NONLINEAR AGGREGATION-DIFFUSION EQUATIONS: RADIAL
SYMMETRY AND LONG TIME ASYMPTOTICS
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
Abstract. We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs.
This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially
symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations
tools allow us to show the existence of global minimizers among these equilibria. Finally, in the
particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria
for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations
as t → ∞.
1. Introduction
The evolution of interacting particles and their equilibrium configurations has attracted the
attention of many applied mathematicians and mathematical analysts for years. Continuum description of interacting particle systems usually leads to analyze the behavior of a mass density
ρ(t, x) of individuals at certain location x ∈ Rd and time t ≥ 0. Most of the derived models
result in aggregation-diffusion nonlinear partial differential equations through different asymptotic
or mean-field limits [59, 12, 23]. The different effects reflect that equilibria are obtained by competing behaviors: the repulsion between individuals/particles is modeled through nonlinear diffusion
terms while their attraction is integrated via nonlocal forces. This attractive nonlocal interaction
takes into account that the presence of particles/individuals at a certain location y ∈ Rd produces
a force at particles/individuals located at x ∈ Rd proportional to −∇W (x − y) where the given
interaction potential W : Rd → R is assumed to be radially symmetric and increasing consistent
with attractive forces. The evolution of the mass density of particles/individuals is given by the
nonlinear aggregation-diffusion equation of the form:
∂t ρ = ∆ρm + ∇ · (ρ∇(W ∗ ρ))
x ∈ Rd , t ≥ 0
(1.1)
with initial data ρ0 ∈ L1+ (Rd ) ∩ Lm (Rd ). We will work with degenerate diffusions, m > 1, that
appear naturally in modelling repulsion with very concentrated repelling nonlocal forces [59, 12].
These models are ubiquitous in mathematical biology where they have been used as macroscopic
descriptions for collective behavior or swarming of animal species, see [53, 13, 54, 55, 67, 16] for
instance, or more classically in chemotaxis-type models, see [60, 44, 40, 39, 11, 10, 20] and the
references therein.
1 Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom. Email: [email protected].
2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria. E-mail:
[email protected].
3 Dipartimento di Ingegneria, Università degli Studi di Napoli “Parthenope”, 80143 Italia.
E-mail:
[email protected].
4 School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160 USA.
E-mail: [email protected].
1
2
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
On the other hand, this family of PDEs is a particular example of nonlinear gradient flows in
the sense of optimal transport between mass densities, see [2, 26, 27]. The main implication for
us is that there is a natural Lyapunov functional for the evolution of (1.1) defined on the set of
centered mass densities ρ ∈ L1+ (Rd ) ∩ Lm (Rd ) given by
Z
Z
1
1
ρm (x) dx +
ρ(x)W (x − y)ρ(y) dx dy
(1.2)
E[ρ] =
m − 1 Rd
2 Rd ×Rd
Z
Z
ρ(x) ≥ 0 ,
ρ(x) dx = M ≥ 0 ,
xρ(x) dx = 0 ,
Rd
Rd
being the last integral defined in the improper sense. Therefore, if the balance between repulsion
and attraction occurs, these two effects should determine stationary states for (1.1) including the
stable solutions possibly given by local (global) minimizers of the free energy functional (1.2).
Many properties and results have been obtained in the particular case of Newtonian attractive
potential due to its applications in mathematical modeling of chemotaxis [60, 44] and gravitational
collapse models [61]. In the classical 2D Keller-Segel model with linear diffusion, it is known that
equilibria can only happen in the critical mass case [9] while self-similar solutions are the long time
asymptotics for subcritical mass cases [11, 18]. For supercritical masses, all solutions blow up in
finite time [40]. It was shown in [48, 19] that degenerate diffusion with m > 1 is able to regularize
the 2D classical Keller-Segel problem, where solutions exist globally in time regardless of its mass,
and each solution remain uniformly bounded in time. For the Newtonian attraction interaction
in dimension d ≥ 3, the authors in [8] show that the value of the degeneracy of the diffusion that
allows the mass to be the critical quantity for dichotomy between global existence and finite time
blow-up is given by m = 2 − 2/d. In fact, based on scaling arguments it is easy to argue that for
m > 2 − 2/d, the diffusion term dominates when density becomes large, leading to global existence
of solutions for all masses. This result was shown in [63] together with the global uniform bound
of solutions for all times.
However, in all cases where the diffusion dominates over the aggregation, the long time asymptotics of solutions to (1.1) have not been clarified, as pointed out in [7]. Are there stationary
solutions for all masses when the diffusion term dominates? And if so, are they unique up to
translations? Do they determine the long time asymptotics for (1.1)? Only partial answers to
these questions are present in the literature, which we summarize below.
To show the existence of stationary solutions to (1.1), a natural idea is to look for the global
minimizer of its associated free energy functional (1.2). For the 3D case with Newtonian interaction
potential and m > 4/3, Lions’ concentration-compactness principle [52] gives the existence of a
global minimizer of (1.2) for any given mass. The argument can be extended to kernels that are no
more singular than Newtonian potential in Rd at the origin, and have slow decay at infinity. The
existence result is further generalized by [4] to a broader classes of kernels, which can have faster
decay at infinity. In all the above cases, the global minimizer of (1.2) corresponds to a stationary
solution to (1.1) in the sense of distributions. In addition, the global minimizer must be radially
decreasing due to Riesz’s rearrangement theorem.
Regarding the uniqueness of stationary solutions to (1.1), most of the available results are for
Newtonian interaction. For the 3D Newtonian potential with m > 4/3, for any given mass, the
authors in [50] prove uniqueness of stationary solutions to (1.1) among radial functions, and their
method can be generalized to the Newtonian potential in Rd with m > 2 − 2/d. For the 3D case
with m > 4/3, [62] show that all compactly supported stationary solutions must be radial up
to a translation, hence obtaining uniqueness of stationary solutions among compactly supported
functions. The proof is based on moving plane techniques, where the compact support of the
stationary solution seems crucial, and it also relies on the fact that the Newtonian potential in
3D converges to zero at infinity. Similar results are obtained in [22] for 2D Newtonian potential
with m > 1 using an adapted moving plane technique. Again, the uniqueness result is based on
showing radial symmetry of compactly supported stationary solutions. Finally, we mention that
3
uniqueness of stationary states has been proved for general attracting kernels in one dimension in
the case m = 2, see [17]. To the best of our knowledge, even for Newtonian potential, we are not
aware of any results showing that all stationary solutions are radial (up to a translation).
Previous results show the limitations of the present theory: although the existence of stationary
states for all masses is obtained for quite general potentials, their uniqueness, crucial for identifying the long time asymptotics, is only known in very particular cases of diffusive dominated
problems. The available uniqueness results are not very satisfactory due to the compactly supported restriction on the uniqueness class imposed by the moving plane techniques. And thus,
large time asymptotics results are not at all available due to the lack of mass confinement results
of any kind uniformly in time together with the difficulty of identifying the long time limits of
sequences of solutions due to the restriction on the uniqueness class for stationary solutions.
If one wants to show that the long time asymptotics are uniquely determined by the initial mass
and center of mass, a clear strategy used in many other nonlinear diffusion problems, see [70] and
the references therein, is the following: one first needs to prove that all stationary solutions are
radial up to a translation in a non restrictive class of stationary solutions, then one has to show
uniqueness of stationary solutions among radial solutions, and finally this uniqueness will allow
to identify the limits of time diverging sequences of solutions, if compactness of these sequences
is shown in a suitable functional framework. Let us point out that comparison arguments used
in standard porous medium equations are out of the question here due to the lack of maximum
principle by the presence of the nonlocal term.
In this work, we will give the first full result of long time asymptotics for a diffusion dominated
problem using the previous strategy without smallness assumptions of any kind. More precisely,
we will prove that all solutions to the 2D Keller-Segel equation with m > 1 converge to the global
minimizer of its free energy using the previous strategy. The first step will be to show radial
symmetry of stationary solutions to (1.1) under quite general assumptions on W and the class of
stationary solutions. Let us point out that standard rearrangement techniques fail in trying to
show radial symmetry of general stationary states to (1.1) and they are only useful for showing
radial symmetry of global minimizers, see [22]. Comparison arguments for radial solutions allow
to prove uniqueness of radial stationary solutions in particular cases [50, 46]. However, up to our
knowledge, there is no general result in the literature about radial symmetry of stationary solutions
to nonlocal aggregation-diffusion equations.
Our first main result is that all stationary solutions of (1.1) are radially decreasing up to translation by a fully novel application of continuous Steiner symmetrization techniques for the problem
(1.1). Continuous Steiner symmetrization has been used in calculus of variations [15] for replacing rearrangement inequalities [14, 49, 56], but its application to nonlinear nonlocal aggregationdiffusion PDEs is completely new. Most of the results present in the literature using continuous
Steiner symmetrization deal with functionals of first order, i.e. functionals involving a power of
the modulus of the gradient of the unknown, see [43, Section II] and [42, 15], while in our case
the functional (1.2) is purely of zeroth order. The decay of the attractive Newtonian potential
interaction term in d ≥ 3 follows from [15, Corollary 2] and [56], which is the only result related
to our strategy. We will construct a curve of measures using continuous Steiner symmetrization
such that the functional (1.2) decays strictly along that curve unless the base point is radially
symmetric. This fact together with the assumption that the base point is a stationary state leads
to a contradiction unless the stationary state is radially symmetric. This first main result is the
content of Section 2 in which we specify the assumptions on the interaction potential and the
notion of stationary solutions in details. We point out that the variational structure of (1.1) is
crucial to show the radially decreasing property of stationary solutions.
The result of radial symmetry for general stationary solutions to (1.1) is quite striking in comparison to other gradient flow models in collective behavior based on the competition of attractive
4
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
and repulsive effects via nonlocal interaction potentials. Actually, it is demonstrated both numerically and asymptotically in [47, 6, 3] that there should be stationary solutions of these fully
nonlocal interaction models which are not radially symmetric despite the radial symmetry of the
interaction potential. Our first main result shows that this break of symmetry does not happen
whenever nonlinear diffusion is chosen to model very strong localized repulsion forces, see [67].
Another consequence of our radial symmetry results is the lack of non-radial local minimizers of
the free energy functional (1.2) which is not all obvious.
We next study more properties of particular radially decreasing stationary solutions. We make
use of the variational structure to show the existence of global minimizers to (1.2) under very general
hypotheses on the interaction potential W . In section 3, we show that these global minimizers are
in fact compactly supported radially decreasing continuous functions. These results fully generalize
the results in [62, 22]. Putting together Sections 2 and 3, the uniqueness and full characterization
of the stationary states is reduced to uniqueness among the class of radial solutions. This result is
known in the case of Newtonian attraction kernels [50].
Finally, we make use of the uniqueness among translations for any given mass of stationary
solutions to (1.1) to obtain the second main result of this work, namely to answer the open problem
of the long time asymptotics to (1.1) with Newtonian interaction in 2D. This is accomplished in
Section 4 by a compactness argument for which one has to extract the corresponding uniform in
time bounds and a careful treatment of the nonlinear terms and dissipation while taking the limit
t → ∞. We do not know how to obtain a similar result for Newtonian interaction in d ≥ 3 due
to the lack of uniform in time mass confinement bounds in this case. We essentially cannot show
that mass does not escape to infinity while taking the limit t → ∞. However, the compactness
and characterization of stationary solutions is still valid in that case.
The present work opens new perspectives to show radial symmetry for stationary solutions to
nonlocal aggregation-diffusion problems. While the hypotheses of our result to ensure existence
of global radially symmetric minimizers of (1.2), and in turn of stationary solutions to (1.1), are
quite general, we do not know yet whether there is uniqueness among radially symmetric stationary
solutions (with a fixed mass) for general non-Newtonian kernels. We even do not have available
uniqueness results of radial minimizers beyond Newtonian kernels. Understanding if the existence
of radially symmetric local minimizers, that are not global, is possible for functionals of the form
(1.2) with radial interaction potential is thus a challenging question. Concerning the long-time
asymptotics of (1.1), the lack of a novel approach to find confinement of mass beyond the usual
virial techniques and comparison arguments in radial coordindates hinders the advance in their
understanding even for Newtonian kernels with d ≥ 3. Last but not least, our results open a window
to obtain rates of convergence towards the unique equilibrium up to translation for the Newtonian
kernel in 2D. The lack of general convexity of this variational problem could be compensated by
recent results in a restricted class of functions, see [25]. However, the problem is quite challenging
due to the presence of free boundaries in the evolution of compactly supported solutions to (1.1)
that rules out direct linearization techniques as in the linear diffusion case [18].
2. Radial Symmetry of stationary states with degenerate diffusion
Throughout this section, we assume that m > 1, and W satisfies the following four assumptions:
(K1) W is attracting, i.e., W (x) ∈ C 1 (Rd \ {0}) is radially symmetric
W (x) = ω(|x|) = ω(r)
and ω 0 (r) > 0 for all r > 0 with ω(1) = 0.
(K2) W is no more singular than the Newtonian kernel in Rd at the origin, i.e., there exists some
Cw > 0 such that ω 0 (r) ≤ Cw r1−d for r ≤ 1.
(K3) There exists some Cw > 0 such that ω 0 (r) ≤ Cw for all r > 1.
5
(K4) Either ω(r) is bounded for r ≥ 1 or there exists Cw > 0 such that for all a, b ≥ 0:
ω+ (a + b) ≤ Cw (1 + ω(1 + a) + ω(1 + b)) .
As usual, ω± denotes the positive and negative part of ω such that ω = ω+ − ω− . In particular,
if W = −N is the attractive Newtonian potential, where N is the fundamental solution of −∆
operator in Rd , then W satisfies all the assumptions.
We denote by L1+ (Rd ) the set of all nonnegative functions in L1 (Rd ). Let us start by defining
precisely stationary states to the aggregation equation (1.1) with a potential satisfying (K1)-(K4).
Definition 2.1. Given ρs ∈ L1+ (Rd ) ∩ L∞ (Rd ) we call it a stationary state for the evolution
1
d
1
d
problem (1.1) if ρm
s ∈ Hloc (R ), ∇ψs := ∇W ∗ ρs ∈ Lloc (R ), and it satisfies
d
in the sense of distributions in R .
d
∇ρm
s = −ρs ∇ψs in R
(2.1)
Let us first note that ∇ψs is globally bounded under the assumptions (K1)-(K3). To see this, a
direct decomposition in near- and far-field sets yields
Z
Z
Z
1
ρ
(y)
dy
+
C
ρs (y) dy
(2.2)
|∇ψs (x)| ≤
|∇W (x − y)|ρs (y) dy ≤ Cw
s
w
d−1
B
Rd
A |x − y|
Z
1
dy kρs kL∞ (Rd ) + Cw kρs kL1 (Rd ) ≤ C(kρs kL1 (Rd ) + kρs kL∞ (Rd ) ) .
≤ Cw
|x
−
y|d−1
A
where we split the integrand into the sets A := {y : |x − y| ≤ 1} and B := Rd \ A, and apply the
assumptions (K1)-(K3).
Under the additional assumptions (K4) and ω(1 + |x|)ρs ∈ L1 (Rd ), we will show that the
potential function ψs (x) = W ∗ ρs (x) is also locally bounded. First, note that (K1)-(K3) ensures
that |ω(r)| ≤ C̃w φ(r) for all r ≤ 1 with some C̃w > 0, where
 2−d
if d ≥ 3
 r
− log(r) if d = 2 .
φ(r) :=
(2.3)

r−1
if d = 1
Hence we can again perform a decomposition in near- and far-field sets and obtain
Z
Z
Z
|ψs (x)| ≤
|W (x − y)|ρs (y) dy ≤ Cw
φ(|x − y|)ρs (y) dy +
ω+ (|x| + |y|)ρs (y) dy
Rd
A
B
Z
≤ Cw
φ(|x − y|)dy kρs kL∞ (Rd ) + Cw (1 + ω(1 + |x|))kρs kL1 (Rd ) + Cw kω(1 + |x|)ρs kL1 (Rd )
A
≤ C(kρs kL1 (Rd ) + kρs kL∞ (Rd ) ) + ω(1 + |x|)kρs kL1 (Rd ) + Cw kω(1 + |x|)ρs kL1 (Rd ) .
(2.4)
Our main goal in this section is the following theorem.
Theorem 2.2. Assume that W satisfies (K1)-(K4). Let ρs ∈ L1+ (Rd )∩L∞ (Rd ) with ω(1+|x|)ρs ∈
L1 (Rd ) be a non-negative stationary state of (1.1) in the sense of Definition 2.1. Then ρs must
be radially decreasing up to a translation, i.e. there exists some x0 ∈ Rd , such that ρs (· − x0 ) is
radially symmetric, and ρs (|x − x0 |) is non-increasing in |x − x0 |.
Before going into the details of the proof, we briefly outline the strategy here. Assume there
is a stationary state ρs which is not radially decreasing under any translation. To obtain a
contradiction, we consider the free energy functional E[ρ] associated with (1.1),
Z
Z
1
1
E[ρ] =
ρm dx +
ρ(W ∗ ρ)dx =: S[ρ] + I[ρ].
(2.5)
m − 1 Rd
2 Rd
We first observe that the energy of the steady state ρs is finite since the potential function ψs =
1,∞
W ∗ ρs ∈ Wloc
(Rd ) satisfies (2.4) with ω(1 + |x|)ρs ∈ L1 (Rd ). Using the assumption that ρs is not
6
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
radially decreasing under any translation, we will apply the continuous Steiner symmetrization to
perturb around ρs and construct a continuous family of densities µ(τ, ·) with µ(0, ·) = ρs , such that
E[µ(τ )] − E[ρs ] < −cτ for some c > 0 and any small τ > 0. On the other hand, using that ρs is a
stationary state, we will show that |E[µ(τ )] − E[ρs ]| ≤ Cτ 2 for some C > 0 and any small τ > 0.
Combining these two inequalities together gives us a contradiction for sufficiently small τ > 0.
Let us characterize first the set of possible stationary states of (1.1) in the sense of Definition
2.1 and their regularity. Parts of these arguments are reminiscent from those done in [62, 22] in
the case of attractive Newtonian potentials.
Lemma 2.3. Let ρs ∈ L1+ (Rd ) ∩ L∞ (Rd ) with ω(1 + |x|)ρs ∈ L1 (Rd ) be a non-negative stationary
state of (1.1) in the sense of Definition 2.1. Then ρs ∈ C(Rd ), and there exists some C =
C(kρs kL1 , kρs kL∞ , Cw , d) > 0, such that
|∇(ρm−1
)| ≤ C
s
in supp ρs .
Proof. We have already checked that under these assumptions on W and ρs , the potential function
1,∞
1
d
m
1
d
ψs ∈ Wloc
(Rd ) due to (2.2)-(2.4). Since ρm
s ∈ Hloc (R ), then ρs is a weak Hloc (R ) solution of
d
∆ρm
s = −∇ · (ρs ∇ψs ) in R
(2.6)
−1,p
Wloc
(Rd )
ρm
s
with right hand side belonging to
for all 1 ≤ p ≤ ∞. As a consequence,
is in fact a
1,p
d
weak solution in Wloc (R ) for all 1 < p < ∞ of (2.6) by classical elliptic regularity results. Sobolev
0,α/m
0,α
d
(Rd ). Let
embedding shows that ρm
s belongs to some Hölder space Cloc (R ), and thus ρs ∈ Cloc
d
d
us define the set Ω = {x ∈ R : ρs (x) > 0}. Since ρs ∈ C(R ), then Ω is an open set and it consists
of a countable number of open possibly unbounded connected components. Let us take any bounded
smooth connected open subset Θ such that Θ ⊂ Ω. Since ρs ∈ C(Rd ), then ρs is bounded away
m
from zero in Θ and thus due to the assumptions on ρs , we have that ∇ρsm−1 = m−1
mρs ∇ρs holds in
the distributional sense in Θ. We conclude that wherever ρs is positive, (2.1) can be interpreted
as
m m−1
∇
ρ
+ ψs = 0 ,
(2.7)
m−1 s
m
in the sense of distributions in Ω. Hence, the function G(x) = m−1
ρsm−1 (x) + ψs (x) is constant in
each connected component of Ω. From here, we deduce that any stationary state of (1.1) in the
sense of Definition 2.1 is given by
ρs (x) =
m−1
(G − ψs )(x)
m
1
m−1
,
(2.8)
+
where G(x) is a constant in each connected component of the support of ρs , and its value may differ
0,1/(m−1)
1,∞
in different connected components. Due to ψs ∈ Wloc
(Rd ), we deduce that ρs ∈ Cloc
(Rd ) if
0,1
d
m ≥ 2 and ρs ∈ Cloc (R ) for 1 < m < 2. Putting together (2.7) and (2.4), we conclude the desired
estimate.
2.1. Some preliminaries about rearrangements. Now we briefly recall some standard notions
and basic properties of decreasing rearrangements for nonnegative functions that will be used later.
For a deeper treatment of these topics, we address the reader to the books [37, 5, 41, 45, 49] or
the papers [64, 65, 66, 57]. We denote by |E|d the Lebesgue measure of a measurable set E in Rd .
Moreover, the set E # is defined as the ball centered at the origin such that |E # |d = |E|d .
A nonnegative measurable function f defined on Rd is called radially symmetric if there is a
nonnegative function fe on [0, ∞) such that f (x) = fe(|x|) for all x ∈ Rd . If f is radially symmetric,
we will often write f (x) = f (r) for r = |x| ≥ 0 by a slight abuse of notation. We say that f is
rearranged if it is radial and fe is a nonnegative right-continuous, non-increasing
function of r > 0.
A similar definition can be applied for real functions defined on a ball BR (0) = x ∈ Rd : |x| < R .
7
We define the distribution function of f ∈ L1+ (Rd ) by
ζf (τ ) = | x ∈ Rd : f (x) > τ |d ,
for all τ > 0.
Then the function f ∗ : [0, +∞) → [0, +∞] defined by
f ∗ (s) = sup {τ > 0 : ζf (τ ) > s} ,
s ∈ [0, +∞),
will be called the Hardy-Littlewood one-dimensional decreasing rearrangement of f . By this definition, one could interpret f ∗ as the generalized right-inverse function of ζf (τ ).
Making use of the definition of f ∗ , we can define a special radially symmetric decreasing function
f , which we will call the Schwarz spherical decreasing rearrangement of f by means of the formula
#
f # (x) = f ∗ (ωd |x|d ) x ∈ Rd ,
(2.9)
d
where ωd is the volume of the unit ball in R . It is clear that if the set Ωf = x ∈ R : f (x) > 0
of f has finite measure, then f # is supported in the ball Ω#
f .
One can show that f ∗ (and so f # ) is equidistributed with f (i.e. they have the same distribution
function). Thus if f ∈ Lp (Rd ), a simple use of Cavalieri’s principle (see e.g. [65, 45]) leads to the
invariance property of the Lp norms:
d
kf kLp (Rd ) = kf ∗ kLp (0,∞) = kf # kLp (Rd )
for all 1 ≤ p ≤ ∞ .
(2.10)
In particular,using the layer-cake representation formula (see e.g. [49]) one could easily infer that
Z ∞
f # (x) =
χ{f >τ }# dτ.
0
Among the many interesting properties of rearrangements, it is worth mentioning the HardyLittlewood inequality (see [37, 5, 45] for the proof): for any couple of nonnegative measurable
functions f, g on Rd , we have
Z
Z
f (x)g(x)dx ≤
f # (x)g # (x)dx.
(2.11)
Rd
Rd
Since in Section 4 we will use estimates of the solutions Keller-Segel problems in terms of their
integrals, let us now recall the concept of comparison of mass concentration, taken from [68], that
is remarkably useful.
Definition 2.4. Let f, g ∈ L1loc (Rd ) be two nonnegative, radially symmetric functions on Rd . We
say that f is less concentrated than g, and we write f ≺ g if for all R > 0 we get
Z
Z
f (x)dx ≤
g(x)dx.
BR (0)
BR (0)
The partial order relationship ≺ is called comparison of mass concentrations. Of course, this
definition can be suitably adapted if f, g are radially symmetric and locally integrable functions on
a ball BR . The comparison of mass concentrations enjoys a nice equivalent formulation if f and g
are rearranged, whose proof we refer to [1, 69]:
Lemma 2.5. Let f, g ∈ L1+ (Rd ) be two nonnegative rearranged functions. Then f ≺ g if and only
if for every convex nondecreasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 we have
Z
Z
Φ(f (x)) dx ≤
Φ(g(x)) dx.
Ω
Ω
From this Lemma, it easily follows that if f ≺ g and f, g ∈ Lp (Rd ) are rearranged and nonnegative, then
kf kLp (Rd ) ≤ kgkLp (Rd ) ∀p ∈ [1, ∞].
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
8
Let us also observe that if f, g ∈ L1+ (Rd ) are nonnegative and rearranged, then f ≺ g if and only
if for all s ≥ 0 we have
Z s
Z s
∗
g ∗ (σ)dσ.
(2.12)
f (σ)dσ ≤
0
0
If f ∈ L1+ (Rd ), we denote by M2 [f ] the second moment of f , i.e.
Z
M2 [f ] :=
f (x)|x|2 dx.
(2.13)
Rd
In this regard, another interesting property which will turn out useful is the following
Lemma 2.6. Let f, g ∈ L1+ (Rd ) with kf kL1 (Rd ) = kgkL1 (Rd ) . If additionally g is rearranged and
f # ≺ g, then M2 [f ] ≥ M2 [g].
Proof.
Let us
consider the sequence of bounded radially increasing functions {ϕn }, where ϕn (x) =
min |x|2 , n is the truncation of the function |x|2 at the level n and define the function
hn = n − ϕn .
Then hn is nonnegative, bounded and rearranged. Thus using the Hardy-Littlewood inequality
(2.11) and [1, Corollary 2.1] we find
Z
Z
Z
f (x) ϕn (x)dx = nkf kL1 (Rd ) −
f (x) hn (x)dx ≥ nkf kL1 (Rd ) −
f # (x) hn (x)dx
d
d
d
R
R
ZR
Z
≥ nkgkL1 (Rd ) −
g(x) hn (x)dx =
g(x) ϕn (x)dx
Rd
Rd
Then passing to the limit as n → ∞ we find the desired result.
2.2. Continuous Steiner symmetrization. Although classical decreasing rearragement techniques are very useful to study properties of the minimizers and for solutions of the evolution
problem (1.1) in next sections, we do not know how to use them in connection with showing that
stationary states are radially symmetric. For an introduction of continuous Steiner symmetrization
and its properties, see [14, 15, 49]. In this subsection, we will use continuous Steiner symmetrization
to prove the following proposition.
Proposition 2.7. Let µ0 ∈ C(Rd ) ∩ L1+ (Rd ), and assume it is not radially decreasing after any
translation. Moreover, assume that there exist some C0 > 0 and m > 1, such that |∇µm−1
| ≤ C0
0
in supp µ0 . Then there exist some δ0 > 0, c0 > 0, C1 > 0 (depending on µ0 and W ) and a function
µ ∈ C([0, δ0 ] × Rd ) with µ(0, ·) = µ0 , such that µ satisfies the following for a short time τ ∈ [0, δ0 ],
where E is as given in (2.5):
E[µ(τ )] ≤ E[µ0 ] − c0 τ,
(2.14)
Z
Di
|µ(τ, x) − µ0 (x)| ≤ C1 µ0 (x)τ
for all x ∈ Rd ,
µ(τ, x) − µ0 (x)dx = 0 for any connected component Di of supp µ0 .
(2.15)
(2.16)
2.2.1. Definitions and basic properties of Steiner symmetrization. Let us first introduce the concept
of Steiner symmetrization for a measurable set E ⊂ Rd . If d = 1, the Steiner symmetrization of
E is the symmetric interval S(E) = {x ∈ R : |x| < |E|1 /2}. Now we want to define the Steiner
symmetrization of E with respect to a direction in Rd for d ≥ 2. The direction we symmetrize
corresponds to the unit vector e1 = (1, 0, . . . , 0), although the definition can be modified accordingly
when considering any other direction in Rd .
Let us label a point x ∈ Rd by (x1 , x0 ), where x0 = (x2 , . . . , xd ) ∈ Rd−1 and x1 ∈ R. Given
any measurable subset E of Rd we define, for all x0 ∈ Rd−1 , the section of E with respect to the
direction x1 as the set
Ex0 = {x1 ∈ R : (x1 , x0 ) ∈ E} .
9
Then we define the Steiner symmetrization of E with respect to the direction x1 as the set S(E)
which is symmetric about the hyperplane {x1 = 0} and is defined by
S(E) = (x1 , x0 ) ∈ Rd : x1 ∈ S(Ex0 ) .
In particular we have that |E|d = |S(E)|d .
Now, consider a non-negative function µ0 ∈ L1 (Rd ), for d ≥ 2. For all x0 ∈ Rd−1 , let us consider
the distribution function of µ0 (·, x0 ), i.e. the function
ζµ0 (h, x0 ) = |Uxh0 |1
for all h > 0, x0 ∈ Rd−1 ,
where
Uxh0 = {x1 ∈ R : µ0 (x1 , x0 ) > h}.
(2.17)
Then we can give the following definition:
Definition 2.8. We define the Steiner symmetrization (or Steiner rearrangement) of µ0 in the
direction x1 as the function Sµ0 = Sµ0 (x1 , x0 ) such that Sµ0 (·, x0 ) is exactly the Schwarz rearrangement of µ0 (·, x0 ) i.e. (see (2.9))
Sµ0 (x1 , x0 ) = sup {h > 0 : ζµ0 (h, x0 ) > 2|x1 |} .
As a consequence, the Steiner symmetrization Sµ0 (x1 , x0 ) is a function being symmetric about
the hyperplane {x1 = 0} and for each h > 0 the level set
{(x1 , x0 ) : Sµ0 (x1 , x0 ) > h}
is equivalent to the Steiner symmetrization
S({(x1 , x0 ) : µ0 (x1 , x0 ) > h})
which implies that Sµ0 and µ0 are equidistributed, yielding the invariance of the Lp norms when
passing from µ0 to Sµ0 , that is for all p ∈ [1, ∞] we have
kSµ0 kLp (Rd ) = kµ0 kLp (Rd ) .
Moreover, by the layer-cake representation formula, we have
Z ∞
Sµ0 (x1 , x0 ) =
χS(U h0 ) (x1 ) dh .
0
x
(2.18)
Now, we introduce a continuous version of this Steiner procedure via an interpolation between a
set or a function and their Steiner symmetrizations that we will use in our symmetry arguments
for steady states.
Definition 2.9. For an open set U ⊂ R, we define its continuous Steiner symmetrization M τ (U )
for any τ ≥ 0 as below. In the following we abbreviate an open interval (c − r, c + r) by I(c, r), and
we denote by sgn c the sign of c (which is 1 for positive c, −1 for negative c, and 0 if c = 0).
(1) If U = I(c, r), then
(
τ
M (U ) :=
I(c − τ sgn c, r)
I(0, r)
for 0 ≤ τ < |c|,
for τ ≥ |c|.
τ
N
τ
(2) If U = ∪N
i=1 I(ci , ri ) (where all I(ci , ri ) are disjoint), then M (U ) := ∪i=1 M (I(ci , ri )) for
τ
0 ≤ τ < τ1 , where τ1 is the first time two intervals M (I(ci , ri )) share a common endpoint.
Once this happens, we merge them into one open interval, and repeat this process starting from
τ = τ1 .
N
(3) If U = ∪∞
i=1 I(ci , ri ) (where all I(ci , ri ) are disjoint), let UN = ∪i=1 I(ci , ri ) for each N > 0,
τ
∞
τ
and define M (U ) := ∪N =1 M (UN ).
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
10
See Figure 1 for illustrations of M τ (U ) in the cases (1) and (2). Also, we point out that
case (3) can be seen as a limit of case (2), since for each N1 < N2 one can easily check that
M τ (UN1 ) ⊂ M τ (UN2 ) for all τ ≥ 0. Moreover, according to [15], the definition of M τ (U ) can be
extended to any measurable set U of R, since
∞
\
U=
On \ N,
n=1
being On ⊃ On+1 n = 1, 2, . . . , open sets and N a nullset.
t
t
t1
x
x
c1
c
c2
Figure 1. Illustrations of M τ (U ) when U is a single open interval (left), and
when U is the union of two open intervals (right).
In the next lemma we state four simple facts about M τ . They can be easily checked for case (1)
and (2) (hence true for (3) as well by taking the limit), and we omit the proof.
Lemma 2.10. Given any open set U ⊂ R, let M τ (U ) be defined in Definition 2.9. Then
(a)
(b)
(c)
(d)
M 0 (U ) = U , M ∞ (U ) = S(E).
|M τ (U )| = |U | for all τ ≥ 0.
If U1 ⊂ U2 , we have M τ (U1 ) ⊂ M τ (U2 ) for all τ ≥ 0.
M τ has the semigroup property: M τ +s U = M τ (M s (U )) for any τ, s ≥ 0 and open set U .
Once we have the continuous Steiner symmetrization for a one-dimensional set, we can define
the continuous Steiner symmetrization (in a certain direction) for a non-negative function in Rd .
Definition 2.11. Given µ0 ∈ L1+ (Rd ), we define its continuous Steiner symmetrization S τ µ0 (in
direction e1 = (1, 0, · · · , 0)) as follows. For any x1 ∈ R, x0 ∈ Rd−1 , h > 0, let
Z ∞
S τ µ0 (x1 , x0 ) :=
χM τ (U h0 ) (x1 )dh,
0
x
where Uxh0 is defined in (2.17).
For an illustration of S τ µ0 for µ0 ∈ L1 (R), see Figure 2.
Using the above definition, Lemma (2.10) and the representation (2.18) one immediately has
S 0 µ0 = µ0 ,
S ∞ µ0 = S µ0 .
Furthermore, it is easy to check that S τ µ0 = µ0 for all τ if and only if µ0 is symmetric decreasing
about the hyperplane H = {x1 = 0}. Below is the definition for a function being symmetric
decreasing about a hyperplane:
Definition 2.12. Let µ0 ∈ L1+ (Rd ). For a hyperplane H ⊂ Rd (with normal vector e), we say µ0
is symmetric decreasing about H if for any x ∈ H, the function f (τ ) := µ0 (x + τ e) is rearranged,
i.e. if f = f # .
11
µ0
S t µ0
x1
Figure 2. Illustrations of µ0 and S τ µ0 (for a small τ > 0).
Next we state some basic properties of S τ without proof, see [15, 41, 43] for instance.
Lemma 2.13. The continuous Steiner symmetrization S τ µ0 in Definition 2.11 has the following
properties:
(a) For any h > 0, |{S τ µ0 > h}| = |{µ0 > h}|. As a result, kS τ µ0 kLp (Rd ) = kµ0 kLp (Rd ) for all
1 ≤ p ≤ +∞.
(b) S τ has the semigroup property, that is, S τ +s µ0 = S τ (S s µ0 ) for any τ, s ≥ 0 and non-negative
µ0 ∈ L1 (Rd ).
Lemma 2.13 immediately implies that S[S τ µ0 ] is constant in τ , where S[·] is as given in (2.5).
2.2.2. Interaction energy under Steiner symmetrization. In this subsection, we will investigate
I[S τ µ0 ]. It has been shown in [15, Corollary 2] and [49, Theorem 3.7] that I[S τ µ0 ] is nonincreasing in τ . Indeed, in the case that µ0 is a characteristic function χΩ0 , it is shown in [56] that
I[S τ µ0 ] is strictly decreasing for τ small enough if Ω0 is not a ball. However, in order to obtain
(2.14) for a strictly positive c0 , some refined estimates are needed, and we will prove the following:
Proposition 2.14. Let µ0 ∈ C(Rd ) ∩ L1+ (Rd ). Assume the hyperplane H = {x1 = 0} splits the
mass of µ0 into half and half, and µ0 is not symmetric decreasing about H. Let I[·] be given in
(2.5), where W satisfies the assumptions (K1)-(K3). Then I[S τ µ0 ] is non-increasing in τ , and
there exists some δ0 > 0 (depending on µ0 ) and c0 > 0 (depending on µ0 and W ), such that
I[S τ µ0 ] ≤ I[µ0 ] − c0 τ
for all τ ∈ [0, δ0 ].
The building blocks to prove Proposition 2.14 are a couple of lemmas estimating how the interaction energy between two one-dimensional densities µ1 , µ2 changes under continuous Steiner
symmetrization for each of them. That is, we will investigate how
Z
IK [µ1 , µ2 ](τ ) :=
(S τ µ1 )(x)(S τ µ2 )(y)K(x − y)dxdy
(2.19)
R×R
changes in τ for a given one dimensional kernel K to be determined. We start with the basic case
where µ1 , µ2 are both characteristic functions of some open interval.
Lemma 2.15. Assume K(x) ∈ C 1 (R) is an even function with K0 (x) < 0 for all x > 0. For
i = 1, 2, let µi := χI(ci ,ri ) respectively, where I(c, r) is as given in Definition 2.9. Then the
following holds for the function I(τ ) := IK [µ1 , µ2 ](τ ) introduced in (2.19):
+
+
(a) ddτ I(0) ≥ 0. (Here ddτ stands for the the right derivative.)
(b) If in addition sgn c1 6= sgn c2 , then
d+
I(0) ≥ cw min{r1 , r2 }|c2 − c1 | > 0,
dτ
(2.20)
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
12
1|
, r1 + r2 + |c2 − c1 |].
where cw is the minimum of K0 (r) for r ∈ [ |c2 −c
2
Proof. By definition of S τ , we have S τ µi = χM τ (I(ci ,ri )) for i = 1, 2 and all τ ≥ 0. If sgn c1 = sgn c2 ,
the two intervals M τ (I(ci , ri )) are moving towards the same direction for small enough τ , during
d
which their interaction energy I(τ ) remains constant, implying dτ
I(0) = 0. Hence it suffices to
focus on sgn c1 6= sgn c2 and prove (2.20).
Without loss of generality, we assume that c2 > c1 , so that sgn c2 − sgn c1 is either 2 or 1. The
definition of M τ gives
Z r1 +c1 −τ sgn c1 Z r2 +c2 −τ sgn c2
K(x − y)dydx
I(τ ) =
−r1 +c1 −τ sgn c1
r1 Z r2
Z
=
−r1
−r2
−r2 +c2 −τ sgn c2
K(x − y + (c1 − c2 ) + τ (sgn c2 − sgn c1 ))dydx.
Taking its right derivative in τ yields
d+
I(0) = (sgn c2 − sgn c1 )
dτ
Z
r1
Z
r2
−r2
−r1
K0 (x − y + (c1 − c2 ))dydx.
+
Let us deal with the case r1 ≤ r2 first. In this case we rewrite ddτ I(0) as
Z
d+
I(0) = (sgn c2 − sgn c1 )
K0 (x − y)dxdy,
dτ
Q
(2.21)
where Q is the rectangle [−r1 , r1 ] × [−r2 + (c2 − c1 ), r2 + (c2 − c1 )], as illustrated in Figure 3. Let
Q− = Q ∩ {x − y > 0}, and Q+ = Q ∩ {x − y < 0}. The assumptions on K imply K0 (x − y) < 0 in
Q− , and K0 (x − y) > 0 in Q+ .
y
c2 −c1
{
2
D
r2 + c 2 − c 1
y=x
r2
Q̃+
−r1
Q
Q−
r1
x
−r2 + c2 − c1
Figure 3. Illustration of the sets Q, Q− , Q̃+ and D in the proof of Lemma 2.15.
+
1
Let Q̃+ := Q+ ∩ {y ≤ r2 }, and D := [−r1 , r1 ] × [r2 + c2 −c
2 , r2 + (c2 − c1 )]. (Q̃ and D are the
yellow set and green set in Figure 3 respectively). By definition, Q̃+ and D are disjoint subsets of
Q+ , so
!
Z
Z
Z
d+
0
0
0
I(0) ≥ (sgn c2 − sgn c1 )
K (x − y)dxdy + K (x − y)dxdy + K (x − y)dxdy . (2.22)
dτ
Q−
Q̃+
D
{z
}
|
{z
} |
{z
} |
≤0
≥0
>0
We claim that Q− K0 (x − y)dxdy + Q̃+ K0 (x − y)dxdy ≥ 0. To see this, note that Q− ∪ Q̃+ forms a
rectangle, whose center has a zero x-coordinate and a positive y-coordinate. Hence for any h > 0,
the line segment Q̃+ ∪ {x − y = −h} is longer than Q− ∪ {x − y = h}, which gives the claim.
R
R
13
Therefore, (2.22) becomes
d+
I(0) ≥ (sgn c2 − sgn c1 )
dτ
Z
D
K0 (x − y)dxdy ≥
Z
D
K0 (x − y)dxdy ≥ |D| min K0 (x − y)
(x,y)∈D
Note that D is a rectangle with area r1 (c2 − c1 ), and for any (x, y) ∈ D, we have y − x ∈
[|c2 − c1 |/2, r1 + r2 + |c2 − c1 |]. This finally gives
d+
|K0 (r)|.
I(0) ≥ r1 (c2 − c1 )
min
|c2 −c1 |
dτ
,r1 +r2 +|c2 −c1 |]
r∈[
2
Similarly, if r1 > r2 , then I 0 (0) can be written as (2.21) with Q̃ defined as [−r1 + (c2 − c1 ), r1 +
(c2 − c1 )] × [−r2 , r2 ] instead, and the above inequality would hold with the roles of r1 and r2
interchanged. Combining these two cases, we have
d+
I(0) ≥ cw min{r1 , r2 }|c2 − c1 |
dτ
for sgn c1 6= sgn c2 ,
1|
where cw is the minimum of |K0 (r)| for r ∈ [ |c2 −c
, r1 + r2 + |c2 − c1 |].
2
The next lemma generalizes the above result to open sets with finite measures.
Lemma 2.16. Assume K(x) ∈ C 1 (R) is an even function with K0 (r) < 0 for all r > 0. For open
sets U1 , U2 ⊂ R with finite measure, let µi := χUi for i = 1, 2, and I(τ ) := IK [µ1 , µ2 ](τ ) is as
defined in (2.19). Then
d
(a) dτ
I(τ ) ≥ 0 for all τ ≥ 0;
(b) In addition, assume that there exists some a ∈ (0, 1) and R > max{|U1 |, |U2 |} such that
|U1 ∩ ( |U21 | , R)| > a, and |U2 ∩ (−R, − |U22 | )| > a. Then for all τ ∈ [0, a/4], we have
where cw
d+
1
I(τ ) ≥
cw a3 > 0,
dτ
128
is the minimum of |K0 (r)| for r ∈ [ a4 , 4R].
(2.23)
Proof. It suffices to focus on the case when U1 , U2 both consist of a finite disjoint union of open
intervals, and for the general case we can take the limit. Recall that S τ µi = χM τ (Ui ) for i = 1, 2
and all τ ≥ 0.
To show (a), due to the semigroup property of S τ in Lemma 2.13, all we need to show is
≥ 0. By writing U1 , U2 each as a union of disjoint open intervals and expressing I(τ ) a
sum of the pairwise interaction energy, (a) immediately follows from Lemma 2.15(a).
d+
dτ I(0)
We will prove (b) next. First, we claim that
a
|U1 | a
A1 (τ ) := M τ (U1 ) ∩
+ , R >
2
4
4
a
for all τ ∈ [0, ].
4
(2.24)
|U1 |
To see this, note that A1 (0) > 3a
4 due to the assumption |U1 ∩ ( 2 , R)| > a. Since each interval
τ
in M (U1 ) moves with speed either 0 or ±1 at each τ , we know A01 (τ ) ≥ −2 for all τ , yielding the
claim. (Similarly, A2 (τ ) := |M τ (U2 ) ∩ (−R, − |U22 | − a4 )| > a4 for all τ ∈ [0, a4 ].)
Now we pick any τ0 ∈ [0, a4 ], and we aim to prove (2.23) at this particular time τ0 . At τ = τ0 ,
1 1
1 1
1
write M τ0 (U1 ) := ∪N
k=1 I(ck , rk ), where all intervals I(ck , rk ) are disjoint, and none of them share
common endpoints – if they do, we merge them into one interval.
Note that for every x ∈ M τ0 (U1 )∩( |U21 | + a4 , R), x must belong to some I(c1k , rk1 ) with a/4 ≤ c1k ≤
R + |U1 |/2. Otherwise, the length of I(c1k , rk1 ) would exceed |U1 |, contradicting Lemma 2.10(a).
We then define
n
o
a
I1 := 1 ≤ k ≤ N1 : ≤ c1k ≤ R + |U1 |/2 .
4
14
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
P
Combining the above discussion with (2.24), we have k∈I1 |I(c1k , rk1 )| ≥ a/4, i.e.
X
a
rk1 ≥ .
8
(2.25)
k∈I1
2 2
2
Likewise, let M τ0 (U2 ) := ∪N
k=1 I(ck , rk ), and denote
P by I2 the set of indices k such that
a
2
−R − |U2 |/2 ≤ ck ≤ − 4 , and similarly we have k∈I2 rk2 ≥ a/8.
The semigroup property of M τ in Lemma 2.10 gives that for all s > 0,
1 1
1
M τ0 +s (U1 ) = M s (M τ0 (U1 )) = M s (∪N
k=1 I(ck , rk )).
Since none of the intervals I(c1k , rk1 ) share common endpoints, we have
N1
1 1
s
1 1
1
M s (∪N
k=1 I(ck , rk )) = ∪k=1 M (I(ck , rk ))
A similar result holds for M
τ0 +s
for sufficiently small s > 0.
(U2 ), hence we obtain for sufficiently small s > 0:
I(τ0 + s) = IK [χM τ0 (U1 ) , χM τ0 (U2 ) ](s) =
N2
N1 X
X
k=1 l=1
IK [χI(c1k ,rk1 ) , χI(c2l ,rl2 ) ](s).
Applying Lemma 2.15(a) to the above identity yields
N2
N1 X
X
X X d
d+
d
I(τ0 ) =
IK [χI(c1k ,rk1 ) , χI(c2l ,rl2 ) ]
IK [χI(c1k ,rk1 ) , χI(c2l ,rl2 ) ]
. (2.26)
≥
dτ
ds
ds
s=0
s=0
{z
}
k=1 l=1
k∈I1 l∈I2 |
=:Tkl
Next we will obtain a lower bound for Tkl . By definition of I1 and I2 , for each k ∈ I1 and l ∈ I2
we have that c1k ≥ a4 and c2l ≤ − a4 , hence |c2l − c1k | ≥ a2 . Thus Lemma 2.15(b) yields
a
Tkl ≥ cw min{rk1 , rl2 }|c2l − c1k | ≥ cw min{rk1 , rl2 } for k ∈ I1 , l ∈ I2 ,
2
0
a
where cw = minr∈[ 4 ,4R] |W (r)|. (Here we used that for k ∈ I1 , l ∈ I2 , we have rk1 +rl2 +|c2l −c1k | ≤
|U1 |/2 + |U2 |/2 + (R + |U1 |/2) + (R + |U2 |/2) ≤ 4R, due to the assumption R > max{|U1 |, |U2 |}.)
Plugging the above inequality into (2.26) and using min{u, v} ≥ min{u, 1} min{v, 1} for u, v > 0,
we have
d+
acw X X
I(τ0 ) ≥
min{rk1 , 1} min{rl2 , 1}
dτ
2
k∈I1 l∈I2
!
!
X
acw X
1
2
=
min{rk , 1}
min{rl , 1}
2
k∈I1
l∈I2
(
)
(
)
X
X
acw
≥
min 1,
rk1 min 1,
rl2
2
k∈I1
l∈I2
n a o2
acw
1
3
≥
min 1,
≥
cw a ,
2
8
128
here we applied (2.25) in the second-to-last inequality, and used the assumption a ∈ (0, 1) for the
last inequality. Since τ0 ∈ [0, a/4] is arbitrary, we can conclude.
Now we are ready to prove Proposition 2.14.
Proof of Proposition 2.14. Since µ0 ∈ C(Rd ) ∩ L1+ (Rd ) is not symmetric decreasing about H =
{x1 = 0}, we know that there exists some x0 ∈ Rd−1 and h > 0, such that Uxh0 := {x1 ∈ R :
µ0 (x1 , x0 ) > h} has finite measure, and its difference from (−|Uxh0 |/2, |Uxh0 |/2) has nonzero measure.
For R > 0, a > 0, define
B1R,a = (x0 , h) ∈ Rd−1 × (0, +∞) : Uxh0 ∩ (|Uxh0 |/2, R) > a, |x0 | ≤ R ,
15
B2R,a = (x0 , h) ∈ Rd−1 × (0, +∞) : Uxh0 ∩ (−R, −|Uxh0 |/2) > a, |x0 | ≤ R .
Our discussion above yields that at least one of B1R,a and B2R,a is nonempty when R is sufficiently
large and a > 0 sufficiently small (hence at least one of them must have nonzero measure by
continuity of µ0 ). Indeed, using the fact that H splits the mass of µ0 into half and half, we can
choose R sufficiently large and a > 0 sufficiently small (both of them depend on µ0 only), such
that both B1R,a and B2R,a have nonzero measure in Rd−1 × (0, +∞).
√
Now, let us define a one-dimensional kernel Kl (r) := − 21 W ( r2 + l2 ). Note that for any l > 0,
the kernel Kl ∈ C 1 (R) is even in r, and Kl0 (r) < 0 for all r > 0. By definition of S τ , we can rewrite
I[S τ µ0 ] as
Z
Z
Z
τ
I[S µ0 ] = −
χM τ (U h1 ) (x1 )χM τ (U h2 ) (y1 ) K|x0 −y0 | (|x1 − y1 |)dx1 dy1 dx0 dy 0 dh1 dh2 .
x0
(R+ )2 R2(d−1) R2
y0
Thus using the notation in (2.19), I[S τ µ0 ] can be rewritten as
Z
Z
I[S τ µ0 ] = −
IK|x0 −y0 | [χU h1 , χU h2 ](τ ) dx0 dy 0 dh1 dh2 ,
(R+ )2
R2(d−1)
x0
y0
and taking its right derivative (and applying Lemma 2.16(a)) yields
Z
Z
d
d+
IK
[χ h1 , χ h2 ](τ ) dy 0 dh2 dx0 dh1 .
− I[S τ µ0 ] ≥
R,a
R,a
dτ
dτ |x0 −y0 | Ux0 Uy0
(x0 ,h1 )∈B1
(y 0 ,h2 )∈B2
(2.27)
By definition of B1R,a and B2R,a , for any (x0 , h1 ) ∈ B1R,a and (y 0 , h2 ) ∈ B2R,a , we can apply
Lemma 2.16(b) to obtain
1
d+
IK
[χ h1 , χ h2 ](τ ) ≥
cw a3 for any τ ∈ [0, a/4],
dτ |x0 −y0 | Ux0 Uy0
128
0
where cw is the minimum of |K|x
0 −y 0 | (r)| in [a/4, 4R]. By definition of Kl (r), we have
p
r
0
0
1
2
0
0 2 p
K|x
.
0 −y 0 | (r) =
2 W ( r + |x − y | )
r2 + |x0 − y 0 |2
Using |x0 | ≤ R and |y 0 | ≤ R (due to definition of B1 , B2 ), we have √
r ∈ [a/4, 4R], hence cw ≥
a
40R
minr∈[ a4 ,4R] W 0 (r).
r
r 2 +|x0 −y 0 |2
≤
(2.28)
a
20R
for all
Plugging (2.28) (with the above cw ) into (2.27) finally yields
−
d+
1
I[S τ µ0 ] ≥
|B R,a ||B2R,a | min
W 0 (r)a4 > 0
dτ
6000 1
r∈[ a
4 ,4R]
hence we can conclude the desired estimate.
for all τ ∈ [0, a/4],
2.2.3. Proof of Proposition 2.7. In the statement of Proposition 2.7, we assume that µ0 is not radially decreasing up to any translation. Since Steiner symmetrization only deals with symmetrizing
in one direction, we will use the following simple lemma linking radial symmetry with being symmetric decreasing about hyperplanes. Although the result is standard (see [34, Lemma 1.8]), for
the sake of completeness we include here the details of the proof.
Lemma 2.17. Let µ0 ∈ C(Rd ). Suppose for every unit vector e, there exists a hyperplane H ⊂ Rd
with normal vector e, such that µ0 is symmetric decreasing about H. Then µ0 must be radially
decreasing up to a translation.
Proof. For i = 1, . . . , d, let ei be the unit vector with i-th coordinate 1 and all the other coordinates
0. By assumption, for each i, there exists some hyperplane Hi with normal vector ei , such that µ0
is symmetric decreasing about Hi . We then represent each Hi as {(x1 , . . . , xd ) : xi = ai } for some
ai ∈ R, and then define a ∈ Rd as a := (a1 , . . . , ad ). Our goal is to prove that µ0 (· − a) is radially
decreasing.
16
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
We first claim that µ0 (x) = µ0 (2a−x) for all x ∈ Rd . For any hyperplane H ⊂ Rd , let TH : Rd →
R be the reflection about the hyperplane H. Since µ0 is symmetric with respect to H1 , . . . , Hd , we
have µ0 (x) = µ0 (THi x) for x ∈ Rd and all i = 1, . . . , d, thus µ0 (x) = µ0 (TH1 . . . THd x) = µ0 (2a−x).
d
The claim implies that every hyperplane H passing through a must split the mass of µ0 into half
and half. Denote the normal vector of H by e. By assumption, µ0 is symmetric decreasing about
some hyperplane H 0 with normal vector e. The definition of symmetric decreasing implies that
H 0 is the only hyperplane with normal vector e that splits the mass into half and half, hence H 0
must coincide with H. Thus µ0 is symmetric decreasing about every hyperplane passing through
a, hence we can conclude.
Proof of Proposition 2.7. Since µ0 is not radially decreasing up to any translation, by Lemma 2.17,
there exists some unit vector e, such that µ0 is not symmetric decreasing about any hyperplane
with normal vector e. In particular, there is a hyperplane H with normal vector e that splits the
mass of µ0 into half and half, and µ0 is not symmetric decreasing about H. We set e = (1, 0, . . . , 0)
and H = {x1 = 0} throughout the proof without loss of generality.
Note that if we set µ(τ, ·) = S τ µ0 , then it directly satisfies (2.14) for a short time, since I[S τ µ0 ]
is decreasing at least linearly for a short time by Proposition 2.14, and we also have S[S τ µ0 ] is
constant in τ . However, S τ µ0 does not satisfy (2.15) and (2.16). To solve this problem, we will
modify S τ µ0 into S̃ τ µ0 , where we make the set Uxh0 := {x1 ∈ R : µ0 (x1 , x0 ) > h} travels at speed
v(h) rather than at constant speed 1, with v(h) given by


h ≥ h0 ,
1
m−1
(2.29)
v(h) :=
h

0 < h < h0 ,

h0
for some sufficiently small constant h0 > 0 to be determined later. More precisely, we define
µ(τ, ·) = S̃ τ µ0 as
Z ∞
S̃ τ µ0 (x1 , x0 ) :=
χM v(h)τ (U h0 ) (x1 )dh
x
0
with v(h) as in (2.29), and our goal is to show that such µ(τ, ·) satisfies (2.14), (2.15) and (2.16)
for small enough τ . For an illustration on the difference between S τ µ0 and S̃ τ µ0 , see Figure 4.
µ0
h0
S t µ0
S̃ t µ0
Figure 4. A sketch on µ0 (grey), S τ µ0 (blue) and S̃ τ µ0 (red dashed) for a small
τ > 0.
Let us first prove that for any h0 > 0, µ(τ, ·) satisfies (2.15) and (2.16) for τ ∈ [0, δ1 ], where
δ1 = δ1 (m, h0 , C0 ) > 0. To show (2.16), note that the assumption |∇(µ0m−1 )| ≤ C0 directly leads to
the following: for any x, y ∈ Rd with µ0 (x) ≥ h > 0 and µ0 (y) = 0, we have that |x−y| ≥ hm−1 /C0 .
This implies that for any connected component Di ⊂ supp µ0 ,
dist ({µ0 > h} ∩ Di , ∂Di ) ≥
hm−1
C0
for all h > 0.
(2.30)
17
Now define Di,x0 as the one-dimensional set {x1 ∈ R : (x1 , x0 ) ∈ Di }. The inequality (2.30) yields
M v(h)τ (Uxh0 ∩ Di,x0 ) ⊂ Di,x0
for all x0 ∈ Rd−1 , h > 0, τ ≤
hm−1
,
C0 v(h)
and note that for any h > 0, we have hm−1 /(C0 v(h)) ≥ hm−1
/C0 by definition of v(h). Using the
0
τ
v(h)τ
above equation, the definition of S̃ and the fact that M
is measure-preserving, we have that
(2.16) holds for all τ ≤ hm−1
/C0 .
0
Next we prove (2.15). Let us fix any y = (y1 , y 0 ) ∈ Rd , and denote h = µ0 (y). Using |∇µm−1
|≤
0
C0 , we have that for any λ > 1,
dist(y1 , Uyλh
0 ) ≥
for all τ ≤
So we have y1 6∈ M v(λh)τ Uyλh
0
(λm−1 −1)hm−1
0
C0 λm−1
(λm−1 − 1)hm−1
.
C0
(λm−1 −1)hm−1
,
C0 v(λh)
due to the fact that v(λh) ≤ (λh/h0 )
that µ0 (y) = h, the following holds for all λ > 1:
S̃ τ [µ0 ](y) ≤ λµ0 (y)
m−1
which is uniformly bounded below by
for all h. By definition of S̃ τ and the fact
for all τ ≤
(λm−1 − 1)hm−1
0
.
C0 λm−1
Note that there exists c1m > 0 only depending on m, such that λm−1 − 1 ≥ c1m (λ − 1) for all
1 < λ < 2. Hence for all 1 < λ < 2 we have
S̃ τ [µ0 ](y) − µ0 (y) ≤ (λ − 1)µ0 (y)
for τ =
c1m hm−1
0
(λ − 1),
C0 2m−1
and this directly implies
S̃ τ [µ0 ](y) − µ0 (y) ≤
C0 2m−1
µ0 (y)τ
c1m h0m−1
c
Similarly, for any 0 < η < 1 we have dist(y1 , (Uyηh
0 ) ) ≥
as above gives us
S̃ τ [µ0 ](y) ≥ ηµ0 (y)
for all τ ≤
for all τ ≤
c1m hm−1
0
.
C0 2m−1
(1−η m−1 )hm−1
,
C0
(2.31)
and an identical argument
(1 − η m−1 )hm−1
0
.
C0 η m−1
Now we let c2m > 0 be such 1 − η m−1 ≥ c2m (1 − η) for all 21 < η < 1. Hence we have S̃ τ [µ0 ](y) −
c2 hm−1
µ0 (y) ≥ −(1 − η)µ0 (y) for τ = m 0
(1 − η), which implies
C0
S̃ τ [µ0 ](y) − µ0 (y) ≥ −
C0
µ0 (y)τ
2
cm hm−1
0
for all τ ≤
c2m hm−1
0
.
2C0
(2.32)
Combining (2.31) and (2.32) together, we have that for any h0 > 0, (2.15) holds for some C1 for
all τ ∈ [0, δ1 ], where both C1 > 0 and δ1 > 0 depend on C0 , h0 and m.
Finally, we will show that (2.14) holds for µ(τ ) = S̃ τ [µ0 ] if we choose h0 > 0 to be sufficiently
small. First, we point out that S[S̃ τ µ0 ] is not preserved for all τ . This is because when different
level sets are moving at different speed v(h), we no longer have that M v(h1 )τ (Uxh01 ) ⊂ M v(h2 )τ (Uxh02 )
for all h1 > h2 . Nevertheless, we claim it is still true that
S[S̃ τ µ0 ] ≤ S[µ0 ] for all τ ≥ 0.
τ
v(h)τ
(2.33)
To see this, note that the definition of S̃ and the fact that M
is measure preserving give us
τ
{S̃ µ0 > h} ≤ |{µ0 > h}| for all h > 0, τ ≥ 0,
R
R
regardless of the definition of v(h). This implies that f (S̃ τ µ0 (x))dx ≤ f (µ0 (x))dx for any
convex increasing function f , yielding (2.33). Indeed, using the regularity assumption |∇µm−1
|≤
0
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
18
C0 and the particular v(h) in (2.29), one can show that S[S̃ τ µ0 ] is preserved for small enough τ ,
but we will not pursue in this direction since (2.33) is sufficient for us.
Due to (2.33) and the fact that E[·] = S[·] + I[·], in order to prove (2.14), it suffices to show
I[S̃ τ µ0 ] ≤ I[µ0 ] − c0 τ
for τ ∈ [0, δ0 ], for some c0 > 0 and δ0 > 0.
(2.34)
τ
Recall that Proposition 2.14 gives that I[S µ0 ] ≤ I[µ0 ] − cτ for τ ∈ [0, δ] with some c > 0 and
δ > 0. As a result, to show (2.34), all we need is to prove that if h0 > 0 is sufficiently small, then
cτ
for all τ .
(2.35)
I[S̃ τ µ0 ] − I[S τ µ0 ] ≤
2
To show (2.35), we first split S τ µ0 as the sum of two integrals in h ∈ [h0 , ∞) and h ∈ [0, h0 ):
Z ∞
Z h0
τ
0
S µ0 (x1 , x ) =
χM τ (U h0 ) (x1 )dh +
χM τ (U h0 ) (x1 )dh =: f1 (τ, x) + f2 (τ, x).
(2.36)
h0
x
x
0
We then split S̃ τ µ0 similarly, and since v(h) = 1 for all h > h0 we obtain
Z h0
S̃ τ µ0 (x1 , x0 ) = f1 (τ, x) +
χM v(h)τ (U h0 ) (x1 )dh =: f1 (τ, x) + f˜2 (τ, x).
x
0
(2.37)
For any τ ≥ 0, we have kf1 (τ, ·)kL∞ (Rd ) ≤ kµ0 kL∞ (Rd ) , while kf2 (τ, ·)kL∞ (Rd ) and kf˜2 (τ, ·)kL∞ (Rd )
are both bounded by h0 . As for the L1 norm, we have that kf1 (τ, ·)kL1 (Rd ) ≤ kµ0 kL1 (Rd ) , and
Z
kf2 (τ, ·)kL1 (Rd ) = kf˜2 (τ, ·)kL1 (Rd ) =
min{µ0 (x), h0 }dx =: mµ0 (h0 ),
Rd
where mµ0 (h0 ) approaches 0 as h0 & 0.
d
Also, since v(h) ≤ 1, we know that for each τ ≥ 0, there is a transport map T (τ,
R ·) : [0, ∞)×R →
˜
˜
R
R with supx∈Rd |T (τ, x)−x| ≤ 2τ , such that T (τ, ·)#f2 (τ, ·) = f2 (τ, ·) (that is, f2 (τ, x)ϕ(x)dx =
f2 (τ, x)ϕ(T (τ, x))dx for any measurable function ϕ). Indeed, since the level sets of f2 are traveling at speed 1 and the level sets of f˜2 are traveling with speed v(h), for each τ we can find a
transport plan between them with maximal displacement L∞ distance at most 2τ in its support.
Since these densities are both in L∞ , there is an optimal transport map for the ∞-Wasserstein
such that |T (τ, x) − x| ≤ 2τ , see [29].
d
Using the decompositions (2.36), (2.37) and the definition of I[·], we obtain, omitting the τ
dependence on the right hand side,
Z
Z
τ
τ
˜
S̃
µ
]
−
I[S
µ
]
≤
f
(W
∗
f
)dx
−
f
(W
∗
f
)dx
I[
2
1
2
1
0
0 {z
}
|
=:A1 (τ )
Z
Z
1 ˜
˜
+ f2 (W ∗ f2 )dx − f2 (W ∗ f2 )dx,
2
|
{z
}
=:A2 (τ )
and we will bound A1 (τ ) and A2 (τ ) in the following. For A1 (τ ), denote Φ(τ, ·) =: W ∗ f1 (τ, ·), and
using the L∞ , L1 bounds on f1 and the assumptions (K2),(K3), we proceed in the same way as in
(2.4) to obtain that k∇ΦkL∞ (Rd ) ≤ C = C(kµ0 kL∞ (Rd ) , kµ0 kL1 (Rd ) , Cw , d).
Using that T (τ, ·)#f2 (τ, ·) = f˜2 (τ, ·), we can rewrite A1 (τ ) as
Z
A1 (τ ) = f2 (x) Φ(x) − Φ(T (τ, x)) dx
≤ kf2 (τ )kL1 (Rd ) sup |Φ(x) − Φ(T (τ, x))| ≤ mµ0 (h0 )k∇ΦkL∞ (Rd ) 2τ
x∈Rd
≤ mµ0 (h0 )C(kµ0 kL∞ (Rd ) , kµ0 kL1 (Rd ) , Cw , d)τ,
19
where the coefficient of τ can be Rmade arbitrarilyRsmall by choosing h0 sufficiently small. To control
A2 (τ ), we first use the identity f (W ∗ g)dx = g(W ∗ f )dx to bound it by
Z
Z
Z
Z
˜
˜
˜
˜
A2 (τ ) ≤ f2 (W ∗ f2 )dx − f2 (W ∗ f2 )dx + f2 (W ∗ f2 )dx − f2 (W ∗ f2 )dx ,
and both terms can be controlled in the same way as A1 (τ ), since both Φ2 := W ∗f2 and Φ̃2 := W ∗f˜2
satisfy the same estimate as Φ. Combining the estimates for A1 (τ ) and A2 (τ ), we can choose h0 > 0
sufficiently small, depending on µ0 and W , such that equation (2.35) would hold for all τ , which
finishes the proof.
2.3. Proof of Theorem 2.2.
Proof. Towards a contradiction, assume there is a stationary state ρs that is not radially decreasing.
Due to Lemma 2.3, we have that ρs ∈ C(Rd ) ∩ L1+ (Rd ), and |∇ρm−1
| ≤ C0 in supp ρs for some
s
C0 > 0. This enables us to apply Proposition 2.7 to ρs , hence there exists a continuous family of
µ(τ, ·) with µ(0, ·) = ρs and constants C1 > 0, c0 > 0, δ0 > 0, such that the following holds for all
τ ∈ [0, δ0 ]:
E[µ(τ )] − E[ρs ] ≤ −c0 τ,
(2.38)
Z
Di
|µ(τ, x) − ρs (x)| ≤ C1 ρs (x)τ
for all x ∈ Rd ,
µ(τ, x) − ρs (x)dx = 0 for any connected component Di of supp ρs .
(2.39)
(2.40)
Next we will use (2.39) and (2.40) to directly estimate E[µ(τ )] − E[ρs ], and our goal is to show that
there exists some C2 > 0, such that
E[µ(τ )] − E[ρs ] ≤ C2 τ 2 for τ sufficiently small.
(2.41)
We then directly obtain a contradiction between (2.38) and (2.41) for sufficiently small τ > 0.
Let g(τ, x) := µ(τ, x) − ρs (x). Due to (2.39), we have |g(τ, x)| ≤ C1 ρs (x)τ for all x ∈ Rd and
τ ∈ [0, δ0 ]. Hence supp g(τ ) ⊂ supp ρs for such τ . Using the notation g(τ, x), we have the following
(where in the integrand we omit the x dependence, due to space limitations)
Z
Z
((ρs + g(τ ))m − ρm
1
s )
E[µ(τ )] − E[ρs ] =
dx +
(ρs + g(τ )) W ∗ (ρs + g(τ )) − ρs (W ∗ ρs )dx
m−1
2
m
Z
Z
ρm
g(τ )
1
s
=
1+
− 1 dx + g(τ )(W ∗ ρs ) + g(τ )(W ∗ g(τ ))dx
m
−
1
ρ
2
s
supp ρs
{z
}
|
:=T (τ,x)
(2.42)
Recall that for all |a| < 1/2, we have the elementary inequality
(1 + a)m − 1 − ma ≤ C(m)a2 for some C(m) > 0.
Since for all x ∈ supp ρs and τ ≤ min{δ0 , C1 /2} we have |g(τ, x)/ρs (x)| ≤ 21 , we can replace a
1
by g(x)/ρs (x) in the above inequality, then multiply m−1
ρm
s to both sides to obtain the following
(with C2 (m) = C(m)/(m − 1)):
T (τ, x) − m g(τ, x)ρs (x)m−1 ≤ C2 (m)ρm−2
g(τ )2 .
s
m−1
Applying this to (2.42), we have the following for all τ ≤ min{δ0 , C1 /2}:
Z
Z
1
m m−1
E[µ(τ )] − E[ρs ] ≤ ρ
+
W
∗
ρ
dx
+
g(τ
)(W
∗
g(τ
))dx
g(τ
)
s
s
m−1
2
supp ρs
Z
+ C2 (m) ρm−2
g(τ )2 dx
s
=: I1 + I2 + I3 .
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
20
m
ρsm−1 + W ∗ ρs = Ci in each connected
Since ρs is a steady state solution, from (2.8) we have m−1
component Di ⊂ supp ρs , hence I1 ≡ 0 for all τ ∈ [0, δ0 ] due to (2.40) and the definition of g(τ, ·).
For I2 and I3 , since |g(τ, x)| ≤ C1 ρs (x)τ for τ ∈ [0, δ0 ], we directly have
Z
1
I2 ≤ C12 τ 2 |ρs (W ∗ ρs )|dx ≤ M τ 2 ,
2
Z
2 2
2
I3 ≤ C2 (m)C1 τ
ρm
s dx ≤ M τ ,
for some
on kρs kL1 (Rd ) , kρs kL∞ (Rd ) , m and d, and putting them together finally
M > 0 depending
gives E[µ(τ )] − E[ρs ] ≤ 2M τ 2 for all τ ≤ min{δ0 , C1 /2}, hence we can conclude.
3. Existence of global minimizers
In Section 2, we showed that if ρs ∈ L1+ (Rd ) ∩ L∞ (Rd ) is a stationary state of (1.1) in the sense
of Definition 2.1 and it satisfies ω(1 + |x|)ρs ∈ L1 (Rd ), then it must be radially decreasing up to
a translation. This section is concerned with the existence of such stationary solutions. Namely,
under (K1)-(K4) and one of the extra assumptions (K5) or (K6) below, we will show that for
any given mass, there indeed exists a stationary solution satisfying the above conditions. We will
generalize the arguments of [22] to show that there exists a radially decreasing global minimizer ρ
of the functional (2.5) given by
Z
Z Z
1
1
ρm dx +
W (x − y) ρ(x)ρ(y) dx dy
E[ρ] =
m − 1 Rd
2 Rd Rd
over the class of admissible densities
Z
YM := ρ ∈ L1+ (Rd ) ∩ Lm (Rd ) : kρkL1 (Rd ) = M,
R2
xρ(x) dx = 0, ω(1 + |x|) ρ(x) ∈ L1 (Rd ) ,
and with the potential satisfying at least (K1)-(K4). Note that the condition on the zero center of
mass has to be understood in the improper integral sense, i.e.
Z
Z
xρ(x) dx = lim
xρ(x) dx = 0
R2
R→∞
|x|<R
since we do not assume that the first moment is bounded in the class YM . In order to avoid loss of
mass at infinity, we need to assume some growth condition at infinity. In this section, we will obtain
the existence of global minimizers under two different conditions related to the works [52, 4, 22],
and show that such global minimizers are indeed L1 and L∞ stationary solutions. Namely, we
assume further that the potential W satisfies at infinity either the property
(K5)
lim ω+ (r) = +∞,
r→+∞
or
(K6)
lim ω+ (r) = ` ∈ (0, +∞) where the non-negative potential K := ` − W is such that, in the
r→+∞
case m > 2, K ∈ Lp̂ (Rd \ B1 (0)), for some 1 ≤ p̂ < ∞, while for the case 2 − (2/d) < m ≤ 2
we will require that K ∈ Lp,∞ (Rd \ B1 (0)), for some 1 ≤ p < ∞. Moreover, there exists an
α ∈ (0, d) for which m > 1 + α/d and
K(τ x) ≥ τ −α K(x),
∀τ ≥ 1, for a.e. x ∈ Rd .
(3.1)
Here, we denote by Lp,∞ (Rd ) the wea-Lp or Marcinkiewicz space of index 1 < p < ∞. In
particular, the attractive Newtonian potential (which is the fundamental solution of −∆ operator
in Rd ) is covered by these assumptions: for d = 1, 2 it satisfies (K5), whereas for d ≥ 3 it satisfies
(K6) with α = d − 2.
21
Notice that the subadditivity-type condition (K4) allows to claim that E[ρ] is finite over the
class YM : indeed if we split the W into its positive part W+ and negative part W− as done in the
bound of ψs in Section 2, the integral with kernel W− is finite by the HLS inequality, see (3.2)
below, while by (K4) we infer
Z Z
Z Z
W+ (x − y) ρ(x)ρ(y) dx dy =
ω+ (|x − y|) ρ(x)ρ(y) dx dy
Rd Rd
Rd |x−y|≥1
Z
≤ CM 2 + 2M
ω(1 + |x|)ρ(x)dx.
Rd
3.1. Minimization of the Free Energy functional. The existence of minimizers of the functional E can be proven with different arguments according to the choice between condition (K5) or
(K6): indeed, (K5) produces a quantitative version of the mass confinement effect while (K6) does
it in a nonconstructive way. For such a difference, we first briefly discuss the case when condition
(K6) is employed, as it can be proven by a simple application of Lion’s concentration-compactness
principle [52] and its variant in [4].
Theorem 3.1. Assume that conditions (K1)-(K4) and (K6) hold. Then for any positive mass M ,
there exists a global minimizer ρ0 , which is radially symmetric and decreasing, of the free energy
functional E in YM . Moreover, all global minimizers are radially symmetric and decreasing.
e + ` M 2 , where
Proof. We write E[ρ] = E[ρ]
2
Z
Z Z
1
e = 1
E[ρ]
ρm dx −
K(x − y) ρ(x)ρ(y) dx dy,
m − 1 Rd
2 Rd Rd
being the kernel K nonnegative and radially decreasing; furthermore condition (K3) implies K ∈
Lp,∞ (B1 (0)), where p = d/(d − 2). Then we are in position to apply [4, Theorem 1] for m > 2 and
[52, Corollary II.1] for 2 − (2/d) < m ≤ 2 to get the existence of a radially decreasing minimizer
ρ0 ∈ YM of Ee (and then of E). Moreover, since K is strictly radially decreasing, all global minimizers
are radially decreasing.
When considering the presence of condition (K5) a direct control of the mass confinement
phenomenon is possible and the concentration-compactness principle is not applicable. Then we
first prove the following Lemma, which provides a reversed Riesz inequality, allowing to reduce the
study the minimization of E to the set of all the radially decreasing density in YM .
Lemma 3.2. Assume that conditions (K1)-(K5) hold and take a density ρ such that
ρ ∈ L1+ (Rd ), ω(1 + |x|) ρ(x) ∈ L1 (Rd ).
Then the following inequality holds:
Z Z
Z
I[ρ] =
W (x − y)ρ(x)ρ(y)dx dy ≥
R2
R2
R2
Z
R2
W (x − y)ρ# (x)ρ# (y)dx dy = I[ρ# ]
and the equality occurs only if and only if ρ is a translate of ρ# .
Proof. The proof proceeds exactly as in [21, Lemma 2], up to replacing the function k(r) defined
there by the function

if r ≤ r0
 −ω(r)
κ(r) =
Rr

1+r 2
−ω(r0 ) − r0 ω 0 (s) 1+s02 ds
if r > r0 ,
being r0 > 0 fixed.
Theorem 3.3. Assume that (K1)-(K5) hold, then the conclusions of Theorem 3.1 remain true.
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
22
Proof. We follow the main lines of [22, Theorem 2.1]. By Lemma 3.2 we can restrict ourselves to
consider only radially decreasing densities ρ. In order to show that I[ρ] is bounded from below,
we first argue in the case d ≥ 3. Thanks to conditions (K1)-(K2) we have
Z Z
Z Z
Z Z
ρ(x)ρ(y)
ρ(x)ρ(y)
dx dy ≥ −C
dx dy.
W (x − y)ρ(x)ρ(y)dx dy ≥ −C
d−2
d−2
Rd Rd |x − y|
Rd Rd
Rd |x−y|≤1 |x − y|
Now we observe that
2d
<m
d+2
= 2, then by the classical HLS and Lp interpolation inequalities, we find
1<
d+2
+ d+2
2d + 2d
Z Z
2(1−α)
I[ρ] =
W (x − y)ρ(x)ρ(y)dx dy ≥ −Ckρk2L2d/(d+2) (Rd ) ≥ −Ckρk2α
L1 (Rd ) kρkLm (Rd ) ,
and
d−2
d
where α =
Rd Rd
1
d+2
m−1 m 2d
(3.2)
− 1 . Then by (3.2) we find that
1
2(1−α)
2α
kρkm
kρkLm (Rd )
(3.3)
Lm (Rd ) − CM
m−1
where we notice that m > 2(1 − α) if and only if m > 2 − d2 , which is the diffusion dominated
regime we work with. Then by (3.3) we can find a constant C1 > 0 and a sufficiently large constant
C2 such that
E[ρ] ≥ −C1 + C2 kρkm
Lm (Rd ) .
E[ρ] ≥
Concerning the case d = 2, we observe that conditions (K1)-(K2) yields
Z Z
Z
Z
W (x − y)ρ(x)ρ(y)dx dy ≥ −C
dx
log(|x − y|)ρ(x)ρ(y)dxdy
R2 R2
R2
|x−y|≤1
Z
Z
≥ −C
dx
log(|x − y|)ρ(x)ρ(y)dxdy
R2
R2
and we can use the classical log-HLS inequality and the arguments of [22] to conclude.
Concerning the mass confinement, due to (K5) and the same arguments in [22], see also Lemma
4.16, allow us to show
Z
C
ρ(x) dx ≤
−→ 0 .
R→∞
ω(R)
|x|>R
Finally, we should check that the interaction potential W is lower semicontinuous as shown in
[22, page8]. Indeed, the only technical point to verify in this more general setting relates to the
control of the truncated interaction potential Aε for d ≥ 3. Notice that we can estimate due to
(2.3)
Z Z
Z Z
ρ(x)ρ(y)
ε
|A [ρ]| := W (x − y)ρ(x)ρ(y)dxdy ≤ C
dxdy
d−2
Rd |x−y|≤ε
Rd |x−y|≤ε |x − y|
Z Z
=C
ρ(x) (χBε (0) N )(x − y) ρ(y) dxdy.
Rd
Rd
Now recall that the Newtonian potential
Z
Ψρ (x) =
Rd
ρ(y)
dy
|x − y|d−2
is well defined for a.e. x ∈ Rd and is in L1loc (Rd ), see [33, Theorem 2.21], then for a.e. x ∈ Rd we
have χBε (0) N ∗ ρ → 0 as ε → 0. Moreover, by the HLS inequality we have
with
ρ(x)(χBε (0) N ∗ ρ)(x) ≤ ρ(x)Ψρ (x) ∈ L1loc (Rd )
2(1−α)
kρ(x)Ψρ kL1 (Rd ) ≤ Ckρk2α
L1 (Rd ) kρkLm (Rd ) .
Then Lebesgue’s dominated convergence theorem allows to conclude that Aε [ρ] → 0 as ε → 0.
This convergence is uniform taken on a minimizing sequence ρn .
23
Now, all ingredients are there to argue as in [22] showing that E achieves its infimum in the class
of all radially decreasing densities in YM .
A useful result, which will be used in the next arguments, regards the behavior at infinity of the
so called W -potential, namely the function
Z
ψf (x) =
W (x − y)f (y)dy.
Rd
Following the blueprint of [28, Lemma 1.1], we have the following result.
Lemma 3.4. Assume that (K1)-(K5) hold, and let
f ∈ L1 (Rd ) ∩ L∞ (Rd \ B1 (0)), ω(1 + |x|) f (x) ∈ L1 (Rd ).
Then
ψf (x)
→
W (x)
Z
f (y)dy
Rd
as |x| → +∞.
Proof. As in Chae-Tarantello [28], we first set
Z
Z
1
ψf (x)
−
f (y)dy =
[ω(|x − y|) − ω(|x|)] f (y)dy
σ(x) :=
ω(|x|)
ω(|x|) Rd
Rd
(3.4)
so that our aim will be to show that σ(x) → 0 as |x| → ∞. Assume that |x| > 2. We then write
σ(x) = σ1 (x) + σ2 (x) + σ3 (x),
where σi , i = 1, 2, 3, are defined by breaking the integral on the right hand side of (3.4) into:
D1 = {y : |x − y| < 1} , D2 = {y : |x − y| > 1, |y| ≤ R} and D3 = {y : |x − y| > 1, |y| > R}
respectively, where R > 2 is a fixed constant. Recall that (K2) implies |ω(r)| ≤ Cφ(r) for r ≤ 1,
with φ given in (2.3). Thus, we have
Z
1
|σ1 (x)| ≤
|ω(|x − y|) − ω(|x|)| |f (y)|dy
ω(|x|) |x−y|<1
Z
Z
C
≤
φ(|x − y|) |f (y)|dy +
|f |dy
ω(|x|) |x−y|<1
|y|>|x|−1
Z
Ckf kL∞ (Rd \B1 (0)) kφkL1 (B1 (0))
≤
+
|f |dy ,
ω(|x|)
|y|>|x|−1
where we used f ∈ L∞ (Rd \ B1 (0)) and |x| > 2 in the last inequality. This means that σ1 (x) → 0
as |x| → ∞. Moreover, we notice that
Z
1
|ω(|x − y|) − ω(|x|)| |f (y)|dy .
|σ2 (x)| ≤
ω(|x|) {y∈Rd :|x−y|>1, |y|<R}
Since by property (K3) we can estimate in the region D2
|ω(|x − y|) − ω(|x|)| ≤ C |x − y| − |x| ≤ C|y| ≤ CR ,
such that
|σ2 (x)| ≤ C
R
kf kL1 (Rd ) ,
ω(|x|)
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
24
which implies that also σ2 (x) → 0 as |x| → +∞. As for σ3 , for x such that |x| > R, using
(K4)-(K5) we write
Z
1
|ω(|x − y|) − ω(|x|)| |f (y)|dy
|σ3 (x)| ≤
ω(|x|) {y∈Rd :|x−y|>1, R<|y|<|x|}
Z
Z
1
+
ω(|x − y|)|f (y)|dy +
|f (y)|dy
ω(|x|) {y∈Rd :|x−y|>1, |y|>|x|}
|y|>R
Z
Z
ω(2|x|)
|f (y)|dy + 2
|f (y)|dy
≤
ω(|x|) |y|>R
|y|>R
Z
Cw
+
[1 + ω(1 + |x|) + ω(1 + |y|)] |f (y)|dy
ω(|x|) {y∈Rd :|x−y|>1, |y|>|x|}
Z
Z
1
1
|f (y)|dy +
ω(1 + |y|)|f (y)|dy
≤C 1 +
ω(|x|)
ω(|x|) Rd
|y|>R
Z
→C
|f (y)|dy
|y|>R
as |x| → +∞, for any fixed R > 1. Hence letting R → +∞ we get σ3 (x) → 0.
In case of assumption (K6), we prove the following lemma, weaker than Lemma 3.7, but enough
for our purposes.
Lemma 3.5. Assume (K1)-(K4) and (K6) hold, and let K := ` − W be as defined in (K6). Then
the following holds for any radially decreasing f ∈ L1+ (Rd ):
Z
K(x − y)f (y)dy = 0,
(3.5)
lim
|x|→∞
and
Z
Rd
where c := 2−α
R
B1 (0)
Rd
K(x − y)f (y)dy ≥ cK(x) for all |x| > 1,
(3.6)
f (y)dy > 0, with α > 0 as given in (K6).
Proof. Since both f and K are radially symmetric, we define f¯, K̄ : [0, +∞) → R such that
f¯(|x|) = f (x), K̄(|x|) = K(x). Note that limr→∞R f¯(r) = limr→∞ K̄(r) = 0 due to (K1), (K6) and
the assumption on f . To prove (3.5), we break Rd K(x − y)f (y)dy into the following three parts
with |x| > 1 and control them respectively by:
Z
K(x − y)f (y)dy ≤ kKkL1 (B1 (0)) f¯(|x| − 1),
|y|>
|x|
2 ,|x−y|≤1
Z
|y|>
and
|x|
2 ,|x−y|>1
K(x − y)f (y)dy ≤ K̄(1)
Z
|y|≤
|x|
2
K(x − y)f (y)dy ≤ K̄
Z
|x|
2
|y|>
|x|
2
f (y)dy,
kf kL1 .
Since all the three parts tend to 0 as |x| → ∞, we obtain (3.5). To show (3.6), we use K, f ≥ 0 to
estimate
Z
Z
Z
K(x − y)f (y)dy ≥
K(x − y)f (y)dy ≥ K̄(|x| + 1)
f (y)dy
|y|≤1
Rd
≥
|x| + 1
|x|
B1 (0)
−α
K̄(|x|)
Z
B1 (0)
f (y)dy ≥ cK(x)
for any |x| > 1,
where
R we apply (K6) to obtain the third inequality, and in the last inequality we define c :=
2−α B1 (0) f (y)dy > 0.
25
Using similar arguments as in [22], we are able to derive the following result, which indeed gives
a natural form of the Euler-Lagrange equation associated to the functional E:
Theorem 3.6. Assume that (K1)-(K4) and either (K5) or (K6) hold. Let ρ0 ∈ YM be a global
minimizer of the free energy functional E. Then for some positive constant D[ρ0 ], we have that ρ0
satisfies
m m−1
ρ
+ W ∗ ρ0 = D[ρ0 ] a.e. in supp(ρ0 )
(3.7)
m−1 0
and
m m−1
ρ
+ W ∗ ρ0 ≥ D[ρ0 ] a.e. outside supp(ρ0 )
m−1 0
where
m−2
2
G[ρ0 ] +
kρ0 km
D[ρ0 ] =
Lm (Rd ) .
M
M (m − 1)
As a consequence, any global minimizer of E verifies
m m−1
ρ
= (D[ρ0 ] − W ∗ ρ0 )+ .
(3.8)
m−1 0
We now turn to show compactness of support and boundedness of the minimizers.
Lemma 3.7. Assume that (K1)-(K4) and either (K5) or (K6) hold and let ρ0 ∈ YM be a global
minimizer of the free energy functional E. Then ρ0 is compactly supported.
Proof. By Theorems 3.1 and 3.3, ρ0 is radially decreasing under either set of assumptions. In
addition, under the assumption (K5), Lemma 3.4 gives that
(W ∗ ρ0 )(x)
→ kρkL1 (Rd ) as |x| → ∞,
W (x)
hence combining this with (K5) gives us (W ∗ ρ0 )(x) → +∞ as |x| → ∞. It implies that the right
hand side of (3.8) must have compact support, hence ρ0 must have compact support too.
Under the assumption (K6), towards a contradiction, suppose ρ0 does not have compact support.
Then ρ0 must be strictly positive in Rd since it is radially decreasing. We can then write (3.7) as
m m−1
ρ
− K ∗ ρ0 = C a.e. in Rd
m−1 0
for some C ∈ R, where K := ` − W is as given in (K6). Indeed, C must be equal to 0, since both
ρ0 (x) and (K ∗ ρ0 )(x) tend to 0 as |x| → ∞, where we used (3.5) on the latter convergence. Thus
1
1/(m−1)
m − 1
m−1
m−1
(K ∗ ρ0 )(x)
cK(x)
for a.e. |x| > 1,
(3.9)
ρ0 (x) =
≥
m
m
R
where we applied (3.6) to obtain the last inequality, with c := 2−α B1 (0) ρ0 (y)dy > 0. Due to the
R
assumptions (3.1) and α < m−1
in (K6), we have |x|>1 K(x)1/(m−1) dx = +∞. Combining this
d
with (3.9) leads to ρ0 6∈ L1 (Rd ), a contradiction.
Lemma 3.8. Assume that (K1)-(K4) and either (K5) or (K6) hold and let ρ0 ∈ YM be a global
minimizer of the free energy functional E. Then ρ0 ∈ L∞ (Rd ).
Proof. By Theorem 3.1, Theorem 3.3 and Lemma 3.7, ρ0 is radially decreasing and has compact
support say inside the ball BR (0). Let us first concentrate on the proof under assumption (K5).
For notational simplicity in this proof, we will denote by kρ0 km the Lm (Rd )-norm of ρ0 .
We will show that ρ0 ∈ L∞ (Rd ) by different arguments in several cases:
1
2
Case A: d ≤ 2. Since ρ0 is supported in BR (0), we can then find some Cw
and Cw
, such that
1
2
W ≥ −Cw N − Cw in B2R (0). Hence for any r < R, we have
1
2
1
2
−(ρ0 ∗ W )(r) ≤ −(ρ0 ∗ (−Cw
N − Cw
))(r) ≤ Cw
(ρ0 ∗ N )(r) + Cw
kρ0 k1 ,
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
26
thus recalling (2.4)
1
2
1
2
e
(ρ0 ∗ W − )(r) ≤ Cw
(ρ0 ∗ N )(r) + Cw
kρ0 k1 + (ρ0 ∗ W + )(r) ≤ Cw
(ρ0 ∗ N )(r) + Cw
kρ0 k1 + C.
Then by equation (3.8) it will be enough to show that the Newtonian potential ρ0 ∗ N is bounded
in BR (0) for d = 1, 2. In d = 1, this is trivial. In d = 2 it follows from [36, Lemma 9.9] since we
have that ρ0 ∗ N ∈ W 2,m (BR (0)), then Morrey’s Theorem yields ρ0 ∗ N ∈ L∞ (BR (0)).
Case B: d ≥ 3 and m > d/2. In this case we get W − ≤ Cw N in the whole Rd for some
constant Cw , so we have for r > 0
(ρ0 ∗ W − )(r) ≤ Cw (ρ0 ∗ N )(r).
Then using Sobolev’s embedding theorem again, we easily argue that for m > d/2 we find (ρ0 ∗
d
∞
d
W − )(r) ∈ L∞
loc (R ), hence ρ0 ∈ L (R ) by (3.8) again.
Case C: d ≥ 3 and 2 − d2 < m ≤ d/2. We aim to prove that ρ0 (0) is finite which is sufficient
for the boundedness of ρ0 since ρ0 is radially decreasing. This is done by an inductive argument.
To begin with, note that we have kρ0 km < ∞ by the proof of Lemma 3.2. Combining this with
the fact that ρ0 is radially decreasing, we have that ρ0 (r)m |B(0, r)| ≤ kρ0 km
m < ∞, which leads to
the basis step of our induction
ρ0 (r) ≤ C(d, m, kρ0 km )r−d/m for all r > 0.
We set our first exponent p̃ = −d/m. For the induction step, we claim that if ρ0 (r) ≤ C1 (1 + rp )
with −d < p < 0, then it leads to the refined estimate
(
p+2
if p 6= −2
C2 (1 + r m−1 )
ρ0 (r) ≤
(3.10)
1
m−1
C2 (1 + | log r|
) if p = −2,
where C2 depends on d, m, ρ0 , W and C1 .
Indeed, taking into account (K2) and (K5), the compact support of ρ0 together with the fact
that N > 0 for d ≥ 3, we deduce that W ≥ −Cw,d N for some constant depending on W and d.
As a result, we have, for r ∈ (0, 1),
Z 1
− (ρ0 ∗ W )(r) ≤ Cw,d (ρ0 ∗ N )(r) = Cw,d (ρ0 ∗ N )(1) −
∂r (ρ0 ∗ N )(s)ds .
(3.11)
r
We can easily bound (ρ0 ∗ N )(1) by some C(d, kρ0 km ). To control
− ∂r (ρ0 ∗ N )(s) =
R1
r
∂r (ρ0 ∗ N )(s)ds, recall that
M (s)
M (s)
=
,
|∂B(0, s)|
σd sd−1
where M (s) is the mass of ρ0 in B(0, s). By our induction assumption, we have
d
Z s
sd+p
s
p
d−1
+
.
M (s) ≤
C1 (1 + t )σd t dt = C1 σd
d
d+p
0
Combining this with (3.12), we have
−∂r (ρ0 ∗ N )(s) ≤ C1
s
s1+p
+
d d+p
,
so we get, for p 6= −2,
Z 1
1
1
2
2+p
∂r (ρ0 ∗ N )(s)ds ≤ C1
−
(1 − r ) +
(1 − r
) .
2d
(d + p)(2 + p)
r
Plugging it into the right hand side of (3.11) yields
−(ρ0 ∗ W )(r) ≤ C(d, m, kρ0 km , Cw,d , C1 )(1 + r2+p ),
(3.12)
27
and using this inequality in the Euler-Lagrange Equation (3.8) leads to (3.10). Moreover, in the
case p = −2, we have instead the inequality
Z 1
1
1
(1 − r2 ) −
log r .
−
∂r (ρ0 ∗ N )(s)ds ≤ C1
2d
d−2
r
Now we are ready to apply the induction starting at p̃ = −d/m to show ρ0 (0) < ∞. We will
show that after a finite number of iterations our induction arrives to
ρ0 (r) ≤ C(1 + ra )
(3.13)
p+2
m−1 ,
for some a > 0, which then implies that ρ0 (0) < ∞. Let g(p) :=
p with positive slope, and let us denote g (n) (p) =: (g ◦ g · · · ◦ g)(p).
{z
}
|
which is a linear function of
n iterations
Subcase C.1: m = d/2.- In this case, we have pe = −2 and by (3.10) we obtain
1
ρ0 (r) ≤ C2 (1 + | log r| m−1 ) ≤ C2 (1 + r−1 )
hence applying the first inequality in (3.10) for p = −1 gives us (3.13) with a = 1/(m − 1).
Then it remains to consider the case m < d/2. Notice that −d < pe < −2. By (3.10) we get, for
all r ∈ (0, 1),
ρ0 (r) ≤ C2 (1 + rg(ep) ).
(3.14)
Then we must consider three cases. We point out that in all the cases we need to discuss the
possibility of g (n) (p) = −2 for some n: if this happens, the logarithmic case occurs again and the
result follows in a final iteration step as in Subcase C.1.
Subcase C.2: m = 2 and m < d/2.- In this case, we have g(p) = p + 2, hence g (n) (p) = p + 2n,
then g (n) (p) > p for all n ∈ N and
lim g (n) (p) = +∞.
n→∞
Therefore we have g (n) (e
p) > 0 for some finite n, whence iterating (3.14) n times we find ρ0 (0) < ∞.
Subcase C.3: m > 2 and m < d/2.- In this case, p = 2/(m − 2) is the only fixed point for the
2
linear function g(p). For all p < m−2
we have g(p) > p which implies g (n) (p) > p for all n ∈ N.
Notice that
2
p(m − 2) − 2
g (n) (p) =
+
,
(3.15)
m − 2 (m − 2)(m − 1)n
so the point p = 2/(m − 2) is attracting in the sense that
lim g (n) (p) =
n→∞
2
,
m−2
2
and the convergence is uniform with respect to p. Since m−2
> 0, it again implies that g (n) (p) > 0
for some finite n. Then choosing p = pe, we have g (n) (e
p) > 0 for some n, then (3.14) implies
ρ0 (0) < ∞ again.
Subcase C.4: m < min(2, d/2).- In this case, the only fixed point
2
, then by (3.15)
g(p) > p for any p > m−2
lim g (n) (p) = +∞ for any p >
n→∞
2
m−2
is unstable, and we have
2
.
m−2
2
Indeed, we have pe > −d > m−2
, where in the last step we used m > 2 − d2 as well as 1 < m < 2.
Hence we again obtain g (n) (e
p) > 0 for some finite n, which finishes the last case.
Let us finally turn back to the proof if we assume (K6) instead of (K5). Notice first that the proof
of the Case C can also be done as soon as the potential W satifies the bound W ≥ −Cw,d (1 + N )
28
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
for some Cw,d > 0. This is trivially true regardless of the dimension if the potential satisfies (K6)
instead of (K5).
Finally, it is interesting to derive some regularity properties of a minimizer ρ0 , as in [22]. Since
W may not be the classical Newtonian kernel, we are led to prove a nice regularity for the W potential ψρ0 (x) which can be transferred to ρ0 via equation (3.7) in the support of ρ0 . Note
that (3.8) ensures that ρ0 satisfies equation (2.1) in the sense of distributions and, as shown in
1,∞
(2.2)-(2.4), we find that ψρ0 ∈ Wloc
(Rd ), and thus using the regularity arguments of the proof of
Lemma 2.3 again together with the compact support property, we finally have ρ0 ∈ C 0,α (Rd ) with
α = 1/(m − 1).
We can summarize all the results in this section in the following theorem.
Theorem 3.9. Assume that conditions (K1)-(K4) and either (K5) or (K6) hold. Then for any
positive mass M , there exists a global minimizer ρ0 of the free energy functional E (2.5) defined
in YM , which is radially symmetric, decreasing, compactly supported, Hölder continuous, and a
stationary solution of (1.1) in the sense of Definition 2.1.
Putting together the previous theorem with the uniqueness of radial stationary solutions for the
attractive Newtonian potential proved in [46, 22], we obtain the following result.
Corollary 3.10. In the particular case of the attractive Newtonian potential W (x) = −N (x), the
global minimizer obtained in Theorem 3.9 is unique among all stationary solutions in the sense of
Definition 2.1.
4. Long-time Asymptotics
We now consider the particular case of (1.1) given by the Keller Segel model in two dimensions
with nonlinear diffusion as
∂t ρ =
∆ρm − ∇ · (ρ∇N ∗ ρ) ,
(4.1)
where m > 1 and the logarithmic interaction kernel is defined as
1
log |x| .
N (x) = −
2π
This system is also referred to as the parabolic-elliptic Keller-Segel system with nonlinear diffusion,
since the attracting potential c = N ∗ρ solves the Poisson equation −∆c = ρ. It corresponds exactly
to the range of diffusion dominated cases as discussed in [19] since solutions do not show blow-up
and are globally bounded. We will show based on the uniqueness part in Section 2 that not only the
solutions to (4.1) exist globally and are uniformly bounded in time in L∞ , but also the solutions
achieve stabilization in time towards the unique stationary state for any given initial mass.
The main tool for analyzing stationary states and the existence of solutions to the evolutionary
problem is again the following free energy functional
Z Z
Z
ρm
1
E[ρ](t) =
dx +
ln |x − y|ρ(x)ρ(y)dx dy .
(4.2)
4π R2 R2
R2 m − 1
A simple differentiation formally shows that E is decaying in time along the evolution corresponding
to (4.1), namely
d
E[ρ](t) = −D[ρ](t)
dt
which gives rise to the following (free) energy - energy dissipation inequality for weak solutions
Z t
E[ρ](t) +
D[ρ]dτ ≤ E[ρ0 ]
(4.3)
0
29
for nonnegative initial data ρ0 (x) ∈ L1 ((1 + ln(1 + |x|2 ))dx) ∩ Lm (R2 ). The entropy dissipation is
given by
Z
D[ρ] =
ρ|∇h[ρ]|2 dx ,
R2
where here and in the following we use the notation
m m−1
ρ
− N ∗ ρ.
h[ρ] =
m−1
We shall note that h corresponds to δE
δρ and that in particular the evolutionary equation (4.1) can
be stated as ∂t ρ = ∇ · (ρ∇h[ρ]). Thus, this equation bears the structure of being a gradient flow of
the free energy functional in the sense of probability measures, see [2, 8, 10, 26] and the references
therein.
We first prove the global well-posedness of weak solutions satisfying the energy inequality (4.3)
in the next subsection as well as global uniform in time estimates for the solutions. In the second
subsection, we used the uniform in time estimates together with the uniqueness of the stationary
states proved in Section 2 to derive the main result of this section regarding long time asymptotics
for (4.1).
4.1. Global well-posedness of the Cauchy problem. In this section we analyze the existence
and uniqueness of a bounded global weak solution for initial data in L1log (R2 ) ∩ L∞ (R2 ), where
here and in the following we denote
L1log (R2 ) = L1 ((1 + ln(1 + |x|2 ))dx) .
Assuming to have a sufficiently regular solution with the gradient of the chemotactic potential
being uniformly bounded, Kowalczyk [48] derived a priori bounds in L∞ with respect to space
and time for the Keller-Segel model with nonlinear diffusion on bounded domains. These a priori
estimates have been improved and extended to the whole space by Calvez and Carrillo in [19]. We
shall demonstrate here how these a priori estimates of [19] can be made rigorous when starting
from an appropriately regularized equation leading to the following theorem.
Theorem 4.1 (Properties of weak solutions). For any nonnegative initial data ρ0 ∈ L1log (R2 ) ∩
L∞ (R2 ), there exists a unique global weak solution ρ to (4.1), which satisfies the energy inequality
(4.3) with the energy being bounded from above and below in the sense that
E∗ ≤ E[ρ](t) ≤ E[ρ0 ]
for some (negative) constant E∗ . In particular ρ is uniformly bounded in space and time
sup kρ(t, ·)kL∞ (R2 ) ≤ C ,
t≥0
where C depends only on the initial data. Moreover the log-moment grows at most linearly in time
Z
N (t) =
ln(1 + |x|2 )ρ(t, x)dx ≤ N (0) + Ct ,
R2
where again C depends only on the initial data.
We shall also state the existence result for radial initial data that was obtained in [50] and [46]
for higher dimensions and the Newtonian potential. Similar methods can be applied in the case
d = 2 considered here:
Theorem 4.2 (Properties of radial solutions). Let ρ0 ∈ L1log (R2 ) ∩ L∞ (R2 ) be nonnegative and
radially symmetric.
(a) Then the corresponding unique weak solution of (4.1) remains radially symmetric for all
t > 0.
(b) If ρ0 is compactly supported, then the solution remains compactly supported for all t > 0.
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
30
(c) If ρ0 is moreover monotonically decreasing, then the solution remains radially decreasing
for all t > 0.
In the remainder of this section we carry out the proof of the existence of a bounded global weak
solution to (4.1) as stated in Theorem 4.1. We therefore introduce the following regularization of
(4.1)
∂t ρε = ∆(ρm
(4.4)
ε + ερε ) − ∇ · (ρε ∇Nε ∗ ρε ) ,
where m > 1 and the regularized logarithmic interaction potential is defined as
1
log(|x|2 + ε2 ) .
Nε (x) = −
4π
Moreover we have for the derivatives
1
1
x
ε2
,
∆N
=
−
∇Nε = −
= −Jε
ε
2π |x|2 + ε2
π (|x|2 + ε2 )2
satisfying
kJε kL1 (R2 ) = 1 .
The regularization in (4.4) was used by Bian and Liu [7], who studied the Keller-Segel equation with nonlinear diffusion and the Newtonian potential for d ≥ 3, which has been modified
accordingly for the logarithmic interaction kernel in d = 2. The additional linear diffusion term in
(4.4) removes the degeneracy and the regularized logarithmic potential Nε possesses a uniformly
bounded gradient, such that the local well posedness of (4.4) is a standard result for any ε > 0. We
shall note that a slightly different regularization for such nonlinear diffusion Keller-Segel type of
equations has been introduced by Sugiyama in [63], which also yields the existence and uniqueness
of a global weak solution. The advantage of the regularization in (4.4) resembling the one in [7]
is the fact that the regularized problem satisfies a free energy inequality, that in the limit gives
exactly (4.3), whereas in [63] the dissipation term could only be retained with a factor of 3/4.
We point out that in the case d = 2 other a priori estimates are available than in higher space
dimensions leading to a different proof for global well posedness of the Cauchy problem for (4.4)
and the limit ε → 0 compared to [7].
4.1.1. Global well posedness of the regularized Cauchy problem. To derive a priori estimates for the
regularized problem (4.4) we use the iterative method used by Kowalczyk [48] based on employing
test functions that are powers of ρε,k = (ρε − k)+ for some k > 0. When testing (4.4) against
pρp−1
ε,k for any p ≥ 2, we obtain:
Z
d
ρp dx
dt R2 ε,k
Z
Z
p
4(p − 1)
p−1
m−1
2
2
=−
(mρε
+ ε)|∇ρε,k | dx + p
(ρε,k + k)(∇Nε ∗ ρε ) · ∇ρε,k
dx
(4.5)
p
2
2
R
R
Z
Z
p
4(p − 1)
2
≤−
m
ρm−1
|∇ρε,k
|2 dx +
(∇Nε ∗ ρε ) · ((p − 1)∇ρpε,k + kp∇ρp−1
ε
ε,k )dx
p
2
2
R
R
Z
p
4(p − 1)
2
≤−
mk m−1 k∇ρε,k
k2L2 +
Jε ∗ ρε ((p − 1)ρpε,k + kpρp−1
ε,k )dx
p
R2
Z
p
4(p − 1)
p−1
2
≤−
mk m−1 k∇ρε,k
k2L2 +
(Jε ∗ ρε,k + k)((p − 1)ρpε,k + kpρε,k
)dx
p
R2
Z
Z
Z
p
4(p − 1)
p+1
p
p−1
m−1
2
2
2
≤−
mk
k∇ρε,k kL2 + C(p − 1)
ρε,k dx + Ckp
ρε,k dx + k p
ρε,k
dx , (4.6)
p
R2
R2
R2
where for estimating the integrals involving convolution terms we used the inequality
Z
1 1 1
+ + = 2, where p, q, r ≥ 1 ,
(4.7)
f (x)(g ∗ h)(x)dx ≤ Ckf kLp kgkLq khkLr ,
p q
r
31
see e.g. Lieb and Loss [49]. Closing the estimate (4.6) would yield a bound for ρε,k in L∞ (0, T ; Lp (R2 ))
and thus also for ρε ∈ L∞ (0, T ; Lp (R2 )), since
Z
Z
Z
Z
p
p−1
p
ρε dx ≤ k
ρε dx +
(ρε − k) dx + C(p, k)
kdx
R2
{ρε <k}
{ρε ≥k}
{ρε ≥k}
Z
≤
ρpε,k dx + (k p−1 + C(p, k))M .
R2
Kowalczyk proceeded from (4.5) with the assumption corresponding to k∇Nε ∗ ρε kL∞ ≤ C.
Observe that it would be sufficient to prove ρε ∈ L∞ (0, T ; Lp (R2 )) for some p > 2 implying
∆Nε ∗ ρε ∈ L∞ (0, T ; Lp (R2 )) and hence the uniform boundedness of the gradient term by Sobolev
imbedding. Calvez and Carrillo [19] circumvent this assumption and derive the bound by using an
equi-integrability property in the inequality (4.6). Hence, in order to being able to follow the ideas
of [19] for the regularized problem, we need to derive the corresponding energy inequality for the
latter.
Proposition 4.3. For any finite time T > 0 the solution ρε to the Cauchy problem (4.4) supplemented with initial data ρ0 ∈ L1log (R2 ) ∩ L∞ (R2 ) satisfies the energy inequality
Z t
Eε [ρε ](t) +
Dε [ρε ](t)dt ≤ Eε [ρ0 ] + εC(1 + t)t ,
(4.8)
0
for a positive constant C = C(M, kρ0 k∞ ) and 0 ≤ t ≤ T , where Eε is an approximation of the free
energy functional in (4.2):
Z m
ρε
ρε
− Nε ∗ ρε dx
Eε [ρε ] =
m−1
2
R2
and Dε the corresponding dissipation
Z
Dε [ρε ](t) =
ρε |∇hε [ρε ]|2 dx
with
hε [ρε ] =
R2
In particular, we obtain equi-integrability
Z
lim sup
k→∞ t∈[0,T ]
R2
m
∇ρm−1 − ∇Nε ∗ ρε .
m−1 ε
(ρε − k)+ dx = 0 .
m
m−1
m−1 ρε
− Nε ∗ ρε we obtain
2
Z m
d
4
m2 2
m−1
Eε (t) +
ρε ∇ρε
− ∇Nε ∗ ρε dx + ε
∇ρε dx
dt
m−1
m R2
2
Z R
Z
=ε
∇Nε ∗ ρε · ∇ρε dx = ε
ρε (Jε ∗ ρε )dx ≤ εkρε k2L2 (R2 ) ,
Proof. Testing (4.4) with
Z
R2
R2
where we have used (4.7) and the fact that kJε kL1 (R2 ) = 1. Hence we need to
R derive an a priori
bound for ρε in L2 (R2 ). We use the estimate (4.6) for p = 2 and bound R2 ρ3ε,k dx using the
Gagliardo-Nirenberg inequality (see for instance [35], [58]) as follows:
kρε,k k3L3 (R2 )
≤
≤
Ck∇ρε,k k2L2 (R2 ) kρε,k kL1 (R2 )
CM k∇ρε,k k2L2 (R2 ) .
Then by (4.6) and interpolation of the L2 -integral, we have
Z
Z
Z
d
2
m−1
2
3
2
ρ dx ≤ −2mk
k∇ρε,k kL2 + C
ρε,k dx + 3k
ρε,k dx
dt R2 ε,k
R2
R2
≤ −(2mk m−1 − CM )k∇ρε,k k2L2 + Ck 2 M.
Hence, choosing k large enough and recalling m > 1, we can conclude integrating in time that
kρε (t, ·)k2L2 (R2 ) ≤ C(1 + t)
for some constant C = C(M, kρ0 kL∞ (R2 ) ), which implies the stated energy inequality.
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
32
In order to obtain a priori bounds and in particular the equi-integrability property, we need to
bound the energy functional also from below. The difference to the corresponding energy functional
for the original model (4.1) lies only in the regularized interaction kernel. Since clearly for all x ∈ R2
we have ln(|x|2 + ε2 ) ≥ 2 ln |x|, we obtain
Z Z
Z
1
ρm
ε
dx +
ρε (x) ln(|x − y|2 + ε2 )ρε (y)dxdy
Eε [ρε ] =
m
−
1
8π
2
2
2
ZR ZR
ZR
1
ρm
ε
dx +
ρε (x) ln |x − y|ρε (y)dxdy = E[ρε ]
≥
4π R2 R2
R2 m − 1
Following [19] we can estimate further using the logarithmic Hardy-Littlewood-Sobolev inequality
Z
M
Eε [ρε ] ≥ E[ρε ] ≥ − C(M ) +
Θ(ρε ) dx ,
(4.9)
8π
R2
where C(M ) is a constant depending on the mass M and
Θ(ρ) :=
M
ρm
−
ρ log ρ.
m − 1 8π
Now it is easy to verify there is a constant κ = κ(m, M ) > 1 for which
Θ(ρ) ≥ 0
for ρ ≥ κ,
such that
Z
Θ− (ρε )dx =
R2
Z
1≤ρε ≤κ
Θ− (ρε )dx ≤
M2
log κ ,
8π
implying in particular
Eε [ρε ] ≥ E[ρε ] ≥ −
M2
M
C(M ) −
ln κ =: E∗ .
8π
8π
(4.10)
We therefore find from (4.8), (4.9) and (4.10) that
Z
Θ+ (ρε (t))dx ≤ C + εCT 2 ,
R2
with C = C(m, kρ0 kL1 (R2 ) , kρ0 kL∞ (R2 ) ) being a constant independent of t. Since Θ+ is superlinear
at infinity, we obtain the equi-integrability as in Theorem 5.3 in [19].
The equi-integrability from Proposition 4.3 allows to close the estimate (4.6) analogously to
Lemma 3.1 of [19] leading to a bound for ρε in L∞ (0, T ; Lp (R2 )). Moreover, using Moser’s iterative
methods of Lemma 3.2 in [19] we finally get a bound for ρε in L∞ (0, T ; L∞ (R2 )). In order to
avoid mass loss at infinity typically the boundedness of the second moment of the solution is
employed. We here however demonstrate that the bound of the log-moment provides sufficient
compactness, having the advantage of less restrictions on the initial data. We therefore denote for
the regularization
Z
Nε (t) =
ln(1 + |x|2 )ρε (t, x)dx .
R2
The following lemma is now obtained following the ideas of [19]:
Lemma 4.4. The solution ρε to (4.4) for a nonnegative initial data ρ0 ∈ L1log (R2 ) ∩ L∞ (R2 )
satisfies for any T > 0:
sup kρε (t, ·)kL∞ (R2 ) + kNε (t)kL∞ (0,T ) ≤ C(1 + T + T 2 ) ,
t∈[0,T ]
where the constant C depends on the initial data.
33
Proof. Computing formally the evolution of the log-moment in (4.4) in a similar fashion to [20],
we find for the test function φ(x) = ln(1 + |x|2 ) after integrating by parts
Z
Z
Z
d
Nε =
∂t ρε φdx = −
ρε ∇hε [ρε ] · ∇φdx + ε
ρε ∆φdx
dt
R2
R2
R2
Z
Z
Z
1
1
≤
ρε |∇φ|2 dx +
ρε |∇hε [ρε ]|2 dx + ε
ρε ∆φdx .
2 R2
2 R2
R2
Computing the derivatives of φ we see
2x ≤ 1,
|∇φ| = 1 + |x|2 |∆φ| =
4
≤ 4.
(1 + |x|2 )2
We thus obtain
d
1
Nε ≤ ((1 + 8ε)M + Dε [ρε ]) .
dt
2
Integration in time and making use of the energy - energy dissipation inequality (4.8) and the
uniform bound on Eε from below in (4.10) gives
1
Nε (t) ≤ Nε (0) + (1 + 8ε)M t + E (ρ0 ) − E∗ + εC(1 + t)t ≤ C(1 + t + t2 )
2
The argument can easily be made rigorous by using compactly supported approximations of φ on
R2 as test functions, see e.g. also [11]. The proof is concluded by referring to Lemma 3.2 in [19]
for the proof of uniform boundedness of ρε .
Remark 4.5.
(i) The fact that the uniform bound of ρε grows linearly with time originates from the term of
order ε in the energy inequality for the regularized equation. Hence the bound on the energy
and therefore the modulus of equi-continuity for the regularized problem are depending on
time. However, for the limiting equation (4.1) this term vanishes and the energy is decaying
for all times, which allows to deduce uniform boundedness of the solution to (4.1) globally
in time and space, see also [19, Lemma 5.7].
(ii) The log-moment of ρε grows at most linearly in time. The same statement is true for the
limiting function. Hence it is only possible to guarantee confinement of mass for finite
times. This property allowing for compactness results will in the following be used to pass
to the limit in the regularized problem. Due to the growth of the bound with time it cannot
be employed for the long-time behavior. Hence different methods will be required.
4.1.2. The limit ε → 0. In order to deduce the global well posedness of the Cauchy problem for
(4.1) it remains to carry out the limit ε → 0. Knowing that the solution remains uniformly bounded
and having the bounds from the energy inequality, we obtain weak convergence properties of the
solution. In order to pass to the limit with the nonlinearities and in the entropy inequality, strong
convergence results will be required. The following lemma summarizes the uniform bounds we
obtain from Proposition 4.3 and Lemma 4.4:
Lemma 4.6. Let ρε be the solution as in Proposition 4.3, then we obtain the following uniform in
ε bounds
kρε kL∞ (0,T ;L1log (R2 )) + kρε kL∞ ((0,T )×R2 ) ≤ C ,
√
√
k ρε ∇Nε ∗ ρε kL2 ((0,T )×R2 ) + k∇Nε ∗ ρε kL∞ ((0,T )×R2 ) + εk∇ρε kL2 ((0,T )×R2 ) ≤ C ,
1
k∂t ρε kL2 (0,T ;H −1 (R2 )) + kρqε kL2 (0,T ;H 1 (R2 )) ≤ C
for any q ≥ m − ,
2
where C depends on m, q, ρ0 and T .
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
34
Proof. The uniform bounds of the L1log (R2 )- and L∞ (R2 )-norms follow from the conservation of
mass and Lemma 4.4. The convolution term
Z
y−x
1
ρε (t, y)dy
∇Nε ∗ ρε = −
2π R2 |y − x|2 + ε2
can be estimated as follows:
2π|∇Nε ∗ ρε | ≤ kρε kL∞ (R2 )
The bound of
√
Z
|x−y|≤1
1
dy + M ≤ C .
|x − y|
(4.11)
ρε ∇Nε ∗ ρε in L2 ((0, T ) × R2 ) follows now easily by using the conservation of mass.
The basic L2 -estimate corresponding to (4.5) for p = 2 and k = 0 implies after integration in
time
Z TZ
Z
Z
Z TZ
1
1
2
2
2
ρm−1
|∇ρε |2 dxdt
|∇ρε | dxdt − m
ρ dx ≤
ρ dx − ε
ε
2 R2 ε
2 R2 0
0
R2
0
R2
Z TZ
+
(Jε ∗ ρε )ρ2ε dxdt .
0
R2
Using the above a priori estimates we can further bound employing the inequality in (4.7)
εk∇ρε k2L2 ((0,T )×R2 )
≤
1
2
Z
R2
ρ20 dx + C
Z
T
Z
0
R2
ρ3ε dxdt ≤ C .
m−1/2
Since m > 1, the conservation of mass and the uniform boundedness of ρε give ρε
R2 ). For the gradient we now use the bound on the entropy dissipation (4.8)
k∇ρm−1/2
k2L2 ((0,T )×R2 )
ε
(m − 1/2)2
≤ 2
m2
Z
0
T
in L2 ((0, T )×
√
Dε [ρε ]dt + k ρε ∇Nε ∗ ρε k2L2 ((0,T )×R2 )
≤ C + Ck∇Nε ∗ ρε k2L∞ ((0,T )×R2 ) M T .
!
(4.12)
The bound for ∇ρq follows easily by rewriting
m−
q
1
2
q−m+ 12
∇ρqε = ρε
m− 21
∇ρε
and using the uniform boundedness of ρε .
It thus now remains to derive the estimate for the time derivative. Using the previous estimates
we have for any test function φ ∈ L2 (0, T ; H 1 (R2 )),
Z Z
Z Z
Z TZ
T
T
m
∂ ρ φ dx dt ≤
|∇(ρε + ερε ) · ∇φ| dx dt +
|ρε (∇Nε ∗ ρε ) · ∇φ| dx dt
0 R2 t ε
0
R2
0
R2
≤ k∇ρm
ε kL2 ((0,T )×R2 ) + εk∇ρε kL2 ((0,T )×R2 )
√
√
+k ρε |∇Nε ∗ ρε |kL∞ ((0,T )×R2 ) T M k∇φkL2 ((0,T )×R2 )
≤ Ck∇φkL2 ((0,T )×R2 ) .
We now use these bounds to derive weak convergence properties. The Dubinskii Lemma (see
Lemma 4.22 in the Appendix) can be applied to obtain the strong convergence locally in space,
which can be extended to global strong convergence using the boundedness of the log-moment.
35
Lemma 4.7. Let ρε be the solution as in Proposition 4.3. Then, up to a subsequence,
ρε
→
ρ
in
ρpε
√
ρε
*
ρp
√
ρ
in
ε∇ρε
→
→
0
Lq ((0, T ) × R2 )
for any 1 ≤ q < ∞ ,
1
L2 (0, T ; H 1 (R2 )) for any m − ≤ p < ∞,
2
L2 ((0, T ) × R2 ) ,
in
in
2
2
2
L ((0, T ) × R ; R )
(4.13)
(4.14)
(4.15)
(4.16)
Proof. Since {ρε }ε are uniformly bounded in Lq ((0, T ) × R2 ) for any 1 ≤ q ≤ ∞, we obtain from
the reflexivity of the Lebesgue spaces for 1 < q < ∞ the weak convergence
ρε * ρ
in Lq ((0, T ) × R2 )
for any 1 < q < ∞ .
(4.17)
Moreover due to the uniform bounds from Lemma 4.6
k∂t ρε kL2 (0,T ;H −1 (R2 )) + kρrε kL2 (0,T ;H 1 (R2 )) ≤ C
for any r ≥ m − 21 , we can apply the Dubinskii Lemma stated in the Appendix to derive
ρε → ρ
in Lr ((0, T ) × BR (0))
for any 2m ≤ r < ∞ and any R > 0 .
The boundedness of the log-moment N (t) allows to extend the strong convergence to the whole
space, since for any 1 ≤ q < ∞ we have
Z TZ
Z TZ
C(1 + T )
ln(1 + |x|2 )
ρqε dxdt ≤ kρε kq−1
ρ dxdt ≤
→ 0,
L∞ ((0,T )×R2 )
2) ε
ln(1
+
R
ln(1
+ R2 )
0 |x|>R
0 |x|>R
as R → ∞. Due to the weak lower semi-continuity of the Lq -norm we can now conclude with
(4.17) that also
Z
Z
q
ρ (t, x)dx ≤ lim inf
ρqε (t, x)dx → 0
as R → ∞ for all q ≥ 1 .
ε>0
|x|>R
|x|>R
Hence we can extend the strong convergence locally in space to strong convergence in R2 :
ρε → ρ
in Lr ((0, T ) × R2 )
for any 2m ≤ r < ∞ .
Additionally the strong convergence in L1 ((0, T ) × R2 ) can be deduced using the bound from the
energy as stated in Lemma 4.21 in the Appendix. Interpolation now yields (4.13).
m−1/2
The weak convergence of ρε
in L2 (0, T ; H 1 (R2 )) holds due to its uniform boundedness given
by inequality (4.12) and the reflexivity of the latter space, where the limit is identified arguing by
the density of spaces. Due to the uniform boundedness of ρε this assertion can be extended to any
finite power bigger than m − 1/2.
√
Since moreover ρε is uniformly bounded in L2 ((0, T ) × R2 ) we have the weak convergence
√
towards ρ in L2 ((0, T ) × R2 ), where again the limit is identified by using the a.e. convergence of
ρε from the strong convergence above. To see (4.15) we rewrite
Z TZ
√
√
√ √
k ρε − ρk2L2 ((0,T )×R2 ) =
(ρε − 2 ρε ρ + ρ)dx dt
R2
0
Z
T
Z
=
0
R2
(ρε − ρ)dx dt − 2
Z
0
T
Z
R2
√
√
√
ρ ( ρε − ρ)dx dt .
The first integral vanishes and the second one converges to 0 due to the weak convergence of
√
√
ρε * ρ in L2 ((0, T ) × R2 ).
√
Finally the convergence in (4.16) is a direct consequence of the bound εk∇ρε kL2 ((0,T )×R2 ) ≤ C
in Lemma 4.6.
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
36
These convergence results from Lemma 4.7 are sufficient to obtain the weak convergence of the
√
nonlinearities ρε ∇hε [ρε ] and ρε ∇hε [ρε ] in L2 ((0, T ) × R2 ), which allow to pass to the limit in the
weak formulation and to deduce the weak lower semicontinuity of the entropy dissipation term:
Lemma 4.8. Let ρε and ρ be as in Lemma 4.7. Then
√
√
ρε ∇hε [ρε ] * ρ ∇h[ρ]
in L2 ((0, T ) × R2 ; R2 )
ρε ∇hε [ρε ] * ρ ∇h[ρ]
2
2
2
in L ((0, T ) × R ; R ) .
(4.18)
(4.19)
Proof. Due to (4.14) and (4.15) it remains to verify
√
√
ρε ∇Nε ∗ ρε * ρ ∇N ∗ ρ in L2 ((0, T ) × R2 ; R2 ) .
√
Due to Lemma 4.6, we have the weak convergence of ρε ∇Nε ∗ ρε in L2 ((0, T ) × R2 ; R2 ). In order
2
to identify the limit we consider for a φ ∈ L ((0, T ) × R2 ; R2 ):
Z TZ
√
√
( ρε ∇Nε ∗ ρε − ρ ∇N ∗ ρ) · φ dxdt
R2
T Z
0
Z
√
√
( ρε − ρ)(∇Nε ∗ ρε ) · φ dxdt +
=
R2
0
Z
T
Z
+
0
R2
√
Z
0
T
Z
R2
√
ρ(∇(Nε − N ) ∗ ρε ) · φ dxdt
ρ ∇N ∗ (ρε − ρ) · φ dxdt
(4.20)
The first term converges to zero using (4.15), since by (4.11) it is bounded by
√
√
√
√
k ρε − ρkL2 ((0,T )×R2 ) k∇Nε ∗ ρε kL∞ ((0,T )×R2 ) kφkL2 ((0,T )×R2 ) ≤ Ck ρε − ρkL2 ((0,T )×R2 ) → 0 .
For the second term we first use the Cauchy-Schwarz inequality
Z TZ
√
√
ρ(∇(Nε − N ) ∗ ρε ) · φ dxdt ≤ M T kφkL2 ((0,T )×R2 ) k∇(Nε − N ) ∗ ρε kL∞ ((0,T )×R2 ) .
0
R2
To see that this convolution term vanishes we bound further
Z x−y
x−y
|∇(Nε − N ) ∗ ρε | = −
ρ
(y)dy
ε
2
2
2
|x − y| + ε
|x − y|
R2
Z
Z ∞
|x − y|
1
≤ ε2 kρε kL∞ (R2 )
dy = εC
ds ≤ εC → 0
2 + ε2 )|x − y|2
2+1
(|x
−
y|
s
2
R
0
uniformly in x, t, where we substituted s = |x − y|/ε. For the remaining term in (4.20) we proceed
changing the order of integration, where we again skip the dependence of ρε and φ on t in the
following:
Z TZ
√
ρ (∇N ∗ (ρε − ρ)) · φ dxdt
0
R2
Z T Z p
p
Z p
p
p
1
x−y
=
ρε (y) − ρ(y)
ρε (y) + ρ(y)
ρ(x)
·
φ(x)dx
dydt
2π 0 R2
|x − y|2
R2
Z p
p
p
1 √
1
√
≤
k ρε − ρkL2 ((0,T )×R2 ) ρ
(·)
+
ρ(·)
ρ(x)
|φ(x)|dx
ε
2
2π
|x − ·|
R2
L ((0,T )×R2 )
To prove that this integral vanishes in the limit, due to (4.15) it suffices to show that
2
Z T Z p
Z p
p
1
ρε (y) + ρ(y)
ρ(x)
|φ(x)|dx dydt ≤ C .
|x − y|
0
R2
R2
37
We shall therefore split the integral into two parts and consider first
!2
1
|φ(x)|dx dydt
|x − y|
0
R2
|x−y|≤1
! Z
!
Z
Z TZ
1
1
2
(ρε (y) + ρ(y))
|φ| (x)
dx
ρ(x)
dx dydt
≤2
|x − y|
|x − y|
0
R2
|x−y|≤1
|x−y|≤1
Z Z Z
T
|φ|2 (x)
≤ CkρkL∞ ((0,T )×R2 ) kρkL∞ ((0,T )×R2 ) + kρε kL∞ ((0,T )×R2 )
dydxdt
0
R2 |x−y|≤1 |x − y|
!
Z
Z TZ
1
2
|φ| (x)
≤C
dy dxdt ≤ Ckφk2L2 ((0,T )×R2 )
|x
−
y|
2
0
R
|x−y|≤1
Z
T
Z
p
Z
p
ρε (y) + ρ(y)
p
ρ(x)
It remains to bound the integral for |x − y| > 1:
Z
T
0
≤2
Z
p
Z
p
ρε (y) + ρ(y)
R2
Z
|x−y|>1
T
kφk2L2 (R2 )
0
≤ 2M
p
ρ(x)
Z
Z
0
Z
(ρε (y) + ρ(y))
ρ(x)
R2
T
kφk2L2 (R2 )
!2
1
|φ(x)|dx dydt
|x − y|
|x−y|>1
Z
R2
1
dxdydt
|x − y|2
(ρε (y) + ρ(y)) dydt = 4M 2 kφk2L2 ((0,T )×R2 ) .
Proof of Theorem 4.1. The convergence property of the nonlinearity in (4.19) and the weak convergence of the time derivative due to Lemma 4.6 allow to pass to the limit in the weak formulation of
the Cauchy problem for (4.1), where the linear diffusion term vanishes due to (4.16). The uniqueness of the solution is implied from Theorem 1.3 and Corollary 6.1 of [25], where we shall not go
further into detail here.
It thus remains to pass to the limit in the energy inequality. Since the energy dissipation is
weakly lower semicontinuous due to (4.18), we get
Z
0
T
D[ρ](t)dt ≤ lim inf
ε>0
T
Z
0
Dε [ρε ](t)dt .
In order to obtain the energy inequality (4.3) in the limit ε → 0 it thus remains to show Eε [ρε ](t) →
E[ρ](t) for t ∈ [0, T ]. Lemma 4.21 and the uniform bounds on ρε in Lemma 4.6 directly imply the
strong convergence of ρε in L∞ (0, T ; Lm (R2 )). It is therefore left to prove the convergence for the
convolution term and we rewrite
Z
Z Z
|x − y|2 + ε2
−4π
(ρε Nε ∗ ρε − ρN ∗ ρ)dx =
ρε (x)ρε (y) ln
dxdy
|x − y|2
R2
R2 R2
Z Z
+2
(ρε (x)(ρε (y) − ρ(y)) + ρ(y)(ρε (x) − ρ(x))) ln |x − y|dxdy .
R2
R2
We split the domain of integration and first analyze the case |x − y| ≥ 1. In this domain
Z
Z
ρε (x)ρε (y) ln
R2
|x−y|≥1
|x − y|2 + ε2
dxdy ≤
|x − y|2
Z
R2
Z
|x−y|≥1
ρε (x)ρε (y) ln(1 + ε2 )dxdy ≤ ε2 M 2 → 0
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
38
and using the Cauchy-Schwarz inequality we obtain moreover
!2
Z Z
ρε (x)(ρε (y) − ρ(y)) ln |x − y|dxdy
R2
|x−y|≥1
2
Z
ρε (x) ln |x − y|dx dy
≤ kρε − ρkL1 (R2 )
|ρε (y) − ρ(y)| |x−y|≥1
R2
Z Z
≤ 2M kρε − ρkL1 (R2 )
(ln(1 + |x|) + ln(1 + |y|))ρε (x)(ρε (y) + ρ(y))dxdy
Z
R2
2
|x−y|≥1
≤ 4M (N (t) + Nε (t))kρε − ρkL1 (R2 ) ≤ C(1 + T )kρε − ρkL∞ (0,T ;L1 (R2 )) → 0
We now turn to the integration domain |x − y| < 1 and estimate
Z
Z 1
Z Z
r 2 + ε2
|x − y|2 + ε2
∞
2
dxdy
≤
kρ
k
ρ
(y)
r ln
drdy
ρε (x)ρε (y) ln
ε L (R )
ε
2
|x − y|
r2
R2
0
R2 |x−y|<1
Z 1
r2 + ε2
≤ CM
r ln
dr
r2
0
√
converges
Within the domain ε < |x − y| = r < 1, the integrand is bounded by ln(1 + ε), which
√
to 0 as ε → 0. For the integral over the remainder of the domain 0 < |x − y| = r < ε, we perform
the substitution r = εs and split the integral once more, leading to
Z √ε
Z 1
Z 1/√ε
r 2 + ε2
1
1
2
2
r ln
dr
=
ε
s
ln
1
+
s
ln
1
+
ds
+
ε
ds
r2
s2
s2
0
1
0
On the other hand we have
Z Z
≤ ε2 C + εC → 0 .
ρε (x)|ρε (y) − ρ(y)| | ln |x − y||dxdy
Z
Z
≤ kρε kL∞ (R2 ))
|ρε (y) − ρ(y)|
| ln |x − y||dxdy
R2
|x−y|<1
R2
≤ Ckρε − ρkL1 (0,T ;L1 (R2 )) → 0 .
|x−y|<1
This proves the convergence of the entropy, which together with the weak lower semicontinuity
of the entropy-dissipation leads to the desired energy-energy dissipation inequality (4.3) for the
limiting solution ρ.
4.2. Long-Time Behavior of Solutions. Our main result of Section 2 together with the uniqueness argument for radial stationary solutions to (4.1) of [46] and the characterization of global
minimizers in [22] and Corollary 3.10 leads to the following result:
Theorem 4.9. There exists a unique stationary state ρM of (4.1) with mass M and zero center
of mass in the sense of Definition 2.1 with the property ρM ∈ L1log (R2 ). Moreover, ρM is compactly
supported, bounded, radially symmetric and non-increasing. Moreover, the unique stationary state
is characterized as the unique global minimizer of the free energy functional (4.2) with mass M .
As a consequence, all stationary states of (4.1) in the sense of Definition 2.1 with mass M are
given by translations of the given profile ρM :
S = ρM (x − x0 ) such that x0 ∈ R2 .
Remark 4.10. As in [46, Corollary 2.3] we have the following result comparing the support and
height for stationary states with different masses based on a scaling argument: Let ρ1 be the radial
solution with unit mass. Then the radial solution with mass M is of the form
1
m−2
ρM (x) = M m−1 ρ1 (M − 2(m−1) x) .
For two stationary states ρM1 and ρM2 with masses M1 > M2 the following relations hold:
39
(a) If m > 2, then ρM1 has a bigger support and a bigger height than ρM2 .
(b) If m = 2, then all stationary states have the same support.
(c) If 1 < m < 2, then ρM1 has smaller support and bigger height than ρM2 .
We will study now the long time asymptotics for the global weak solutions ρ of (4.1) that
according to the entropy inequality in Theorem 4.1 satisfy
Z ∞
D[ρ](t)dt ≤ E[ρ0 ] .
lim E[ρ](t) +
t→∞
0
Since the entropy is bounded from below, this implies for the entropy dissipation
Z ∞
D[ρ](s)ds = 0 .
lim
t→∞
t
Let us therefore now consider the sequence
ρk (t, x) = ρ(t + tk , x)
on (0, T ) × R2
for some tk → ∞ ,
for which we obtain
Z
0
=
lim
k→∞
∞
tk
D[ρ](t)dt ≥ lim
k→∞
Z
T
0
Thus D[ρk ] → 0 in L1 (0, T ), or equivalently
√
k ρk |∇h[ρk ]| k2L2 ((0,T )×R2 ) → 0
D[ρ](t + tk )dt ≥ 0 .
as k → ∞ .
The proof of convergence towards the steady state will be based on weak lower semicontinuity of
the entropy dissipation. Assume that ρk * ρ in L∞ (0, T ; L1 (R2 )∩Lm (R2 )), then we have to derive
p
√
k ρ|∇h[ρ]|kL2 ((0,T )×R2 ) ≤ lim inf k ρk |∇h[ρk ]|kL2 ((0,T )×R2 ) = 0 .
k→∞
2
Since the L -norm is weakly lower semicontinuous, it therefore remains to show similarly as in
Lemma 4.8
p
√
ρk ∇h[ρk ] * ρ∇h[ρ]
in L2 ((0, T ) × R2 ) .
From there it can be deduced that ρ is the stationary state ρM with M = kρ0 kL1 (R2 ) by the
uniqueness theorem 4.9, if we can guarantee that no mass gets lost in the limit.
The main difficulty for passing to the limit in the long-time behavior lies in obtaining sufficient
compactness avoiding the loss of mass at infinity. Hence, a crucial part for identifying the limiting solution ρM as the stationary state with the same mass is proving the weak convergence in
L1 ((0, T ) × R2 ). Moreover, if we want to choose the right limiting profile among the stationary
states in S, we need to show the conservation of some symmetry. In fact, it is easy to check that
the center of mass should formally be preserved by the evolution due to the antisymmetry of the
gradient of the Newtonian potential. However, we lack control of the tail of the solutions since the
bound on the logarithmic moment of the density is time dependent. We therefore need to develop
uniform in time bounds for larger moments of ρ(t). The main theorem of this section is:
Theorem 4.11. Let ρ be the weak solution to (4.1) given in Theorem 4.1 with nonnegative initial
data ρ0 ∈ L1 ((1 + |x|2 )dx) ∩ L∞ (R2 ). Then, as t → ∞, ρ(·, t) converges to the unique stationary
state with the same mass and center of mass as the initial data, i.e., to
Z
1
xρ0 (x) dx ,
ρcM := ρM (x − xc )
where xc =
M R2
with M = kρ0 kL1 (R2 ) , ensured by Theorem 4.9. More precisely, we have
lim kρ(t, ·) − ρcM (·)kLq (R2 ) → 0
t→∞
for all 1 ≤ q < ∞ .
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
40
Our aim is to show that the second moment of solutions to (4.1) is uniformly bounded in time
for all t ≥ 0. This in turn shows easily that the first moment is preserved in time for all t ≥ 0, as
we will prove below. Recall that by (2.13) we denote by M2 [f ] the second moment of f ∈ L1+ (Rd ).
We first derive rigorously the evolution of the second moment in time:
Z tZ
tM 2
(4.21)
M2 [ρ(t, ·)] − M2 [ρ(0, ·)] = 4
ρm dx dt −
2π
0
R2
starting from the regularized system (4.4).
problem, we obtain
Z
d
M2 [ρε ] = 4
(ρm
ε + ερε )dx −
dt
R2
Z
= 4
(ρm
ε + ερε )dx −
R2
Computing the second moment of the regularized
Z Z
1
x · (x − y)
dx dy
ρε (x, t)ρε (y, t)
π R2 R2
|x − y|2 + ε2
Z Z
1
|x − y|2
dx dy .
ρε (x, t)ρε (y, t)
2π R2 R2
|x − y|2 + ε2
The strong convergence in (4.13) allows to pass to the limit ε → 0 in the first integral. In order
to pass to the limit in the convolution term we need to split the domain in three parts as follows.
First of all, we have
Z Z
Z Z
1
1
|x − y|2
dxdy
=
ρε (t, x)ρε (t, y)
ρε (t, x)ρε (t, y)dxdy − R1ε (t)
2π R2 |x−y|>1
|x − y|2 + ε2
2π R2 |x−y|>1
where
R1ε (t)
ε2
2π
=
Z Z
ρε (t, x)ρε (t, y)
|x−y|>1
1
M2
dxdy ≤ ε2
→0
2
2
|x − y| + ε
2π
√
as ε → 0 uniformly in time. Similarly for ε < |x − y| ≤ 1 we have
Z Z
Z Z
1
|x − y|2
1
ρ
(t,
x)ρ
(t,
y)
dxdy
=
ρε (t, x)ρε (t, y)dxdy − R2ε (t),
ε
ε
2π R2 √ε<|x−y|≤1
|x − y|2 + ε2
2π R2 √ε<|x−y|≤1
where
R2ε (t)
=
≤
ε2
2π
1
dxdy
2 + ε2
|x
−
y|
ε<|x−y|≤1
Z
r
C0 M
ε2 C0 M √
dr = ε2
(ln(1 + ε2 ) − ln(ε + ε2 )) ≤ Cε → 0
2 + ε2
r
2
ε<r≤1
Z
R2
Z
√
ρε (t, x)ρε (t, y)
∞
as
√ ε → 0 uniformly in time, where we let C0 = supε>0 kρε kL . Finally for the domain 0 < |x−y| ≤
ε we obtain
Z Z
|x − y|2
C0 M √
R3ε (t) =
ρε (t, x)ρε (t, y)
dxdy ≤
|B ε | ≤ εC → 0
2
2
√
|x − y| + ε
2π
R2 0<|x−y|< ε
as ε → 0 uniformly in time. Thus combining all integrals we obtain
Z Z
Z Z
|x − y|2
1
1
ρε (x, t)ρε (y, t)
dx dy =
ρε (t, x)ρε (t, y)dxdy − Rε
2π R2 R2
|x − y|2 + ε2
2π R2 |x−y|>√ε
Z Z
1
M2
→
ρε (t, x)ρε (t, y)dxdy =
,
2π R2 R2
2π
where we denoted Rε = R1ε + R2ε − R3ε → 0 as ε → 0. The argument can easily be made rigorous
by using compactly supported approximations of |x|2 on R2 as test functions, see e.g. also [11].
We finally obtain (4.21) by integrating in time.
Now, we want to compare general solutions to (4.1) with its radial solutions. In order to do this
we will make use of the concept of mass concentration, which has been recalled in (2.4), and used
for instance in [46, 32] for classical applications to Keller-Segel type models.
Following exactly the same proof as in [46], the following two results hold for the solutions of
(4.1). The first result says that for two radial solutions, if one is initially “more concentrated”
41
than the other one, then this property is preserved for all time. The second result compares a
general (possibly non-radial) solution ρ(t, ·) with another solution µ(t, ·) with initial data ρ# (0, ·),
i.e., the decreasing rearrangement of the initial data for ρ(t, ·), and it says that the symmetric
rearrangement of ρ(t, ·) is always “less concentrated” than the radial solution µ(t, ·). This result
generalizes the results from [32] to nonlinear diffusion with totally different proofs. We also refer the
interested reader to the survey [69] for a general exposition of the mass concentration comparison
results for local nonlinear parabolic equations and to the recent developments obtained in [71], [72]
in the context of nonlinear parabolic equations with fractional diffusion.
Proposition 4.12. Let m > 1 and f, g be two radially symmetric solutions to (4.1) with f (0, ·) ≺
g(0, ·). Then we have f (t, ·) ≺ g(t, ·) for all t > 0.
Proposition 4.13. Let m > 1 and ρ be a solution to (4.1), and let µ be a solution to (4.1) with
initial condition µ(0, ·) = ρ# (0, ·). Then we have that µ(t, ·) remains radially symmetric for all
t ≥ 0, and in addition we have
ρ# (t, ·) ≺ µ(t, ·)
for all t ≥ 0.
Now we are ready to bound the second moment of solutions in the two-dimensional case: we
will show that if ρ(t, ·) is a solution to (4.1) with M2 [ρ0 ] finite, then M2 [ρ(t)] must be uniformly
bounded for all time.
Theorem 4.14. Let ρ0 ∈ L1 ((1 + |x|2 )dx) ∩ L∞ (R2 ). Let ρ(t, ·) be the solution to (4.1) with initial
data ρ0 . Then we have that
M2 [ρ(t)] ≤ M2 [ρ0 ] + C(kρ0 kL1 )
for all t ≥ 0.
Proof. Recalling that ρM is the unique radially symmetric stationary solution with the same mass
as ρ0 and zero center of mass, we let ρM,λ := λ2 ρM (λx) with some parameter λ > 1. Since
ρ0 ∈ L1 (R2 ) ∩ L∞ (R2 ), we can choose a sufficiently large λ such that ρ#
0 ≺ ρM,λ . Note that λ > 1
also directly yields that ρM ≺ ρM,λ .
Let µ(t, ·) be the solution to (4.1) with initial data ρM,λ . Combining Proposition 4.12 and
Proposition 4.13, we have that
ρ# (t, ·) ≺ µ(t, ·)
for all t ≥ 0.
It then follows from (2.10) that
Z
Z
Z
ρm (t, x)dx =
[ρ# ]m (t, x)dx ≤
R2
R2
R2
µm (t, x)dx
for all t ≥ 0.
(4.22)
Now using the computation of the time derivative of M2 [ρ(t)] in (4.21), where ρ(·, t) is a solution
to (4.1), we get
Z tZ
tM 2
.
(4.23)
M2 [ρ(t)] − M2 [ρ0 ] = 4
ρm (t, x)dx dt −
2π
0
R2
Since µ(t, ·) is also a solution to (4.1), (4.23) also holds when ρ is replaced by µ. Combining this
fact with (4.22), we thus have
M2 [ρ(t)] − M2 [ρ0 ] ≤ M2 [µ(t)] − M2 [µ(0)] ≤ M2 [µ(t)].
(4.24)
Finally, it suffices to show M2 [µ(t)] is uniformly bounded for all time. Since ρM is a stationary
solution and we have ρM ≺ ρM,λ , it follows from Proposition 4.12 that ρM ≺ µ(t, ·) for all t ≥ 0,
hence we have M2 [ρM ] ≥ M2 [µ(t)] due to Lemma 2.6. Plugging this into (4.24) yields
M2 [ρ(t)] ≤ M2 [ρ0 ] + M2 [ρM ]
for all t ≥ 0,
where M2 [ρM ] is a constant only depending on the mass M := kρ0 kL1 (R2 ) , which can be computed
as follows: using Remark 4.10, we know the support of ρM is given by the ball centered at 0 of
m−2
radius R(M ) = C0 M 2(m−1) (where C0 is the radius of the support for the stationary solution with
2m−3
unit mass), hence M2 [ρM ] ≤ M R(M )2 ≤ C02 M m−1 .
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
42
Remark 4.15. It is interesting the contrast of our last result showing uniform-in-time bounds for
the second moment for m > 1 finite, with the results in [30, 31] where the case m → ∞ limit of
the gradient flow is analysed. In the “m = ∞” case, the second moment of any solution is actually
decreasing in time, leading to the result that all solutions converge towards the global minimizer with
some explicit rate. As mentioned in the introduction, a result of this sort for any other potential
rather than the attractive logarithmic potential is lacking.
As already mentioned above, a key ingredient in the proof of Theorem 4.11 is the confinement
of mass, which is first now obtained as follows:
Lemma 4.16. Let ρ be a global weak solution as in Theorem 4.1 with mass M with initial data
ρ0 ∈ L1 ((1 + |x|2 )dx) ∩ L∞ (R2 ) and consider as above the sequence {ρk }k∈N = {ρ(· + tk , ·)}k∈N in
(0, T ) × R2 . Then there exists a ρ ∈ L1 ((0, T ) × R2 ) ∩ Lm ((0, T ) × R2 ) and a subsequence, that we
denote with the same index without loss of generality, such that:
in L1 ((0, T ) × R2 ) ∩ Lm ((0, T ) × R2 )
ρk (t, x) * ρ(t, x)
as k → ∞.
Proof. Due to the entropy being uniformly bounded from below and by the entropy inequality
(4.2), we have ρk ∈ L∞ ((0, T ); Lm (R2 )). Using Theorem 4.14, we deduce that
M2 [ρk (t)] ≤ M2 [ρ0 ] + C(kρ0 kL1 (R2 ) )
∞
for all k ∈ N and 0 ≤ t ≤ T.
m
(4.25)
2
Since {ρk }k∈N are also uniformly bounded in L (0, T ; L (R )) we obtain equi-integrability and
can therefore apply the Dunford-Pettis theorem (see Theorem 4.20 in Appendix) to obtain the
weak convergence in L1 ((0, T ) × R2 ) ∩ Lm ((0, T ) × R2 ).
In order to obtain weak lower semicontinuity of the entropy dissipation term, we need additional
convergence results. These are derived from the following uniform bounds:
Lemma 4.17. Let ρ be a global weak solution as in Theorem 4.1 with mass M and consider as
above the sequence {ρk }k∈N = {ρ(· + tk , ·)}k∈N in (0, T ) × R2 . Then
kρk kL∞ (0,T ;L1 (R2 )) + kρk kL∞ ((0,T )×R2 ) ≤ C
√
k ρk ∇N ∗ ρk kL2 ((0,T )×R2 ) + k∇N ∗ ρk kL∞ ((0,T )×R2 ) ≤ C
k∂t ρk kL2 (0,T ;H −1 (R2 )) + kρqk kL2 (0,T ;H 1 (R2 )) ≤ C
for any q ≥ m −
1
.
2
Proof. The bounds are obtained from the energy-energy dissipation inequality (4.3) in an analogous
way to the ones given in Lemma 4.6 with the only difference concerning the replacement of Nε by
N , which however makes no difference in the estimate (4.11).
Using these estimates the following convergence properties can be derived in an analogous way
to the proof of Lemma 4.7.
Lemma 4.18. Let the assumptions of Lemma 4.16 hold. Then, up to subsequences that we denote
with the same index,
ρk → ρ
ρpk
√
* ρ
ρk →
in
p
p
in
ρ
Lq ((0, T ) × R2 ))
2
1
for any 1 ≤ q < ∞ ,
2
L (0, T ; H (R ))
in
2
2
L ((0, T ) × R ) .
for any m −
1
2
≤ p < ∞,
These convergence results from Lemma 4.18 and Lemma 4.16 are sufficient to obtain the weak
√
convergence of the nonlinearities ρk ∇h[ρk ] and ρk ∇h[ρk ] in L2 ((0, T ) × R2 ), which allows to
deduce the weak lower semicontinuity of the entropy dissipation term and to pass to the limit in
the weak formulation of (4.1) in the same way as in the proof of Lemma 4.8.
43
Lemma 4.19. Let ρk and ρ be as in Lemma 4.18. Then
p
√
ρk ∇h[ρk ] * ρ ∇h[ρ]
in L2 ((0, T ) × R2 ; R2 )
in L2 ((0, T ) × R2 ; R2 ) .
ρk ∇h[ρk ] * ρ ∇h[ρ]
This enables us to close the proof of convergence towards the set of stationary states.
Proof of Theorem 4.11. Let us first notice that ρ ∈ L∞ ((0, T ) × R2 ) due to the first convergence in
Lemma (4.18) and the uniform in time bound on the weak solutions in Theorem 4.1. Due to the
weak lower semicontinuity of the L2 ((0, T ) × R2 )-norm and the bound from below of the entropy
as done in Proposition 4.3 implies that D[ρk ] → 0 in L1 (0, T ), and as consequence
p
√
k ρ|∇h[ρ]|k2L2 ((0,T )×R2 ) ≤ lim inf k ρk |∇h[ρk ]|k2L2 ((0,T )×R2 ) = 0 .
k→∞
Thus ρ solves
ρ|∇h[ρ]|2 = 0
a.e. in (0, T ) × R2 .
(4.26)
Moreover, due to the convergence properties in Lemmas 4.18 and 4.19 the limiting density ρ is
a weak distributional solution to (4.1) with test functions is L2 (0, T ; H 1 (R2 )). Due to (4.26), we
get that ρ∇h[ρ] = 0 a.e. in (0, T ) × R2 and thus ∂t ρ = 0 in L2 (0, T ; H −1 (R2 )). This yields that
ρ(t, x) ≡ ρ(x) does not depend on time.
Due to the convergence properties in Lemma 4.18, the uniform bound on the second moment
(4.25) together with Lemma 4.21 in the Appendix, we can deduce that ρ ∈ L1 ((1 + |x|2 )dx) and
that ρk → ρ in L∞ (0, T ; L1 (R2 )). In particular, ρ has mass M .
Putting together all the properties of ρ just proved together with the fact that ∇ρm ∈ L2 (R2 )
due to Lemma (4.18), we infer that ρ corresponds to a steady state of equation (4.1) in the sense
of Definition 2.1. The uniqueness up to translation of stationary states in Theorem 4.9 shows that
ρ is a translation of ρM , and thus ρ ∈ S. In fact, we have shown that the limit of all convergent
sequences {ρk }k∈N must be a translation of ρM . This in turn shows that the set of accumulation
points of any time diverging sequence belongs to S.
Finally, in order to identify uniquely the limit, we take advantage of the translational invariance.
We first remark that the center of mass of the initial data is preserved for all time due to the antisymmetry of ∇N . Due to Proposition 4.14, all time diverging sequences have uniformly bounded
second moments, thus since ρ is an accumulation point of a sequence ρk , by Lemma 4.21 we have
Z
Z
Z
x(ρ(x)
xρ(x)dx
=
−
x
M
−
ρ
(t,
x))dx
−
x(ρ
(t,
x)
−
ρ
(x))dx
c
k
k
0
2
2
R
R
R2
Z
1/2
≤
|x||ρ(x) − ρk (t, x)|dx ≤ M2 [|ρk (t) − ρ|]1/2 kρk (t) − ρkL1 (R2 )
R2
1/2
≤ Ckρk (t) − ρkL∞ (0,T ;L1 (R2 )) → 0.
Hence all accumulation points of the sequences have the same center of mass as the initial data.
Then, all possible limits reduce to the translation of ρM to the initial center of mass as desired. Appendix
Theorem 4.20 (Dunford-Pettis Theorem). Let (X, Σ, µ) be a probability space and F be a bounded
subset of L1 (µ). Then F is equi-integrable if and only if F is a relatively compact subset in L1 (µ)
with the weak topology.
J. A. CARRILLO1 , S. HITTMEIR2 , B. VOLZONE3 , Y. YAO4
44
Lemma 4.21. Let (fε ) be a sequence of nonnegative functions uniformly bounded in the space
L∞ (0, T ; L1log (R2 ) ∩ L∞ (R2 )) with kfε kL1 (R2 ) = kf kL1 (R2 ) = M . Assume moreover that fε → f
a.e. in R2 × (0, T ). Then, f ∈ L∞ (0, T ; L1log (R2 ) ∩ L∞ (R2 )) and
fε → f
in L∞ (0, T ; L1 (R2 )) .
The same result holds by replacing the logarithmic moment by the second moment, i.e., by replacing
L1log (R2 ) by L1 ((1 + |x|2 )dx) everywhere.
Proof. A similar argument was used in the proof of Proposition 2.1 in [38]. First observe that by
the Fatou lemma, for any m > 1
Z
Z
Z
m
m
f dx = sup
lim fε dx ≤ lim inf sup
fεm dx ≤ C.
sup
(0,T )
(0,T )
ε→0
ε→0
(0,T )
Let now L > 1 and gε = min{fε , L}. Then gε → g = min{f, L} a.e. Let moreover R > 1, then by
the dominated convergence, it holds for sufficiently small ε > 0 on the ball BR (0):
Z
C
|gε − g|dx ≤ m−1 .
sup
L
(0,T ) BR (0)
and we obtain
Z
Z
|fε − f |dx ≤ sup
sup
(0,T )
(0,T )
BR (0)
≤ sup
(0,T )
≤
BR (0)
Z
|fε − gε |dx + sup
(0,T )
Z
BR (0)
+
C
Lm−1
(0,T )
1
(fε − L)dx +
{fε ≥L}∩BR (0)
Z
fεm
sup
dx
m−1
(0,T ) BR (0) L
|gε − g|dx + sup
BR (0)
Z
Lm−1
Z
+ sup
(0,T )
Z
+ sup
(0,T )
BR (0)
{f ≥L}∩BR (0)
m
|g − f |dx
(f − L)dx
f
3C
dx ≤ m−1 .
Lm−1
L
Using additionally the confinement of mass from the bound on the log-moment, we obtain
Z
Z
ln(1 + |x|2 )
C
sup
|fε − f |dx ≤
|fε − f |dx ≤
→0
as R → ∞.
2
ln(1 + R2 )
(0,T ) {|x|>R}
R2 ln(1 + R )
Since L > 1 is arbitrary and m > 1, this shows that fε → f strongly in L∞ (0, T ; L1 (R2 )). The
proof in case we replace L1log (R2 ) by L1 ((1 + |x|2 )dx) is done analogously.
For the proof of the following Dubinskii Lemma we refer to [24] or Theorem 12.1 in [51]:
Lemma 4.22. Let Ω ⊂ R2 be bounded with ∂Ω ∈ C 0,1 and let {fε }, 0 < ε < 1, satisfy
k∂t fε kL1 (0,T ;(H s (Ω))0 ) + kfεp kLq (0,T ;H 1 (Ω)) ≤ C,
for some p ≥ 1, q ≥ 1 and s ≥ 0. Then {fε } is relatively compact in Lpl (0, T ; Lr (Ω)) for any
r < ∞ and l < q.
Acknowledgments
JAC was partially supported by the Royal Society by a Wolfson Research Merit Award and the EPSRC
grant EP/K008404/1. SH acknowledges support by the Austrian Science Fund via the Hertha-Firnberg
project T-764, and the previous funding by the Austrian Academy of Sciences ÖAW via the New Frontiers
project NST-0001. BV is partially supported by the INDAM-GNAMPA project 2015 “Proprietà qualitative
di soluzioni di equazioni ellittiche e paraboliche” (ITALY). YY was partially supported by the NSF grant
DMS-1565480. YY wants to thank Almut Burchard for a helpful discussion.
45
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