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) ∂ t = ∂ 2 ρ α  ∂ρ ∂ x 2 + χ ∂ � ρ ∂ � � ∂ x W ∗ ρ ρ ( x ) dx = 1 ,   ∂ x  R W ( x ) = | x | γ  α ≥ 1 , γ ∈ ( − 1 , 1) . ,   γ The free energy writes: 1 � ρ ( x ) α dx + χ �� ρ ( x ) | x − y | γ ρ ( y ) dxdy F [ ρ ] = α − 1 2 γ R R × R

Interacting particles in 1D Geometric numerical analysis Continuation after BU Cumulative distribution function � x X ( m ) = M − 1 ( m ) , M ( x ) = ρ ( y ) dy , X : (0 , 1) → R , X ր −∞ Alternative formulation of the energy F [ ρ ] = G [ X ]: 1 � ( X ′ ( m )) 1 − α dm + χ �� (0 , 1) 2 | X ( m ) − X ( m ′ ) | γ dmdm ′ G [ X ] = α − 1 2 γ (0 , 1)

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 ] ∂ t X = −∇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 [ λ X ] = λ 1 − α G [ 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. � log( X ′ ( m )) dm + χ �� (0 , 1) 2 log | X ( m ) − X ( m ′ ) | dmdm ′ G [ X ] = − 2 (0 , 1) − 1 + χ � � G [ λ X ] = G [ X ] + log λ 2 Consequence: − 1 + χ � � ∇ X · G [ X ] = 2 − 1 + χ � � X · ( − ∂ t X ) = 2 d � 1 � 1 − χ � � 2 | X ( t ) | 2 = dt 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 � 1 � 2 | X ( t ) − X ∗ | 2 ≤ 0 dt If in addition G is uniformly convex: D 2 G ≥ ν Id , then � � 1 � d 2 | X ( t ) − X ∗ | 2 ≤ − ν | X ( t ) − X ∗ | 2 dt Surprisingly, the same holds true here. If X ∗ is a critical point of the energy, then d � 1 � 2 | X ( t ) − X ∗ | 2 ≤ 0 dt 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 ! . . . G rescaled [ Y ] = G [ Y ] + 1 2 | Y | 2 There exists a critical point if χ < 2: − 1 + χ ∇G [ Y ∗ ] + Y ∗ = 0 , � � + | Y ∗ | 2 = 0 2 Theorem (C, Carrillo) In the sub-critical case χ < 2 d dt | Y ( t ) − Y ∗ | 2 ≤ − 2 | Y ( t ) − Y ∗ | 2 d ρ ( t ) , ρ ∗ ) 2 ≤ − 2 W (ˆ ρ ( t ) , ρ ∗ ) 2 dt W (ˆ 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 [ λ X ] = λ 1 − α G [ X ] Consequence: X · ∇G [ X ] = (1 − α ) G [ X ] X · ( − ∂ t X ) = (1 − α ) G [ X ] d � 1 � 2 | X ( t ) | 2 = ( α − 1) G [ X ] ≤ ( α − 1) G [ X 0 ] dt Singularity if G [ X 0 ] < 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 χ c ( α ) > 0 such that: • if χ < χ c the energy F is everywhere positive. • if χ > χ c there exists a cone of negative energy. The density blows-up in finite time if F [ ρ 0 ] < 0 . The case F [ ρ 0 ] ≥ 0 and χ > χ c is open.

Interacting particles in 1D Geometric numerical analysis Continuation after BU Towards long time asymptotics (critical case) Assume χ = χ c . For any stationary state ρ ∗ it holds that 1 d dt W ( ρ ( t ) , ρ ∗ ) 2 ≤ ( α − 1) F [ ρ ( t )] . (1) 2 In particular the second moment σ ( t ) 2 = | x | 2 ρ ( t , x ) dx satisfies � � σ ( t ) 2 d � = ( α − 1) F [ ρ ( t )] ≥ 0 . (2) dt 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 d 2 dt 2 ω ( t ) = ( α + 1)( α − 1) H [ˆ ρ ( t )] ≤ 0 . ω ( t )

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 N − 1 + χ X i +1 − X i � log | X j − X i | � 2 (∆ m ) 2 � � G [ X ] = − ∆ m log i =1 i � = j N +∆ m | X i | 2 . � 2 i =1 − → preserves the homogeneity of the problem. Theorem (Blanchet-C-Carrillo) χ c = 2(1 − ∆ m ) − 1 . The critical parameter is ˜ • If χ > ˜ χ c the solution of the discrete gradient flow blows-up in finite time (meaning that ∃ i 0 : X i 0 +1 − X i 0 = 0 after finite time or a finite number of steps). • If χ < ˜ χ c 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 2 4 1.8 3 1.6 2 1.4 1 1.2 1 0 0.8 −1 0.6 −2 0.4 −3 0.2 0 −4 −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Convergence of the solution towards the unique stationary state in self-similar variables when χ < 2. 2 4 1.8 3 1.6 2 1.4 1 1.2 1 0 0.8 −1 0.6 −2 0.4 −3 0.2 0 −4 −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 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. 4 4.5 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 subcritical case intermediate case 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 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 X 0 ∈ 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 A ≤ | X i ( t ) − X j ( t ) | 1 ( ∀ t > 0) ( ∀ ( i , j ) ∈ I ) ≤ A � 2 β ( T − t ) Idea of the proof: decompose the particles into inner particles and outer 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 p = 2 − α � ρ 0 � L p < C , 1 + γ > 1 • Finite-time blow-up if � | x | 2 ρ 0 ( x ) dx < C , R OR � (1 − α ) / 2 � 1 − α − 1 1 � �� | x | 2 ρ 0 ( x ) dx C + F [ ρ 0 ] < 0 . γ R