GLOBAL-IN-TIME WEAK MEASURE SOLUTIONS, FINITE-TIME
AGGREGATION AND CONFINEMENT FOR NONLOCAL
INTERACTION EQUATIONS
J. A. CARRILLO
1
, M. DIFRANCESCO
2
, A. FIGALLI
3
, T. LAURENT
4
AND D. SLEP
ˇ
CEV
5
Abstract. In this pape r, we provide a well-posedness theory for weak measure solu-
tions of the Cauchy problem for a family of nonlocal interaction equations. These equa-
tions are continuum models for interacting particle systems with attractive/repulsive
pairwise interaction potentials. The main phenomenon of interest is that, even with
smo oth initial data, the solutions can concentrate mass in finite time. We develop an
existence theory that enables one to go beyond the blow-up time in classical norms and
allows for solutions to form atomic parts of the measure in finite time. The weak mea-
sure solutions are shown to be unique and exist globally in time. Moreover, in the case
of sufficiently attractive potentials, we show the finite time total collapse of the solution
onto a single point, for compactly supported initial measures. Finally, we give condi-
tions on compensation between the attraction at large distances and local repulsion of
the potentials to have global-in-time confined systems for compactly supported initial
data. Our approach is based on the theory of gradient flows in the space of probability
measures endowed with the Wasserstein metric. In addition to classical tools, we exploit
the stability of the flow with respect to the transportation distance to greatly simplify
many problems by reducing them to questions about particle approximations.
1. Introduction
We consider a mass distribution of particles, µ ≥ 0, interacting under a continuous
interaction potent ial, W . The ass ociated interaction energy is defined as
W[µ] :=
1
2
Z
R
d
Z
R
d
W (x −y) dµ(x) dµ(y). (1.1)
Our paper is devoted to the class of continuity equations of the form
∂µ
∂t
= div
∇
δW
δµ
µ
= div [(∇W ∗ µ)µ] x ∈ R
d
, t > 0. (1.2)
The equation is typically coupled with an initial datum
µ(0) = µ
0
. (1.3)
The velocity field in the continuity equation, −(∇W ∗ µ)(t, x), represents the combined
contributions, at the point x, of the interaction through the potential W with particles
at all other points.
The choice of W depends on the phenomenon studied. For instance in population
dynamics, one is interested in the description of the evolution of a density of individuals.
Very often the interaction b e tween two individuals only depends on the distance between
them. This suggests a choice of W as a radial function, i.e. W (x) = w(|x|). Moreover,
a choice of w such that w
0
(r) > 0 corresponds to an attractive force among the particles
(or individuals), whereas w
0
(r) < 0 models a repulsive force.
Date: 16th April 2009.
1
2 J.A. CARRILLO, M. DIFRANCESCO, A. FIGALLI, T. LAURENT AND D. SLEP
ˇ
CEV
Equation (1.2) arises in several applications in physics and biology. Simplified inelastic
interaction models for granular media were considered in [4, 18] with W = |x|
3
/3 and
[40, 26] with W = |x|
α
, α > 1. Such models usually lead to convex attractive potentials.
Mathematical modelling of the collective b ehavior of individuals, such as swarming,
has also been treated by continuum models steaming from discrete particle models [30,
14, 38, 31, 39, 15, 33, 13, 21, 22, 16, 17]. Typical examples of interaction potentials
appearing in these works are the attractive Morse potential W (x) = −e
−|x|
, attractive-
repulsive Morse potentials W (x) = −C
a
e
−|x|/`
a
+ C
r
e
−|x|/`
r
, W (x) = −e
−|x|
2
, W (x) =
−C
a
e
−|x|
2
/`
a
+ C
r
e
−|x|
2
/`
r
, or W being the characteristic function of a set in R
d
. A
major issue is the possibility of a finite time blow-up of initially regular solutions, which
occurs when w is attractive enough near r = 0. In particular, the solution c an aggregate
(collapse) part (or all) of its mass to a point in finite time. Blow-up producing potentials
feature a suitable singularity in their second derivative at r = 0. Typically, the potential
is of the form W (x) ≈ |x|
1+α
with 0 ≤ α < 1, see [25, 7, 6, 5] in case of the Lipschitz
singularity. Related questions with diffusion added to the system have be en tackled in
[9, 27, 28, 29].
Finally, another source of models with interaction potential appear in the modelling of
cell movement by chemotaxis. In fact, the c lassical Patlak-Keller-Segel [35, 24] system,
see [12, 10, 11], corresponds to the choice of the Newtonian potential in R
2
as interaction,
W =
1
2π
log |x| with linear diffusion. In the case without diffusion, a notion of weak mea-
sure solutions was introduced in [36] for which the author proved global-in-time existence,
although uniqueness is lacking.
Given a continuous potential W , thanks to the structure of (1.2), we can assume
without loss of generality that the following basic assumption holds:
(NL0) W is continuous, W (x) = W (−x), and W (0) = 0.
We will say that W is a locally attractive potential if it verifies (NL0) and:
(NL1) W is λ–convex for a certain λ ∈ R, i.e. W (x) −
λ
2
|x|
2
is convex.
(NL2) There exists a constant C > 0 such that
W (z) ≤ C(1 + |z|
2
), for all z ∈ R
d
.
(NL3) W ∈ C
1
(R
d
\ {0}).
(NL4) W has local minimum at x = 0.
We will say that the potential is a pointy locally attractive potential if it satisfies (NL0)-
(NL4) and it has a Lipschitz singularity at the origin. In case the potential is continuously
differentiable at the origin, we will speak about a smooth potential.
Remark 1.1. Assumptions (NL0)-(NL1) imply that
W (x) ≥
λ
2
|x|
2
, (1.4)
since 0 ∈ ∂W (0) and W (0) = 0. Hypotheses (NL1)-(NL3) imply a growth control on
the gradient of W . More precisely, using the convexity of x 7→ V (x) := W (x) −
λ
2
|x|
2
and
the quadratic growth of W (x), there exists K > 0 such that
∇V (x) · p ≤ V (x + p) − V (x) ≤ K(1 + |x|
2
+ |p|
2
)
NONLOCAL INTERACTION EQUATIONS 3
for any x 6= 0. Now, taking the supremum among all vectors p such that |p| = max{|x|, 1},
we get |∇V (x)| ≤ K(2 + 2|x|) from which
|∇W (x)| ≤ 2K + (2K + |λ|) |x|. (1.5)
Let us also remark that (NL1) together with (NL3) imply that if the potential is not
differentiable at the origin, then it has at most a Lipschitz singularity at the origin.
Examples of locally attractive potentials neither pointy nor smooth are the ones with a
local behavior at the origin like |x|
1+α
, with 0 < α < 1.
The first problem we treat in this paper is to give a well-posedness theory of weak
measure solutions in the case of locally attractive potentials. Due to the possible concen-
tration of solutions in a finite time, one has to allow for a concept of weak solution in a
(nonnegative) measure sense. Our work fills in an important gap in the present studies
of the equation. Simplistically speaking: [2, 3] provide a good theory for weak measure
solutions for potentials which do not produce blow-up in finite time. On the other hand
in the works that study potentials that do produce blow-up [25, 7, 6, 5] the notion of the
solution breaks down at the blow-up time.
Before discussing the main results of this work, we introduce the concept of weak
measure solution to (1.2). A natural way to introduce a concept of weak measure solution
is to work in the space P(R
d
) of probability measures on R
d
. Since the class of equation
described here doe s not feature mass–threshold phenomena, we normalize the mass to 1
without loss of generality. Following the approach developed in [2, 3], we shall consider
weak measure solutions which additionally belong to the metric space
P
2
(R
d
) :=
µ ∈ P(R
d
) :
Z
R
d
|x|
2
dµ(x) < +∞
of probability measures with finite second moment, e ndowed with the 2–Wasserstein
distance d
W
; see the next section.
Definition 1.2. A locally absolutely continuous curve µ : [0, +∞) 3 t 7→ P
2
(R
d
) is said
to be a weak measure solution to (1.2) with initial datum µ
0
∈ P
2
(R
d
) if and only if
∂
0
W ∗ µ ∈ L
1
loc
((0, +∞); L
2
(µ(t))) and
Z
+∞
0
Z
R
d
∂ϕ
∂t
(x, t) dµ(t)(x) dt +
Z
R
d
ϕ(x, 0) dµ
0
(x) =
Z
+∞
0
Z
R
d
Z
R
d
∇ϕ(x, t) · ∂
0
W (x −y) dµ(t)(x) dµ(t)(y) dt,
for all test functions ϕ ∈ C
∞
c
([0, +∞) × R
d
).
In this definition, ∂
0
W (x) denotes the element of minimal norm in the subdifferential
of W at x. In particular, thanks to (NL3) and (NL4), we show
(∂
0
W ∗ µ)(x) =
Z
y6=x
∇W (x −y) dµ(y).
Here, the absolute continuity of the curve of measures means that its metric derivative is
integrable, see next section. Let us point out that the nonlinear term is well-defined due
to the integrability of the velocity field against the weak measure solution.
4 J.A. CARRILLO, M. DIFRANCESCO, A. FIGALLI, T. LAURENT AND D. SLEP
ˇ
CEV
The main idea to construct weak measure solutions to (1.2) is to use the interpretation
of these equations as gradient flows in the space P
2
(R
d
) of the interaction potential
functional (1.1) with respect to the transport distance d
W
. Such an interpretation turns
out to be extremely well-adapted to proving uniqueness and stability results for gradient
flow solutions compared to other strategies. This basic intuitive idea, introduced in [34]
for the porous medium equation and generalized to a wide class of equations in [19], was
made completely rigorous for a large class of equations in [2, 3] including some particular
instances of (1.2). Let us point out that the solutions we eventually construct, called
gradient flow solutions, satisfy more properties than just being weak measure solutions.
For these solutions, we are able to obtain the existence, uniqueness and d
W
-stability.
Let us remark that the well-posedness theory of gradient flow solutions in the space of
probability me asures is developed in [2, 3] in case the potential W is smooth and convex.
Here, we mainly focus on generalizing this theory to allow Lipschitz singularities at the
origin. In the case of locally attractive potentials, the technical point to deal with is the
characterization of the subdifferential and its element of minimal norm. Moreover, we
generalize this gradient flow theory allowing a negative quadratic behaviour at infinity.
This fact introduces certain technical difficulties at the level of coercivity and lower semi-
continuity of the functional defining the variational scheme. The well-posedness theory
of gradient-flow solutions is the goal of Section 2.
One of the key properties of the constructed solutions is the stability with respect to
d
W
: given two gradient flow solutions µ
1
t
and µ
2
t
,
d
W
(µ
1
t
, µ
2
t
) ≤ e
−λt
d
W
(µ
1
0
, µ
2
0
)
for all t ≥ 0. This stability result is not only useful for showing uniqueness but it is mainly
a tool for approximating general solutions by particle ones. In fact, the previous estimate
can be considered as a proof of the convergence of the continuous particle method for this
equation on bounded time intervals. This is very much in the spirit of early works in the
convergence of particle approximations to Vlasov-type equations in kinetic theory [32, 37].
Section 3 is devoted to show qualitative properties of the approximate solutions ob-
tained by the variational scheme as in [23]. More precisely, we prove that particles
remain particles at the level of a discrete variational scheme, provided the time step is
small enough. In particular, this shows that the gradient flow solution starting from a fi-
nite number of particles remains at any time a finite number of particles, whose positions
are determined by an ODE system. Although the fact that this construction give the
solutions for finite number of particles can be directly checked on the solution concept,
it is quite interesting to prove it directly at the variational scheme level, as it shows its
suitability as a numerical scheme.
Section 4 is devoted to the question of finite-time blow-up of solutions. For a radially
symmetric attractive potential, i.e. W (x) = w(|x|), w
0
(r) > 0 for r > 0, the numb er
T (ε
1
) :=
Z
ε
1
0
dr
w
0
(r)
, ε
1
> 0 (1.6)
can be thought as the time it takes for a particle obeying the ODE
˙
X = −∇W (X) to
reach the origin if it start at a distance ε
1
from the origin. This number quantifies the
attractive strength of the potential: the smaller T (ε
1
) is, the more attractive the potential
NONLOCAL INTERACTION EQUATIONS 5
is. It was shown in [6, 7, 8] that if T (ε
1
) = +∞ for some (or equivalently for all) ε
1
> 0,
then solutions of (1.2) starting with initial data in L
p
will stay in L
p
for all time, whereas
if T (ε
1
) < +∞ for some ε
1
> 0, then compactly supported solutions will leave L
p
in finite
time (this result holds in the class of potentials which does not oscillate pathologically
around the origin). In Section 4, thanks to our developed existence theory, we are able
to obtain further understanding of the nature of the blow-up: loosely speaking, we prove
that if the potential is attractive enough (i.e. T (ε
1
) < +∞ for some ε
1
> 0) then solutions
of (1.2) starting with measure initial data will concentrate to a single Delta Dirac in finite
time. We refer to this phenomena as finite time total collapse.
We will say that W is a locally attractive non-Osgood potential if in addition to (NL0)-
(NL4), it satisfies the finite time blow-up condition:
(NL-FTBU) W is radial, i.e. W (x) = w(|x|), W ∈ C
2
(R
d
\{0}) with w
0
(r) > 0
for r > 0 and satisfying the following monotonicity condition: either (a) w
0
(0
+
) >
0, or (b) w
0
(0
+
) = 0 with w
00
(r) monotone decreasing on an interval (0, ε
0
).
Moreover, the potential satisfies the integrability condition
Z
ε
1
0
1
w
0
(r)
dr < +∞, for some ε
1
> 0. (1.7)
Let us point out that the condition of monotonicity of w
00
(r) is not too restrictive. It
is actually automatically satisfied by any potential who satisfies (1.7) and whose sec ond
derivative doesn’t oscillate badly at the origin, as in [6, 7] (more comments on this as-
sumption are done in Section 4). Examples of this type of potentials are the ones having
a local behavior at the origin like w
0
(r) ' r
α
with 0 < α < 1 or w
0
(r) ' r log
2
r.
The proof is based on showing a finite-time total collapse result for the particles ap-
proximation independent of the number of particles possibly depending on the initial
support. This fact, together with the convergence of the particle approximation, leads to
the finite-time aggregation onto a single particle with the total mass of the system. This
is the main technical novelty of our approach to blow-up.
Let us remark that the blow-up of the solution in L
p
norms will in general happen before
the total aggregation/collapse onto a single point. The transition from the first L
∞
-blow-
up to the total collapse can be very complicated. For instance one could have multiple
points of aggregation onto Dirac deltas interacting between them and with smooth parts
of the measure in a challenging evolution before the total aggregation onto a single point.
This is also explained in Section 4. This complex behavior was already encountered in
[36] in the case of the chemotaxis model without diffusion, but his notion of solution lacks
of uniqueness and stability. Many problems on the details of the blowup in (1.2) and the
interaction of delta masses with surrounding absolutely-continuous-measure part remain.
The last section is devoted to proving confinement of solutions for attractive/repulsive
potentials which are radial and increasing outside a ball, that is for ones that satisfy
(NL-RAD) There exists R
a
≥ 0 such that W (x) = w(|x|) for |x| ≥ R
a
, and
w
0
(r) ≥ 0 for r > R
a
.
We say that a potential is strongly-attractive-at-infinity if in addition to (NL0)-(NL4)
and (NL-RAD) it satisfies the strong confinement condition:
