Entropy Dissipation and Long-Range Interactions

Reynaud, Alexandre; Desvillettes, Laurent; Villani, Cédric; Wennberg, Bernt

doi:10.1007/s002050000083

Cited by 252 publications

(413 citation statements)

References 24 publications

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“…where we have used the change of variable from cancellation lemmas in [1] (which is possible since b has its support included in [0, π/2]). The variable φ σ (v, v * ) denotes the inverse application of v → v ′ keeping v * and σ frozen (it is given explicitly in [1]).…”

Section: Proof Of the Stability Estimatesmentioning

confidence: 99%

Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

Desvillettes

Mouhot

2009

Arch Rational Mech Anal

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Abstract. In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to the one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non cutoff and non mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.

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Section: Proof Of the Stability Estimatesmentioning

confidence: 99%

Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

Desvillettes

Mouhot

2009

Arch Rational Mech Anal

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“…This theory provided tools which allowed one to handle numerous new questions; as an example let us note the regularizing effect of the operator Q + [87]. Especially the case without cut-off has been treated, with the so-called grazing collisions leading to the Landau operator; see Degond and Lucquin [41], Alexandre et al [2] and the references in [116]. Stationary Boltzmann equations are also important; see Arkeryd and Nouri [5] for weak solutions in a slab, Ukai and Asano [115] for strong solutions in an exterior domain with given Maxwellian at infinity and nearly compatible boundary conditions, or Liu and Yu [93] and the references therein for shock profiles.…”

Section: −β|K|mentioning

confidence: 99%

“…We present them in section 4, which includes Vlasov equations of plasma physics, scattering models (including microscopic chemotaxis modeling) and the Boltzmann equation, but no physics background is assumed from the reader. The last section deals with Its solution is given by (2) f (t, x, ξ) = f 0 (x − ξt, ξ).…”

Section: Introductionmentioning

confidence: 99%

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Mathematical tools for kinetic equations

Perthame

2004

Bull. Amer. Math. Soc.

113

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Abstract. Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, their range of application has been considerably extended. First introduced as a means to unify various perspectives on fluid mechanics, they are now used in plasma physics, semiconductor technology, astrophysics, biology.... They all are characterized by a density function that satisfies a Partial Differential Equation in the phase space.This paper presents some of the simplest tools that have been devised to study more elaborate (coupled and nonlinear) problems. These tools are basic estimates for the linear first order kinetic-transport equation. Dispersive effects allow us to gain time decay, or space-time L p integrability, thanks to Strichartz-type inequalities. Moment lemmas gain better velocity integrability, and macroscopic controls transform them into space L p integrability for velocity integrals.These tools have been used to study several nonlinear problems. Among them we mention for example the Vlasov equations for mean field limits, the Boltzmann equation for collisional dilute flows, and the scattering equation with applications to cell motion (chemotaxis).One of the early successes of kinetic theory has been to recover macroscopic equations from microscopic descriptions and thus to be able theoretically to compute transport coefficients. We also present several examples of the hydrodynamic limits, the diffusion limits and especially the recent derivation of the Navier-Stokes system from the Boltzmann equation, and the theory of strong field limits.

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“…When singular (nonlinear) kernels are concerned (non cutoff Boltzmann kernel, or Landau kernel), the derivation of the corresponding kinetic equation seems completely open, though the equations themselves have been extensively studied (cf. for example [1,2,7] and the references therein for recent results on these equations).…”

Section: Introductionmentioning

confidence: 99%