Tanaka Theorem for Inelastic Maxwell Models

Bolley, François; Carrillo, José A.

doi:10.1007/s00220-007-0336-x

Cited by 26 publications

(39 citation statements)

References 30 publications

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“…The mathematical theory is treated in e.g. [3,2,4], where, for example, it is proven that this equation generates a nonlinear semigroup S NL t f in := f t for any f in ∈ P q (R d ), q ≥ 2. Notice that unlike the classical Boltzmann equation the kinetic energy is not conserved.…”

Section: Inelastic Collisions With Thermal Bathmentioning

confidence: 99%

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

Mischler

Mouhot

Wennberg

2013

Probab. Theory Relat. Fields

132

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This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.

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Section: Inelastic Collisions With Thermal Bathmentioning

confidence: 99%

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

Mischler

Mouhot

Wennberg

2013

Probab. Theory Relat. Fields

132

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“…Needless to say, the introduction of inelasticity through a constant coefficient of normal restitution α ≤ 1, while keeping the independence of the collision rate with the relative velocity, opens up new perspectives for exact results, including the elastic case (α = 1) as a special limit. This justifies the growing interest in the so-called inelastic Maxwell models (IMM) by physicists and mathematicians alike in the past few years [3,4,5,6,7,8,10,11,12,13,14,15,19,23,25,26,28,32,33,34,35,36,38,40,42,50,57,62,63,68,69,71,73,75,78]. Furthermore, it is interesting to remark that recent experiments [56] for magnetic grains with dipolar interactions are well described by IMM.…”

Section: Introductionmentioning

confidence: 98%

Hydrodynamics of Inelastic Maxwell Models

Garzó

Santos

2011

Math. Model. Nat. Phenom.

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Abstract. An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier-Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman-Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman-Enskog-like method and generalized (tensorial) transport coefficients are obtained.

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“…As in the elastic case [20,29], a significant way of overcoming the above problem is to apply a mean-field approach whereby the collision frequency is replaced by an effective quantity independent of the relative velocity. This defines the so-called inelastic Maxwell model (IMM), which has received much attention in the last few years, especially in the applied mathematics literature (see, for instance, [2,3,4,6,7,8,10,11,12,13,14,17,18,21,22,23,24,26,27] and the review papers [5,9,16,19,25]).…”

mentioning

confidence: 99%

Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state

Santos¹,

Garzó²

2012

Granular Matter

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The collisional rates associated with the isotropic velocity moments V 2r and the anisotropic momentsare exactly derived in the case of the inelastic Maxwell model as functions of the exponent r , the coefficient of restitution α, and the dimensionality d. The results are applied to the evolution of the moments in the homogeneous free cooling state. It is found that, at a given value of α, not only the isotropic moments of a degree higher than a certain value diverge but also the anisotropic moments do. This implies that, while the scaled distribution function has been proven in the literature to converge to the isotropic self-similar solution in well-defined mathematical terms, nonzero initial anisotropic moments do not decay with time. On the other hand, our results show that the ratio between an anisotropic moment and the isotropic moment of the same degree tends to zero.

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Tanaka Theorem for Inelastic Maxwell Models

Cited by 26 publications

References 30 publications

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

Hydrodynamics of Inelastic Maxwell Models

Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state

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