Mathematical modelling of collective behavior

Mathematical modelling of collective behavior Young-Pil Choi Fakultät für Mathematik Technische Universität München This talk is based on joint works with José A. Carrillo, Maxime Hauray, and Samir Salem Mathematical Topics in Kinetic Theory University of Cambridge May 13, 2016 1 / 29

Introduction Singular communication weights Normalized communication weights Sharp sensitivity regions Swarming in Nature 2 / 29

Individual Based Models Figure: 3 Zones 3 / 29

Cucker-Smale model Let (x i, v i ) be the position and velocity of i-th particle. Then Cucker-Smale model reads as dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N 1 ψ(x j x i )(v j v i ), ψ(x) := N (1 + x 2, β 0. ) β/2 j=1 Definition: Let P := {(x i, v i )} N i=1 be an N-body interacting system. Then P exhibits a flocking if and only if the following two relations hold: lim v i (t) v j (t) = 0 and sup x i (t) x j (t) <, 1 i, j N. t 0 t< Property: Conservation of momentum. Flocking estimate: If ψ(r) dr = ( β 1), then we have sup max x i (t) x j (t) < and max v i (t) v j (t) Ce Ct for t 0. 0 t< 1 i,j N 1 i,j N Ref.- Cucker-Smale(2007), Ha-Tadmor(2008), Ha-Liu(2009), Carrillo-Fornasier-Rosado-Toscani(2010),... 4 / 29

Numerical Simulation by Sergio Pérez 5 / 29

Kinetic Cucker-Smale equation When N, the macroscopic observables for Cucker-Smale model can be calculated from the velocity moments of the density function f = (x, v, t) which is a solution to the following Vlasov-type equation: Flocking: tf + v x f + v [F (f )f ] = 0, (x, v) R d R d, F (f )(x, v, t) := ψ(x y)(w v)f (y, w, t) dydw. R d R d v v c 2 f dxdv 0 as t where v c = v f dxdv f dxdv. Several drawbacks: - Collision between individuals - Dynamics away from equilibrium - Vision geometrical constraints -... Ref.- Cucker-Dong(2010), Park-Kim-Ha(2010), Motsch-Tadmor(2011), Ahn-Choi-Ha-Lee(2012), Agueh-Illner-Richardson(2011),... 6 / 29

Collision avoidance between individuals 7 / 29

Previous work Cucker-Smale model with collision avoiding forces: dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N ψ(x i x j )(v j v i ) 1 N 1 K(x i x j ), ψ(x) = N N (1 + x 2 ) β/2. j i j i }{{}}{{} Alignment Attraction/Repulsion Ref.- Cucker-Dong(2010), Ha-Park-Kim(2010), Yang-Chen(2014),... 8 / 29

Numerical simulation K(x) = kφ(x) α x 2 2 where φ = δ 0. 9 / 29

Numerical simulation K(x) = kφ(x) α x 2 2 where φ = δ 0. by Sergio Pérez 9 / 29

Singular communication weights Cucker-Smale model with singular communication weights: dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N ψ(x j x i )(v j v i ), ψ(x) = ψ s (x) := 1 N x α, α > 0. j=1 (1) 10 / 29

Singular communication weights Cucker-Smale model with singular communication weights: dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N ψ(x j x i )(v j v i ), ψ(x) = ψ s (x) := 1 N x α, α > 0. j=1 (1) Flocking estimate: Let α 1. Suppose that the initial data satisfies v 0 v c(0) l < 1 ψ(s) ds. 2 2 x 0 x c (0) l Then there exist positive constants c 1 > c 0 > 0 such that x(t) x c(0) l [c 0, c 1 ] and v(t) v c(0) l v 0 v c(0) l e ψ(2c1)t. Ref.- Ahn-Choi-Ha-Lee(2012) 10 / 29

Sharp condition to avoid collision Finite-time collision estimate(α < 1): Let d = 1. If we consider the two-particle system, then the following statements are equivalent: 1. There exists a time t 0 < such that φ(t 0 ) = φ(t 0 ) = 0. 2. Initial data satisfy φ(0) = 2Ψ(φ(0)), where Ψ(s) := 1 1 α s1 α is a primitive of φ and φ(t) = x 1 (t) x 2 (t). Non collision estimate(α 1): Suppose that the initial data (x 0, v 0 ) are non-collisional, i.e., they satisfy x i0 x j0 for 1 i j N. Then the system (1) admits a unique smooth solution. Moreover, the trajectories of this solution are also non-collisional, i.e., x i (t) x j (t) for 1 i j N, t 0. Ref.- Peszek(2014), Carrillo-C.-Mucha-Peszek(work in progress) 11 / 29

Cucker-Smale model with singular communication weights Main equation: tf + v x f + v (F (f )f ) = 0, (x, v) R d R d, t > 0, (2) where F denotes the alignment force between particles: F (f )(x, v, t) = ψ(x y)(w v)f (y, w) dydw, ψ(x) = 1 with α (0, d 1). x α Definition: For a given T (0, ), f is a weak solution of (3) on the time-interval [0, T ) if and only if the following condition are satisfied: (i) f L (0, T ; (L 1 + Lp )(R d R d )) C([0, T ]; P 1 (R d R d )), (ii) f satisfies the equation (3) in the weak sense. 12 / 29

Local existence and uniqueness of weak solutions Theorem(Carrillo-C.-Hauray, 2014): Suppose that f 0 is compactly supported in velocity and (α + 1)p < d, and the initial data f 0 satisfies f 0 (L 1 + L p )(R d R d ) P 1 (R d R d ). Then there exist T > 0 and unique weak solution f to the system (3) in the sense of Definition 1 on the time interval [0, T ]. Furthermore, if f i, i = 1, 2 are two such solutions to (3), then we have the following d 1 -stability estimate. d dt d 1(f 1 (t), f 2 (t)) Cd 1 (f 1 (t), f 2 (t)), for t [0, T ], where C is a positive constant. 13 / 29

Idea of Proof (Regularization) The approximate solutions are obtained by solving: tf ε + v x f ε + v [F ε ] (f ε)f ε = 0, (x, v) R d R d, t > 0, F ε (f ε)(x, v, t) := ψ ε(x y)(w v)f ε(y, w, t)dydw, R d R d f ε(x, v, 0) =: f 0 (x, v). where ψ ε := ψ θ ε. (Support estimate of f ε in velocity) Consider the forward bi-characteristics (X ε(s; 0, x, v), V ε(s; 0, x, v)) satisfying the following ODE system: dx ε(s) = V ε(s), ds dv ε(s) = ψ ε (X ε(s) y) (w V ε(s))f ε(y, w, s)dydw. ds R d R d Set Ω ε(t) and R v ε (t) the v-projection of compact suppf ε(, t) and maximum value of v in Ω ε(t), respectively: Ω ε(t) := {v R d : (x, v) R d R d such that f ε(x, v, t) 0}, Rε v (t) := max v. v Ω ε(t) Then we have R v ε (t) R v ε (0) = R v 0 := max v Ω(0) v. 14 / 29

Idea of Proof(Conti.) (Uniform bound estimate): Let f ε be the solution to regularized equation. Then there exists a T > 0 such that the uniform L 1 L p -estimate of f ε sup f ε L 1 L p C, t [0,T ] holds, where C is a positive constant independent of ε. (f ε is a Cauchy sequence): Let f ε and f ε be two solutions of the regularized equation. Then there exists C independent of ε and ε such that holds for all t 0. d dt d 1(f ε(t), f ε (t)) C(d 1 (f ε(t), f ε (t)) + ε + ε ), (Passing to the limit ε 0): The limit curve of measure f obtained from the above is a solution of the equation (3). Open question: Rigorous derivation of the mean-field limit 15 / 29

Further extensions: Flocking models with singular kernels Main equation: tf + v x f + v (F (f )f ) = 0, (x, v) R d R d, t > 0, (3) where F denotes the alignment force between particles: F (f )(x, v, t) = ψ(x y) v φ(v w)f (y, w, t)dydw. R d R d Here, the potential function φ(v) for the velocity coupling is given by Two different cases: φ(v) = 1 β v β, where β > 0. Singular communication weight and super-linear velocity coupling: ψ(x) = ψ 1 (x) := 1 x α with α (0, d 1) and β 2. Regular communication weight and sub-linear velocity coupling: ( ) 3 d ψ(x) = ψ 2 (x) 0 symmetric, with ψ 2 (L loc Lip loc ) (Rd ) and β, 2. 2 16 / 29

Normalized communication weights Cucker-Smale model: dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N 1 ψ(x j x i )(v j v i ), ψ(x) = N (1 + x 2 ) β/2. j=1 Cucker-Smale model with normalized communication weights: dx i dt = v i, t > 0, i = 1,, N, dv i dt = 1 N k=1 ψ(x i x k ) N ψ(x j x i )(v j v i ). j=1 Motsch-Tadmor(2011) ψ 2 (r) dr = = Flocking Ref.- Tadmor-Tan (2014): ψ(r) dr = = Flocking 17 / 29

Numerical simulation by Sergio Pérez 18 / 29

Interacting diffusing particle systems and kinetic equation Interacting diffusing particle system: dx i = v i dt, t > 0, i = 1,, N, dv i = 1 N k=1 ψ(x i x k ) N ψ(x j x i )(v j v i )dt + 2dB i. j=1 19 / 29

Interacting diffusing particle systems and kinetic equation Interacting diffusing particle system: Kinetic model: dx i = v i dt, t > 0, i = 1,, N, dv i = where ũ F is given by with 1 N k=1 ψ(x i x k ) N ψ(x j x i )(v j v i )dt + 2dB i. j=1 tf + v x F + v ((ũ F v)f ) = v F, x R d, v R d, ũ F (x, t) := (ψ bf )(x, t) (ψ a F )(x, t), a F (x, t) := F (x, v, t) dv and b F (x, t) := vf (x, v, t) dv. R d R d 19 / 29

Perturbation framework Nonlinear Fokker-Planck equation: Define the perturbation f = f (x, v, t) by tf + v x F = v ( v F + (v u F )F ). F = M + ) 1 Mf, M = M(v) = exp ( v 2 (2π) d/2 2 The equation for the perturbation f satisfies tf + v x f + u F v f = Lf + Γ(f, f ), (4) where the linear part Lf and the nonlinear part Γ(f, f ) are given by Lf := 1 ( ( )) ( ) f 1 v M v and Γ(f, f ) = u F M M 2 vf + v M, respectively. Γ(f, f ) is not bilinear! Ref.- Duan-Fornasier-Toscani(2010), Karper-Mellet-Trivisa(2013) 20 / 29

Global classical solutions and time-asymptotic behavior Theorem(C. 2015): Let d 3 and s 2[d/2] + 2. Suppose F 0 M + Mf 0 0 and f 0 H s ɛ 0 1. Then we have the global existence of the unique classical solution f (x, v, t) to the equation (4) satisfying f C([0, ); H s (R d R d )), F M + Mf 0, and t t f (t) 2 H s + c 1 x (a, b) 2 H 0 s 1 ds + c 1 k x l v {I P}f 2 µds c 2 f 0 2 H s, 0 0 k+l s for some positive constants c 1, c 2 > 0. Furthermore, if f 0 L 2 v (L 1 ) is bounded, we have ( ) f (t) H s C f 0 H s + f 0 L 2 v (L 1 ) (1 + t) d 4, t 0, where C is a positive constant independent of t. Idea of proof: Careful analysis of the nonlinear dissipation term + Coercivity of the linear operator L + Hypocoercivity for a modified linearized Cauchy problem with a non-homogeneous microscopic source. 21 / 29

Collective behavior models with sharp sensitivity regions 22 / 29

Collective behavior models at particle and kinetic levels Particle system: dx i dt = v i, t > 0, i = 1,, N, dv i dt = m j 1 K(vi )(x i x j )(v j v i ), j i where 1 K(vi ) is the indicator function on the set K(v i ). Here {x i } N i=1, {v i } N i=1, {m i } N i=1 are the position, velocity, and weight of i-th particle, respectively. Kinetic equation: As a mean-field limit, we consider tf + v x f + v [F (f )f ] = 0, (x, v) R d R d, t > 0, (5) where F (f ) is a alignment force of velocities given by F (f )(x, v, t) = 1 K(v) (x y)(w v)f (y, w) dydw. R d R d 23 / 29

Main assumptions (H1): K(v) is globally compact, i.e., K(v) is compact and there exists a compact set K such that K(v) K for all v R d. (H2): There exists a family of closed sets v Θ(v) and a constant C such that: - K(v) Θ(v), for all v R d, - Θ(v) ε,+ Cε, for all ε (0, 1), - K(v) K(w) Θ(v) C v w,+, for v, w R d, - Θ(w) Θ(v) C v w,+, for v, w R d. Figure: A sketch of the sets K, K ε,+, and K ε, 24 / 29

Examples A ball with a fixed radius: K(v) B(0, r) where B(0, r) := {x R d : x r} with r > 0. A ball with a radius depending on the speed: K(v) = B(0, r( v )) where r : R + R + is bounded and Lipschitz. Θ(v) = K(v) 25 / 29

Examples A vision cone: K(v) = C(r, v, θ( v )) := { ( ) } x v x : x r and θ( v ) cos 1 θ( v ), x v with 0 < θ(z) C (R) satisfying θ(z) = π for 0 z 1, θ(z) is decreasing for z 1, and θ(z) θ > 0 as z +. { C(r, v, θ( v )) R(v) if v (1/2, 1), Θ(v) := C(r, v, θ( v )) else, where R(v) = [a(v), b(v)] with a(v) = r v v, b(v) = 2r( v 1) v v. 26 / 29

Global weak solutions & stability Definition: For a given T (0, ), f is a weak solution of (5) on the time-interval [0, T ) if and only if the following condition are satisfied: (i) f L (0, T ; (L 1 + L )(R d R d )) C([0, T ]; P 1 (R d R d )), (ii) f satisfies the equation (5) in the weak sense. Theorem(Carrillo-C.-Hauray-Salem 2016): Given an initial data satisfying f 0 (L 1 + L )(R d R d ) P 1 (R d R d ), (6) and assume further that f 0 is compactly supported in velocity. Then there exists a positive time T > 0 such that the system (5) with the sensitivity region set-valued function K(v) satisfying (H1)-(H2) admits a unique weak solution f L (0, T : (L 1 + L )(R d R d )), which is also compactly supported in velocity. Furthermore if f i, i = 1, 2 are two such solutions to the system (5) with initial data f i (0) satisfying (6), we have d 1 (f 1 (t), f 2 (t)) d 1 (f 1 (0), f 2 (0))e Ct, for t [0, T ], where C is a positive constant that depends only on the L (R d R d (0, T )) norm of f 1. Weak-strong stability! 27 / 29

Further extensions Second-order model: tf + v x f + v (F (f )f ) = 0, (x, v) R d R d, t > 0, where the force term F (f ) can be chosen from two different types: ψ(x y)1 K(v) (x y)h(w v)f (y, w) dydw F (f )(x, v, t) := x ϕ(x y)1 K(v) (x y)f (y, w) dydw (Cucker-Smale) (Attractive-Repulsive). Here ψ, h, and ϕ denote the communication weight, velocity coupling, and interaction potential, respectively. First-order model: tρ + x (ρu) = 0, x R d, t > 0, where the vector field u is given by u(x, t) := 1 K(w(x)) (x y) x ϕ(x y)ρ(y) dy, R d and ϕ W 1, (R d ) and w is a given orientational field satisfying w W 1, (R d ) and w w 0 > 0. 28 / 29

Thank you for your attention 29 / 29