A stochastic  particle numerical method    for 3D Boltzmann equations without cutoff

Fournier, Nicolas

doi:10.1090/s0025-5718-01-01339-4

Cited by 39 publications

(47 citation statements)

References 16 publications

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“…In [34] Graham and Méléard obtained a rate of convergence (of order 1/N for the ℓ-th marginal) on any bounded finite interval of the N -particle system to the deterministic Boltzmann dynamics in the case of Maxwell molecules under Grad's cut-off hypothesis. Lastly, in [29,30] Fournier and Méléard obtained the convergence of the Monte-Carlo approximation (with numerical cutoff) of the Boltzmann equation for true Maxwell molecules with a rate of convergence (depending on the numerical cutoff and on the number N of particles).…”

Section: 3mentioning

confidence: 99%

Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

161

292

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Abstract. This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42,53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.Motivated by the inspirative paper by Grünbaum [35], we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

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Section: 3mentioning

confidence: 99%

Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

161

292

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“…We follow the line of Tanaka [15] (see also [9]), who was dealing with the Maxwellian case, that is Φ ≡ 1.…”

Section: Coupling Boltzmann Processesmentioning

confidence: 99%

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Fournier

Guérin

2008

J Stat Phys

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Abstract. We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka [15]: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other.

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“…This speed of convergence in l/l is known for the Maxwell cross section I/o 2 , see for example [7] for a proof in the 3D case.…”

Section: Numerical Studymentioning

confidence: 78%

“…The natural Interpretation of the nonlinearity in (1.14) leads to a simple mean field interacting System but a physical Interpretation of the equation leads also naturally to a binary mean field interacting particle System. In both cases, these n-particle Systems are pure-jump Markov processes with values in (2R 2 ) n and with generators defined for φ € C 6 ((2R 2 ) n ) by (3)(4)(5)(6)(7)(8)(9)(10) for the simple mean-field interacting System and by \ ( φ(νη + e " 7 ' (Vi -"· z) + ej -7ife -Vi ' z}) -φ(υη} } αζ (3 · η) for the binary mean-field interacting System. In these formulas, v n = (t/i, ..., v n ) denotes the generic point of (M*) n and ei : h 6 M 2 t-> e-t .h = (0, ..., 0, /i, 0, ..., 0) 6 (lR 2 ) n with h at the t'-th place.…”

Section: J-ir J-irmentioning

confidence: 99%

Monte-Carlo approximations for 2d homogeneous Boltzmann equations without cutoff and for non Maxwell molecules

Fournier¹

2001

McMa

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In [10], using a typically probabilistic Substitution in the Boltzmann equation, we extend Tanaka's probabilistic Interpretation [19] to much more general spatially homogeneous Boltzmann equations, i.e. homogeneous Boltzmann equations without cutoff and for non Maxwell molecules.In this paper we show how this Interpretation allows us to build some approximating cutoff interacting particle Systems, and to derive some Monte-Carlo algorithms for the Simulation of Solutions of the Boltzmann equation.

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A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Cited by 39 publications

References 16 publications

Kac’s program in kinetic theory

Kac’s program in kinetic theory

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Monte-Carlo approximations for 2d homogeneous Boltzmann equations without cutoff and for non Maxwell molecules

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