Table Of ContentEULERIAN DYNAMICS WITH
A COMMUTATOR FORCING II: FLOCKING
ROMAN SHVYDKOY AND EITAN TADMOR
Abstract. We continue our study of one-dimensionalclass of Euler equations, introduced
7
1 in [ST2016], driven by a forcing with a commutator structure of the form [Lφ,u](ρ) =
0 φ∗(ρu)−(φ∗ρ)u, where u is the velocity field and φ belongs to a rather general class of
2 influence or interaction kernels.
n In this paper we quantify the large-time behavior of such systems in terms of fast
a flocking for two prototypical sub-classes of kernels: bounded positive φ’s, and singular
J φ(r) = r−(1+α) of order α ∈ [1,2) associated with the action of the fractional Laplacian
6 L = −(−∂ )α/2. Specifically, we prove fast velocity alignment as the velocity u(·,t) ap-
2 φ xx
proaches a constant state, u→u¯, with exponentially decaying slope and curvature bounds
] |ux(·,t)|∞+|uxx(·,t)|∞ .e−δt. Thealignmentisaccompaniedbyexponentiallyfastflocking
P
of the density towards a fixed traveling state ρ(·,t)−ρ∞(x−u¯t)→0.
A
.
h
t
a
m Contents
[
1. Introduction and statement of main results. 1
1 2. Flocking with smooth positive kernels 5
v
3. Flocking with singular kernels 8
0
1 References 18
7
7
0
.
1 1. Introduction and statement of main results.
0
7 1.1. Flocking hydrodynamics. In this paper we continue our study initiated in [ST2016],
1
of Eulerian dynamics driven by forcing with a commutator structure. In the one-dimensional
:
v case, the dynamics of a velocity u : Ω×R 7→ R is governed by the system of form
i +
X
ρ +(ρu) = 0,
t x
r (1.1)
a
(cid:26) u +uu = [L ,u](ρ),
t x φ
where the commutator on the right, [L ,u](ρ) := L (ρu) − L (ρ)u, involves a convolution
φ φ φ
kernel
(1.2) L (f) := φ(|x−y|)(f(y)−f(x))dy.
φ
ZR
Date: January 27, 2017.
1991 Mathematics Subject Classification. 92D25, 35Q35,76N10.
Key words and phrases. flocking, alignment, fractional dissipation, Cucker-Smale,Mortsch-Tadmor,crit-
ical thresholds.
Acknowledgment. Researchwas supported in part by NSF grants DMS16-13911,RNMS11-07444(KI-
Net) and ONR grant N00014-1512094 (ET) and by NSF grant DMS 1515705 (RS). RS thanks CSCAMM
forthe hospitalityfor hisMarch2016visitwhichinitiated this project. Bothauthorsthank the Institute for
Theoretical Studies (ITS) at ETH-Zurich for the hospitality.
1
2 ROMANSHVYDKOYAND EITANTADMOR
Themotivationfor(1.1)comesfromthehydrodynamicdescriptionofalarge-crowddynamics
driven by Cucker-Smale agent-based model
x˙ = v ,
i i
(1.3) 1 N (x ,v ) ∈ Ω×R, i = 1,2,...,N.
i i
v˙ = φ(|x −x |)(v −v ),
i i j j i
N
Xj=1
Here, φ is a positive, bounded influence function which models the binary interactions
among agents in Ω. We focus our attention on the periodic or open line setup, Ω = T,R.
For large crowds, N ≫ 1, the dynamics can be encoded in terms of the empirical dis-
tribution f = 1 N δ (x) ⊗ δ (v), so that its limiting moments lead to a density,
N N i=1 xi vi
P
ρ(x,t) = lim f (x,v,t)dv, and momentum, ρu(x,t) = lim vf (x,v,t)dv, governed
N N
N→∞ZR N→∞ZR
by (1.1), [HT2008, CCP2017].
ρ +(ρu) = 0,
t x
(1.4) (x,t) : Ω×[0,∞).
ut +uux = φ(|x−y|)(u(y,t)−u(x,t))ρ(y,t)dy
ZR
The other important limit of such systems — their large time behavior for t ≫ 1, is
described by the flocking phenomenon. To this end, let us introduce the set of flocking state
solutions, consisting of constant velocities, u¯, and traveling density waves ρ¯= ρ (x−tu¯),
∞
(1.5) F = {(u¯,ρ¯) : u¯ ≡ constant,ρ¯(x,t) = ρ (x−tu¯)}.
∞
We say that a solution (u(·,t),ρ(·,t)) converges to a flocking state (u¯,ρ¯) ∈ F in space X×Y
if
ku(·,t)−u¯k +kρ(·,t)−ρ¯(·,t)k → 0, as t → ∞.
X Y
This represents the process of alignment where the diameter of velocities tends to zero
(1.6a) V(t) := max |u(x,t)−u(y,t)| → 0, as t → ∞.
x,y∈suppρ(·,t)
In particular, there is a fast alignment if the flocking convergence rate is exponential. In the
present case of symmetric interactions, the conservation of averaged mass and momentum,
1 1
M(t) := ρ(x,t)dx ≡ M , P(t) := (ρu)(x,t)dx ≡ P
0 0
2π ZT 2π ZT
implies that a limiting flocking velocity, provided it exists, is given by u¯ = P /M .
0 0
Remark 1.1. In the case when the dynamics of (1.4) takes place over the line Ω = R as in
[HT2008, TT2014], the flocking phenomenon assumes a compactly supported initial config-
uration with finite initial velocity variation,
D := max |x−y| < ∞, V := max |u (x)−u (y)| < ∞.
0 0 0 0
x,y∈supp(ρ0) x,y∈supp(ρ0)
It requires that, in addition to (1.6a), the flow remains compactly supported
(1.6b) D(t) 6 D < ∞, D(t) := max |x−y|
∞
x,y∈suppρ(·,t)
This reflects the corresponding flocking behavior in the agent-based Cucker-Smale model,
max |x (t)−x (t)| 6 D and max |v (t)−v (t)| → 0 as t → ∞, [HT2008, MT2014].
i j ∞ i j
16i,j6N 16i,j6N
EULERIAN DYNAMICS WITH A COMMUTATOR FORCING 3
1.2. Smooth solutions must flock. The flocking hydrodynamics of (1.4) for bounded
positive φ’s follows, as long as they admit global smooth solutions. Indeed, the statement
that “smooth solutions must flock” holds in the general setup of positive kernels whether
symmetric or not [ISV2016, Lemma 3.1], [TT2014, Theorem 2.1]. For the sake of complete-
ness we include below the proof of flocking along the lines of [MT2014, theorem 2.3] which
T
is stated in the following lemma for the periodic case Ω = .
Lemma 1.2 (Smooth solutions must flock). Let (ρ,u) be a smooth solution of the one-
dimensional system
ρ +(ρu) = 0,
t x
(1.7) (x,t) : T×[0,∞),
ut +uux = k(x,y,t)(u(y,t)−u(x,t))ρ(y,t)dy
ZR
with strictly positive kernel, ι (t) = inf k(x,y,t) > 0. Then there is a flocking alignment
k
x,y∈T
t
V(t) 6 V(0)exp −M ι (τ)dτ , V(t) = max|u(x,t)−u(y,t)|.
k
(cid:26) Z (cid:27) x,y∈T
τ=0
In particular, the case of symmetric interaction (1.4) admits fast alignment,
(1.8) V(t) 6 V(0)e−Mιφt, ι := minφ(|x|).
φ
x∈T
Proof. Let x (t) be a point where u = u(x (t),t) = minu, and x (t) be a point where
− − − +
u = u(x (t),t) = maxu. Then the maximal value does not exceed,
+ +
d
u = k(x ,y,t)(u(y)−u )ρ(y,t)dy 6 ι (u(y)−u )ρ(y,t)dy.
+ + + k +
dt ZT ZT
Similarly, we have the lower bound
d
u > ι (u(y)−u )ρ(y,t)dy.
− k −
dt ZT
SubtractingthelatterimpliesthatthevelocitydiameterV(t) = max|u(x,t)−u(y,t)|satisfies
x,y∈T
d
V(t) 6 −ι MV(t), V(t) = u (t)−u (t)
k + −
dt
and the result readily follows. (cid:3)
We demonstrate the generality of lemma 1.2 with the following two examples.
Example 1.3 (an example on non-symmetric kernel). The Mostch-Tadmor model [MT2011]
uses an adaptive normalization, where the pre-factor 1/N on the right of (1.3) is replaced
by 1/ φ(|x − x |), leading to the flocking hydrodynamics with non-symmetric kernel
j i j
k(x,y,Pt) = φ(|x−y|)/(φ∗ρ)(x,t). The lower-bound
φ(|x−y|) ι
k(x,y,t) = > φ , I = maxφ(|x|).
φ
(φ∗ρ)(x,t) IφM x∈T
shows that flocking holds for positive, bounded φ’s, with exponential rate dictated by the
condition number of φ but otherwise independent of the total mass, V(t) 6 V(0)e−(ιφ/Iφ)t.
4 ROMANSHVYDKOYAND EITANTADMOR
Example 1.4 (an example of unbounded kernels). The fractional Laplacian L := L is
α φα
associated with the singular periodized kernels
1
(1.9) φ (x) = , for 0 < α < 2.
α
|x+2πk|1+α
Xk∈Z
Since the argument of lemma 1.2 does not use local integrability, it applies in the present
setting with ι = inf φ (|x|) > 0, leading to fast alignment (1.8).
α x α
We close this subsection by noting that the the extension of lemma 1.2 to the case of open
R
space Ω = was proved in [TT2014]. To this end one restricts attention to the dynamics
over {suppρ(·,t)}: the growth of the velocity diameter V(t) := max |u(x,t)−u(y,t)|,
x,y∈suppρ(·,t)
d
V(t) 6 −ι MV(t), V(t) = max u(x,t)− min u(y,t),
k
dt x∈suppρ(·,t) y∈suppρ(·,t)
d
is coupled with the obvious bound on the growth of the density support, D(t) 6 V(t)D(t).
dt
Assume that φ is decreasing so that ι > φ(D(t)). It implies a decreasing free energy E(t) :=
φ
D(t)
V(t) + φ(τ)dτ 6 E , and fast alignment follows with a finite diameter, D(t) 6 D ,
τ=0 0 ∞
dictatedRby
D∞
D(t) 6 D , M φ(s)ds = V .
∞ 0
Z
D0
Thus, inthecaseofopenspace, Ω = R, compactness of{suppρ(·,t)}requiresafinitevelocity
∞
variationV < φ(s)ds. Inparticular, anunconditional flocking followsforglobalφ’swith
0
Z
D0
unbounded integrable tails. Of course, in the periodic settings, compactness of the support
of ρ is automatic.
1.3. Statement of main results. Lemma 1.2 tells us that for positive φ’s, the question
of flocking is reduced to the question of global regularity. The latter question — the global
regularity of (1.4), was addressed in our previous study [ST2016] in the larger context of
three classes of interaction kernels. Namely, for bounded φ’s, global regularity follows for
sub-critical initial datasuch thatu′(x)+φ∗ρ (x) > 0. Forsingular kernels φ (x) := |x|−(1+α)
0 0 α
corresponding to L = −(−∆)α/2, global regularity follows for α ∈ [1,2). Finally, global
φ
regularity also holds in the limiting case α = 2 which corresponds to the Navier-Stokes
equations with L = ∆, and we recall that the global regularity for the cases α ∈ [1,2] is
φ
independent of a critical threshold requirement. A main feature of the forcing in all three
cases is their commutator structure in (1.1) which yields is the conservative transport of the
first-order quantity u +L (ρ), [CCTT2016, ST2016]
x φ
(1.10) e +(ue) = 0, e := u +L (ρ).
t x x φ
In this paper, we complement our earlier study of global regularity with the flocking
behavior for these classes of interaction kernels. In particular, we make a more precise
flocking statement, where fast alignment max |u(x,t)−u(y,t)| . e−δt is strengthened
x,y∈suppρ(·,t)
to an exponential decay of slope and curvature of the velocity |u | + |u | . e−δt, and
x ∞ xx ∞
the flocking itself is proved in the strong sense of exponential convergence to one of the
flocking states F. We treat here the flocking behavior in the two cases of positive φ’s, and
of fractional φ , α ∈ [1,2). The limiting case of Navier-Stokes equations L = ∆ does not
α φ
EULERIAN DYNAMICS WITH A COMMUTATOR FORCING 5
seem to satisfy fast alignment due to lack of non-local interactions. Its large-time behavior
remains open.
We begin with the case of a bounded positive kernel, and general density with a possibility
of vacuum. The result is proved in both periodic and open line domains.
Theorem 1.5 (Bounded positive kernels). Consider the system (1.4) with bounded positive
kernel φ ∈ W2,∞(Ω), where Ω = T or R. For any initial conditions (u ,ρ ) ∈ W2,∞ ×
0 0
(W1,∞ ∩L1) which satisfies the sub-criticality condition,
(1.11) u′ +φ∗ρ > 0,
0 0
there exists a unique global solution (ρ,u) ∈ L∞([0,∞);W2,∞ × (W1,∞ ∩ L1)). Moreover,
for fixed β < 1 there exists C,δ > 0 (depending on β) such that the velocity satisfies the fast
alignment estimate
(1.12) |u (t)| +|u (t)| 6 Ce−δt,
x ∞ xx ∞
and there is an exponential convergence towards the flocking state (u¯,ρ¯) ∈ F, where u¯ =
P /M and ρ¯= ρ (x−tu¯) ∈ W1,∞,
0 0 ∞
(1.13) |u(t)−u¯|W2,∞ +|ρ(t)−ρ¯(t)|Cβ 6 Ce−δt, t > 0.
Next we turn to the case of singular kernels, φ (x) = |x|−(1+α), 1 6 α < 2, in the periodic
α
setting Ω = T, and no-vacuum condition ρ > 0. The latter two are necessary to maintain
0
uniform parabolicity of the system.
Theorem 1.6 (Singular kernels of fractional order α ∈ [1,2)). Consider the system (1.4)
with singular kernel φ (x) = |x|−(1+α), 1 6 α < 2 on the periodic torus T. For any initial
α
condition (u ,ρ ) ∈ H3 ×H2+α away from the vacuum there exists a unique global solution
0 0
(ρ,u) ∈ L∞([0,∞);H3 ×H2+α). Moreover, for fixed s < 3 there exists C,δ > 0 (depending
on s) such that the velocity satisfies the fast alignment estimate,
(1.14) |u (t)| +|u (t)| +|u (t)| 6 Ce−δt,
x ∞ xx ∞ xxx 2
and there is an exponential convergence towards the flocking state (u¯,ρ¯) ∈ F, where u¯ =
P /M and ρ¯= ρ (x−tu¯) ∈ H3,
0 0 ∞
(1.15) |u(t)−u¯|H3 +|ρ(t)−ρ¯(t)|Hs 6 Ce−δt, t > 0.
Remark 1.7. (On the singular case of fractional order α ∈ (0,1)) We recently learned that
shortly after our release of [ST2016] another approach to the regularity of (1.4) with sin-
gular kernels φ appeared in the work of T. Do et. al. [DKRT2017]. Their alternative
α
approach, based on the propagation of properly-tuned modulus of continuity along the lines
of [KNV2008], covers the regularity of (1.4) which is treated as critical system for the range
α
α ∈ (0,1). Although our arguments of regularity in [ST2016] can be adapted to the respec-
tive range, it is not clear whether the flocking statement in theorem Theorem 1.6 survives
for 0 < α < 1 with either one of these approaches.
2. Flocking with smooth positive kernels
Our starting point is the conservative transport (1.10). In the case of positive mollifier we
have [L ,u](ρ) = [φ∗,u](ρ) and (1.10) yields
φ
(2.1) e +(ue) = 0, e := u′(x,t)+φ∗ρ(x,t)
t x
6 ROMANSHVYDKOYAND EITANTADMOR
Hence the positivity e > 0 propagates in time. It follows that (1.4) admits global smooth
0
solutions under the critical threshold condition e (·) > 0, see [CCTT2016] for details. The
0
flocking of these solutions, which is guaranteed by Lemma 1.2, is quantified in the following
two lemmas in terms of constants C,δ > 0, depending on M,|φ|W1,∞,|u0|W1,∞,minφ > 0
and mine > 0.
0
Lemma 2.1. Suppose e > 0 on Ω. There exist constants C,δ > 0 such that
0
|u (·,t)| 6 Ce−δt.
x ∞
Proof. We rewrite (2.1) as a logistic equation along characteristics x˙(t) = u(x(t),t) with
non-autonomous threshold h := φ∗ρ,
D
(2.2) e = (h−e)e, h = φ∗ρ.
Dt
D
Here and below, denotes differentiation along generic particle path {x(t) = x(t;x )}
0
Dt
initiated at x(t = 0;x ) = x . Hence, in view of the bound Mι 6 h(x,t) 6 MI ,
0 0 φ φ
D
(MI −e)e > e > (Mι −e)e.
φ φ
Dt
Since e is uniformly bounded from above and away from zero, it follows that there exists a
0
time t > 0 such that the quantity e(t) = e(t;x ) remains likewise bounded from above and
0 0
below uniformly for all initial conditions e = e(t = 0;x ),
0 0
(2.3) C := 2MI > e(t) > Mι /2 =: c > 0, t > t .
0 φ φ 0 0
Let us now write the equation for φ∗ρ by convolving the mass equation φ∗ρ +φ∗(ρu) = 0:
t x
(2.4a) (φ∗ρ) +u(φ∗ρ) = −[φ′∗,u]ρ,
t x
where the commutator is given by
[φ′∗,u]ρ = φ′(|x−y|) u(x,t)−u(y,t) ρ(y,t)dy.
ZT
(cid:0) (cid:1)
In view of Lemma 1.2, and the fact that φ ∈ W1,∞ we have
(2.4b) |[φ′∗,u]ρ| 6 CM|φ|W1,∞e−δt.
In what follows we denote by E = E(t) a generic exponentially decaying quantity, so (2.4)
D
reads h = E, uniformly over all initial conditions x ∈ Ω. Let us rewrite this equation for
0
Dt
the difference h = e−u :
x
D
u = −u e+E.
x x
Dt
Recall that for large enough time, t > t we have the positive boundedness 0 < c < e < C
0 0 0
uniformly over initial conditions. This readily implies
|u (x(t;x ),t)| 6 E(t)
x 0
uniformly over x . Since at any time t characteristics cover all Ω we arrive at the desired
0
bound |u (t)| 6 E(t). (cid:3)
x ∞
EULERIAN DYNAMICS WITH A COMMUTATOR FORCING 7
Solving the density equation along characteristics we obtain
t
(2.5) ρ(x(t;x ),t) = ρ (x )exp − u (x(s;x ),s)ds .
0 0 0 x 0
(cid:26) Z (cid:27)
0
So, in view of Lemma 2.1 the density enjoys a pointwise global bound (which is not given a
priori)
(2.6) sup|ρ(·,t)| < ∞.
∞
t>0
Next we establish a second round of estimates in higher order regularity in order to get a
control over ρ and then prove flocking of the density.
x
Lemma 2.2. There exist C,δ > 0 such that
|u (·,t)| 6 Ce−δt.
xx ∞
Proof. Let us write the equations for e and h = φ′ ∗ρ:
x x
D
e = −u e +(h −e )e
x x x x x
Dt
D
h = φ′′(|x−y|) u(x,t)−u(y,t) ρ(y,t)dy.
x
Dt ZT
(cid:0) (cid:1)
Clearly, the right hand side of h -equation is exponentially decaying, = E. Subtracting the
x
two we obtain the equation for u :
xx
D
(2.7) u = −u e−u e +E.
xx xx x x
Dt
Note that clearly, h is a bounded function, hence u e = Eu +E. So,
x x x xx
D
(2.8) u = −u e−Eu +E.
xx xx xx
Dt
Once again, the positive boundedness of e in (2.3), implies that for large enough time,
D
(2.9) u = −u e+E, t > t ,
xx xx 0
Dt
and the lemma follows. (cid:3)
Let’s write the equation for ρ :
x
D
(2.10) ρ = −2u ρ −u ρ = Eρ +E
x x x xx x
Dt
This shows that ρ is uniformly bounded. We are now ready to prove Theorem 1.5. We state
x
the last computation in a lemma as it will also be used ad verbatim in the next section.
Lemma 2.3. Let (u,ρ) ∈ W2,∞×W1,∞ be any solution pair to (1.4). For fixed β < 1 there
exist C,δ > 0 (depending on β) and a flocking pair (u¯,ρ¯) ∈ F, ρ¯∈ W1,∞, such that,
(2.11) |ρ(·,t)−ρ¯(·,t)| 6 Ce−δt, t > 0.
Cβ
Thus, F contains all limiting states of the system (1.4).
8 ROMANSHVYDKOYAND EITANTADMOR
Proof. The velocity alignment goes to its natural limit u¯ = P/M. Denote ρ(x,t) := ρ(x+
tu¯,t). Then ρ satisfies
e
ρ +(u−u¯)ρ +u ρ = 0,
t x x
e
where all the u’s are evaluated at x + tu¯. According to the established bounds we have
e e e
|ρ | < Ce−δt. This proves that ρ(t) is Cauchy as t → ∞, and hence there exists a unique
t ∞
limiting state, ρ (x), such that
∞
e e
|ρ(·,t)−ρ (·)| < C e−δt.
∞ ∞ 1
Shifting x this can be expressed in terms of ρ and ρ¯(·,t) = ρ (x−tu¯)
e ∞
|ρ(·,t)−ρ¯(·,t)| < C e−δt.
∞ 1
We also have ρ¯ ∈ W1,∞ from weak-star compactness. The statement of the lemma follows
by interpolation of this exponential bound with the C1-bound of ρ. (cid:3)
3. Flocking with singular kernels
3.1. Uniform bounds on density, and velocity alignment. The results of this section
lead towards the first step in the proof of Theorem 1.6. However, we state them in such
generality since they hold for a much broader class of kernels satisfying the following three
properties.
(i) Boundedness (away from the origin): for any r > 0,
ι (r) := inf φ(|x|) < ∞;
φ
|x|<r
(ii) Positivity: ι (2π) = inf φ(|x|) > 0;
φ x
(iii) Singularity : limsup rι (r) = ∞.
r→0 φ
This class of kernels which was already identified in [ST2016, Section 3.2], includes the
singular periodized kernels associated with −(−∂ )α/2,
xx
1
(3.1) φ (x) = for 0 < α < 2.
α
|x+2πk|1+α
Xk∈Z
The case of local Laplacian L = ∂ is not included.
xx
We recall that due to the positivity (ii), lemma Lemma 1.2 applies — as noted in Example
1.4, fast flocking holds irrespective of (lack of) local integrability
(3.2) V(t) 6 V(0)e−Mιφt.
As before, our starting point is the conservative transport (1.10) involving L ≡ L ,
α φα
(3.3) e +(ue) = 0, e = u +L (ρ).
t x x α
Paired with the mass equation we find that the ratio q = e/ρ satisfies the transport equation
D
(3.4) q = q +uq = 0.
t x
Dt
Starting from sufficiently smooth initial condition with ρ away from vacuum we can assume
0
that
(3.5) Q = |q(t)| = |q | < ∞.
∞ 0 ∞
EULERIAN DYNAMICS WITH A COMMUTATOR FORCING 9
Lemma 3.1. Fix α ∈ [1,2) and consider the pair (u,ρ) which is a smooth solution to
(1.1),(1.2) subject to initial density ρ away from vacuum. Then there are positive constants
α 0
c = c(M,Q,φ), C = C(M,Q,φ) such that
(3.6) 0 < c 6 ρ(x,t) 6 C, x ∈ T, t > 0.
Remark 3.2. The upper bound and the weaker lower-bound ρ(·,t) & 1/t was established for
the CS hydrodynamics (1.3) along the line of the 1D decay for commutator forcing [ST2016].
We include both cases for completeness.
Proof. Let us recall that the density equation can be rewritten as
(3.7) ρ +uρ = −qρ2 +ρL (ρ).
t x α
Let us evaluate at a point where the maximum ρ is reached at x . We obtain
+ +
d
ρ = −q(x ,t)ρ2 +ρ φ(|z|)(ρ(x +z,t)−ρ )dz
dt + + + +Z + +
6 Qρ2 +ρ φ(|z|)(ρ(x +z,t)−ρ )dz
+ + + +
Z
|z|<r
6 Qρ2 +ι (r)ρ (M−rρ ) = Qρ2 +ι (r)Mρ −rι (r)ρ2.
+ φ + + + φ + φ +
In view of assumption (iii) we can pick r large enough to satisfy rι (r) > Q + 1, while
φ
according to (i) ι (r) itself remains finite. We thus achieve inequality
φ
d
ρ 6 −ρ2 +ι (r)Mρ ,
dt + + φ +
which establishes the upper bound by integration.
As to the lower bound we argue similarly. Let ρ and x the minimum value of ρ and a
− −
point where such value is achieved. We have
d
ρ > −Qρ2 +ρ φ(|z|)(ρ(x +z,t)−ρ )dz
dt − − −ZT − −
> −Qρ2 +ι (2π)ρ (M−2πρ ) = −(Q+2πι (2π))ρ2 +ι (2π)Mρ .
− φ − − φ − φ −
In view of (ii) the linear term on the right hand side has a positive coefficient. This readily
implies the uniform lower bound. (cid:3)
Subsequently we focus solely on the critical case α = 1. The case 1 < α < 2 requires the
same prerequisites as established in this section, however, incorporating it into the existing
proof below would actually require fewer steps due to excess of dissipation. We will skip
those for the sake of brevity of what will already be a technical exposition.
3.2. An initial approach towards Theorem 1.6. In this section we specialize on the
case of critical singular kernel, and establish uniform control on the first order quantities
|ρ | , |u | . We assume that we start with initial condition (u ,ρ ) ∈ H3(T). As shown in
x ∞ x ∞ 0 0
[ST2016] there exists a global solution to (1.1) in the same space. So, we can perform all of
the computations below as classical.
First, let us recite one argument from [ST2016]. Recall that the density ρ satisfies the
following parabolic form of the density equation, expressed in terms of L ≡ −Λ,
1
f(x)−f(y)
(3.8) ρ +uρ +eρ = −ρΛ(ρ),, Λ(f) = p.v. dy.
t x ZR |x−y|2
10 ROMANSHVYDKOYAND EITANTADMOR
Similarly, one can write the equation for the momentum m = ρu:
(3.9) m +um +em = −ρΛ(m).
t x
In both cases the drift u and the forcing eρ or em are bounded a priori due to the maximum
principle and Lemma 3.1. Moreover, the diffusion operator has kernel
1
K(x,h,t) = ρ(x) .
|h|2
Using lower bound on the density from Lemma 3.1 we conclude that the kernel falls under
the assumptions of Schwab and Silverstre [SS2016] uniformly on the time line. A direct
application of [SS2016] implies that there exists an γ > 0 such that
|ρ|Cγ(T×[t+1,t+2)) 6 C(|ρ|L∞(t,t+2) +|ρe|L∞(t,t+2))
(3.10) |m|Cγ(T×[t+1,t+2)) 6 C(|m|L∞(t,t+2) +|me|L∞(t,t+2))
|u|Cγ(T×[t+1,t+2)) 6 C(|u|L∞(t,t+2),|ρ|L∞(t,t+2)),
holds for all t > 0. Since the right hand sides are uniformly bounded on the entire line we
have obtained uniform bounds on Cγ-norm starting from time t = 1. Since we are concerned
with long time dynamics let us reset initial time to t = 1, and allow ourselves to assume that
Cγ-norms are bounded from time t = 0.
Lemma 3.3. We make the same assumptions stated in Theorem 1.6. Then the following
uniform bound holds
(3.11) sup|ρ (·,t)| < ∞.
x ∞
t>0
Proof. The argument goes verbatim as presented in [ST2016, Section 6.2] with all the con-
stants involved being uniform in view of the established bounds above. We recall the penul-
timate inequality
d
|ρ′|2 6 c +c |ρ′|2 −c Dρ′(x),
1 2 3
dt
where all the quantities are evaluated at a point of maximum of |ρ′|, and where
|ρ′(x)−ρ′(x+z)|2
Dρ′(x) = dz.
ZR |z|2
Using the nonlinear maximum bound from [CV2012]
|ρ′|3
Dρ′(x) > c ∞ > c |ρ′|3 ,
4 |ρ| 5 ∞
∞
we can further hide the quadratic term into dissipation to obtain
d
(3.12) |ρ′|2 6 c −c Dρ′(x) 6 c −c |ρ′|3.
6 7 6 8
dt
This enables us to conclude the Lemma. (cid:3)
We turn to study the flocking behavior in the singular case. To this end, we first prepare
preliminary estimates on the dissipation terms to be encountered in the sequel. Below is
an improvement on nonlinear maximum principle bound of [CV2012] in the case of small
amplitudes. As a byproduct we obtain a trilinear estimate that will be used in the sequel.
