The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

Golse, François; Saint‐Raymond, Laure

doi:10.1007/s00222-003-0316-5

Cited by 274 publications

(280 citation statements)

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“…While the reference [18] treated the case of bounded collision kernels, the theorem above was later extended to all hard cutoff potentials in the sense of Grad -which includes the case of hard spheres considered in this paper -in [19]. The arguments in [18,19] have been recently refined by Levermore and Masmoudi [25] to encompass both soft as well as hard potentials, under a cutoff assumption more general than that proposed by Grad in [20].…”

Section: Family Of Renormalized Solutions Of the Boltzmann Equation (mentioning

confidence: 99%

“…Controlling the high speed tail of (fluctuations of) the distribution function is an essential step in the derivation of fluid dynamic limits of the Boltzmann equation, and involves rather technical estimates based on the entropy and entropy production estimates (16) together with the dispersion effects of the streaming operator Sh ∂ t + v · ∇ x (see [4,18,19]). …”

Section: The Conservation Lawsmentioning

confidence: 99%

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From the Kinetic Theory of Gases to Continuum Mechanics

Golse

2011

AIP Conference Proceedings

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Abstract. Recent results on the fluid dynamic limits of the Boltzmann equation based on the DiPerna-Lions theory of renormalized solutions are reviewed in this paper, with an emphasis on regimes where the velocity field behaves to leading order like that of an incompressible fluid with constant density.Keywords: Hydrodynamic limits, Kinetic models, Boltzmann equation, Entropy production, Euler equations, Navier-Stokes equations PACS: 47.45-n, 51.10.+y, 51.20.+d, 47.10.ad In memory of Carlo Cercignani (1939Cercignani ( -2010 Relating the kinetic theory of gases to their description by the equations of continuum mechanics is a question that finds its origins in the work of Maxwell [30]. It was subsequently formulated by Hilbert as a mathematical problem -specifically, as an example of his 6th problem on the axiomatization of physics [21]. In Hilbert's own words "Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes which lead from the atomistic view to the laws of motion of continua". Hilbert himself studied this problem in [22] as an application of his theory of integral equations. The present paper reviews recent progress on this problem in the past 10 years as a consequence of the DiPerna-Lions global existence and stability theory [12] for solutions of the Boltzmann equation. This Harold Grad lecture is dedicated to the memory of Carlo Cercignani, who gave the first Harold Grad lecture in the 17th Rarefied Gas Dynamics Symposium, in Aachen (1990), in recognition of his outstanding influence on the mathematical analysis of the Boltzmann equation in the past 40 years. THE BOLTZMANN EQUATION: FORMAL STRUCTUREIn kinetic theory, the state of a monatomic gas at time t and position x is described by its velocity distribution function F ≡ F(t, x, v) ≥ 0. It satisfies the Boltzmann equationassuming that gas molecules behave as perfectly elastic hard spheres of diameter d. In this formula, we have denotedMolecular interactions more general than hard sphere collisions can be considered by replacing d 2 2 |(v − v * ) · ω| with appropriate collision kernels of the form b(|v − v * |, | v−v * |v−v * | · ω|). In this paper, we restrict our attention to the case of hard sphere collisions to avoid dealing with more technical conditions on the collision kernel. Properties of the collision integralWhile the collision integral is a fairly intricate mathematical expression, the formulas (1) entail remarkable symmetry properties. As a result, the collision integral satisfies, for each continuous, rapidly decaying f ≡ f (v), the identities

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Section: Family Of Renormalized Solutions Of the Boltzmann Equation (mentioning

confidence: 99%

Section: The Conservation Lawsmentioning

confidence: 99%

From the Kinetic Theory of Gases to Continuum Mechanics

Golse

2011

AIP Conference Proceedings

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“…These scaling assumptions correspond exactly to the invariance scaling for the incompressible Navier-Stokes motion equation. Theorem 1.11 (F. Golse-L. Saint-Raymond [38,40]) Let F ε be a family of renormalized solutions of the Cauchy problem for the Boltzmann equation with initial data…”

Section: Incompressible Navier-stokes Limitmentioning

confidence: 99%

Fluid Dynamic Limits of the Kinetic Theory of Gases

Golse

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…The Boltzmann equation specifies the evolution of a one-molecule probability-density distribution in position/momentum space [12,23]. The Boltzmann equation in fact encapsulates all conventional continuum models, in the sense that with appropriate scalings of the macroscopic length and time scales, limit solutions of the Boltzmann equation correspond to solutions of these continuum models [3,16,21,31,32]. Numerical approximation of the Boltzmann equation poses fundamental complications, however, on account of the high dimensional setting of the equation: for a problem with d spatial dimensions, the corresponding position/momentum domain of the Boltzmann equations is 2d dimensional.…”

Section: Introductionmentioning

confidence: 99%