Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Cortez, Roberto; Fontbona, Joaquín

doi:10.1007/s00220-018-3101-4

Cited by 17 publications

(43 citation statements)

References 32 publications

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“…Corresponding particle approximations (including a rate of convergence) was studied by Fournier, Mischler [FM16] for hard potentials and by Xu [Xu18] for soft potentials. Similar results for Maxwellian molecules, but with another particle system, were also obtained in [CF18]. The precise formulation of hard and soft potentials are introduced in the next section.…”

Section: The Enskog Processsupporting

confidence: 67%

The Enskog process for hard and soft potentials

Friesen

Rüdiger

Sundar

2019

Nonlinear Differ. Equ. Appl.

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The density of a moderately dense gas evolving in a vacuum is given by the solution of an Enskog equation. Recently we have constructed in [ARS17] the stochastic process that corresponds to the Enskog equation under suitable conditions. The Enskog process is identified as the solution of a McKean-Vlasov equation driven by a Poisson random measure. In this work, we continue the study for a wider class of collision kernels that includes hard and soft potentials. Based on a suitable particle approximation of binary collisions, the existence of an Enskog process is established.Let us briefly comment on particular examples of collision kernels Bp|v´u|, nq in dimension d " 3. Boltzmann's original model was first formulated for (true) hard spheres where Bp|u´v|, nqdn " |pu´v, nq|dn.A transformation in polar coordinates to a system where the center is in u`v 2 and e 3 " p0, 0, 1q is parallel to u´v, i.e. e 3 |u´v| " u´v, leads to Bp|u´v|, nqdn " |pu´v, nq|dn " |u´v| sinˆθ 2˙c osˆθ 2˙d θdφ,where θ P p0, πs is the angle between u´v and u ‹´v‹ and φ P p0, 2πs is the longitude angle, see Tanaka [Tan79] or Horowitz and Karandikar [HK90]. More generally, many results rely on Grad's angular cut-off assumption where it is supposed that ż S d´1Bp|v´u|, nqdn ă 8,

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Section: The Enskog Processsupporting

confidence: 67%

The Enskog process for hard and soft potentials

Friesen

Rüdiger

Sundar

2019

Nonlinear Differ. Equ. Appl.

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“…As remarked above, our estimates (Theorem 1.1, 1.3, 1.7) improve this rate; this improvement is due to the improvement of Theorem 1.6 over the corresponding estimate in [30], and this will be discussed further below. More recently, [8] proved a chaoticity estimate for Maxwell molecules in d = 3, measured in the L 2 (P) norm of Wasserstein 2 distance (1.16), and with an almost optimal rate N ǫ−1/3 , which is almost completely analagous to Theorem 1.1.…”

Section: Probabilistic Techniques For the Kac Process And Boltzmann Ementioning

confidence: 84%

“…From existence and uniqueness, we can consider the Boltzmann equation as describing a non-linear semigroup of flow operators on (φ t ) t≥0 on ∪ k>2 S k . To prove Proposition 1, Norris [33] introduces a family of random linear operators E st , and develops a representation formula in terms of these operators, which will be reviewed in Sections 4,8. Cruicial to the proof are estimates for the operator norms of E st , which are obtained by Grönwallstyle estimates. As a result, the constant C depends badly on the terminal time t fin , with a priori exponential growth.…”

Section: Introduction and Mainmentioning

confidence: 99%

Pathwise convergence of the hard spheres Kac process

Heydecker¹

2019

Ann. Appl. Probab.

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We derive two estimates for the deviation of the N -particle, hardspheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a k th moment, k > 2. Our approach is similar to Kac's proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.

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“…Then, Fournier-Mischler [13] proved the propagation of chaos at rate N −1/4 for the Nanbu system and for hard potentials without cutoff (γ ∈ [0, 1] and ν ∈ (0, 1)). Finally, as mentioned in Section 1.5, Cortez-Fontbona [4] used two coupling techniques for Kac's binary interaction system and obtained a uniform in time estimate for the Boltzmann equation with Maxwell molecules (γ = 0) under some suitable moments assumptions on the initial datum. Let us mention that the time-uniformity uses the recent nice results of Rousset [27].…”

Section: Resultsmentioning

confidence: 99%

“…It is thus impossible to simulate it directly. For this reason, we will study a truncated version of Nanbu's particle system applying a cutoff procedure as [13], who were studying the Nanbu system for hard potentials and Maxwell molecules, and [4], who were dealing with the Kac system for Maxwell molecules. Our particle system with cutoff corresponds to the generator L N,K defined, for any bounded Lipschitz function φ : (R 3 ) N → R and v = (v 1 , ..., v N ) ∈ (R 3 ) N , by (1.12) with G defined by (1.4).…”

Section: The Particle Systemmentioning

confidence: 99%

Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials

Xu¹

2018

Ann. Appl. Probab.

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We prove a strong/weak stability estimate for the 3D homogeneous Boltzmann equation with moderately soft potentials (γ ∈ (−1, 0)) using the Wasserstein distance with quadratic cost. This in particular implies uniqueness in the class of all weak solutions, assuming only that the initial condition has a finite entropy and a finite moment of sufficiently high order. We also consider the Nanbu N -stochastic particle system which approximates the weak solution. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.

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Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Cited by 17 publications

References 32 publications

The Enskog process for hard and soft potentials

The Enskog process for hard and soft potentials

Pathwise convergence of the hard spheres Kac process

Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials

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