≤ (h − γ) Cj + C
+ γ C(j + 1) +
h
h
(
)
√
√
≤ C hj + hj + (j + 1)h + h(j + 1)
√
≤ C(τ + τ ) =: q(τ ).
τ ↓0
Clearly, q(τ ) → 0.
All that remains to be demonstrated is Claim (49). Set ζk,j (v) := Uk+j (v)
∫ − Uk (v), and let us
denote by ∇Φl the map pushing Fl forward to Fl−1 , l = 1, 2, ·.. Subtracting RN Fk+i (v)ζk,j (v) dv
from the identity
∫
∫
Fk+i−1 (v)ζk,j (v) dv =
Fk+i (v)ζ(∇Φk+i (v)) dv (i = 1, ·.., j),
RN
one obtains
RN
∫
∫
RN
(Fk+i − Fk+i−1 )(Uk+j − Uk ) dv =
RN
Fk+i [ζk,j − ζk,j ◦ ∇Φk+i ] dv.
Summing up from i = 1 to j, we get from the last equality
∫
Fk+j (v)(Uk+j (v) − Uk (v)) dv
RN
j ∫
∑
=
≤
N
i=1 R
j
∑
∫
√
Fk+i
1
|∇ζk,j (sv + (1 − s)∇Φk+i )|ds
√
Fk+i (v − ∇Φk+i (v)) dv
0
(50)
W2 (Fk+i−1 , Fk+i )
i=1
∫
×
0
1 (∫
RN
)1/2
|∇ζk,j (sv + (1 − s)∇Φk+i )|2 Fk+i (v) dv
ds.
In order to estimate the integral over RN in the last expression, we make use of a lemma by
Otto (cf. [23], Appendix Lemma 3). By using the L∞ estimate on Fk (cf. Corollary 3.2) and
the definition of push-forward, the hypothesis of the lemma,3
∫
∫
s(y)η(∇ψ(y)) dy ≤ K
η(x) dx
∀ 0 ≤ η ∈ C0 (RN ),
RN
RN
|2 ,
is evidently satisfied for s = Fk+i , η = |∇ζk,j and ψ = Φk+i We may thus deduce
∫
∫
2
Fk+i (v)|∇ζk,j (sv + (1 − s)∇Φk+i (v))| dv ≤ K
|∇ζk,j |2 dv (i = 1, ·.., j)
RN
RN
∫
=K
|∇Uk+j (v) − ∇Uk (v)|2 dv
N
[R
]
≤ K ∥∇Uk+j ∥2L2 (RN ) + ∥∇Uk ∥2L2 (RN ) .
3
Observe that ∇ζ ◦ (s id + (1 − s)∇Φ) is indeed well-defined almost everywhere in RN , since Φ is a uniformly
convex function, and because the map v 7→ s v + (1 − s)∇Φ(v) is injective.
22
Inserting this inequality into (50), one has
∫
Fk+j (v)(Uk+j (v) − Uk (v)) dv
RN
≤K
j
∑
[
]
W2 (Fk+i−1 , Fk+i ) ∥∇Uk+j ∥L2 (RN ) + ∥∇Uk ∥L2 (RN ) . (51)
i=1
We next use our knowledge that the optimizer F (λk−1 ) in the constrained manifold is the minimizer of the unconstrained problem of finding minP2ac IFk−1 ,λk−1 [F ]: F (λk−1 ) = Fk . In view of
the weak formulation (32) of the Euler-Lagrange equation corresponding to the unconstrained
minimization problem for IFk−1 ,λk−1 [F ] (with the pertinent modifications p = pk , α(z) = z m ,
F ∗ = Fk , and Φ(v) = λk−1 |v|2 ), one sees that
∫
∫
(w − v) · ξ(w) dpk (v, w) + h
[∇Uk + 2λk−1 v] · ξ(v) dv = 0
RN ×RN
RN
for all ξ ∈ Cc∞ (RN ; RN ), where pk denotes the joint probability measure with marginals Fk−1 ,
Fk , respectively. Rewriting this identity as
∫
∫
∫
h
∇Uk · ξ(v) dv = −2λk−1 h v · ξ(v) dv +
(v − w) · ξ(w) dpk (v, w)
RN
RN ×RN
we perform the following estimates:
∫
2
2
h ∇Uk · ξ dv RN
∫
2 (∫
2
2
≤ 4λk−1 h v · ξ(v) dv +
supp
ξ
≤ 4λ2k−1 h2 C + W22 (Fk−1 , Fk )
) (∫
|v − w| dpk (v, w)
2
RN ×RN
∥Fk ∥L∞ ∥ξ∥2L2 ,
RN
)
2
Fk (w)|ξ(w)| dw
where we used Hölder’s inequality and the marginal property of pk in the first step, and the fact
that ξ is compactly supported in the second (cf. also [29]). The latter estimate, together with
the boundedness of ξ, implies that ∇Uk belongs to L2 (RN ):
∥∇Uk ∥2L2 (RN ) ≤
C 2
W (Fk−1 , Fk ) + 4λ2k−1 C.
h2 2
Plugging this into (51), one sees that
∫
Fk+j (v)(Uk+j (v) − Uk (v)) dv
RN
j
C∑
≤
W2 (Fk+i−1 , Fk+i ) [W2 (Fk−1 , Fk ) + W2 (Fk+j−2 , Fk ) + 2λk,j ] ,
h
i=1
where λk,j := max{λk−1 , λk+j−1 }.
Next, we claim there exists a positive constant C independent of k, j, and h such that
n−j−1
∑ ∫
(Fk+j − Fk )(Uk+j − Uk ) dv
R
k=1
n−j−1
∑
≤
k=1
×
N
C
[W2 (Fk−1 , Fk ) + W2 (Fk+j−1 , Fk+j ) + 2λk,j ]
h
j
∑
W2 (Fk+i−1 , Fk+i ) + Cjh(n − j − 1).
i=1
23
(52)
Since h(n − j − 1) ≤ T = hn, the only inequalities we still have to prove are
n−j−1
∑
j
∑
k=1
i=1
[W2 (Fk−1 , Fk ) + W2 (Fk+j−1 , Fk+j )]
and
2λk,j
j
∑
W2 (Fk+i−1 , Fk+i ) ≤ Cjh
W2 (Fk+i−1 , Fk+i ) ≤ C
√
jh.
i=1
The latter inequality follows at once by the generalized Young inequality (48) and the uniform
bound on λk,j , and the former by rearranging the sums in the left-hand side and estimating
from above by the squares of the quadratic Wasserstein distances.
Claim (49) is thus proved. This brings the proof to its conclusion.
Remark 5.5 (Convergence in Lm ((0, T ) × RN )). With the preceding lemma at hand, one has
that F h → F in Lm+1 ((0, T ) × RN ) and almost everywhere (up to a subsequence). In order to
see that {F h }h converges to F in Lm ((0, T ) × RN ) as well, we apply the interpolation inequality
for Lebesgue spaces (cf. [18]) for the case of the Lebesgue space indices 1 < m < m + 1:
h
F − F Lm (RN )
s
≤ F h − F 1
L (RN )
1−s
h
F − F m+1
L
(RN )
1−s
≤ 2s F h − F m+1
L
(RN )
,
s = 1/m2 ,
where for the last inequality we used the fact that F h (t, ·) and F (t, ·) are probability density
functions. If m = 1, Hölder’s inequality provides the answer.
As a consequence of the above remark, one obtains a subsequence hj → 0+ for which
∥F hj (t, ·)∥Lm (RN ) → ∥F (t, ·)∥Lm (RN ) in L1 (0, T ) and for a.e. t ∈ (0, T ).
Also,
( from (52)) and Proposition 4.4 we deduce that the sequence
L2 (0, T ) × RN , thus
{
(53)
(
)}
is bounded in
∇v F hj
)m
(
)
(
⇀ ∇v F m weakly in L2 (0, T ) × RN ; RN .
∇ v F hj
(54)
Following arguments which are by now standard (cf. for example [23, 27, 29]), one can now use
Proposition 4.6 along with (53) and (54) to complete the proof of Theorem 1.2.
Acknowledgments
A. T. warmly thanks the Research Institute for Mathematical Sciences at Kyoto University for
its kind hospitality during his stay there in February 2010.
M. W. gratefully acknowledges financial support by the JSPS Postdoctoral Fellowship P09024.
He would like to thank Marco Di Francesco for initially posing the problem of constrained
optimization and steepest descent in the quadratic Wasserstein space to him. He furthermore
thanks Wilfrid Gangbo and Turkay Yolcu for stimulating discussions on the topic during the
IPAM Optimal Transport Program in 2008. Parts of this work contain material from M. W.’s
PhD thesis, which was supervised by Peter A. Markowich.
24
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