Particle scheme for the 1D Keller-Segel equation Vincent Calvez - - PowerPoint PPT Presentation

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Particle scheme for the 1D Keller-Segel equation

Vincent Calvez

CNRS, ENS de Lyon, France

Journ´ ees ANR TOMMI, Grenoble, Octobre 2013

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Contents

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Contents

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Diffusive-Interacting particles in 1D1D

We consider a continuum of particles that diffuse and interact pairwise with an attracting potential W .

• nonlinear diffusion (porous-medium type)
• nonlocal interaction (power-law)

       ∂ρ ∂t = ∂2ρα ∂x2 + χ ∂ ∂x

• ρ ∂

∂x W ∗ ρ

• ,
• R

ρ(x) dx = 1 W (x) = |x|γ γ , α ≥ 1 , γ ∈ (−1, 1) . The free energy writes: F[ρ] = 1 α − 1

• R

ρ(x)α dx + χ 2γ

• R×R

ρ(x)|x − y|γρ(y) dxdy

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Cumulative distribution function

M(x) = x

−∞

ρ(y) dy , X(m) = M−1(m) , X : (0, 1) → R , X ր Alternative formulation of the energy F[ρ] = G[X]: G[X] = 1 α − 1

• (0,1)

(X ′(m))1−α dm+ χ 2γ

• (0,1)2 |X(m)−X(m′)|γ dmdm′

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Gradient flow interpretation

Claim [Jordan, Kinderlehrer & Otto]: the Keller-Segel system is the gradient flow of the energy G[X] ∂tX = −∇G[X] Key observations:

• The functional G[X] is not convex
• Each contribution is homogeneous (resp. 1 − α and γ).

If α − 1 + γ = 0 the two contributions have the same homogeneity: the competition is fair. G[λX] = λ1−αG[X] ∀λ > 0

Interacting particles in 1D Geometric numerical analysis Continuation after BU

The logarithmic case α = 1, γ = 0

The functional is almost zero-homogeneous. G[X] = −

• (0,1)

log(X ′(m)) dm + χ 2

• (0,1)2 log |X(m) − X(m′)| dmdm′

G[λX] = G[X] +

• −1 + χ

2

• log λ

Consequence: ∇X · G[X] =

• −1 + χ

2

• X · (−∂tX) =
• −1 + χ

2

• d

dt 1 2|X(t)|2

• =
• 1 − χ

2

• Singularity if χ > 2: blow-up!

Interacting particles in 1D Geometric numerical analysis Continuation after BU

The logarithmic case, ctd.

• Given the gradient flow of a convex energy G, and a critical

point ∇G[X ∗] = 0, then d dt 1 2|X(t) − X ∗|2

• ≤ 0

If in addition G is uniformly convex: D2G ≥ νId, then d dt 1 2|X(t) − X ∗|2

• ≤ −ν|X(t) − X ∗|2
• Surprisingly, the same holds true here. If X ∗ is a critical point of

the energy, then d dt 1 2|X(t) − X ∗|2

• ≤ 0

Problem: there exists a critical point only when χ = 2 . . .

Interacting particles in 1D Geometric numerical analysis Continuation after BU

The logarithmic case, ctd.

. . . Rescale space/time x = (√1 + t)y! Grescaled[Y ] = G[Y ] + 1 2|Y |2 There exists a critical point if χ < 2: ∇G[Y ∗] + Y ∗ = 0 ,

• −1 + χ

2

• + |Y ∗|2 = 0

Theorem (C, Carrillo)

In the sub-critical case χ < 2 d dt |Y (t) − Y ∗|2 ≤ −2|Y (t) − Y ∗|2 d dt W (ˆ ρ(t), ρ∗)2 ≤ −2W (ˆ ρ(t), ρ∗)2 Explanation: the interaction part (concave) is ”digested” by the diffusion contribution (Jensen’s inequality).

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Fair competition – blow-up

Assume α − 1 + γ = 0, α > 1. The functional is (1 − α)-homogeneous. G[λX] = λ1−αG[X] Consequence: X · ∇G[X] = (1 − α)G[X] X · (−∂tX) = (1 − α)G[X] d dt 1 2|X(t)|2

• = (α − 1)G[X] ≤ (α − 1)G[X0]

Singularity if G[X0] < 0: blow-up!

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Fair competition – critical parameter

In the case of fair competition the dichotomy is the following:

Theorem (adapted from Blanchet-Carrillo-Lauren¸ cot)

Assume 1 < α < 2, γ = 1 − α. There exists χc(α) > 0 such that:

• if χ < χc the energy F is everywhere positive.
• if χ > χc there exists a cone of negative energy. The density

blows-up in finite time if F[ρ0] < 0. The case F[ρ0] ≥ 0 and χ > χc is open.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Towards long time asymptotics (critical case)

Assume χ = χc. For any stationary state ρ∗ it holds that 1 2 d dt W (ρ(t), ρ∗)2 ≤ (α − 1)F[ρ(t)] . (1) In particular the second moment σ(t)2 =

• |x|2ρ(t, x) dx satisfies

d dt σ(t)2 2

• = (α − 1)F[ρ(t)] ≥ 0 .

(2) We face the following alternative:

• If the second moment converges then we have lim F[ρ(t)] = 0

(ground state).

• If the second moment diverges then some power of the

standard deviation ω(t) = σ(t)α+1 satisfies d2 dt2 ω(t) = (α + 1)(α − 1) ω(t) H[ˆ ρ(t)] ≤ 0 .

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Contents

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Geometric numerical analysis

Main ideas:

• discretize the positions of the particles with respect to the

partial mass {X(i∆m), i = 1 . . . N}

• discretize the functional G while preserving its homogeneity
• perform a finite dimensional euclidean gradient flow of G
• prove similar results (global existence, long time asymptotics,

blow-up) using gradient flow + homogeneity arguments. Benefits:

• many proofs are easily transported at the numerical level
• boundary conditions are simply X 0 = X N+1 = ±∞.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Space discretization

The functional G is discretized using finite differences G[X] = −∆m

N−1

• i=1

log

• X i+1 − X i

+ χ 2 (∆m)2

i=j

log |X j − X i| +∆m 2

N

• i=1

|X i|2 . − → preserves the homogeneity of the problem.

Theorem (Blanchet-C-Carrillo)

The critical parameter is ˜ χc = 2(1 − ∆m)−1.

• If χ > ˜

χc the solution of the discrete gradient flow blows-up in finite time (meaning that ∃i0 : X i0+1 − X i0 = 0 after finite time or a finite number of steps).

• If χ < ˜

χc the solution of the rescaled gradient flow converges towards a unique stationary state at exponential rate.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Numerical illustrations

−3 −2 −1 1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −4 −3 −2 −1 1 2 3 4

Convergence of the solution towards the unique stationary state in self-similar variables when χ < 2.

−3 −2 −1 1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −4 −3 −2 −1 1 2 3 4

Blow-up of the discrete gradient flow when χ > 2.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Some insights on the dynamics in small dimension

N = 3: selection of the critical number of particles.

subcritical case intermediate case

supercritical case

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Dynamics in the case of N particles

work in collaboration with Thomas Gallou¨ et.

Theorem (C-Gallou¨ et)

Stability There exists a ”basin of attraction” Ω(N) such that if X0 ∈ Ω(N) then X(t) blows-up in finite time with the minimal number of particles. Rigidity Denote by I the set of blowing-up particles. There exists a constant A such that (∀t > 0) (∀(i, j) ∈ I) 1 A ≤ |Xi(t) − Xj(t)|

• 2β(T − t)

≤ A Idea of the proof: decompose the particles into inner particles and

• uter particles, and derive conditions to isolate the inner set.

Rigidity is proved by induction.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Investigation of the attraction-dominating case

Assume γ < 1 − α. It is a 1D model for the standard Keller-Segel equation in 3D.

• Global existence if

ρ0Lp < C , p = 2 − α 1 + γ > 1

• Finite-time blow-up if
• R

|x|2ρ0(x) dx < C , OR C

• 1

1 − α − 1 γ

R

|x|2ρ0(x) dx (1−α)/2 + F [ρ0] < 0 .

Interacting particles in 1D Geometric numerical analysis Continuation after BU

The N-particle scheme

Work in collaboration with Lucilla Corrias (α = 1, γ < 0).

Theorem (C-Corrias)

• Global existence if
• h

|γ|

N−1

• i=1

(Xi+1 − Xi)γ

• ≤ C(N)
• Finite time blow-up if

|X0|2 < 1 ∆m 1 2 − 2

γ +1 N−2/γ(N − 1)

(N + 1)− 2

γ +1 .

OR |X0|2 < (∆m)2(N − 1) exp

• −

2 ∆m(N − 1)G[X0] + 2 γ

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Sharpness of BU criteria

0.5 1 0.2 0.4 0.6 0.8

u v

0.5 1 0.2 0.4 0.6 0.8

u v

0.5 1 0.2 0.4 0.6 0.8

u v

0.5 1 0.5 1

u v

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Contents

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Aim and principle

Several attempts to define reasonable solutions after blow-up time. [Vel´ azquez (2004)], [Dolbeault-Schmeiser (2009)] Common strategy:

• introduce a regularized problem (yieldings global existence),

for instance W (z) = log |z| ← − Wε(z) = log(|z| + ε) .

• prove tight convergence of the solution {ρε(t)} and the

interaction (Kε ∗ ρε(t))ρε(t) towards a pair of measures ρ(t) and m(t). The defect measure ν(t) = m(t) − (K0 ∗ ρ(t))ρ(t) is diagonal.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Theorem (Devys, Dolbeault-Schmeiser, after Poupaud)

The family {ρε(t)} converges tightly (up to extraction) to a measure valued solution ρ(t). There exists a defect measure ν(t) such that (ρ, ν) is a distribution solution of ∂ρ ∂t + χ ∂ ∂x j[ρ, ν] − ∂2ρ ∂x2 = 0 The convective flux is given by

• φ(x)j[ρ, ν] dx = − 1

2π

• x=y

φ(x) − φ(y) x − y ρ(t, x)ρ(t, y) dxdy − 1 2π

• ∂xφ(x)ν(t, x) dx .

The defect measure ν is nonnegative and diagonal, ν(t, x) ≤

• a∈Sat(ρ(t)

(ρ(t)(a))2δ(x − a) .

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Particle scheme in 2D [Haskovec-Schmeiser (2009)]

Discrete model: interacting stochastic particles. Distinction between two kinds of particles:

• light particles (infinitesimal fraction of the mass): stochastic
• heavy particles (aggregate, mass > 8π): deterministic

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Transient strong formulation in 1D

Assume the following decomposition (basically between two ”big” collision events) ρ = ρ +

• Mk(t) δ(x − xk(t)) ,

where f is regular enough. The defect measure has the following form ν(t, x) =

• νk(t)δ(x − xk(t)) ,

0 ≤ νk ≤ M2

k

Plugging this ansatz into the weak formulation yields                    ∂ρ ∂t − χ ∂ ∂x

• ρ ∂

∂x W ∗ ρ

• −

χ π ∂ ∂x

• ρ Mk

x − xk

• − ∂2

∂x2 ρ = 0 dMk dt = χ π Mk + 1 ∂ ∂x ρ(t, x+

k ) − ∂

∂x ρ(t, x−

k )

• dxk

dt = −χ ∂ ∂x (W ∗ ρ) (xk) − χ π

• l=k

Ml xk − xl

Interacting particles in 1D Geometric numerical analysis Continuation after BU

dMk dt = χ π Mk + 1 ∂ ∂x ρ(t, x+

k ) − ∂

∂x ρ(t, x−

k )

• Claim

The regular part ρ vanishes at (xk).

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Cut & Paste numerical scheme

[Anne Devys’ PhD thesis] Deterministic numerical scheme: time-implicit numerical scheme. Procedure: Accurate condition to detect plateaus and collision between Dirac masses.

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Numerical simulations

Aggregation into a single plateau

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Numerical simulations, ctd.

Complicated transient before final aggregation

Interacting particles in 1D Geometric numerical analysis Continuation after BU

Conclusion

One dimensional (deterministic) particle schemes have many benefits:

• They capture the dynamics of the PDE
• Boundary conditions are transparent
• Accuracy of the scheme in the concentrated zone
• After BU, the regular part is clearly separated from the

aggregated part Drawbacks: Heavy computational cost (see Carrillo and Moll 2010) THANK YOU FOR YOUR ATTENTION!1

1Thanks to Anne Devys for drawing the nicest figures of this talk