Propagation of chaos for system of vortices in 2D M. Hauray, in collaboration with N. Fournier and S. Mischler. Univ. Aix-Marseille Rennes, Centre Lebesgue, April 2013 M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 1 / 38
An overview of the problem. 1 Limits of N particles distributions. 2 Particles systems towards McKean-Vlasov non-linear eq. 3 Dissipation of entropy and uniform smoothness estimates. 4 Propagation of regularity in the limit. 5 Conclusion : results on propagation of chaos. 6 M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 2 / 38
An overview of the problem. The Navier-Stokes equation in 2D In 2D, the NS equation ∂ t u + u · ∇ u = −∇ p + ν ∆ u , divu = 0 , +I.C. is oftently rewritten in terms of vorticity ω = ∇ ⊥ · u = ∂ 1 u 2 − ∂ 2 u 1 � ∂ t ω + u · ∇ ω = ν ∆ ω + I.C. , (1) x ⊥ u ( t , x ) = K ∗ ω = 2 π | x | 2 ∗ ω x ⊥ 2 π | x | 2 is the Biot-Savard kernel K ∈ L 2 , ∞ . where K ( x ) = Well-posedness theory : Leray ( u 0 ∈ L 2 ), Giga-Miyakawa-Osada or Ben-Artzi ( ω 0 ∈ L 1 ), Cannone-Planchon or Meyer ( u 0 ∈ some Besov space), Gallagher-Gallay ( ω 0 measure) and many others... Less is known for the Euler equation ( ν = 0) : Yudovich (well-posed if ω ∈ L ∞ ), Delort (Existence if ω 0 positive measure), Scheffer, Schnirelman, De Lellis-Szekelyhidi (non-uniqueness). M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 3 / 38
An overview of the problem. The Vortex approximation Idea : Approximate a “continuous” vorticity profile by a some of N Dirac masses, with position X i and strength a i N ∈ R . The Euler Equation is transformed in a system of ODEs, and NS2D in a system of SDEs � 1 � � ∀ i ≤ N , dX i = a j K ( X i − X j ) dt + σ dB i (2) N j � = i sometimes called Helmholtz-Kirchhoff system (if ν = 0). Justification : Simulation of decaying 2D Turbulence Theoritical justification given by Marchioro-Pulvirenti and Gallay. Well-posedness of the N vortex system : ν = 0: Marchioro-Pulvirenti (OK for a.e. initial positions and vortices strengths). ν > 0. Takanobu ( a i > 0), Osada ( a i ∈ R ), Fontbana-Martinez... Simplification: From now, a i = 1 for all i . M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 4 / 38
An overview of the problem. Numerical applications. A simulation by Chorin in the ’70. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 5 / 38
An overview of the problem. The question of convergence as N → + ∞ . A natural question. NS2D : Positive answer (for σ large enough) given by Osada in the ’80. Euler: Very difficult. In the viscous case, the difficulty is the singularity of the drift. Goals of the talk : Review the general procedure (with an analyst? point if view). Explain some improvements we introduced. State and comment the result for the vortex system. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 6 / 38
Limits of N particles distributions. Limits of symmetric (exchangeable) N particles distributions M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 7 / 38
Limits of N particles distributions. Two possible representations. Here and below : E = R d or C ([0 , + ∞ ) , R d ) (Polish space). Analyst: Let F N be a sequence of symmetric proba on P ( E N ). Probabilist: Let X N = ( X N 1 , . . . , X N N ) be a sequence of exchangeable R. V. What are the possible limit points? 1 : with empirical measures. N X := 1 µ N � F N ¯ with law δ X N N i i =1 converge to some R.V. f in P ( E ), with law ¯ π ∈ P ( P ( E )). 2 : with infinite sequence of R.V. F N seen as probabilities on E ∞ . They can converges towards some π ∈ P sym ( E ∞ ). In both cases, tightness is equivalent to tightness of L ( X N 1 ). M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 8 / 38
Limits of N particles distributions. The two representations are the same. Notations : Marginals of π ∈ P sym ( E ∞ ) are denoted by π N (law of the N first RV). � π N := ρ ⊗ N π ( d ρ ) ∈ P ( E N ) . For ¯ π ∈ P ( P ( E )), ¯ We can construct the following maps between P ( P ( E )) and P sym ( E ∞ ). R ρ ⊗∞ π ( d ρ ) � π ¯ − − − − → π ∞ := ¯ : P sym ( E ∞ ) P ( P ( E )) : S { Limits of π N } ← − − − − π Theorem (De Finetti - Hewitt & Savage) R ◦ S = Id P sym ( E ∞ ) , S ◦ R = Id P ( P ( E )) and S is univalent. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 9 / 38
Limits of N particles distributions. The algebraic relation R ◦ S = Id P sym ( E ∞ ) . In fact, we can compute for instance with j = 2 � ρ ⊗ 2 ¯ π N ) 2 ( ¯ := π N ( d ρ ) � ( µ N X ) ⊗ 2 π N ( d X N ) = 1 � �� � � π N ( d X N ) = δ X i ⊗ δ X j + δ X i ⊗ δ X i N 2 i � = j i N − 1 π 2 + 1 N π 1 δ X 1 = X 2 = N ⇂ ⇂ � � R ◦ S ( π ) = π 2 2 Do it for all j ∈ N and get R ◦ S ( π ) = π . M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 10 / 38
Limits of N particles distributions. S ◦ R = Id P ( P ( E )) is a consequence of concentration. Here concentration means : Glivenko-Cantelli theorem or empirical law of large number . Theorem (Varadarajan) If the ( X i ) i ∈ N are i.i.d with law ρ , then µ N X goes in law towards the constant ρ . In other words, S ( ρ ∞ ) = limits of ρ ⊗ N = δ ρ � ( ρ ′ ) ⊗∞ δ ρ ( ρ ′ ) = ρ ∞ , but since R ( δ ρ ) = we get S [ R ( δ ρ )] = δ ρ �� � � �� And by linearity and continuity S R δ ρ π ( d ρ ) = δ ρ π ( d ρ ) To remember : Concentration implies that for N large, ρ ⊗ N and ρ ⊗ N have almost 1 2 disjoints supports. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 11 / 38
Limits of N particles distributions. Two equivalent descriptions of convergence. Going back to the original problem, we can give two equivalent definitions of convergence for F N ∈ P sym ( E N ). F N ⇀ π ∈ P sym ( E ∞ ), (usual sense for product space) F N ∀ j ∈ N , ⇀ π j , j F N = L ( µ N ¯ X ) ⇀ ¯ π ∈ P ( P ( E )). Or better, the RV µ N X goes in law toward some RV ρ ∈ P ( E ). M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 12 / 38
Limits of N particles distributions. Chaotic sequences We call F N a chaotic sequence if the limit is an extremal point. Corollary (of the previous theorem) For π ∈ P sym ( E ∞ ) π = ρ ∞ ⇐ ⇒ π 2 = ρ ⊗ 2 . “There cannot be three particles correlations if there is no two-particles correlations.” Exercice : Find a counter-example if N = + ∞ is replaced by N = 3. Definition For ρ ∈ P ( E ) , F N is a ρ -chaotic sequence if one of the three (equivalent) statements is true : i) µ N X goes in law towards ρ , F N ⇀ ρ ⊗ j , ii) ∀ j ∈ N , j F N 2 ⇀ ρ ⊗ 2 . iii) M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 13 / 38
Limits of N particles distributions. Propagation of chaos Definition G N ( t ) dynamical flow of a N particle system. G ∞ ( t ) “flow” the unique expected (non-linear) limit. Preservation of chaos holds in that case if with for all t F N ( t ) = F N (0) ◦ G N ( − t ) , ρ ( t ) = G ∞ ( t )( ρ 0 ) ρ 0 − chaotic F N (0) is ⇓ F N ( t ) is ρ ( t ) − chaotic Even better Definition (Prop. of chaos II) Trajectorial POC holds if for X N that are ρ -chaotic, then the trajectories X N ([0 , ∞ )) are X ([0 , ∞ )) -chaotic, where X stands for the unique solution of the expected non linear limit SDE. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 14 / 38
Particles systems towards McKean-Vlasov non-linear eq. Particles systems towards McKean-Vlasov non-linear eq. M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 15 / 38
Particles systems towards McKean-Vlasov non-linear eq. A stochastic interacting particle system. N vortices interacting via a 2 particles kernel b ( x , y ). Important : b ( x , x ) = 0 . � 1 � � ∀ i ≤ N , dX i = b ( X i , X j ) dt + σ dB i (3) N j � = i = b ( X i , µ N X ) dt + σ dB i What is the expected limit? If all the µ N X remains close to the law ρ ( t ) of X 1 ( t ) (i.e. the independence is approximately preserved in time ?), the X i will look as N ind. copies of d X ( t ) = b ( X ( t ) , ρ ( t )) dt + σ dB . (4) where ρ ( t ) is the law of X ( t ). M. Hauray (UAM) Chaos for 2D vortex systems Rennes, Centre Lebesgue, April 2013 16 / 38
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