On Pointwise Exponentially Weighted Estimates for the Boltzmann Equation

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

doi:10.1137/18m1213191

Cited by 11 publications

(6 citation statements)

References 38 publications

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“…L ∞ v moments) were considered in [15,16] for the homogeneous (cutoff) equation, and pointwise exponential decay was established in [17]. The latter result was extended to the non-cutoff homogeneous equation in [18]. It should be noted that the conditional assumptions (1.6) are not necessary in the space homogeneous case, because the mass M f (t) and energy E f (t) are conserved by the evolution of the equation, and the entropy H f (t) is nonincreasing.…”

Section: Related Workmentioning

confidence: 99%

Velocity decay estimates for Boltzmann equation with hard potentials

Cameron

Snelson

2020

Nonlinearity

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We establish pointwise polynomial decay estimates in velocity space for the spatially inhomogeneous Boltzmann equation without cutoff, in the case of hard potentials (γ + 2s > 2), under the assumption that the mass, energy, and entropy densities are bounded above, and the mass density is bounded below. These estimates are self-generating, i.e. they do not require corresponding decay assumptions on the initial data. Our results extend the recent work of Imbert et al [2020 J. École Polytech. 7 143-83], which addressed the case of moderately soft potentials (γ + 2s ∈ [0, 2]).

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Section: Related Workmentioning

confidence: 99%

Velocity decay estimates for Boltzmann equation with hard potentials

Cameron

Snelson

2020

Nonlinearity

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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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“…x L 1 (1 + |v| q ). Another line of research opened by [60] consists in establishing exponential Gaussian pointwise decay by maximum principle arguments (see also [22,13,61]). But these works assumes exponential integral moments whose propagation in time is not known, therefore it is not clear how to use them in this context.…”

Section: Weak Harnack Inequality and Local Hölder Regularitymentioning

confidence: 99%

De Giorgi-Nash-Moser and H{ö}rmander theories: new interplay

Mouhot

2018

Preprint

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We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows Hölder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of Hörmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent extensions of the De Giorgi-Nash-Moser theory to hypoelliptic equations of Kolmogorov (kinetic) type with rough coefficients. These equations combine a first-order skewsymmetric operator with a second-order elliptic operator involving derivatives in only certain variables, and with rough coefficients. We then discuss applications to the Boltzmann and Landau equations in kinetic theory and present a program of research with some open questions.

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On Pointwise Exponentially Weighted Estimates for the Boltzmann Equation

Cited by 11 publications

References 38 publications

Velocity decay estimates for Boltzmann equation with hard potentials

Velocity decay estimates for Boltzmann equation with hard potentials

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

De Giorgi-Nash-Moser and H{ö}rmander theories: new interplay

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