The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions

0.21 MB·English

by  Hongjie Dong

THE AGGREGATION EQUATION WITH POWER-LAW
KERNELS: ILL-POSEDNESS, MASS CONCENTRATION
AND SIMILARITY SOLUTIONS
1
1
0 HONGJIE DONG
2

n
Abstract. We study the multidimensional aggregationequation
a
u +div(uv)=0,v =−∇K∗uwithinitialdatainP (Rd)∩L (Rd).
J t 2 p
We prove that with biological relevant potential K(x) = |x|, the
2
1 equationis ill-posedin the criticalLebesguespace Ld/(d−1)(Rd)in
the sense that there exists initial data in P2(Rd)∩Ld/(d−1)(Rd)
P] such that the unique measure-valued solution leaves Ld/(d−1)(Rd)
A immediately. Wealsoextendthisresulttomoregeneralpower-law
kernels K(x) = |x|α, 0 < α < 2 for p = p := d/(d+α−2), and
. s
h prove a conjecture in [5] about instantaneous mass concentration
at for initial data in P2(Rd)∩Lp(Rd) with p<ps. Finally, we char-
m acterize all the “first kind” radially symmetric similarity solutions
[ in dimension greater than two.

2
v
2
3 1. Introduction
2
2 In this paper we consider the multidimensional aggregation equation
.
5
0 ut +div(uv) = 0, v = −∇K ∗u (1.1)
0
1 for x ∈ Rd and t > 0 with the initial data
:
v u(0,x) = u (x), x ∈ Rd.
i 0
X
Here d ≥ 2, u ≥ 0, K is the interaction potential, and ∗ denotes the
r
a spatial convolution. This equation arises in various models for bio-
logical aggregation and problems in granular media; see, for instance,
[13, 8, 12]. The problems of the well-posedness in different spaces,
finite-time blowups, asymptotic behaviors of solutions of this equation,
as well as the equation with an additional dissipation term, have been
studied extensively by a number of authors; see [12, 6, 3, 9, 2, 10, 11,
Date: January 13, 2011.
1991 Mathematics Subject Classification. 35B40, 35K55,92B05.
Key words and phrases. aggregation equation, ill-poshness, instantaneous mass
concentration, similarity solutions.
HongjieDongwaspartiallysupportedbytheNationalScienceFoundationunder
agreement No. DMS-0800129.
1
2 H.DONG
5, 7] and reference therein. We refer the reader to [4] for a nice review
about recent progress on the aggregation equation.
In [5] Bertozzi, Laurent and Rosado studied comprehensively the L
p
theory for the aggregation equation (1.1). Among some other results,
they considered radially symmetric kernels where the singularity at
the origin is of order |x|α for some α > 2 − d, and proved the local
well-posedness of (1.1) in P (Rd) ∩ L (Rd) for any p > p , where
2 p s
p = d/(d+α−2)(see below for the definition of thespace P (Rd)). In
s 2
the biological relevant case K(x) = |x|, they showed that solutions can
concentrate mass instantaneously for initial data in P (Rd) ∩L (Rd)
2 p
for any p < p . It remains unknown if (1.1) is well-posed in the critical
s
space P (Rd)∩L (Rd). Another interesting open question is whether
2 ps
one can show a similar instantaneous mass concentration phenomenon
for the equation with general power-law potential |x|α. The authors
conjectured in [5] that the answer to the second question is positive.
The aim of the current paper is to answer these questions. For
the first question, we shall construct radially symmetric initial data in
P (Rd)∩L (Rd), such that the unique measured-valued solution
2 d/(d−1)
leaves L (Rd) immediately for t > 0; see Theorem 2.1. This re-
d/(d−1)
sult implies that (1.1) is ill-posed in P (Rd) ∩ L (Rd), and the
2 d/(d−1)
well-posedness result for p > d/(d − 1) obtained in [5] is sharp. For
the second question, we show that, for any α ∈ (0,2) and any p < p ,
s
there exists radially symmetric initial data in P (Rd) ∩ L (Rd) such
2 p
that the solution concentrates mass at the origin instantaneously. In
other words, a Dirac delta appears immediately in the solution. There-
fore, we settle down the aforementioned conjecture in [5]. We also
prove that, for any α ∈ (0,2), (1.1) is ill-posed in P (Rd)∩L (Rd) by
2 ps
constructing initial datainP (Rd)∩L (Rd)such that anyweakly con-
2 ps
tinuous measured-valued solution, if exists, leaves L (Rd) immediately
ps
for t > 0.
The proofs use some ideas in [5] by considering the flow map driven
by the velocity field v. Roughly speaking, there are two steps in the
proofs. Inthefirststep, wefindasuitablerepresentationofthevelocity
field v in the polar coordinates, and prove the monotonicity, positivity
and asymptotics of the corresponding kernel. For K(x) = |x|, these
have already been established in [5] (Lemma 2.3). More delicate anal-
ysis is needed for general power-law potential K(x) = |x|α (see Lemma
4.4). In the second step, we deduce certain positive lower bounds for
the velocity (Lemmas 2.5, 4.6 and 4.7). Combined with the monotonic-
ity of the velocity in time, we then reduce the problems to study the
dynamics of solutions to some ordinary differential equations. In the
THE AGGREGATION EQUATION WITH POWER-LAW KERNELS 3
case p < p , it is shown that the flow map reaches the origin in a short
s
time, which generates a Dirac delta. While in the critical case p = p ,
s
a Dirac delta may not develop shortly, but the flow map makes the
mass concentrate quickly enough near the origin such that the solution
u leaves L (Rd) immediately.
ps
Wealso consider profiles ofsimilarity solutionsto(1.1)attheblowup
time with the potential K(x) = |x|, which conserve mass. This type of
solutions is an example of “first-kind” similarity solutions; see [1]. In
[2], Bertozzi, Carrillo and Laurent constructed radially symmetric first-
kind similarity solutions in thedimension oneandtwo, andproved that
in any odd dimension d ≥ 3 such solutions cannot exist with support
on open sets. By observing certain concavity property of the kernel
in the polar coordinates, in Section 3 we characterize all the radially
symmetric first-kind similarity solutions in the dimension d ≥ 3.
We finish the Introduction by fixing some notation. Most notation
in this paper are chosen to be compatible with those in [5]. For r > 0,
let
B = {x ∈ Rd : |x| < r}, S = {x ∈ Rd : |x| = r}.
r r
By ω we mean the surface area of the unit sphere S in Rd. We denote
d 1
P(Rd) to be the set of all probability measures on Rd, and P (Rd)
2
to be the set of all probability measures on Rd with bounded second
moment:
P (Rd) := µ ∈ P(Rd) : |x|2dµ(x) < ∞ .
2
(cid:26) ZRd (cid:27)
2. Ill-posedness when K(x) = |x|
In this section, we prove the following result, which reads that with
potential function K(x) = |x|, the aggregation equation (1.1) is ill-
posed in P ∩L (Rd).
2 d/(d−1)
Theorem 2.1. Let K(x) = |x|, k ∈ ((d−1)/d,1) and
L
u (x) = 1 ∈ P (Rd)∩L (Rd), (2.1)
0 |x|d−1(−log|x|)k |x|≤1/2 2 d/(d−1)
where
L = |x|−d+1(−log|x|)−kdx
Z|x|≤1/2
is a normalization constant. Let (µ ) be the unique measure-
t t∈(0,∞)
valued solution to the aggregation equation (1.1). Then for any t > 0
the density of µ , if exists, is not in L (Rd).
t d/(d−1)
4 H.DONG
We notethat for K(x) = |x|α,α ≥ 1 theglobal existence andunique-
ness of a weakly continuous measure-valued solution was proved in [7].
Moreover, for α < 2, any measure-valued solution will eventually col-
lapse to a Dirac delta at the center of mass in a finite time. In [5], it
was also proved that if u ∈ P then the measure-valued solution stays
0 2
in P .
2
Definition 2.2. Let µ ∈ P(Rd) be a radially symmetric probability
measure. We define µˆ ∈ P([0,+∞)) by
µˆ(I) = µ({x ∈ Rd : |x| ∈ I})
for all Borel sets I in [0,+∞)).
We reformulate some results in [5, Sect. 4] as the following lemmas.
Lemma 2.3. Let K(x) = |x|, µ ∈ P(Rd) be a radially symmetric
measure. Then for any x 6= 0, we have
∞ x
(∇K ∗µ)(x) = ψ(ρ/|x|)dµˆ(ρ) ,
|x|
Z0
where ψ : [0,∞) → R is a function defined by
1−ρy
ψ(ρ) = – 1 dσ(y).
|e −ρy|
ZS1 1
Moreover, ψ is continuous, positive, non-increasing on [0,∞), and
d−1
ψ(0) = 1, lim ψ(ρ)ρ = .
ρ→∞ d
Lemma 2.4. Let K(x) = |x|, and (µ ) be a radially symmetric
t t∈[0,∞)
weakly continuous measured-valued solution of the aggregation equation
(1.1). Then the vector field
v(t,x) = −(∇K ∗µ )1
t x6=0
is continuous on [0,∞)×(Rd \{0}). For any t ≥ 0, we have
µ = X#µ ,
t t 0
where, for each x ∈ Rd, X = X (x) is an absolutely continuous func-
t t
tion on [0,∞) satisfying
d
X (x) = v(t,X (x)) for a.e. t ∈ (0,∞);
t t
dt

X (x) = x,
 0
and X# means the push-forward of a measure by the map X . More-
t  t
over, for any x 6= 0, X (x) = R (|x|)x/|x|, where R is an absolutely
t t t
THE AGGREGATION EQUATION WITH POWER-LAW KERNELS 5
continuous, non-negative and non-increasing function in t ∈ [0,∞),
and
µˆ = R#µˆ .
t t 0
Consequently, for any x 6= 0,
∞ x
(∇K ∗µ )(x) = ψ(R (ρ)/|x|)dµˆ (ρ) ,
t t 0
|x|
Z0
and |(∇K ∗µ )(x)| is non-decreasing in t.
t
Next we establish a point-wise lower bound of the velocity v at t = 0.
Lemma 2.5. Let K(x) = |x| and u be the initial data defined in (2.1)
0
with k ∈ ((d−1)/d,1):
L
u (x) = 1 .
0 |x|d−1(−log|x|)k |x|≤1/2
Then there exists a constant δ > 0 such that
1
|(∇K ∗u )(x)| ≥ δ |x|(−log|x|)1−k (2.2)
0 1
for any x ∈ Rd satisfying 0 < |x| < 1/2.
Proof. Clearly, we have
uˆ (ρ) = Lω (−logρ)−k1 .
0 d ρ≤1/2
By the positivity and continuity of |v(0,·)| in Rd \ {0}, it suffices to
prove (2.2) for x ∈ Rd satisfying 0 < |x| < 1/4. It follows from Lemma
2.3 that for any ρ ∈ (|x|,1/2),
ψ(ρ/|x|) ≥ δ |x|/ρ
0
foraconstantδ > 0independent of|x|. Thus, foranyx ∈ Rd satisfying
0
0 < |x| < 1/4,
1/2
|(∇K ∗u )(x)| ≥ ψ(ρ/|x|)uˆ (ρ)dρ
0 0
Z|x|
1/2 dρ
≥ δ Lω |x| (−logρ)−k
0 d
ρ
Z|x|
δ Lω
= 0 d|x| (−log|x|)1−k −(log2)1−k
1−k
≥ δ |x|(−lo(cid:0)g|x|)1−k, (cid:1)
1
since 0 < |x| < 1/4. The lower bound (2.2) is proved. (cid:3)
We are now ready to prove Theorem 2.1.
6 H.DONG
Proof of Theorem 2.1. By Lemma 2.4, for each x ∈ Rd, |v(t,x)| is non-
decreasing in t. Therefore, from the lower bound of |v(x,0)| (2.2) we
infer that for any r ∈ (0,1/2),
d
R (r) ≤ −δ R (r)(−logR (r))1−k. (2.3)
t 1 t t
dt
Solving the ordinary differential inequality (2.3) gives
(−logR (r))k ≥ (−logr)k +kδ t. (2.4)
t 1
From (2.4), we obtain
(−logR (r))k kδ t
t 1
≥ 1+ ,
(−logr)k (−logr)k
and by Taylor’s formula, for t sufficiently small,
(−logR (r)) δ t
t 2
≥ 1+ ,
(−logr) (−logr)k
where δ = δ /2. We thus obtain
2 1
R (r) ≤ re−δ2t(−logr)1−k. (2.5)
t
Nowwesupposethat, forsomet > 0, µ hasadensity functionu(t,x) ∈
t
L (Rd). By H¨older’s inequality, for any r ∈ (0,1/2)
d/(d−1)
µ (B ) = u(t,x)dx
t Rt(r)
ZBRt(r)
d−1
d
d 1
≤ ud−1(t,x)dx |BRt(r)|d
ZBRt(r) !
≤ ku(t,·)k R (r). (2.6)
Ld/(d−1)(Rd) t
On the other hand, by the definitions of µ and R ,
t t
r
µ (B ) ≥ µ (B ) = Lω (−logρ)−kdρ. (2.7)
t Rt(r) 0 r d
Z0
Note that the above inequality is strict only when there is a mass
concentration before time t. We combine (2.6), (2.7) and (2.5) to get
µ (B )
ku(t,·)k ≥ t Rt(r)
Ld/(d−1)(Rd) R (r)
t
r(−logρ)−kdρ
≥ Lω 0 . (2.8)
d re−δ2t(−logr)1−k
R
THE AGGREGATION EQUATION WITH POWER-LAW KERNELS 7
However, by L’Hospital’s rule,
r(−logρ)−kdρ
lim 0
rց0 re−δ2t(−logr)1−k
R
(−logr)−k
= lim
rց0 e−δ2t(−logr)1−k 1+δ2t(1−k)(−logr)−k
= ∞.
(cid:0) (cid:1)
Thisgivesacontradictionto(2.8)sinceweassumeu(t,·) ∈ L (Rd).
d/(d−1)
(cid:3)
The theorem is proved.
3. Similarity solutions
In this section, we consider the problem of similarity solutions to the
aggregation equation. This problem is closely related to the blowup
profile for (1.1) at the blowup time. Let us consider mass-conserving
similarity solutions
1 x
u(t,x) = u (3.1)
R(t)d 0 R(t)
(cid:18) (cid:19)
to the aggregation equation (1.1) with interaction kernel K(x) = |x|.
These solutions are “first-kind” similarity solutions, while “second-
kind” similarity solutions do not conserve mass.
If u is a radially symmetric first-kind similarity solution given by
(3.1), then by the homogeneity it is easily seen that
x
v(t,x) = v , v = −∇|x|∗u .
0 0 0
R(t)
(cid:18) (cid:19)
Moreover, it was proved in [2] that R(t) must be a linear function and
on the support of u
0
v = −λx (3.2)
0
for some constant λ > 0. In one space dimension, the authors of [2]
constructed a first-kind similarity solution
1 x
u(t,x) = U ,
T∗ −t T∗ −t
(cid:18) (cid:19)
where U is the uniform distribution on [−1,1]. In any dimension, there
is a first-kind similarity measure-valued solution which is a single delta
ona sphere withradius shrinking linearly intime[2, Remark 3.8]. Such
solution is called single delta-ring solution. In two space dimension, a
two delta-ring solution was constructed in the same paper, i.e., µˆ =
0
m δ + m δ for some 0 < ρ < ρ < ∞ and m ,m > 0. On the
1 ρ1 2 ρ2 1 2 1 2
other hand, for any odd dimension d ≥ 3, by using a relation between
8 H.DONG
K(x) and the Newtonian potential they proved the non-existence of
radially symmetric first-kind similarity solutions with support on open
sets.
Wecharacterizealltheradiallysymmetricfirst-kindsimilaritymeasure-
valued solutions in the following theorem, which in particular implies
that in dimension three and higher there cannot exist first-kind simi-
larity solutions with support on open sets or multi delta rings.
Theorem 3.1 (Characterization of similarity solutions). Let d ≥ 3
and K(x) = |x|. Then any radially symmetric first-kind similarity
measure-valued solution is of the form
1 x
µ (x) = µ ,
t R(t)d 0 R(t)
(cid:18) (cid:19)
where
µˆ = m δ +m δ (3.3)
0 0 0 1 ρ1
for some constants m ,m ≥ 0 and ρ > 0.
0 1 1
For the proof, first we recall that by Lemma 2.3 for any radially
symmetric measure-valued solution µ and t > 0,
t
∞ x
v(t,x) = − φ(|x|/ρ)dµˆ (ρ) , (3.4)
t
|x|
Z0
where φ : [0,∞) → R is a function defined by
r −y
φ(r) = – 1 dσ(y). (3.5)
|re −y|
ZS1 1
The proof of Theorem 3.1 relies on the following observation.
Lemma 3.2. i) Let d ≥ 4 and K(x) = |x|. Then the function φ defined
by (3.5) is C2 on [0,∞) and satisfies
φ(0) = 0, lim φ(r) = 1, φ′(r) > 0 on [0,∞), (3.6)
r→∞
φ′′(r) < 0 on (0,∞).
ii) Let d = 3 and K(x) = |x|. Then the function φ defined by (3.5)
is C1 on [0,∞) and satisfies (3.6). Moreover, it is concave on [0,∞),
linear on [0,1] and strictly concave on (1,∞).
Suppose for the moment that Lemma 3.2 is verified. We prove The-
orem 3.1 by a contradiction argument. Suppose (µ ) is a radially sym-
t
metric first-kind similarity measure-valued solution such that there are
two numbers
0 < ρ < ρ , ρ ,ρ ∈ suppµˆ .
1 2 1 2 0
THE AGGREGATION EQUATION WITH POWER-LAW KERNELS 9
Denote w(ρ) := |v (ρe )|. By (3.4),
0 1
∞
w(ρ ) = φ(ρ /ρ)dµˆ (ρ), k = 1,2.
k k 0
Z0
It follows from Lemma 3.2 that
ρ
1
φ(ρ /ρ) ≥ φ(ρ /ρ),
1 2
ρ
2
and the inequality is strict for all ρ in a small neighborhood of ρ
1
because ρ /ρ > 1. Since ρ ∈ suppµˆ , we get
2 1 1 0
ρ
1
w(ρ ) > w(ρ ).
1 2
ρ
2
On the other hand, from (3.2) w is a linear function on suppµˆ :
0
ρ
1
w(ρ ) = w(ρ ).
1 2
ρ
2
Therefore, we reach a contradiction.
We finish this section by proving Lemma 3.2.
Proof of Lemma 3.2. We rewrite (3.5) as
ω π (r−cosθ)(sinθ)d−2
d−1
φ(r) = dθ, (3.7)
ω A(r,θ)
d Z0
where
A(r,θ) = (1+r2 −2rcosθ)1/2.
For r ∈ [0,1)∩(1,∞), a direct computation gives
ω π 1 (r −cosθ)2
φ′(r) = d−1 (sinθ)d−2 − dθ
ω A A3
d Z0 (cid:18) (cid:19)
ω π (sinθ)d
d−1
= dθ, (3.8)
ω A3
d Z0
and
3ω π (sinθ)d(r −cosθ)
φ′′(r) = − d−1 dθ. (3.9)
ω A5
d Z0
Integration by parts yields
3ω π r(sinθ)d (sinθ)d+1 ′
φ′′(r) = − d−1 − dθ
ω A5 (d+1)A5
d Z0 (cid:0) (cid:1)
3ω π r(sinθ)d 5(sinθ)d+1rsinθ
d−1
= − − dθ
ω A5 (d+1)A7
d Z0
3ω π r(sinθ)d 5
= − d−1 r2 +1−2rcosθ− sin2θ dθ.
ω A7 d+1
d Z0 (cid:18) (cid:19)
10 H.DONG
Case 1: d ≥ 4. In this case, we have
5
r2 +1−2rcosθ − sin2θ ≥ (r −cosθ)2.
d+1
Therefore, we get φ′′ < 0 on (0,1)∩(1,∞). Note that because
|r−cosθ| ≤ A, r|sinθ| ≤ A,
the integrals on the right-hand sides of (3.8) and (3.9) are absolutely
convergent and continuous at r = 1 by the dominated convergence
theorem. Thus (3.8) and (3.9) hold on the whole region [0,∞). So
we conclude φ ∈ C2([0,∞)) and φ′′ < 0 on (0,∞). Moreover, since
φ(0) = 0, φ → 1 as r → ∞ and φ′ > 0 by (3.8), φ is bounded and
non-negative on [0,∞).
Case 2: d = 3. As before, the integral on the right-hand side of (3.8)
is absolutely convergent and continuous at r = 1 by the dominated
convergence theorem. Thus (3.8) holds on the whole region [0,∞), and
we conclude φ ∈ C1([0,∞)) and φ′ > 0 on [0,∞). For d = 3, one can
explicitly compute φ. From (3.7), we have
1 π (r −cosθ)sinθ
φ(r) = dθ
2 A(r,θ)
Z0
1 π cosθsinθ
= ∂ A− dθ
θ
2 A
Z0
1
= (|1+r|−|1−r|)−I, (3.10)
2
where,
1 π cosθsinθ
I := dθ.
2 A
Z0
Integrating by parts and using (3.8), we get
1 π (sin2θ)′ 1 π rsin3θ r
I = dθ = dθ = φ′(r).
4 A 4 A3 2
Z0 Z0
Thus, φ satisfies
r
r − φ′(r) for r ∈ [0,1];
2
φ(r) =
r
1− φ′(r) for r ∈ (1,∞),
 2
and φ(1) = 2/3 by (3.7). Solving this ordinary differential equation,

we obtain
2r/3 for r ∈ [0,1];
φ(r) = 1
1− for r ∈ (1,∞),
 3r2
