Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

Pavić-Čolić, Milana; Tasković, Maja

doi:10.3934/krm.2018025

Cited by 14 publications

(7 citation statements)

References 13 publications

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“…In particular, exponential moments of order γ are shown to generate in a finite time, while Gaussian moments propagate if initially are finite, which holds independently of γ. Moreover, all these result were generalized when the angular part is not integrable (in the angular non-cutoff regime) [41,30,16,37,2].…”

Section: Introductionmentioning

confidence: 88%

On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

Gamba,

Pavić-Čolić

2020

Preprint

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In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad assumption on transition probability rate, that comprises hard potentials for both the relative speed and internal energy with the rate in the interval (0, 2], which is multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ODE theory in Banach spaces, for an initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite k * polynomial moment, with k * depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated to the solution are both generated and propagated uniformly in time.

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Section: Introductionmentioning

confidence: 88%

On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

Gamba,

Pavić-Čolić

2020

Preprint

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“…The generation results were improved to obtain exponential moments of order γ, while Gaussian moments were propagated for any initial data that would have that property, independent of γ. All these results were extended to the angular noncutoff regime (lack of angular integrability) in [21,15] still for hard potentials with γ ∈ (0, 2], and in [7,18] for pseudo-Maxwellian and Maxwellian case (γ = 0). In the later referenced work, these non-Gaussian tailed moments are called Mittag-Leffler moments as in fact the summability of partial sums is shown to converge to an L 1 -Mittag-Leffler function weighted norm for the unique probability density function solving the initial value problem associated to the Boltzmann equation, whose order and rate depend on the initial data as much as on the order of singularity in the angular section.…”

Section: Introductionmentioning

confidence: 92%

On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monatomic Gas Mixtures

Gamba

Pavić-Čolić

2019

Arch Rational Mech Anal

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We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates γ ∈ (0, 1], with a bounded angular section modeled by a bounded function of the scattering angle. The initial data for the vector valued solutions needs to be a vector of nonnegative positive measures that will have finite mass, momentum and strictly positive energy, as well as to have finite L 1 2+2γ (R 3 )-integrability corresponding to a sum across each species of 2 + 2γ-polynomial weighted norms depending in the corresponding mass fraction parameter for each species, referred as to the scalar polynomial moment of order 2 + 2γ. In addition there is no assumption on the finiteness of the initial system associated scalar entropy. This set of initial data assumptions allows for some of the species to be a singular mass initially.The existence and uniqueness proof relies on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with explicit constants that decay with the order of the corresponding scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.

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“…This yields Theorem 1.2 (a). It is worth mentioning that the derivation of exponential bounds for solutions to kinetic equations is a very active field of research nowadays, in particular for Boltzmann equations, see [1,11,10,17,18] and the references therein. Finally, we sketch in Section 5 the computations leading to the explicit solutions given in Proposition 1.1.…”

Section: 2)mentioning

confidence: 99%

The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

Laurençot¹,

Walker²

2021

KRM

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<p style='text-indent:20px;'>The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.</p>

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Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

Cited by 14 publications

References 13 publications

On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monatomic Gas Mixtures

The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

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