On Mittag-Leffler Moments for the Boltzmann Equation for Hard Potentials Without Cutoff

Tasković, Maja; Alonso, Ricardo; Gamba, Irene M.; Pavlović, Nataša

doi:10.1137/17m1117926

Cited by 23 publications

(41 citation statements)

References 31 publications

(63 reference statements)

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“…(v) The non-cutoff Boltzmann equation for hard potentials γ > 0 was studied in [11] where generation of stretched exponential moments of order s = γ was proved, and [13] where propagation of stretched exponential moments was proved depending on the singularity rate of the angular kernel. We extend the work of [13] to include the case γ = 0.…”

Section: The Main Resultsmentioning

confidence: 99%

“…Finiteness of such sums can be studied by proving term-by-term geometric decay, or by showing that partial sums are uniformly bounded. Our proof is inspired by the works [1,13], where the partial sum approach is developed. Moreover, motivated by [13], we exploit the notion of Mittag-Leffler moments, which serve as a generalization of stretched exponential moments and which are very flexible for the calculations at hand.…”

Section: Introductionmentioning

confidence: 99%

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

Pavić-Čolić¹,

Tasković²

2018

Kinetic &Amp; Related Models

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We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its 1-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Pavić-Čolić¹,

Tasković²

2018

Kinetic &Amp; Related Models

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“…Proof. We briefly point out that the main steps in the proofs are adaption of the proof given in [21]. Let us consider polynomial moment m δq [F](t) =: m δq , 0 < δ ≤ 2, q ≥ 0, and derive an ODI for it starting from (6.5) with k = δq.…”

Section: Generation and Propagation Of Exponential Momentsmentioning

confidence: 99%

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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“…We refer to [30] for a review of the classical results, as well as to e.g. [3,5,17,25,27] and references therein. The rough picture is that, for hard potentials (including hard spheres), the space homogeneous Boltzmann equation generates higher-order moments instantaneously as soon as the initial energy is finite.…”

Section: Introductionmentioning

confidence: 99%

Moments and regularity for a Boltzmann equation via Wigner transform

Chen¹,

Denlinger²,

Pavlović³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain R d , d ≥ 2, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform whenWe prove that if α, β are large enough, then it is possible to propagate moments in x and derivatives in v (for instance,The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of f . We also prove a persistence of regularity result for the scale of Sobolev spaces H α,β ; and, continuity of the solution map in H α,β . Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the H-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in H α,β for any α, β > d−1 2 , without any additional regularity or decay assumptions.

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On Mittag-Leffler Moments for the Boltzmann Equation for Hard Potentials Without Cutoff

Cited by 23 publications

References 31 publications

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

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

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

Moments and regularity for a Boltzmann equation via Wigner transform

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