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Particles approximation for Vlasov equation with singular interaction M. Hauray, in collaboration with P.-E. Jabin. Universit e dAix-Marseille Oberwolfach Worshop, December 2013 M. Hauray (UAM) Particles systems towards Vlasov

Particles approximation for Vlasov equation with singular interaction M. Hauray, in collaboration with P.-E. Jabin. Universit´ e d’Aix-Marseille Oberwolfach Worshop, December 2013 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 1 / 27
Outline Introduction of the problem 1 A toy model: the 1D Vlasov-Poisson system. 2 The convergence of particles systems in 3D 3 Some ingredients of the proof. 4 The related problem of stability 5 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 2 / 27
Outline Introduction of the problem 1 A toy model: the 1D Vlasov-Poisson system. 2 The convergence of particles systems in 3D 3 Some ingredients of the proof. 4 The related problem of stability 5 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 2 / 27
Outline Introduction of the problem 1 A toy model: the 1D Vlasov-Poisson system. 2 The convergence of particles systems in 3D 3 Some ingredients of the proof. 4 The related problem of stability 5 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 2 / 27
Outline Introduction of the problem 1 A toy model: the 1D Vlasov-Poisson system. 2 The convergence of particles systems in 3D 3 Some ingredients of the proof. 4 The related problem of stability 5 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 2 / 27
Outline Introduction of the problem 1 A toy model: the 1D Vlasov-Poisson system. 2 The convergence of particles systems in 3D 3 Some ingredients of the proof. 4 The related problem of stability 5 M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 2 / 27
Introduction of the problem Particle systems with singular forces. N particles with masses (or charges) a i / N , positions X i et speed V i in R 2 d [ Z i = ( X i , V i )] interacting through force F � ˙ X i = V i ∀ i ≤ N , � ˙ V i = 1 j � = i a j F ( X i − X j ) + 0 dB i . N Singular forces : Satisfying for some 0 < α < d − 1, F ∈ C 1 b ( R d \{ 0 } ) and : x ( S α -condition) C C F ( x ) ∼ precisely | F ( x ) | ≤ | x | α , |∇ F | ≤ | x | α +1 | x | α +1 x → 0 About the resolution Repulsive case : OK (No collisions). Attractive case : For α = d − 1 ⇒ N -body problem. True collisions are rare, but does non non-collisions singularities are? (Xia) and (Saary) α < 1 : OK by DiPerna-Lions theory. For N large , particles systems should converge towards... M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 3 / 27
Introduction of the problem Particle systems with singular forces. N particles with masses (or charges) a i / N , positions X i et speed V i in R 2 d [ Z i = ( X i , V i )] interacting through force F � ˙ X i = V i ∀ i ≤ N , � ˙ V i = 1 j � = i a j F ( X i − X j ) + 0 dB i . N Singular forces : Satisfying for some 0 < α < d − 1, F ∈ C 1 b ( R d \{ 0 } ) and : x ( S α -condition) C C F ( x ) ∼ precisely | F ( x ) | ≤ | x | α , |∇ F | ≤ | x | α +1 | x | α +1 x → 0 About the resolution Repulsive case : OK (No collisions). Attractive case : For α = d − 1 ⇒ N -body problem. True collisions are rare, but does non non-collisions singularities are? (Xia) and (Saary) α < 1 : OK by DiPerna-Lions theory. For N large , particles systems should converge towards... M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 3 / 27
Introduction of the problem Particle systems with singular forces. N particles with masses (or charges) a i / N , positions X i et speed V i in R 2 d [ Z i = ( X i , V i )] interacting through force F � ˙ X i = V i ∀ i ≤ N , � ˙ V i = 1 j � = i a j F ( X i − X j ) + 0 dB i . N Singular forces : Satisfying for some 0 < α < d − 1, F ∈ C 1 b ( R d \{ 0 } ) and : x ( S α -condition) C C F ( x ) ∼ precisely | F ( x ) | ≤ | x | α , |∇ F | ≤ | x | α +1 | x | α +1 x → 0 About the resolution Repulsive case : OK (No collisions). Attractive case : For α = d − 1 ⇒ N -body problem. True collisions are rare, but does non non-collisions singularities are? (Xia) and (Saary) α < 1 : OK by DiPerna-Lions theory. For N large , particles systems should converge towards... M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 3 / 27
Introduction of the problem An example : Antennae galaxies. M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 4 / 27
Introduction of the problem The Vlasov-“Poisson” equation f ( t , x , v ) is the density of particles and satisfies :  ∂ t f + v · ∇ x f + E ( t , x ) · ∇ v f = 0  (1) � �  E ( t , x ) = Ω F ( x − y ) ρ ( t , y ) dy , ρ ( t , x ) = f ( t , x , v ) dv + initial condition: f (0 , x , v ) = f 0 ( x , v ). x Two particular cases : F ( x ) = ± c | x | d ⇒ E = −∇ V , ∆ V = ± ρ , − : gravitationnal case , +: Coulombian one. About the Resolution Compact school : Pfaffelm¨ oser (’92), Sch¨ affer(’93), H¨ orst (’96). Moment school : Lions-Perthame (’91), Jabin-Illner-Perthame (’99), Pallard (’11). α < 1 : much simpler. In the following, f ( t ) is a compactly supported and strong solution of (1). M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 5 / 27
Introduction of the problem The Vlasov-“Poisson” equation f ( t , x , v ) is the density of particles and satisfies :  ∂ t f + v · ∇ x f + E ( t , x ) · ∇ v f = 0  (1) � �  E ( t , x ) = Ω F ( x − y ) ρ ( t , y ) dy , ρ ( t , x ) = f ( t , x , v ) dv + initial condition: f (0 , x , v ) = f 0 ( x , v ). x Two particular cases : F ( x ) = ± c | x | d ⇒ E = −∇ V , ∆ V = ± ρ , − : gravitationnal case , +: Coulombian one. About the Resolution Compact school : Pfaffelm¨ oser (’92), Sch¨ affer(’93), H¨ orst (’96). Moment school : Lions-Perthame (’91), Jabin-Illner-Perthame (’99), Pallard (’11). α < 1 : much simpler. In the following, f ( t ) is a compactly supported and strong solution of (1). M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 5 / 27
Introduction of the problem The Vlasov-“Poisson” equation f ( t , x , v ) is the density of particles and satisfies :  ∂ t f + v · ∇ x f + E ( t , x ) · ∇ v f = 0  (1) � �  E ( t , x ) = Ω F ( x − y ) ρ ( t , y ) dy , ρ ( t , x ) = f ( t , x , v ) dv + initial condition: f (0 , x , v ) = f 0 ( x , v ). x Two particular cases : F ( x ) = ± c | x | d ⇒ E = −∇ V , ∆ V = ± ρ , − : gravitationnal case , +: Coulombian one. About the Resolution Compact school : Pfaffelm¨ oser (’92), Sch¨ affer(’93), H¨ orst (’96). Moment school : Lions-Perthame (’91), Jabin-Illner-Perthame (’99), Pallard (’11). α < 1 : much simpler. In the following, f ( t ) is a compactly supported and strong solution of (1). M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 5 / 27
Introduction of the problem The Vlasov-“Poisson” equation f ( t , x , v ) is the density of particles and satisfies :  ∂ t f + v · ∇ x f + E ( t , x ) · ∇ v f = 0  (1) � �  E ( t , x ) = Ω F ( x − y ) ρ ( t , y ) dy , ρ ( t , x ) = f ( t , x , v ) dv + initial condition: f (0 , x , v ) = f 0 ( x , v ). x Two particular cases : F ( x ) = ± c | x | d ⇒ E = −∇ V , ∆ V = ± ρ , − : gravitationnal case , +: Coulombian one. About the Resolution Compact school : Pfaffelm¨ oser (’92), Sch¨ affer(’93), H¨ orst (’96). Moment school : Lions-Perthame (’91), Jabin-Illner-Perthame (’99), Pallard (’11). α < 1 : much simpler. In the following, f ( t ) is a compactly supported and strong solution of (1). M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 5 / 27
Introduction of the problem The case of regular interaction forces. Important remark : Under the assumption F (0) = 0, The empirical distribution N Z ( t ) = 1 � µ N a i δ Z i ( t ) N i =1 of the particle system is a solution of the Vlasov eq. (1). ⇒ For smooth F , a theory of measure solutions of the Vlasov eq. is possible Stability of meas. sol ⇒ Convergence of part. systems Theorem (Braun & Hepp ’77, Neunzert & Wick ’79, Dobrushin) Two measures solution µ and ν of the Vlasov eq. satisfy W 1 ( µ ( t ) , ν ( t )) ≤ e (1+2 �∇ F � ∞ ) t W 1 ( µ 0 , ν 0 ) Also CLT available using linearisation of VP, ... W 1 is the order one Monge-Kantorovitch-Wasserstein distance. M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 6 / 27
Introduction of the problem The case of regular interaction forces. Important remark : Under the assumption F (0) = 0, The empirical distribution N Z ( t ) = 1 � µ N a i δ Z i ( t ) N i =1 of the particle system is a solution of the Vlasov eq. (1). ⇒ For smooth F , a theory of measure solutions of the Vlasov eq. is possible Stability of meas. sol ⇒ Convergence of part. systems Theorem (Braun & Hepp ’77, Neunzert & Wick ’79, Dobrushin) Two measures solution µ and ν of the Vlasov eq. satisfy W 1 ( µ ( t ) , ν ( t )) ≤ e (1+2 �∇ F � ∞ ) t W 1 ( µ 0 , ν 0 ) Also CLT available using linearisation of VP, ... W 1 is the order one Monge-Kantorovitch-Wasserstein distance. M. Hauray (UAM) Particles systems towards Vlasov Oberwolfach, Dec. 2013 6 / 27

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