stability and pattern formation in nonlocal interaction models · motivation macroscopic models:...

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Motivation Macroscopic Models: Repulsive-Attractive Potentials 2nd Order models: Stability of Patterns Conclusions

Stability and Pattern formation inNonlocal Interaction Models

J. A. Carrillo

Imperial College London

Kinetic Description of Multiscale Phenomena

ACMAC, Heraklion, 2013

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Outline

1 MotivationCollective Behavior Models

2 Macroscopic Models: Repulsive-Attractive PotentialsA quick reviewRepulsive-Attractive PotentialsStability/Instability of Delta RingsDimensionality of the support for Minimizers

3 2nd Order models: Stability of PatternsStability of flock rings for second order models

4 Conclusions

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Collective Behavior Models

Outline

4 Conclusions

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Individual Based Models (Particle models)

Swarming = Aggregation of agents of similar size and body type generally moving ina coordinated way.

Highly developed s ocial organization: insects (locusts, ants, bees ...), fish, birds,micro-organisms (myxo-bacteria, ...) and artificial robots for unmanned vehicleoperation.

Interaction regions between individualsa

aAoki, Helmerijk et al., Barbaro, Birnir et al.

Repulsion Region: Rk.

Attraction Region: Ak.

Orientation Region: Ok.

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2nd Order Model: Newton’s like equations

D’Orsogna, Bertozzi et al. model (PRL 2006):dxi

dt= vi,

dt= (α− β |vi|2)vi −

∑j 6=i

∇U(|xi − xj|).

Model assumptions:

Self-propulsion and friction termsdetermines an asymptotic speed of√α/β.

Attraction/Repulsion modeled by aneffective pairwise potential U(x).

U(r) = −CAe−r/`A + CRe−r/`R .

One can also use Bessel functions in 2Dand 3D to produce such a potential.

C = CR/CA > 1, ` = `R/`A < 1 andC`2 < 1:

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dt= vi,

∑j 6=i

∇U(|xi − xj|).

Model assumptions:

C = CR/CA > 1, ` = `R/`A < 1 andC`2 < 1:

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dt= vi,

∑j 6=i

∇U(|xi − xj|).

Model assumptions:

C = CR/CA > 1, ` = `R/`A < 1 andC`2 < 1:

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dt= vi,

∑j 6=i

∇U(|xi − xj|).

Model assumptions:

C = CR/CA > 1, ` = `R/`A < 1 andC`2 < 1:

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Model with an asymptotic speed

Typical patterns: milling, double milling or flocking:

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A quick review

Outline

4 Conclusions

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A quick review

Reduction - 1st Order Friction Model:

Edelshtein-Keshet, Mogilner (JMB 2000): Assume the variations of the velocity andspeed are much smaller than spatial variations, then from Newton’s equation:

d2t+ α

dt+∑j 6=i

∇U(|xi − xj|) = 0

so finally, we obtain

dt= −

∑j 6=i

∇U(xi − xj) in the continuum setting V

{∂ρ∂t + div (ρv) = 0v = −∇U ∗ ρ

“repelling/attracting field”: −∇U : Rd → Rd

For which interaction repulsive/attractive potentials do we get convergence towardssome nontrivial steady states/patterns?

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A quick review

d2t+ α

dt+∑j 6=i

∇U(|xi − xj|) = 0

dt= −

∑j 6=i

{∂ρ∂t + div (ρv) = 0v = −∇U ∗ ρ

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A quick review

d2t+ α

dt+∑j 6=i

∇U(|xi − xj|) = 0

dt= −

∑j 6=i

{∂ρ∂t + div (ρv) = 0v = −∇U ∗ ρ

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A quick review

d2t+ α

dt+∑j 6=i

∇U(|xi − xj|) = 0

dt= −

∑j 6=i

{∂ρ∂t + div (ρv) = 0v = −∇U ∗ ρ

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A quick review

d2t+ α

dt+∑j 6=i

∇U(|xi − xj|) = 0

dt= −

∑j 6=i

{∂ρ∂t + div (ρv) = 0v = −∇U ∗ ρ

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A quick review

Formal Gradient FlowBasic Properties

1 Conservation of the center of mass.2 Liapunov Functional: Gradient flow of

F [ρ] = 12

∫∫U(x− y) ρ(x) ρ(y) dxdy

with respect to the Wasserstein distance W2.(C., McCann, Villani; RMI 2003, ARMA 2006).

The macroscopic equation can be rewritten as

∂t(t, x) = div

(ρ(t, x)∇

[δFδρ

(t, x)])

with entropy dissipation:

ddtF [ρ(t)] = −

∫R2ρ(t, x)

∣∣∣∣∇δFδρ (t, x)∣∣∣∣2 dx .

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A quick review

F [ρ] = 12

∂t(t, x) = div

(ρ(t, x)∇

[δFδρ

(t, x)])

ddtF [ρ(t)] = −

∫R2ρ(t, x)

∣∣∣∣∇δFδρ (t, x)∣∣∣∣2 dx .

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A quick review

F [ρ] = 12

∂t(t, x) = div

(ρ(t, x)∇

[δFδρ

(t, x)])

ddtF [ρ(t)] = −

∫R2ρ(t, x)

∣∣∣∣∇δFδρ (t, x)∣∣∣∣2 dx .

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Repulsive-Attractive Potentials

Outline

4 Conclusions

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Repulsive-Attractive Potentials

Summary: Particle Simulations d = 2Potential a = 4,

b = 2.1

Potential a = 4,b = 0.85

Xi = −∑j 6=i

mj ∇U(Xi−Xj)

U(x) =|x|a

a− |x|

b2− d ≤ b < a

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Stability/Instability of Delta Rings

Outline

4 Conclusions

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Radial Setting

The velocity field generated by a spherical shell of radius η is given by:

ω(r, η) = − 1σN

∫∂B(0,1)

∇U(re1 − ηy) · e1 dσ(y),

Under some conditions on the potential U, the function ω ∈ C1(R2+).

The equation ∂ρ∂t (t, x) = div (ρ(t, x) [∇U ∗ ρ] (t, x)) written in radial coordinates is

∂tµ+ ∂r(µv) = 0

v(t, r) =∫ +∞

0ω(r, η)dµt(η) .

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Radial Setting

ω(r, η) = − 1σN

∫∂B(0,1)

∇U(re1 − ηy) · e1 dσ(y),

∂tµ+ ∂r(µv) = 0

v(t, r) =∫ +∞

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Radial Setting

ω(r, η) = − 1σN

∫∂B(0,1)

∇U(re1 − ηy) · e1 dσ(y),

∂tµ+ ∂r(µv) = 0

v(t, r) =∫ +∞

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Spherical shell

A spherical shell (a uniform distribution on the sphere) of radius R (δR from now on)is a stationary state for the aggregation equation if

ω(R,R) = 0.

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Conditions for radial ins/stability

Suppose that δR is a stationary state, ω(R,R) = 0.

Instability: Suppose that one of the following cases is satisfied

ω ∈ C1(R2+) and ∂1ω(R,R) > 0.

ω ∈ C(R2) ∩ C1(R2 \ D) and

lim(r1,r2)6∈D

(r1,r2)→(R,R)

∂1ω(r1, r2) = +∞.

Conclusion: it is not possible for an Lp radially symmetric solution to convergetoward a δR when t→∞.

Stability: If the following conditions hold

∂1ω(R,R) < 0, (fattening instability)∂1ω(R,R) + ∂2ω(R,R) < 0. (shifting instability)

Then if the radial initial data is close enough to δR there is convergence. This isa local result of stability.

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Conditions for radial ins/stability

Suppose that δR is a stationary state, ω(R,R) = 0.

Instability: Suppose that one of the following cases is satisfied

ω ∈ C1(R2+) and ∂1ω(R,R) > 0.

ω ∈ C(R2) ∩ C1(R2 \ D) and

lim(r1,r2)6∈D

(r1,r2)→(R,R)

∂1ω(r1, r2) = +∞.

Conclusion: it is not possible for an Lp radially symmetric solution to convergetoward a δR when t→∞.

Stability: If the following conditions hold

∂1ω(R,R) < 0, (fattening instability)∂1ω(R,R) + ∂2ω(R,R) < 0. (shifting instability)

Then if the radial initial data is close enough to δR there is convergence. This isa local result of stability.

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Power-Law Case

U(x) =|x|a

a− |x|

b2− d < b < a

Theorem: Ins/Stability of Delta Rings withrespect to radial perturbations.

There is a computable value of R suchthat the uniform distribution on thesphere of radius R, δR is an steady state.

If the velocity field generated by δR isstrictly increasing at R then it is unstable.

If the velocity field generated by δR isstrictly decreasing at R then it is locallyasymptotically stable.

��

3− d

2− d

b =(3−d)a−10+7d−d2

a+d−3

(Balagué, C., Laurent, Raoul; Physica D 2013)

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Power-Law Case

U(x) =|x|a

a− |x|

b2− d < b < a

��

3− d

2− d

b =(3−d)a−10+7d−d2

a+d−3

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Power-Law Case

U(x) =|x|a

a− |x|

b2− d < b < a

��

3− d

2− d

b =(3−d)a−10+7d−d2

a+d−3

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Power-Law Case

U(x) =|x|a

a− |x|

b2− d < b < a

��

3− d

2− d

b =(3−d)a−10+7d−d2

a+d−3

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Dimensionality of the support for Minimizers

Outline

4 Conclusions

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Dimensionality of the support

Some simulations with power law potentials of the form

W(x) =|x|a

a− |x|

b, 2− d < b < a

dim=3 dim=2 dim=2 dim=0

b = −0.5 b = 0.5 b = 1.25 b = 2.5a = 5 a = 23 a = 15 a = 5

Local minimizers in 3D for different parameters when b > −1 increases. Thecomputations were done with n = 2, 500 particles.

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Mild versus Strong Repulsive potentialsSupport is essentially 0-dimensional for b > 2.

Let U ∈ C2(RN) be a radially symmetric potential which behaves like −|x|b/b in aneighborhood of the origin with b > 2.

Then a local minimizer of the interaction energy F with respect to W∞ cannot have as-dimensional component for any 1 ≤ s ≤ d.

Dimension of the Support depends on 2− d < b < 2.

Assume that µ is a local minimizer of the interaction energy F with respect to W∞such that U is radial with U(x) ∼ −|x|b near zero and 2− d < b < 2. If µ containss-Haussdorff dimensional connected components in its support, then s > 2− b.

(Balagué, C., Laurent, Raoul; ARMA 2013)

Strategy: Pure variational approach: by contradiction we build better competitors.

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Relevant example: Power-law Potential in 2d

(2, 1.5)

(7, 1.5)

(15, 2.5)

(5, 0.5)

(5, 0.01)

(5, 1.1)

(5, 2.2)

(5, 1.5)

b = aa−1

dt= −

N∑j 6=i

∇U(xi − xj)

The spectral gap of thelinearized stability ofthe Delta ringdepending on Ndisappears as N →∞.

(Kolokonikov, Sun, Uminsky, Bertozzi; Physical Review E 2011)(Bertozzi, von Brecht, Sun, Kolokolnikov, Uminsky; Comm. Math. Sci. 2012)

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(2, 1.5)

(7, 1.5)

(15, 2.5)

(5, 0.5)

(5, 0.01)

(5, 1.1)

(5, 2.2)

(5, 1.5)

b = aa−1

dt= −

N∑j 6=i

∇U(xi − xj)

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(2, 1.5)

(7, 1.5)

(15, 2.5)

(5, 0.5)

(5, 0.01)

(5, 1.1)

(5, 2.2)

(5, 1.5)

b = aa−1

dt= −

N∑j 6=i

∇U(xi − xj)

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Stability of flock rings for second order models

Outline

4 Conclusions

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2nd order models

The Bertozzi-D’Orsogna model:xj = vj

vj = (α− β|vj|2)vj +1N

N∑l=1l 6=j

∇U(xl − xj) , j = 1, . . . ,N,

with α, β > 0.The continuum version of this particle model (C., D’Orsogna, Panferov; KRM 2008)is the kinetic equation

∂f∂t

+ v · ∇xf + divv[(α− β|v|2)vf )]− divv[(∇xU ∗ ρ)f ] = 0,

whereρ(t, x) =

f (t, x, v) dv.

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2nd order models

The Bertozzi-D’Orsogna model:xj = vj

vj = (α− β|vj|2)vj +1N

N∑l=1l 6=j

∇U(xl − xj) , j = 1, . . . ,N,

with α, β > 0.The continuum version of this particle model (C., D’Orsogna, Panferov; KRM 2008)is the kinetic equation

∂f∂t

+ v · ∇xf + divv[(α− β|v|2)vf )]− divv[(∇xU ∗ ρ)f ] = 0,

whereρ(t, x) =

f (t, x, v) dv.

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Asymptotic solutions

Definition

•We call a flock ring, the solution such that {xj}Nj=1 are equally distributed on a circle

with a certain radius, R and {vj}Nj=1 = u0, with |u0| =

√α/β.

•We call a mill ring, the solution such that {xj}Nj=1 are equally distributed on a circle

with a certain radius, R and {vj}Nj=1 =

√α/β x⊥j /|xj| with x⊥j the orthogonal vector.

Figure : Flock and mill ring solutions.

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Analysis of the stability of flock rings (I)

Change of variables to the comoving frame:{yj = xj(t)− u0tzj = vj(t)− u0

, j = 1, . . . ,N,

Then the system readsyj = zj

zj = (α− β|zj|2)(zj + u0) +1N

N∑l=1l 6=j

∇U(yl − yj) , j = 1, . . . ,N.

Write the stationary ring (y0j , z

0j ) = (Reiθj , 0) where θj =

2πjN , for j = 1, . . . ,N.

Consider the following type of perturbations:

yj(t) = Reiθj(1 + hj(t)),N∑

hj(t) =N∑

h′j(t) = 0, with |hj| � 1.

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, j = 1, . . . ,N,

zj = (α− β|zj|2)(zj + u0) +1N

N∑l=1l 6=j

∇U(yl − yj) , j = 1, . . . ,N.

2πjN , for j = 1, . . . ,N.

hj(t) =N∑

h′j(t) = 0, with |hj| � 1.

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, j = 1, . . . ,N,

zj = (α− β|zj|2)(zj + u0) +1N

N∑l=1l 6=j

∇U(yl − yj) , j = 1, . . . ,N.

2πjN , for j = 1, . . . ,N.

hj(t) =N∑

h′j(t) = 0, with |hj| � 1.

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Analysis of the stability of flock rings (II)

Write the matrix of the linearized system for these perturbations

02N Id2N

M −2βU0

where M is symmetric and represents the 2N × 2N Jacobian that results fromlinearizing the first order model, and U0 is the diagonal matrix with N blocks of the2× 2 matrix u0uT

0 along the diagonal.

Instability Result

The linearized second order system around the flock ring solution has an eigenvaluewith positive real part if and only if the linearized first order system around the ringsolution has a positive eigenvalue for all number of particles N.

(Albi, Balagué, C., von Brecht; submitted)Stability Result V (C., Huang, Martin; in preparation) and Stephan Martin’s talk.

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02N Id2N

M −2βU0

Instability Result

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02N Id2N

M −2βU0

Instability Result

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Conclusions

Optimal Transportation Tools can deal with evolutions by PDEs leading toconcentration happening at finite or infinite time.

The dimensionality of the support of local minimizers of the interaction energycan be classified in terms of the repulsion strength of the potential near zero.

The stability and instability of flocks for the second order model is impliedfrom the analysis of the first order model at the particle level. What about thekinetic equation?

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Conclusions

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Conclusions

stability and pattern formation in nonlocal interaction models · motivation macroscopic models:...

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