A Mean Field Equation as Limit of Nonlinear
Diffusions with Fractional Laplacian Operators
Sylvia Serfaty and Juan Luis Vázquez
June 2012

Abstract
In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a“mean field” equation arising in superconductivity and superfluidity. For this equation, we
obtain uniqueness, universal bounds and regularity results. We also show that solutions
with finite second moment and radial solutions admit an asymptotic large time limiting
profile which is a special self-similar solution: the “elementary vortex patch”.

2000 Mathematics Subject Classification: 35K55, 35K65, 76S05.
Keywords and phases. Nonlinear diffusion, fractional Laplacian, hydrodynamic limit, superconductivity, vortex patch, universal bound.

1

Introduction

We consider the class of evolution models with nonlocal effects given by the system(1.1)

ut = ∇ · (u ∇p),

p = Ku.

where u is a function of the variables (x, t) to be thought of as a density or concentration, andtherefore nonnegative, while p is the “pressure” and v = −∇p represent the mass propagation
speed. We postulate that p is related to u via a linear integral operator K which we assume inpractice to be the inverse of a fractional Laplacian, that is, K = (−∆)−s , with s ∈ (0, 1]. Moregeneral integral operators are also now being considered in the literature.
The problem is posed for t > 0 and x ∈ Rn , n ≥ 2. (The case n = 1 is easier and a bit different,
and a short explanation will be given below). Finally, we give initial conditions(1.2)

u(x, 0) = u0 (x),where u0 is a nonnegative integrable function in Rn .
1

x ∈ Rn ,2
Nonlocal diffusive models. These correspond to the equationut = ∇ · (u ∇(−∆)−s u),(1.3)

in the parameter range 0 < s < 1. This equation can be considered as a nonlocal version of thestandard porous medium equation ut = ∆(u2 ). For convenience, we will often refer to it as the
FPME (fractional porous medium equation).
The FPME has been derived starting from the continuity equation∂t u + ∇ · (uv) = 0 ,(1.4)

where u is a density and v is a velocity field, which according to Darcy’s law derives from apressure, v = −∇p (as in the theory of gases in porous media). Physical applications range frommacroscopic evolution of particle systems with short- and long-range interactions [33], to phasesegregation [31, 32], and dislocation dynamics [35, 15]. The existence of a weak solution, therelevant integral estimates, as well as the property of compact support have been established by
Caffarelli and the second author in the paper [20]. The existence is shown by passing to the limit
in the sequence of smooth solutions of approximate problems, and using the compactness of theintegral operators involved in the weak formulation and its energy estimates for s < 1 (thus themethod does not directly apply to s = 1). Note that no uniqueness is proven, and examples areconstructed where the maximum principle does not hold for s > 1/2.
Hydrodynamic or mean field equation. The case s = 1 of equation (1.1)–(1.2)(1.5)

ut = ∇ · (u ∇p),p = (−∆)−1 u ,supplemented with initial data:(1.6)

u(x, 0) = u0 (x) ,is no more a diffusion equation. This equation was first studied in R2 by Lin-Zhang [44]. There,
existence is proven by a vortex point approximation, and uniqueness by estimates in Zygmund
spaces. In dimension 2, the equation is directly related to the Chapman-Rubinstein-Schatzman
mean field model of superconductivity [23] and to E’s model of superfluidity [26], which would
correspond rather to the equation(1.7)

ut = ∇ · (|u|∇p)which coincides with (1.5) when u ≥ 0. That model is a mean field model for the motionof vortices in a superconductor in the Ginzburg-Landau theory. There, u represents the local
vortex-density, and p represents the induced magnetic field in the sample. In this application
the equation is really posed in a bounded domain and for signed “vorticity measures” u (which
can be measures instead of functions), but the study of (1.5) in R2 with u ≥ 0, corresponding to
the situation where the vorticity is positive, is a reasonable start. It can also be viewed as thegradient-flow version of the Euler equation in vorticity form i.e.
ut = ∇ · (u ∇⊥ p)

p = (−∆)−1 u ,3hence also the term hydrodynamic equation. In [2], Ambrosio and the first author studied
the model in a bounded two-dimensional domain (taking into account the possibility of flux ofvortices through the boundary), via a gradient flow approach: the equation is the gradient-flow
for a natural energy (see below) for the 2-Wasserstein distance. There, u is denoted µ, and p isdenoted h. Solutions with u measure (and not L1 ) are also treated there. The equation (1.7) withchanging sign u has been studied by Masmoudi-Zhang [49] in R2 , Ambrosio-Mainini-Serfaty [3]in bounded domains of the plane and in the whole plane, again using a gradient-flow approach.
This equation is also related to nonlocal aggregation models, which have attracted a lot of
attention lately, notably in conjunction with the Keller-Segel model [37], see e.g. [10, 11, 12].
They correspond to (1.1) with K a general integral operator corresponding to an attractive(instead of repulsive) interaction. Particularly relevant to us is the paper of Bertozzi-LaurentLéger [12] which deals mostly with aggregation via the Coulomb kernel, but contains a section(Section 3) that deals with the repulsive case, i.e. exactly the equation (1.5)-(1.6). In addition
to existence and uniqueness results, it makes a detailed analysis of the property of support
propagation and asymptotic behaviour for solutions with compactly supported initial data.
Outline of results. In the present paper we are interested in obtaining Equation (1.5) as thelimit of the nonlocal diffusive equations (1.3) when s → 1; and in expanding the theory for thelimiting equation. We consider the initial-value problem posed in Rn , for all dimensions n ≥ 2,with comments on the easier case n = 1. We develop an existence theory for weak solutions witha nonnegative finite measure as initial data. Such solutions are uniformly bounded for positive
times in the form |u(x, t)| ≤ 1/t. We prove a uniqueness result for bounded solutions.
As another important feature of the theory, we derive the support propagation properties (forgeneral solutions), and compare the result with a similar property for the fractional diffusion
approximation.
Some insight is obtained with the introduction of the special formulation of the equation underconditions of radial symmetry, leading to a non-homogeneous Burgers’ equation for the so-called
mass function. With this Burgers’ equation approach, we recover the existence of a fundamental
solution in the form of a round vortex patch of constant (in space) density, and show that they
are attractors for the evolution of all radial solutions. We also find other explicit radial solutions
that allow to derive a new non-comparison result.
We also study the question of large time behaviour for another class of initial data where therestriction of compact support is dropped, more precisely integrable data with finite second
moment. The method, which is interesting for its own sake, is an energy method that follows
the now well-known entropy - entropy dissipation approach, see [21] and references.
Plan of the paper. The paper is organized as follows: Section 2 is devoted to the definitionof solutions for s < 1 and s = 1 and to recalling some of their properties (conserved quantities,etc). In Section 3 we show the existence of solutions to (1.5) by taking the limit s → 1 insolutions to (1.1). Section 4 is devoted to the proof of uniqueness for bounded solutions. Section
5 contains the universal decay estimate for the solutions. In Section 6 we investigate the theory
in the class of radial solutions via the Burgers’ equation approach. In Section 7 we show that thecompact support property is preserved during the evolution, see also [12]. Section 8 discusses thequestion of large time behaviour for two classes of initial data where the restriction of compact

4
support is dropped: (i) integrable data with finite second moment, (ii) all solutions that areradially symmetric in the space variable.
Open problems. Here is list of the main problems that remain open: obtaining uniquenessresults beyond the class of bounded solutions, analyzing the long-term behaviour of all solutions
with integrable data, and finally treating the case of sign changing solutions.
Acknowledgments. We would like to thank J. A. Carrillo for pointing out to us the reference[12] after a first draft of this paper was written. S. S. thanks the Universidad Autónoma de
Madrid for its hospitality that allowed this work to be completed. S. S. was supported by a
EURYI award, and JLV by Spanish Project MTM2008-06326-C02-01.

2

Definitions and first properties of the solutions

It will be convenient to write for any s ∈ (0, 1] the equation in the form
(2.1)

∂t u = ∇ · (u ∇Ks u),

Ks = (−∆)−s .
1/2

Ks is a positive essentially self-adjoint operator. For convenience, we introduce Hs = Ks(−∆)−s/2 . We write p = Ks u, so that u = (−∆)s p and Hs u = (−∆)s/2 p.

=

We start with the definition of weak solutions. The notation M+ (Rn ) denotes the space of
positive Radon measures on Rn with finite mass.
Definition 2.1. For any s ∈ (0, 1], a weak solution of (2.1) in QT = Rn ×(0, T ) is a nonnegative
1,1
function u(x, t) such that u ∈ L1 (QT ), p = Ks u ∈ L1 ((τ, T ), Wloc
(Rn )), u ∇p ∈ L1 ((τ, T ) × Rn )for all τ > 0, and the identity
ZZ(2.2)(u φt − u ∇p · ∇φ) dxdt = 0
holds for all continuously differentiable test functions φ compactly supported in Q = Rn × (0, T ).
If s < 1 we also assume that u is continuous in QT .
Definition 2.2. For any s ∈ (0, 1], a weak solution of the problem formed by equation (2.1) in
QT = Rn × (0, T ) with initial data µ ∈ M+ (Rn ) is a continuous nonnegative function u(x, t)
1,1
with u ∈ L1 (QT ), p = Ks u ∈ L1 ((0, T ), Wloc
(Rn )), u ∇p ∈ L1 (QT ), and the identity
ZZ
Z
(2.3)
(u φt − u ∇p · ∇φ) dxdt + φ(x, 0) dµ(x) = 0
holds for all continuously differentiable test functions φ in Rn × [0, ∞) such that φ has compact
support in the space variable and vanishes for t ≥ T .
In practice the solutions obtained below have much better properties that can be exploited in
the analysis. In particular they will be uniformly bounded for t ≥ τ > 0 and satisfy energy
inequalities. Following [44] we may call such improved solutions dissipative solutions.

5

2.1

Properties of the solutions in the case s < 1

The existence of weak solutions in the sense of Definition 2.2 is proved in [20] under the assumption that the initial data is actually a bounded function u0 ≥ 0 decaying exponentially
as |x| → ∞, cf. Theorem 4.1 of [20]. It is extended to general nonnegative L1 initial data in[19]. Let us list the main properties of those solutions with attention to the way they dependon s ∈ (0, 1). We point out that we do not use all of them in the sequel.
- Conservation of mass: The solutions exist globally in time and for every t > 0 we have
Z
Z
(2.4)
u(x, t) dx = u0 (x) dx.
- Conservation of positivity: u0 ≥ 0 implies that u(·, t) ≥ 0 for all times.
-Lp estimates: the Lp norm of the constructed solutions does not increase in time, for any
1 ≤ p ≤ ∞.
- A general comparison theorem, i. e., a form of the usual maximum principle, is proven to
be false by constructing counterexamples for s > 1/2, see [20]. However, comparison of solutions
for an integrated version of the equation is established in dimension n = 1 by the method ofviscosity solutions in Biler-Karch-Monneau [15]. We will return to that issue below.
- Finite speed of propagation. If the initial function has compact support so does thesolution at any given time t > 0. Estimates on the growth of the support for very large time are
given in [20] and [21].
-Log-entropy estimate: We haveZZdu(x, t) log u(x, t) dx = − |∇Hs u|2 dx.
(2.5)
dt
therefore, solutions with u0 log u0 ∈ L1 (Rn ) stay in the same space and moreover satisfy
ZZ|∇Hs u|2 dxdt < ∞.
- Energy estimate. We first make the observation that if p = Ks u we have
Z
Z
Z
s/2 2
(2.6)
E(u) := up dx = ((−∆) p) dx = (Hs u)2 dx ≥ 0.
This formal computation is justified for the class of constructed weak solutions. This is the mainenergy we are going to use. We also have formally
Z
Z
Z
2
s/2 2
|∇Hs u| dx = |∇(−∆) p| dx = u(−∆p) dx.
The estimate on the evolution of the energy according to our flow is then
Z
Z
Z
Z
d
2
|Hs u| dx = 2 (Hs u)(Hs u)t dx = 2 (Ks u)ut dx = −2 u|∇Ks u|2 dx.
dt

6
where the last identity is obtained after using the equation and integrating by parts. This isjustified for smooth solutions, while in the generality of weak solutions we getZ
Z t2 ZZ
2
2|Hs u(t1 )|2 dx
u|∇Ks u| dx ≤|Hs u(t2 )| dx + 2
(2.7)
Rn

Rn

t1

- Boundedness: The solutions to (2.1) constructed in [20] have the following boundedness
effect (used in the subsequent paper [19]): For all 0 < s < 1, weak energy solutions in L1 arebounded for all positive times. More precisely, there is a constant C = C(n, s) such that
(2.8)

ku(t)k∞ ≤ C(n, s) t−α ku0 kσL1

Dimensional considerations imply that α = n/(n + 2 − 2s) and σ = (2 − 2s)/(n + 2 − 2s). This
fact is proved in [19]. Notice that as s → 1 we have α → 1 and σ → 0. Once we can control theconstant C(n, s) uniformly in s we can pass to the limit s → 1 to obtain the a priori estimate(already found by Lin and Zhang in their study in dimension n = 2 in [44])(2.9)

u(x, t) ≤ C/t

for the solutions of the limit equation obtained in the limit process, this will be done in the next
section in Theorem 3.1. This just requires a closer inspection of the proof of [19].
- Energy solutions: The basis of the boundedness analysis is a property that goes beyond
the definition of weak solution. We will review the formulas with attention to the constants
that appear since this is not done in [19]. The general energy property is as follows: for any Fsmooth and such that f = F 0 is bounded and nonnegative, we have for every 0 ≤ t1 ≤ t2 ≤ T ,Z
Z
Z t2 ZF (u(t2 )) dx − F (u(t1 )) dx = −∇[f (u)]u∇p dx dt =1
Z t2 tZ−∇h(u)∇(−∆)−s u dx dt
t1

where h is a function satisfying h0 (u) = u f 0 (u). We can write the last integral as a bilinear formZ∇h(u)∇(−∆)−s u dx = Bs (h(u), u)(2.10)and this bilinear form Bs is defined on the Sobolev space W 1,2 (Rn ) by
ZZ
ZZ
1
(2.11) Bs (v, w) = Cn,s∇v(x)∇w(y) dx dy =N−s (x, y)∇v(x)∇w(y) dx dy|x − y|n−2swhere N−s (x, y) = Cn,s |x − y|−(n−2s) is the kernel of operator (−∆)−s . After some integrations
by parts we also haveZZ
1
(2.12)Bs (v, w) = Cn,1−s(v(x) − v(y))(w(x) − w(y)) dx dy|x − y|n+2(1−s)
since −∆N−s = N1−s . It is known [56] that Bs (u, u) is an equivalent norm for the fractional
Sobolev space W 1−s,2 (Rn ) and we will see just below that Cn,1−s ∼ Kn (1 − s) as s → 1, for someconstant Kn depending only on n.

7
Proposition 2.1. The constant C(n, s) in (2.8) is uniformly bounded as s → 1, i.e. for any
solution to (2.1), we have
Cn ku0 kσL1
ku(t)k∞ ≤,tαwith α = n/(n + 2 − 2s) and σ = (2 − 2s)/(n + 2 − 2s).
Proof. It is based on a careful scrutiny of the ideas and proof of paper [19], Section 4.
(i) We have Cn,1−s ∼ Kn (1 − s) as s → 1. Indeed, if 0 < s < 1 we can also use the integral
representationZg(x) − g(z)s
(2.13)(−∆) g(x) = Cn,s P.V.
dz = Ns ∗ g(x)n+2s
Rn |x − z|sis a normalization constant, see forwhere P.V. stands for principal value and Cn,s = 4 πsΓ((n/2)+s)
n/2 Γ(1−s)
example [41, 56]. Note that Cn,s ∼ Kn s as s → 0 and Cn,s ∼ Cn0 (1 − s) as s → 1.
(ii) Next, we need the following result from [50]ZZZ(u(x) − u(y))2n−1lim+ rdxdy = |S | u(x)2 dx.
n+2r
r→0|x − y|Setting r = 1 − s and combining this with (2.12) and Cn,1−s ∼ Kn (1 − s) as s → 1− , we deducethatlim− Bs (u, u) = Kn |Sn−1 |kuk22 .
s→1(iii) We can now revisit the proof of boundedness of Theorem 4.1, in Section 4 of [19], and checkthat the constants in all the arguments are uniform in s as s → 1.
We end this section with some further properties of the solutions.
- Hölder estimates are proven for bounded energy solutions in [19] and hold for all s ∈ (0, 1).
The Hölder exponent has to deteriorate as s → 1 since we do not expect solutions of the limit
equation to be continuous, as can be seen in the case of the “vortex-patch solutions” below.
- More general data. The property of boundedness and the C α regularity are used in [19]to obtain solutions with similar properties for all nonnegative initial data in L1 (Rn ). We will
generalize that result to measures in Section 3 below.

3

Existence of solutions as limits of fractional diffusion

In dimension n = 2, existence and uniqueness of positive L∞ solutions has been proven by Lin
and Zhang in [44] (they also proved existence of positive measure-valued solutions). Existencewith positive initial data of finite energy has also been proven by a gradient flow approach, inbounded domains of the plane in [2], and in all of R2 in [3]. In general dimension, existence canbe obtained by taking the limit s → 1 in the solutions for s < 1 constructed in [20] as we willsee below.

8
We examine here the situation s = 1 as limit of the equations for s < 1. Since all the estimates
above are uniform in s as s → 1, we may write the formal version for s = 1, where the equationis (1.5). Here we need to distinguish the case n ≥ 3 from the case n = 2 which we will examine
separately below. For n ≥ 3, and noting that now Hu = (−∆)1/2 p, the energy isZ
ZZ2
up dx = (Hu) dx =|∇p|2 dx,
(3.1)
E(u) =
Rn

Rn

which is an analogue of the energy of [2] (there, it was set in bounded domains) for which theequation is a Wasserstein gradient flow, and its evolution is given by
Z
dE
= −2
u|∇p|2 dx.
(3.2)
dt
Rn
This is a formal computation that will be justified as a consequence of the passage to the
R limits → 1 in formula (2.7). This energy would not make sense for n = 2 because, unless u = 0,
p = (−∆)−1 u is never of gradient in L2 in dimension 2 (the Green kernel is a logarithm), howevera definition is given below in Section 3.2.

3.1

The existence result by s → 1 limit

We next prove the following general result in all dimensions n ≥ 2.
Theorem 3.1. For every µ ∈ M+ (Rn ), there exists a continuous and nonnegative weak solutionof equation (1.5), with data µ in the sense of Definition 2.2. It can be obtained as the limit s → 1of the solutions of the FPME with 0 < s < 1 with data u0k ∈ L1 (Rn ) ∩ L∞ (Rn ) that approximateµ in the sense of measures. In addition these solutions satisfy(3.3)

ku(t)kL∞ ≤Ctfor some C > 0 depending only on n.
Proof. Existence is proved by passing to the limit s → 1 in the solutions of the FPME (1.1).
The outline is similar to the proof of Theorem 13.1 in [19] where s was considered constant, but
we cannot use the continuity in time of the solution, since it may be lost in the limit s → 1.
Let us consider initial data u0 = µ in M+ (Rn ), and a sequence u0k of nonnegative functions
approaching u0 as measures as k → ∞. (If u0 ∈ L1 we may simply take u0k = u0 .) Let us thenconsider the family of solutions uk (x, t) with parameter sk → 1 of (1.1) given by Theorem 4.1 of[20] or Theorem 13.1 of [19], and let pk (x, t) = (−∆)−s uk (x, t) be the corresponding pressures.
(i) In a first step, we assume that n ≥ 3, and prove that the limit exists (along subsequences
skj → 1) and is a solution of the equation in the sense of Definition 2.1, without discussing thequestion of the initial data. By Proposition 2.1, we have that for any τ > 0,
(3.4)

kuk kL∞ ((0,τ ),L1x ) ≤ C

9where C > 0 only depends on τ . In order to pass to the limit in the weak formulation
ZZ(3.5)(uk φt − uk ∇pk · ∇φ) dxdt = 0
for φ compactly supported in Rn × (0, T ), it suffices to have weak convergence of uk , together
with strong convergence of pk and ∇pk . Let us work on [τ, T ) containing the support of φ. From
(3.4) we have weak convergence (up to subsequence extraction) of uk in Lq ([τ, T ) × BR ) for any
q ≥ 1, R > 0.
- For pk , we may invoke the embedding theorems associated to the Riesz kernels, which tell usthat pk = (−∆)−s uk is uniformly bounded in L∞ ((τ, T ), Lqx (BR )) for all large q and R > 0. Moreprecisely, q > n/(n − 2s), which is uniform in s ≈ 1 if n ≥ 3.
0s ,q
- Moreover, ∇pk has a similar bound in spaces of the form L∞ ((τ, T ), Wloc
) with s0 < 2s − 1and convenient q > n/(n − s). Since we are interested in s → 1, we can restrict to 2s − 1 > 0
so we can take s0 > 0 and both (pk ) and (∇pk ) are a locally compact family with respect to thespace variable.

- We still need compactness in time for pk and ∇pk , and this is obtained from Aubin-Simon’s
results, cf. [6], [55]. First, for pk , we use the facts that
pk is bounded in L∞ ((τ, T ), Lp (BR ))(3.6)∂t pk = (−∆)−sk ∂t uk = (−∆)−sk ∇ · (uk pk )and since uk pk is bounded in Lq (BR ) for any q > 0 and uniformly in t ∈ [τ, +∞), it follows that∂t pk is bounded in L∞ ((τ, T ), W 2sk −1,q (BR )).

(3.7)(3.6) and (3.7) ensure, via Aubin-Simon’s theorem, that pk is compact in C([τ, T ], Lq (BR )), for
any q ≥ 1 and R > 0. The proof for ∇pk is entirely similar: first, from the above, we have0

∇pk is bounded in L∞ ((τ, T ), W sk ,q (BR ))(3.8)∂t ∇pk = (−∆)−sk ∂t uk = ∇(−∆)−sk ∇ · (uk pk )and since uk pk is bounded in Lq (BR ) for any q > 0 and uniformly in t ∈ [τ, T ], it follows that∂t ∇pk is bounded in L∞ ((τ, T ), W 2sk −2,q (BR )).
Again, these suffice to apply Aubin-Simon’s result and obtain that ∇pk is compact in say
C([τ, T ], Lq (BR )) for every τ > 0, T > τ, R > 0, q > 1. The limits after extraction of uk
and pk are denoted u and p.
Taking φ smooth compactly supported in space and vanishing for t ≥ T , it follows from theseconvergence results that
Z ∞Z
Z ∞Z
(3.9)
lim
(uk φt − uk ∇pk · ∇φ) dxdt =(uφt − u∇p · ∇φ) dxdt.
k→∞

0

0

10
(ii) When n = 2 we do not have a uniform bound of the pressures pk in Lq spaces valid for alls ≈ 1, since this is a limit case of the Sobolev embeddings. Fortunately, the pressures do notappear directly in the equation, but only through the pressure gradients for which the estimates
are uniform in the range s ≈ 1. The rest of the argument follows.
(iii) There remains to control terms corresponding to the time interval [0, τ ] in order to complete
the proof that the solution takes on the initial data, as in Definition 2.2. For that we follow [19],proof of Theorem 13.1. We consider φ ∈ C 1 (Rn × [0, ∞)) compactly supported in space andvanishing for t ≥ T . First, let τ > 0 be arbitrary. Applying the result of (3.9) we have that
Z ∞Z
Z ∞Z
(3.10)
lim(uk φt − uk ∇pk · ∇φ) dxdt =(uφt − u∇p · ∇φ) dxdt.
k→∞

τ

τ

In addition, by weak convergence of u0k to u0 , we haveZ
Z Z
0
(3.11)limuk (x)φ(x, 0) dx =u0 (x)φ(x, 0) dx.
k→∞There remains to show that the contributions on the interval [0, τ ] can be made small.
Starting again from Proposition 2.1 we havekuk (t)kL∞ ≤ C(u0 )t−αα = n/(n + 2 − 2sk ),which yields by conservation of mass−α(p−1)kuk (t)kpLp ≤ kuk (t)kL1 kuk (t)kp−1.
L∞ ≤ C(u0 )t

By Riesz embedding we deduceZ
2|∇Hsk uk |2 dx ≤ Ckuk (t)k2Lp ≤ Ct−α(p−1) p ,

1
1 sk
= −
2
p
n

where we recall Hs u = (−∆)−s/2 u for any u, and combining the definitions of α and p, this yields
Z
2−1|∇Hsk uk |2 ≤ Ct n+2−2sk .
Combining with the energy inequality (2.7), we deduce that we also have
Z ∞Z
2
−1
(3.12)
uk |∇pk |2 ≤ Ct n+2−2sk .
t

Rτ R
We may now bound the remaining contribution 0 (uk φt − uk ∇pk · ∇φ) dxdt as follows: fortm = 2−m , we first write
Z tm−1 Z
uk φt ≤ tm−1 kφkC 1 kuk kL∞ ((0,∞),L1 ) ≤ Ctm−1tm

11
and second
Z tm−1 Z

 12 Z tm−1 Z

Z tm−1 Zuk ∇pk · ∇φ ≤ kφkC 1

tm

2

 12

uk |∇pk |

uk
tm

tm
1

1
− 12
n+2−2sk2tm≤ Ctm−1

1

1− 12 + n+2−2s2≤ 2Ctmtm

k1
n+2−2sk= Ctm.

Summing over m we find that the contributions on [0, τ ] tend to 0 as τ tends to 0 and combining
with (3.10)–(3.11) we obtainZ ∞Z

u0k (x)φ(x, 0) dx
Z ∞ZZ(uφt − u∇p · ∇φ) dxdt + u0 (x)φ(x, 0) dx.
=(uk φt − uk ∇pk · ∇φ) dxdt +

lim

k→∞

Z

0

0

as desired.
(iii) Finally, by passing to the limit sk → 1 in (2.8) and using the fact that C(n, s) does notblow up in this limit, we obtain the universal estimate (3.3).
Remark 3.1. The handling of the initial data in the proof above could be replaced by showingthat the initial data is taken via a result of continuity of the solution in Wasserstein metric, see
Lemma 5.2 below.
When sk is taken fixed this is the proof of existence of solutions with measure data that improves
Theorem 10.1 of [19].
Theorem 3.2. For every µ ∈ M+ (Rn ), there exists a continuous and nonnegative weak solutionof FPME, 0 < s < 1, with data µ in the sense of Definition 2.2. It is a bounded energy solution
for all t ≥ τ > 0.
Remark 3.2. We will provide in Proposition 5.1 below another approach to the existence ofsolutions for s = 1 via the vanishing viscosity method.

3.2

Definition of the energy in two space dimensions

R
In dimension 2, even though p = (−∆)−1 u is not in general of finite energy |∇p|2 < ∞, wemay still define an energy by a “renormalization” procedure. Here we reproduce the definitionof [3], which itself follows a similar construction for Ginzburg-Landau in [51] and [13].
R
The first idea is to define the energy in a “renormalized way”, over functions u such that u = kwith k a fixed constant (we may do this without loss of generality since the integral is preservedby the evolution) by

Z
k2
2
(3.13)
E(u) = lim|∇p| −
log R .
R→∞
2π
B(0,R)
R
Contrarily to |∇p|2 , it is possible to make this energy finite even if k 6= 0.

12consider any smooth
positive compactly supported u0 with
RAnother definition is the following:
R
−1
u
=
k,
and
let
p
=
(−∆)
u
.
For
any
u
with
u
=
k such that w := p − p0 is in H 1 (R2 ),00
R2 0
we define the energy byZ
Z
2|∇w| + 2
u0 w(3.14)
E(u) =
R2

R2

R
k2
In fact, expanding w as p − p0 and noting that B(0,R) |∇p0 |2 ∼ 2π
log R + cst as R → ∞, and12using an integration by parts we see that if w ∈ H (R ), the two definitions differ only by theconstant (depending only on the choice of u0 ). So, for energy we may take E.
Formally, along the flow, since wt = pt , we haveZZd∇w · ∇wt + 2
u0 wt
E(u(t)) = 2
dt
2
RZ2
R
Z
w∆pt + 2
u0 pt
= −2
2
R2
Z R
Z
= 2
(w + p0 )ut = 2
put
2
R2
ZR
|∇p|2 u.
= −R2

An alternate approach would have been to view the (−∆)−1 operator as a limit of (−∆ + hI)−1
as h → 0.

4

Uniqueness of bounded solutions for s = 1

In this section we focus on obtaining a uniqueness result for s = 1 for bounded solutions.
Uniqueness is not expected in all situations, see [1]. We also recall that there is no maximum
principle so uniqueness cannot follow from comparison arguments. The proof works only fors = 1; no uniqueness proof is known for the FPME in dimensions n ≥ 2.
A natural space in which to work for such continuity equations is the space of probability measures endowed with the Kantorovich-Rubinstein-Wasserstein distance (commonly called Wasserstein), indeed the PDE is a gradient flow for the 2-Wasserstein distance (point of view which
was exploited in [2, 3]. For completeness, we recall (see e.g. [1]) that the p-Wasserstein distance between two probability measures µ and ν on Rn (and by extension between two positive
measures with same mass) is defined by
ZZ
p
(4.1)
Wp (µ, ν) = inf|x − y|p dγ(x, y)γ∈Γ(µ,ν)Rn ×Rnwhere Γ(µ, ν) is the set of probability measures on Rn × Rn whose first marginal is µ and whose
second marginal is ν. This inf is achieved by an “optimal transport plan”, and when 1 ≤ p < ∞,Wp defines a distance.

13
Theorem 4.1. There exists at most a unique solution of the initial-value problem for Equation(1.5) in L∞ ((0, T ) : L∞ (Rn )).
When n = 2, this theorem recovers with a different method the result on existence and uniqueness of bounded solutions of [44] but without requiring initial compact support.
Proof. The proof is inspired by the method of Loeper [47] (see also [2]). We will need thefollowing result, adapted from [48, Lemma 8.1]. The result there is only stated in dimension
n = 2, we include here a proof in any dimension n.
Lemma 4.1. Let p = (−∆)−1 u. There exists a constant C > 0 depending only on the dimension
n such that∀x, y ∈ Rn , |∇p(x) − ∇p(y)| ≤ C(ku0 kL1 + ku0 k∞ )|x − y| | log |x − y||1|x−y|≤1/e + 1
Proof. Using the integral expression for ∆−1 , we have, for any n ≥ 1,
Z
x
(4.2)
∇p(x) = Cn
K(x − z)u(z) dz
K(x) = n .
|x|RnLet us denote d = |x − y|. First, we treat the case d ≤ 1/e. We may write ∇p(x) − ∇p(y) =Cn (I1 + I2 + I3 ) withZ
Z
(K(x − z) − K(y − z))u(z) dz(K(x − z) − K(y − z))u(z) dz, I2 :=I1 :=B(x,4)\B(x,2d)
Rn \B(x,4)
Z(K(x − z) − K(y − z))u(z) dz.
I3 :=B(x,2d)For z ∈ Rn \B(x, 4), and since d = |x − y| ≤ 1/e we have |K(x − z) − K(y − z)| ≤ C|x − y| forsome constant C. Hence we may write |I1 | ≤ C|x − y|kukL1 .
For z ∈ B(x, 4)\B(x, 2d), by a mean value argument, and since d = |x − y|, we have |K(x −|x−y|z) − K(y − z)| ≤ C |x−z|
n . We may thus writeZdz∞= C|x − y|kukL∞ | log d| = CkukL∞ |x − y|| log |x − y||.
|I2 | ≤ C|x − y|kukLn
B(x,4)\B(x,2d) |z − x|
For z ∈ B(x, 2d) we have |K(x − z) − K(y − z)| ≤ |K(x − z)| + |K(y − z)| = |x−z|1 n−1 + |y−z|1 n−1
and we concludeZdz
dz|I3 | ≤ kukL∞+≤ CkukL∞ d = C|x − y|kukL∞ .
n−1|y − z|n−1
B(x,2d) |x − z|
Combining these estimates on I1 , I2 and I3 we obtain the result in this case.
If d ≥ 1/e, we split the integral into two parts:
Z
I1 :=(K(x − z) − K(y − z))u(z) dz,
Rn \B(x,2d)Z(K(x − z) − K(y − z))u(z) dz.

I2 :=B(x,2d)For the first part, we use again the fact that for z ∈/ B(x, 2d) we have |K(x − z) − K(y − z)| ≤|x−y|
C |x−z|n ≤ C|x − y| since d ≥ 1/e. We thus find |I1 | ≤ C|x − y|kukL1 . For z ∈ B(x, 2d) we argue
just as above for I3 and find |I2 | ≤ C|x − y|kukL∞ . This concludes the proof.

14
We now give the proof of the theorem. Let us consider two bounded solutions u1 , u2 on [0, T ]with initial data u0 , and let p1 , p2 be the corresponding pressures.
We consider the characteristics equation (for i = 1, 2)
dXi(t, x) = −∇pi (t, Xi (t, x))
dt

(4.3)

Xi (0, x) = x.

According to [1] (see also Corollary 3.3 in [47]), if ui solve (1.5), then we haveui = (Xi )# u0 .

(4.4)
We then defineZ
Q(t) =

u0 (x)|X1 (t, x) − X2 (t, x)|2 dx.

We have that Q(0) = 0, and from (4.4), W22 (u1 (t), u2 (t)) ≤ Q(t).
Let us study the evolution of Q. Differentiating Q in time yields
Z0
Q (t) = −2 u0 (x)(X1 (t, x) − X2 (t, x)) · (∇p1 (t, X1 (t, x)) − ∇p2 (t, X2 (t, x))) dx
hence using the Cauchy-Schwarz inequality
1 0
(4.5)
|Q (t)| ≤
2

Z

0

2

1/2 Z

u |X1 − X2 | dx

1/2
u |∇p1 (t, X1 ) − ∇p2 (t, X2 )| dx
0

2

1/2

=Q
On the other hand,

R

Z

0

2

u |∇p1 (t, X1 ) − ∇p2 (t, X2 )| dx

1/2
.

1/2
1/2
1/2
u0 |∇p1 (t, X1 ) − ∇p2 (t, X2 )|2 dx≤ T1 + T2 whereZ
T1 = u0 |∇p1 (t, X1 ) − ∇p1 (t, X2 )|2 dx

and

Z
T2 =

u0 |∇p1 (t, X2 ) − ∇p2 (t, X2 )|2 dx.

Note first that from (4.4)
Z
(4.6)

T2 =

u2 (x, t)|∇p1 (t, x) − ∇p2 (t, x)|2 dx.

But Theorem 4.4. in [47] asserts that, if ω1 and ω2 are two positive measures of same finite total
mass and with L∞ densities, then Ψi = −(∆)−1 ωi satisfy
(4.7)

1

k∇Ψ1 − ∇Ψ2 kL2 (Rn ) ≤ (max(kω1 kL∞ , kω2 kL∞ )) 2 W2 (ω1 , ω2 )

15
Applying this to u1 (t) and u2 (t), using the uniform L∞ bound on these solutions, and inserting
into (4.6), we find|T2 | ≤ max(ku1 (t)k2∞ , ku2 (t)k2∞ )W22 (u1 (t), u2 (t)) ≤ CW22 (u1 (t), u2 (t)).
There remains to bound T1 . Using Lemma 4.1, and denoting M = ku0 kL1 , we may write|T1 | ≤ C(M2+ ku1 (t)k2∞ )Z

u0 (x)|X1 − X2 |2 log2 |X1 − X2 |1|X1 −X2 |≤1/e dxZ+ u0 (x)|X1 − X2 |2 dx .

Let us define f (x) = x log2 (x) for 0 ≤ x ≤ 1/e and f (x) = 1/e for x ≥ 1/e. It is easy to checkthat f is concave, f (x) ≤ x log2 x, and moreover from the above (and the boundedness of u1 )
Z
Z

0
2
|T1 | ≤ C
u (x)f (|X1 − X2 | ) dx + u0 (x)|X1 − X2 |2 dx
Using the concavity of f , by Jensen’s inequality, we have

Z 0
Z
u (x)
Q(t)
2
0
2
|X1 − X2 | dx = M f () ≤ CM Q(t) log2 Q(t).
u (x)f (|X1 − X2 | ) dx ≤ M f
M
M
We conclude that |T1 | ≤ CQ(t)(log2 Q(t) + 1). Combining with the bound for T2 , recalling that
W22 (u1 , u2 ) ≤ Q, and inserting all these bounds into (4.5), we finds1
Q0 (t) ≤ |Q0 (t)| ≤ CQ(t) 1 + log2,Q(t)where C depends on M and the uniform bounds on u1 and u2 .
R
√ dv
We may now conclude by a standard integration argument, and since= log(log v +v 1+log2 v
p
1 + log2 v) (using a change of variables u = log v), that Q(0) = 0 implies Q(t) = 0 for t ∈ [0, T ],and since W22 (u1 (t), u2 (t)) ≤ Q(t) and W2 is a distance, the conclusion follows.

5

The universal bound for s = 1

We now show that the L∞ norm of solutions is decreasing in time. Indeed, the equation can be
written as
ut = −u2 − ∇u · ∇p.
which is the sum of absorption and transport with speed v = −∇p. This expression suggests a
universal decay in time of the form
(5.1)

ku(·, t)k∞ ≤

1
t

16
which we will establish below. This agrees with the limit s → 1 of the smoothing effect of
Proposition 2.1 for s < 1, cf. Theorem 3.1. What is surprising is that the bound is universal
in the sense that it does not depend on any norm of the data. Note that in dimension 2, thisbound appears in [44]. We have a slightly more precise result in terms of the L∞ norm of thedata. The subject was taken up in the recent paper [12], which is very close to the ideas we
explain next in some detail for the reader’s convenience.
Theorem 5.1. For every weak solution u with bounded initial data u0 we have the estimate(5.2)

u(x, t) ≤1
t+τ

where τ = 1/ supx u0 (x). Therefore these solutions coincide (via the uniqueness result forbounded solutions, Theorem 4.1) with those of Theorem 3.1.
Remark 5.1. A similar result holds for the lower bound of the negative values when dealing withsigned solutions.
We start by studying a viscous approximation of the equation (which, by the way, is like therepulsive version of the Keller-Segel equation for chemotaxis [37, 36]).
Proposition 5.1. Let u0 ∈ L∞ (Rn ) ∩ L1 (Rn ) with u0 ≥ 0. There exists a unique solution to∂t uε = ε∆uε + ∇ · (uε ∇pε )pε = (−∆)−1 uε(5.3)
uε (·, 0) = u0
Moreover uε is smooth, satisfies(5.4)

0 ≤ uε (x, t) ≤ ku0 kL∞

and converges locally in Lq (Rn × [0, +∞) for any 1 ≤ q < ∞, to u, the unique solution to(1.5)–(1.6).
We note that this “vanishing viscosity approach” is an alternative way of obtaining existenceof solutions to (1.5)–(1.6).
Proof. This natural “viscosity approximation” was previously considered in Schätzle-Styles [53],for a very similar equation, but in a bounded domain of R2 . Because the proof is tailored for abounded domain, it is not immediate to adapt it to our case. In order to prove the existence ofsolutions to (5.3), we set up a fixed point argument.
Take T > 0 and let
Z∞n
K = v(x, t) ∈ L (R × (0, T )), v ≥ 0 , and ∀t ∈ [0, T ],Z
v(x, t) dx ≤Rn

K is clearly a closed convex subset of the Banach space L∞ (Rn × [0, T ]).

0u (x) dx .
Rn

17
Lemma 5.1. Let u0 ∈ L∞ (Rn ) ∩ L1 (Rn ). For any v ∈ K, letting pv = (−∆)−1 v. Then, we have
∇pv ∈ L∞ (Rn × [0, T ]) and there exists a unique solution u := T (v) over [0, T ] of the problem
∂t u = ε∆u + ∇ · (u∇pv ) ,(5.5)
u(·, 0) = u0 .
That solution satisfies T (v) ∈ K and kT (v)kC 0,α ≤ C. T is continuous and compact from K to
itself.
Proof. First we note that if v ∈ K, then for all t ∈ [0, T ], v(·, t) ∈ L1 ∩ L∞ , which implies that
∇pv ∈ L∞ (Rn × [0, T ]) by Nash-Sobolev embedding, or more precisely, by observingZv(y, t)dy ≤ kvkL∞ (Rn ×[0,T ])|∇pv |(x, t) ≤n−1
Rn |x − y|

Z

dy
+
n−1
B(x,1) |x − y|Zv(y) dyRn≤ C kvkL∞ (Rn ×[0,T ]) + kv(·, t)kL1 (Rn ) ,where C depends only on the dimension. In fact, direct inspection allows to prove in addition
that ∇pv (·, t) ∈ C 0,α (Rn ) for some α > 0.
We are thus led to solving a linear equation of the form∂t u = ε∆u + ∇ · (uf )(5.6)
u(·, 0) = u0
with f ∈ L∞ (Rn × [0, T ], Rn ). This can be solved by a standard convolution argument, see forexample Lemma 12 in [16] which applies exactly (the fact that the dimension is n and not 2 does
not affect at all the argument), and it provides us with a uniqueR solution T (v)R ≥ 0 defined forall positive times, hence in [0, T ]. The fact that for all t ∈ [0, T ], Rn T (v) dx ≤ Rn u0 follows by
integrating in time the equation. On order to show that T (v) ∈ K, there remains to show that
T (v) ∈ L∞ (Rn × [0, T ]).
Let us consider u = T (v) solving (5.6) and let φ : R → R be a smooth function. Using (5.6),
integration by parts and ∆pv = −v, we haveZZdφ(u) = φ0 (u) (ε∆u + ∇ · (u∇pv ))
(5.7)
dt Rn
Z
Z
00
2
= −ε φ (u)|∇u| − uφ00 (u)∇pv · ∇uZZ002
= −ε φ (u)|∇u| − ∇pv · ∇[H(u)]ZZ002= −ε φ (u)|∇u| − vH(u) ,where we define H(s) =v ≥ 0, we findRs0

tφ00 (t) dt. If in addition φ is convex it follows that H ≥ 0 and, sinceZdφ(u) ≤ 0.
dt Rn

18
Taking φ nonnegative convex and supported over [ku0 kL∞ , +∞) we conclude that 0 ≤ u(x, t) ≤ku0 kL∞ for every (x, t) ∈ Rn × [0, T ]. This proves that T (v) ∈ L∞ and it belongs to K. T thus
maps K into itself. It is also easy to see that it is continuous.
Now, standard regularity theory (cf. [39]) gives us Schauder estimateskT (v)kC 0,α (Rn ×[0,T ]) ≤ C kvkL∞ (Rn ×[0,T ]) + kvkL1 (Rn ×[0,T ]) ≤ CT ku0 kL∞ + ku0 kL1 .
Since C 0,α embeds compactly into L∞ , T (K) is thus precompact.
We may now apply Schauder’s fixed point theorem [43] which gives us the existence of a fixed
point for T hence a solution to (5.3), satisfying uε ≥ 0 and kuε kL∞ (Rn ×[0,T ]) ≤ ku0 kL∞ .
Finally once we have established (5.4), convergence of uε to a solution of (1.5) can be provenfollowing the same steps as in the proof of Theorem 3.1 (except replacing sk by 1). The solution
is unique by Theorem 4.1, which implies convergence of the whole sequence uε .
Proof of Theorem 5.1. We first prove the desired estimate for uε solution to (5.3), which issmooth by standard parabolic regularity theory. LetZus consider φ smooth such that φ(u) = 0dφ(uε − M (t))dx, with M some smoothfor u ≤ 0, φ00 ≥ 0, φ0 ≥ 0, and let us computedt
nonnegative function of time. This yields
Z
Z
Z
d
0
0
(5.8)
φ(uε − M (t))dx = φ (uε − M (t))∂t uε dx − M (t) φ0 (uε − M (t)) dx.
dt
Using the equation, we find, after several integrations by parts
Z
Z
Z
0
0
φ (uε − M (t))∂t uε = ε φ (uε − M (t))∆uε − φ00 (uε − M (t))∇uε · uε ∇pεZZ002= −ε φ (uε − M (t))|∇uε | − φ00 (uε − M (t))(uε − M (t))∇uε · ∇pεZ− M (t) φ00 (uε − M (t))∇uε · ∇pεZZ≤ − ∇[H(uε − M (t))] · ∇pε − M (t) φ00 (uε − M (t))∇uε · ∇pε .
RsHere, we have defined again H(s) = 0 tφ00 (t) dt. Using −∆pε =R uε and another integration byparts, we find that the first term on the right-hand side is − H(uε − M (t))uε ≤ 0. Thereremains to estimate in a similar wayZZ00− M (t) φ (uε − M (t))∇uε · ∇pε = −M (t) ∇[φ0 (uε − M (t))] · ∇pεZZ02= −M (t) φ (uε − M (t))uε ≤ −M (t)φ0 (uε − M (t)) ,because φ0 is supported on the positive part of the real line and M (t) ≥ 0. We conclude thatZZ
0
2
φ (uε − M (t))∂t uε dx ≤ −M (t)φ0 (uε − M (t)) dx

19
and combining with (5.8)
Z
Z
d
2
0
(5.9)
φ(uε − M (t))dx ≤ −(M (t) + M (t)) φ0 (uε − M (t)) dx.
dt1
with τ = (sup u0 (x))−1 , we have M 0 (t) = −M 2 (t), and it follows that
Choosing M (t) = t+τZRdφ(uε − M (t)) dx ≤ 0. On the other hand, since u0 ≤ M (0) we have φ(u0 − M (0)) dx ≤ 0,dt
R
and thus we have obtained that φ(uε − M (t)) dx ≤ 0 for all t ≥ 0.

Taking now φ = φn to be a sequence that approximates the function φ0 (x) = sgn+ (x). we
obtain uε (t) ≤ M (t) a.e. for t ≥ 0. Passing to the limit ε → 0, and using Proposition 5.1, wearrive at the result.
Remark 5.2. The estimate |u(x, t)| ≤ t+ku1 k−1 also applies to solutions of every sign which
0 ∞generalizes the case of “dissipative weak solutions” studied in [49].
We conclude this section with a result of continuity of the solutions in Wasserstein distance,
which we do not use however.
Lemma 5.2. The solutions to (1.5) with u0 ∈ M+ (Rn ) satisfying (3.3) are “absolutely continuous” (hence continuous) for the Wq -Wasserstein distance for any q ∈ [1, ∞): more precisely
Rt
Wq (u(t2 ), u(t1 )) ≤ t12 m(s) ds with m ∈ L1 ([0, T ]) for any T > 0.
Proof. By conservation of mass, for every t > 0 we have the two a priori estimates ku(t)k1 ≤ M ,ku(t)k∞ ≤ 1/t. By the Nash-Sobolev embeddings we then get ∇p(·, t) ∈ L∞ (Rn ). A simple
scaling argument then shows that
(5.10)

1/n(n−1)/nk∇p(·, t)k∞ ≤ Ck∆p(·, t)k1 k∆p(·, t)k∞≤ C1 M 1/n t−(n−1)/n

The claimed continuity is a direct consequence of the nature of the equation (1.5) as a continuity
equation, i.e. of the form ∂t u+∇·(uv) = 0. Theorem 8.3.1 in [1] asserts that for such an equation,RT
if 0 kv(t)kLq (u(t)) dt < +∞ then the solution u(t) is absolutely continuous in [0, T ] with values
in the space of measures equipped with the distance Wq . The precise definition of this (cf. [1])Rtis that Wq (u(t1 ), u(t2 )) ≤ t12 m(s) ds with m ∈ L1 ([0, T ]), and it implies continuity in the usual
sense. In the present case, v = ∇p and, from (5.10) we have
Z T

Z T Zkv(t)kLq (u(t)) dt =

0

0

1/q
|∇p(t)| u(t)
dx dt
q

Rn

Z T
≤

1/n−1

Ct
0

1/q

Z
u(t) dx

dt ≤ C 0 M 1/q T 1/n .

Rn

It follows that the result of [1] applies and u(t) is continuous for the Wasserstein distance Wq .

20

6

Radial solutions. Burgers’ equation

In this section we analyze radially symmetric solutions and exhibit the special class of vortexpatch solutions. We use the Burgers’ equation approach via the mass transform, which alreadyappeared in [11] and has precedents in chemotaxis problems, [36]. The main purpose of most of
this section is to fix the ideas and notation that we will use in the last two subsections, and in
the asymptotic analysis of Section 8.2.
Lemma 6.1. Let u be a solution to (1.5) with radially symmetric bounded initial data u0 (r),r = |x|. Then u is radial, and the “mass function” defined byZ r
sn−1 u(s, t) ds(6.1)
M (r, t) =0solves the Burgers’ equation
(6.2)

Mt +

1
rn−1

M Mr = 0 ,which after the change of variables s = rn /n can be written in the standard form Mt +M Ms = 0.
Proof. This follows from elementary computations, or can be found in [11].
Note that (6.2) is now a local equation, contrary to (1.5) and that it looks very much like theequation for chemotaxis, but with the opposite sign. The speed of movement is M/rn−1 = v ≥ 0.
Notice thatMt ≤ 0.

(6.3)

A common problem with Burgers’ equations and other first order conservation laws is that they
may develop shocks in finite time and the uniqueness of solutions needs the concept of entropy
solutions, cf. [42, 38]. This is not the case here, due to the fact that M (r, t) is a monotone
nondecreasing function of r and that the velocity v is positive, so that the characteristic lines
spread. Conversely, it is easy to see that weak solutions M (s, t) of the Burgers equation with
Ms ≥ 0 bounded, give rise to nonnegative, bounded, radial, weak solutions of the hydrodynamic
equation ut = ∇(u∇(−∆)−1 u).
Burgers’ Equation (6.2) has the property of comparison (maximum principle) in the class ofnonnegative and monotone solutions M . Since the equation is also 2-homogeneous in M , BénilanCrandall’s result [8] applies to give the pointwise inequality
M.
t
Using the equation this implies the further inequality (that we already know)
Mt ≥ −

(6.4)

rn−1
1
,
i. e. u(x, t) ≤ .
ttThese inequalities are optimal as we will show by means of the examples of the next section.
(6.5)

Mr ≤

The approach via the mass function M (r, t) will be used below to construct interesting classesof explicit solutions, that serve as prototypes or as counter-examples of the theory.

21

6.1

Solutions in dimension n = 1

The Burgers’ transformation, i. e., the formulation in terms of the mass function, applies easily
when we work in dimension n = 1 and we do not need the condition of radial symmetry to getin a direct way the standard Burgers’ equation(6.6)

Mt + M Mr = 0,

where we are still assuming that we deal with nonnegative solutions u ≥ 0.

6.2

The elementary vortex patch

We next recover the existence of the vortex patch self-similar solution.
Proposition 6.1. The equation (1.5) admits the family of radial “vortex patch solutions” of the
form
(6.7)

u(x, t) =1
χ1/n
t + τ |x|≤R(t+τ )(as well as all their translates) where R > 0, τ ≥ 0 are free parameters.
We skip the proof of this known result, see for instance [12]. This configuration is called in theterminology of transport equations a rarefaction wave. The simplest form of the solution is1u(r, t) = χr≤Rt1/n
t

(6.8)

which formally corresponds to τ = 0. We call this solution the elementary vortex patch. Noticethat it is a solution in the sense of Definition 2.2. It has a Dirac mass as initial data for thedensity, hence it qualifies as the fundamental solution of the problem.
All of the above solutions can also be described as expanding mesas. The use of the term“mesa” appears in some limit cases to nonlinear diffusions, see e.g. [27] and [28] or [59].
Largest solution. Note that in the limit when the initial mass goes to infinity we obtain thesolution1(6.9)
u(x, t) = ,M (r, t) = rn nttthat does not have an interface. This solution is an absolute a priori upper bound for all solutionsof our problem with s = 1. It satisfies equations (6.4) and (6.5) with equality.

6.3

Self-similar solutions

The vortex patch solution can be obtained as the limit of interesting self-similar solutions of the
FPME with 0 < s < 1. Such solutions are constructed in the paper [21] as self-similar solutionsof the form
(6.10)u(x, t) = t−α U (x/tβ )22In order to cancel the factors including t explicitly, we get the condition on the exponents
α + (2 − 2s)β = 1(6.11)

If we also impose conservation of (finite) mass, which amounts to the condition α = nβ, we
arrive at the precise value for the exponents:
β = 1/(n + 2 − 2s),

α = n/(n + 2 − 2s).

We also arrive at the nonlinear, nonlocal elliptic equation for the self-similar profiles(6.12)∇y · (U ∇y (P + a|y|2 )) = 0,

with P = Ks U.

where a = β/2, and β are defined just above. The solution of equation (6.12) is obtained in [21]by reducing it to an obstacle problem for the s-Laplacian operator. On the other hand, Biler,
Imbert and Karch [14] have obtained the explicit formula for a self-similar solution in the form
1−su(x, t; C1 ) = t−α (C1 − k1 x2 t−2α/n )+(6.13)

with k1 , C1 > 0 and α = n/(n + 2 − 2s) as before. These solutions bear a resemblance tothose of the standard porous medium equation [59, 15, 60]. This is why they have been termed
Barenblatt solutions for the FPME.
Selfsimilar solution for s = 1: This approach works for s = 1 and gives α = 1, β = 1/n,
where we replace Ks by K1 = (−∆)−1 . We then recover the vortex patch solution (6.8). It is alsoeasy to see that the family of self-similar solutions (6.13) pass to the limit s → 1 to produce thevortex patch solution (6.8) (after careful selection of the constants). This is an explicit example
of application of Theorem 3.1.
Comment. For the reader’s convenience, we give more details in the case s = 1. The solutionof the obstacle problem has to satisfy U = −∆P = 2na = nβ = 1 in the coincidence set |y| ≤ R,
while ∆P = 0 outside (for |y| ≥ R). Therefore, in the outer region
P = D |y|2−n ,D > 0.

C 1 agreement of the two expressions at y = r gives D = Rn /n(n − 2) and R2 = 2(n − 2)C, andwe recover the solution (6.8).

6.4

Other examples of vortex patches

We may next construct many solutions of the radial problem as rarefaction waves, which are notvery different from the elementary vortex patch. We take an initial configuration formed by aball patch at the origin plus an annular patch around, i.e.,u0 (r) = c1

for 0 ≤ r ≤ R1 ,u0 (r) = c2

for R2 ≤ r ≤ R3 ,

23with c1 , c2 > 0 and 0 < R1 < R2 < R3 , u0 (r) = 0 otherwise. Then
M (r, 0) = (c1 /n)rn= (c1 /n)R1n= (c2 /n)rn + (c1 /n)R1n − (c2 /n)R2n= (c2 /n)(R3n − R2n ) + (c1 /n)R1n

for 0 ≤ r ≤ R1
for R1 ≤ r ≤ R2
for R2 ≤ r ≤ R3
for r ≥ R3

Note that the continuity of M requires that we choose c2 R2n = c1 R1n . At later times the solution
keeps the same shape, i.e. the support of u(r, t) is still the union of a ball and an annulus (bothexpanding). u(r, t) and M (r, t) are explicit, in regions that are separated by interfaces:(i) The first region is an inner ball where u and M have the formu1 (r, t) =

c1
1
=
;
c1 t + 1
t + τ1M1 (r, t) =rn
c1 r n=n(c1 t + 1)
n(t + τ1 )(τ1 = 1/c1 ). The bounding circle (centered at the origin) has radius S1 (t) satisfying S10 (t) =v(S1 (t), t) = M (S1 (t), t)/S1 (t)n−1 with S1 (0) = R1 , hence
1/n

S1 (t) = R1 (c1 t + 1)1/n = R1 c1 (t + τ1 )1/n .
(ii) There is then an annular region with M (r, t) = c1 R1n /n and u(r, t) = 0, bounded by thisinterface and the circle of radius
1/n

S2 (t) = R1 (c1 t + (R2 /R1 )n )1/n = R2 (c2 t + 1)1/n = R2 c2 (t + τ2 )1/n ,τ2 = (R2 /R1 )n τ1 .

(iii) There is then an annular region of positive density (initially R2 < r < R3 ) given by
u2 (r, t) =

c2
1
=
;
c2 t + 1
t + τ2M2 =rn
c2 r n=n(c2 t + 1)
n(t + τ2 )where τ2 = 1/c2 > τ1 . The outer interface of this annulus is the circle of radius S3 (t) =1/n
R3 (c2 t + 1)1/n = R3 c2 (t + τ2 )1/n .
(iv) Finally, in the exterior region, complement of the ball of radius S3 (t), M (r, t) is constant M
and u is zero.
Conclusion. In this example there are two connected components for the density that expand
out with time. The inner patch is unperturbed by the outer mass, while the outer annular patchis spreading out in a self-similar way. The density in both patches is constant in space but
the density outside depends on the density inside, time, and the ratio R2 /R1 . Both densities
approach 1/t in first approximation as t → ∞. Moreover, the gap between the two patches isd(t) = C [(t + τ2 )1/n − (t + τ1 )1/n ]which relatively goes to zero like O(t−(n−1)/n ) as t → ∞.

24

6.5

Lack of comparison principle

This example allows to construct a counterexample to comparison of densities. Let us take forfirst solution u1 (r, t) the above example with parameters 0 < R1 < r2 < R3 . For second solution
u2 (x, t) let us take the elementary vortex patch with center x0 such that |x0 | = (R2 + R3 )/2,
with height equal or less than c2 and initial radius R4 much less than (R3 − R2 )/2. In this way
we have u1 (x, 0) ≥ u2 (x, 0). However, the support of u2 expands from the initial ball BR4 (x0 )while the support of u1 has a “hole” that eventually will “cover” that ball at some time. Hencecomparison of densities does not persist in time.
Note that the lack of a comparison of densities principle was shown for the solutions of the
FPME with s > 1/2 in [20]. The argument there was less explicit due to the lack of explicit
solutions.
Recall that for radial solutions on the other hand we have comparison of the mass functions
M (r, t). A similar result for FPME is known in dimension n = 1 where the approach of integrating in space is viable, [15].

7

Solutions with compact support

We now return to non necessarily radial solutions to (1.5) but impose the condition of compact
support on the data. This short section is essentially contained in [12], where more details canbe found. We have kept the proof since some of the arguments we explain here are used in what
follows.
It is known [20] that the solutions of the FPME for 0 < s < 1 have the property of finite
propagation so that initial data with compact support produces solutions with the same property
for every fixed t > 0. We can check that the calculation of the supersolution used to prove theproperty for s < 1 can be extended to s = 1. However, a comparison argument with explicit
solutions is easier in this case, by using the selfsimilar solutions constructed in Section 6.2. It
works as follows.
Theorem 7.1. Given bounded initial data u0 with compact support in the ball of radius R0 , thenfor every t > 0 the solution u(·, t) is supported in the ball of radius R0 (1 + ku0 k∞ t)1/n .
Proof. We consider the characteristics that according to [1] transport the solution. Using theequation we see that the speed of such lines is given byZu(y, t)(x0 − y)
−1
dy
(7.1)
v(x0 , t) = ∇p(x0 , t) = (∇(−∆) u)(x0 , t) = C1
|x0 − y|nwhere C1 = 1/((n − 2)ωn ), with ωn the volume of the unit ball in Rn . So it is bounded at all x
and t and we conclude that the initial support propagates only to a finite distance in any finite
time interval.
Take now t > 0 fixed and assume that we are at x0 , a point on the free boundary or boundaryof the support Ω(t) = {x : u(x, t) > 0}. We consider an extreme point, i. e., such that |x0 (t)| =sup{|x| : x ∈ ∂Ω(t)}. We may assume after rotation that x0 = (R(t), 0, · · · , 0). We calculate

25
the velocity along the radius direction (which is the velocity with which the boundary separates
from the origin). Then,Zu(y, t)(R(t) − y1 )
(7.2)
v1 (x0 , t) = C1dy,|x0 − y|n
The integrand has a positive sign for every y ∈ Ω(t). Therefore, the worst case is when we
replace u by its maximum estimate given by Theorem 5.1, to get
Z(R − y1 )1dy,|v1 (x0 , t)| ≤ C1
t + τ BR (0) |x0 − y|nwhere τ = 1/ku0 k∞ . Using scaling we get
R(t)|v1 (R(x0 , t)| ≤ C1t+τ

(1 − y1 )dy,n
B1 (0) |e1 − y|Zwith e1 = (1, 0, · · · , 0). Next, we notice that this is the integral that would be obtained for the
self-similar solution of Proposition 6.1. After a calculation in radial coordinates we haveR(t)n(t + τ )
1/nwhich, integrated in time, yields the estimate R(t) ≤ R0 t+τ
, i.e. the result.
τR0 (t) = |v1 (x0 , t)| ≤Remark 7.1. We have really proved the following: Let u be a bounded nonnegative weak solution.
Assume the support of the initial data is contained in a ball of radius R0 . Let U be the vortexpatch solution with same initial L∞ norm and initial support BR0 (0). Then the support of u attime t is contained in the support of U at time t.
We conclude the section with a similar statement for solutions which are not necessarily initiallybounded.
Proposition 7.1. For every initial measure µ ∈ M+ (Rn ) with compact support, there exists a
weak energy solution with compact support at all times and satisfying a bound of the type (3.3).
Proof. The existence of a solution satisfying (3.3) is proven in Theorem 3.1. To check it hascompact support, let us approximate first the initial data Rby smooth initial data with same
compact support and same bound M for their total mass u0 dx. Then, by conservation ofmass, for every t > 0 we have the two a priori estimates ku(t)kL1 ≤ M , ku(t)k∞ ≤ 1/t. By
the Nash-Sobolev embeddings we then get ∇p(·, t) ∈ L∞ (Rn ). A simple scaling argument thenshows that
(7.3)

1/n

(n−1)/nk∇p(·, t)k∞ ≤ Ck∆p(·, t)k1 k∆p(·, t)k∞≤ C1 M 1/n t−(n−1)/nUsing the description by characteristics (valid for the approximating solutions, as we saw in theprevious theorem) and integrating, it follows that the supports of the approximating solutions(compact by the previous theorem) are included in a ball of radius(7.4)

R(t) ≤ R0 + C2 M 1/n t1/n ,where C2 is independent of the approximation (and neither does R0 ). The conclusion follows by
passing to the limit. Note that we estimate the growth of the support like O(t1/n ) as expectedin the best case.

26

8

Asymptotic behaviour

The class of nonnegative solutions with finite mass has a property of simple asymptotic behaviourof self-similar form that is typical of diffusion processes, and also of some conservation laws. We
want to prove that the self-similar vortex patch solutions of Section 6.2 are attractors of thefinite-mass solutions. This has been proven in [12] in the case of compactly supported initial
data. We extend it here to solutions with finite second moment and to all radial solutions.

8.1

Asymptotics by the entropy method

By using the entropy method we can prove asymptotic behaviour for solutions with general data,with the only extra requirement that the second moment of the initial mass distribution has tobe finite. The proof follows the outline of the one done for the diffusive process with s < 1 in
[21].
In order to consider the question of large-time behaviour it is better to switch to the renormalized flow. More specifically, we introduce a change of the space, time and density variables
corresponding to the expected large time behaviour, as follows:(8.1)

y = x/(t + 1)1/n ,τ = log(1 + t),

U (y, τ ) = (t + 1) u(x, t) ,as well as the corresponding change for pressure and velocity:
P (y, τ ) = (t + 1)(n−2)/n p(x, t),V (y, τ ) = (t + 1)(n−1)/n v(x, t) = −∇P (y, τ ).

As a model for the asymptotic behaviour we consider the elementary vortex of Subsection 6.2
with initial height 1, i. e. ,u∗ (x, t) =

1
t+1

for

R(r) = R0 (t + 1)1/n .

x ∈ BR (t),and u∗ (x, t) zero otherwise. We further choose R0 so that the total mass M is the same:ZZu∗ (x, t) dx = u0 (x, t) dx,(i. e., ωn R0n = ku0 k1 = M ). The corresponding renormalized density is(8.2)

U∗ (y, τ ) = χ|y|≤R0 ,and the corresponding pressure, defined by P∗ (τ, y) = (−∆)−1 U∗ (τ, y) satisfiesyP∗ (τ, y) = C − (y 2 /2n) and − ∇P∗ (τ, y) =in BR0 (0).
n
In the proofs below we will consider the solution U (y, τ ) of the renormalized flow associated toa solution u(x, t) via (8.1). Some direct calculations show that U solves(8.3)

Uτ = ∇ · [U (∇P + y/n)],∆P + U = 0,

27where ∇ and ∆ indicate now differentiation w.r.t. y. Note that this is a mass conservation law
Uτ + ∇ · (U~v ) = 0,~v = −∇P − y/n,where the velocity is the sum of the renormalized particle velocity and the confining velocity−y/n produced by the change of variables.
We next state the asymptotic behaviour result.
TheoremR 8.1. Let n ≥ 2 and assume that u0 ∈ L1 (Rn ), u0 ≥ 0, and u0 has a finite second
moment, Rn y 2 u0 (y) dy < ∞. Then, the speed and density converge to equilibrium after normalization, as follows:(i) U stabilizes towards U∗ , the elementary patch with the same mass, in the sense that
(8.4)

lim kU (y, τ ) − U∗ (y)k1 = lim ku(·, t) − u∗ (·, t)k1 = 0 .

τ →∞

t→∞

We also have strong convergence U (y, τ ) → U∗ (y) in all the Lp (Rn ) spaces, 1 ≤ p < ∞, andweak-* convergence in L∞ (Rn ).
(ii) The corresponding velocities converge(8.5)

lim V (y, τ ) −

τ →∞

y
= 0.
n L2 (BR0 )Proof. (i) By displacing the origin of time by one unit, we may assume that our solution satisfies
1, hence U (y, τ ) ≤ 1. The mass of the renormalized solution will
the a priori bound u(x, t) ≤ t+1
be M at all times τ ≥ 0. This time displacement also allows to assume that the initial energy isfinite, i. e., ∇p0 (y) ∈ L2 (Rn ).
The boundedness of the orbit U (·, τ ) in L1 ∩ L∞ implies that as τ → ∞ there is a sequence τksuch that U (·, τk ) converges weakly to some U∞ (y), and also 0 ≤ U∞ (y) ≤ 1.
(ii) If n ≥ 3, we consider the “entropy”(8.6)

1
Ent(U ) :=2

Z(|∇P |2 +Rn1
U y 2 ) dynwhich is the version for the renormalized flow of the energyR (3.1). RecallRthat ∇P = −V and∇ · V = U . Not also that under the stated assumptions |∇P |2 dy = U P dy. It is easily
calculated that the entropy evolves according to the ruleZdy 2
Ent(U (τ )) = −D(U (τ )),
D(U (τ )) := U V −
dy .
dτ
n
(See [21] for a similar entropy calculation for s < 1.)(ii’) If n = 2, this entropy is not finite and its definition needs to be modified according to
Sectionsmooth compactly supported nonnegative function such thatR
R3.2: we set U0 to be any−1
U0 = U , and let P0 = (−∆) U0 . We then define

Z 
1
1
2
2
Ent(U ) :=|∇(P − P0 )| + U0 (P − P0 ) + U y
dy
2
2n
R2

28
which is finite. Computing as in Section 3.2 we find thatdEnt(U (τ )) = −D(U (τ ))dτstill holds, and we can continue the proof in the same way as for n ≥ 3.
(iii) The fact that Ent(U (τ )) is nonincreasing along the orbit of the renormalized flow implies
that there exists the limit(8.7)

E∗ (U ) = lim Ent(U (τ )) .
τ →∞

R∞R
U |V − (y/n)|2 dy dτ is convergent
in an infinite time
Moreover, the integral 1
R as an integral
2interval. Another consequence is that the second moment U (y, τ ) y dy is bounded for all τ(by 2n Ent(U (0)), to be
R precise), and this control of the behaviour as y → ∞ allows to makesure that in the limit U∞ (y) dy = M.
(iv) We now take limits as τk → ∞ not only in the family {U (y, τk )}k , k → ∞, but in thefamily of orbits {U (y, τ + τk )}k , for y ∈ Rn and 0 < τ ≤ T for fixed T > 0; k → ∞. The
compactness (which can be simply reproduced from Theorem 3.1) allows to show that along afurther subsequence that we also call τk → ∞ we get
e (y, τ ),U (y, τ + τk ) → Ue (y, 0) = U∞ (y).
in the weak sense, as well as V (y, τ + τk ) → Ve (y, τ ) strongly. Notice that U
Since (8.7) holds, we also havee (τ )) = E∗ (U ),Ent(Ue (τ )) = 0 .
a constant; and D(UThe last expression means that for a.e τ > 0 and a.e. y we havee (Ve − (y/n)) = 0,Ue=which according to the renormalized evolution equation (8.3) and V = −∇P means that ∂τ U
0. Therefore, the limit orbit is a stationary solution of the renormalized equation.
(v) We claim that our candidate U∗ (y) of (8.2) is the only stationary solution satisfying theRe (y) dy = M and 0 ≤ Ue ≤ 1 and Ve (y) = y/n on the support of Ue.
conditions U
First of all the a priori bound for Ve in L∞ (due to integral kernel estimates on the representation
formula performed in Formula (7.3)) implies that Ve = y/n cannot be true for large y, therefore
e must be compactly supported. Once we know the compact support property, it is alreadyU
e = U∗ (this also follows easily from our analysis ofproven in [12] that the asymptotic limit is U
Theorem 7.1).
(vi) The convergence of U (·, τ ) towards U∗ is at this point weak in L1 . But we also havesupy U (y, τ ) ≤ 1, this unilateral bound allows to improve the weak convergence to strong convergence in the ball BR0 (0), i. e., kU (y, τ ) − U∗ (y)k1 → 0 in that ball. But the mass of U∗

29
contained in that ball is the whole mass, so that U (y, τ ) must converge to zero in the complement of the ball. This proves the main asymptotic formula (8.4). Since the solutions are
uniformly bounded, the convergence in all the Lp spaces also follows for 1 ≤ p < ∞. However,the convergence in L∞ is impossible due to the discontinuous form of the limit function.
(vii) The convergence of the velocities now follows from properties of the convolution formulafor V in terms of U .
(viii) We still have to check that in displacing the origin of times from t = 0 to t = 1 we have notlost the property of finite second moment. Indeed, using the equation and several integrations
by parts, we have
Z
Z
Z
Z
d
2
2
u x dx = ∇ · (u∇p) x dx = −2 x · (u∇p) dx = 2 x · ∇p∆p dx
dt
RRR= −2 ∇p · ∇(x · ∇p) = −2 |∇p|2 − x · ∇|∇p|2
Z
Z2= (n − 2) |∇p| dx = (n − 2) u p dx.
If n = 2 then we are done. If n ≥ 3, we have found that the time derivative of the second
moment is a constant times the energy (3.1) which is itself nonincreasing in time, hence bounded
independently of time. It follows that the second moment remains bounded for all times.

8.2

Asymptotic behaviour for radial solutions

We will perform next the proof that the large-time behaviour of finite-mass solutions is given by
the self-similar vortex patch solutions of Section 6.3 in the case of radially symmetric data, where
the Burgers’ reformulation allows to use familiar tools. We do not need any further restriction
on the class of initial data. A reference for asymptotic convergence of conservation laws is [46].
Section 3 of that paper applies directly in the case n = 1 and it also does when n ≥ 2 after
replacing the space variable r by s. Since we also want to estimate u(r, t), which is a weighted
derivative of the mass function M (r, t), we give the details of the argument.
Theorem 8.2. For any radial solution to (1.5) with integrable initial data, letting M (x, t) be its“mass function” as in (6.1) and assuming maxr≥0 M (r, 0) = M0 , we have 0 ≤ M (rt1/n , t) ≤ rn /n
and
(8.8)

|rn /n − M (rt1/n , t)| → 0

uniformly in ∈ R and t → ∞. Moreover, if u0 is compactly supported, the set of x such that
M (|x|, t) < M0 , i.e. the support of u(x, t), is included in a disk centered at the origin of radius
S(t) such that S(t) ≥ (nM0 t)1/n and(8.9)

S(t) − (nM0 t)1/n → 0 as t → ∞.

30
Proof. (i) We first check the upper bound. First,
since M (r, t) is increasing in r, it has a limitR
limr→∞ = supr M (r, t) which is also equal to u(r, t) rn−1 dr = M0 hence is constant in time.
In view of the universal estimate of Section 5 we have u(x, t) ≤ 1/t so that M (r, t) ≤ rn /nt andsince limr→∞ M (r, t) = M0 , we deduce in view of the above that S(t) ≥ (nM0 t)1/n (this includes
that possibility that S(t)∞. This settles half of the estimates.
(ii) In order to bound the solution from below we consider a vertically displaced function
M1 (r, t) = M (r, t) + ε,

ε > 0.

Then,
11(M1 )t = − n−1 (M1 − ε)(M1 )r ≥ − n−1 M1 (M1 )rr
ri.e. M1 is a supersolution of the original Burgers’ equation. If R0 be determined by the condition
M (R0 , 0) ≥ M0 − ε. R0 depends on ε and may be large but is is finite. Then
M1 (r, 0) ≥ ε

for 0 ≤ r ≤ R0 ,M1 (r, 0) = M0 + ε

for r ≥ R0

We now consider the vortex patch solution u2 (r, t) with initial data τ1 χBR and total mass M0 .
For that we must have Rn /(nτ ) = M0 . We let M2 (r, t) be its mass function. Then
M2 (r, t) =

rn
χ
1/n .
n(t + τ ) {r≤R(1+t/τ ) }If we choose τ > 0 so that ε ≥ R0n /nτ , then we will have M1 (r, 0) ≥ M2 (r, 0). By the maximum
principle for (6.2), we deduce M1 (r, t) ≥ M2 (r, t) for all r ≥ 0 and all t > 0. This means that
M (rt1/n , t) ≥rn
−εn(1 + (τ /t))for r ≤ [n(M0 + ε)(1 + (τ /t))]1/n .

Passing to the limit t → ∞, the term in τ disappears, and letting then ε → 0 we easily arrive at
both estimates (8.8).
(iii) The estimates on the support for compactly supported data are derived in Section 3 of [12],even with rates of convergence.
As a consequence of this result, we have estimates for the radial velocity v = M/rn−1 that we
resemble the velocity of the elementary vortex patch V (r, t) = r/nt. They are as follows: thevelocity satisfies the estimatesv(r, t) ≤rnt

for 0 < r < R(t)and
(8.10)r− t(n−1)/n v(rt1/n , t) → 0nuniformly in r ≤ S(t)/t1/n as t → ∞. We can also give estimates for the density.

31
Corollary 8.1. The density satisfies the estimates(8.11)u(r, t) ≤1
tfor 0 < r < R(t)and t u(rt1/n , t) → χBR1 (0) (r) in L∞ (with weak-* convergence) as t → ∞, where R1 = (nM0 )1/n .
Proof. It is convenient to introduce the rescaled density û(r, τ ) = t u(rt1/n , t). We know that û
is bounded uniformly for large times by Theorem 5.1 and its support is uniformly bounded, by
Theorem 8.2. It therefore converges in the weak-∗ topology of L∞ (Rn ) to a function U withcompact support. Due to the relation between u and M , the limit U must be related to therescaled mass function of the elementary vortex patch by rn−1 U (r) = tM 0 (rt1/n ) = M 0 (r). We
conclude that U (r) is the characteristic function of the ball of radius R1 . Since the limit isunique the convergence does not depend on the subsequence tk → ∞.
Remark 8.1. The example of the radial solution with an expanding ball together with an expanding annulus shows that the convergence of û toward the equilibrium state Û (r) = χBM0 /n (0) (r)cannot happen in the uniform topology.

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Addresses
Sylvia Serfaty:
UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions,
Paris, F-75005 France ;CNRS, UMR 7598 LJLL, Paris, F-75005 France& Courant Institute, New York University, 251 Mercer st, NY NY 10012, USA(e-mail: serfaty@ann.jussieu.fr)
J. L. Vázquez:
Departamento de Matemáticas, Universidad Autónoma de Madrid,
28049 Madrid, Spain.
(e-mail: juanluis.vazquez@uam.es).