Gelfand–Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation

Lerner, Nicolás; Morimoto, Yoshinori; Pravda-Starov, Karel; Xu, Chao-Jiang

doi:10.1016/j.jfa.2015.04.017

Cited by 29 publications

(37 citation statements)

References 39 publications

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“…Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.…”

mentioning

confidence: 81%

“…The procedure is standard, which is similar to that of Theorem 4.2 in [18]. Here, we omit details for simplicity.…”

Section: The Local Existence Of Weak Solutionmentioning

confidence: 99%

“…In order to show Theorem 3.3, we need to develop the regularization method in [18]. The proof can be divided into several steps for clarity.…”

Section: The Local Existence Of Weak Solutionmentioning

confidence: 99%

“…Recently, Lerner-Morimoto-Starov-Xu [18] considered the spatially inhomogeneous non-cutoff Kac equation in the Sobolev space and showed that the Cauchy problem for the fluctuation around the Maxwellian distribution admitted S…”

Section: Introductionmentioning

confidence: 99%

“…Gelfand-Shilov regularity properties with respect to the velocity variable and G 1+ 1 2s Gevrey regularizing properties with respect to the position variable. In [18], the authors conjectured that it remained still open to determine whether the regularity indices 1 + 1 2s is sharp or not. On the other hand, Duan-Liu-Xu [12] and Morimoto-Sakamoto [22] studied the Cauchy problem for the Boltzmann equation with the initial datum belonging to critical Besov space.…”

Section: Introductionmentioning

confidence: 99%

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The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Cao

et al. 2020

Journal of Differential Equations

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In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

show abstract

mentioning

confidence: 81%

“…The procedure is standard, which is similar to that of Theorem 4.2 in [18]. Here, we omit details for simplicity.…”

Section: The Local Existence Of Weak Solutionmentioning

confidence: 99%

“…In order to show Theorem 3.3, we need to develop the regularization method in [18]. The proof can be divided into several steps for clarity.…”

Section: The Local Existence Of Weak Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Cao

et al. 2020

Journal of Differential Equations

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Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

Barbaroux

Hundertmark

Ried

et al. 2017

Arch Rational Mech Anal

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Abstract. It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in Le., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

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Microlocal analysis for Gelfand–Shilov spaces

Rodino

Wahlberg

2023

Annali di Matematica

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We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

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Gelfand–Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation

Cited by 29 publications

References 39 publications

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

Microlocal analysis for Gelfand–Shilov spaces

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