Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

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

doi:10.1016/j.jde.2013.10.001

Cited by 42 publications

(48 citation statements)

References 26 publications

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“…The proof of this Proposition is similar to Lemma 3.5 in [7], Proposition 3.2 in [8] and Section 3 in [3].…”

Section: The Trilinear Estimates For Non Linear Operatormentioning

confidence: 61%

Cauchy Problem for the Spatially Homogeneous Landau Equation with Shubin Class Initial Datum and Gelfand--Shilov Smoothing Effect

Li¹,

Xu²

2019

SIAM J. Math. Anal.

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In this work, we study the nonlinear spatially homogeneous Landau equation with Maxwellian molecules, by using the spectral analysis, we show that the non linear Landau operators is almost linear, and we prove the existence of weak solution for the Cauchy problem with the initial datum belonging to Shubin space of negative index which conatins the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys S 1 2 1 2 -Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with Shubin class initial datum is ultra-analytics and exponential decay for any positive time.We shall study the linearization of the Landau equation (1.1) near the absolute Maxwellian distribution µ(v) = (2π) − 3 2 e − |v| 2 2 . Considering the fluctuation of density distribution function1 since Q L (µ, µ) = 0, the Cauchy problem (1.1) is reduced to the Cauchy problemand the Shubin spaces, for β ∈ R, (see [15], Ch. IV, 25.3),We havewhere H β (R 3 ) is the usual Sobolev spaces. In particular, for β < − 3 2 the Shubin space Q β (R 3 ) contains the probability mesures (see [2, 9, 12, 17] and [8]). See Appendix 6 for more properties of Gelfand-Shilov spaces and the Shubin spaces. 2

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“…The proof of this Proposition is similar to Lemma 3.5 in [7], Proposition 3.2 in [8] and Section 3 in [3].…”

Section: The Trilinear Estimates For Non Linear Operatormentioning

confidence: 61%

Cauchy Problem for the Spatially Homogeneous Landau Equation with Shubin Class Initial Datum and Gelfand--Shilov Smoothing Effect

Li¹,

Xu²

2019

SIAM J. Math. Anal.

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“…More recently, Gelfand-Shilov spaces were used in [9,10] to describe exponential decay and holomorphic extension of solutions to globally elliptic equations, and in [48] in the regularizing properties of the Boltzmann equation. We refer to [52] for a recent overview and for applications in quantum mechanics and traveling waves, and to [78] for the properties of the Bargmann transform on Gelfand-Shilov spaces.…”

Section: Gelfand-shilov Spacesmentioning

confidence: 99%

The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces

Teofanov

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This paper offers a review of the results concerning localization operators on modulation spaces, and related topics. However, our approach, based on the Grossmann-Royer transform, gives a new insight and (slightly) different proofs. We define the Grossmann-Royer transform as interpretation of the Grossmann-Royer operator in the weak sense. Although such transform is essentially the same as the cross-Wigner distribution, the proofs of several known results are simplified when it is used instead of other time-frequency representations. Due to the importance of their role in applications when dealing with ultrafast decay properties in phase space, we give a detailed account on the Gelfand-Shilov spaces and their dual spaces, and extend the Grossmann-Royer transform and its properties in such context. Another family of spaces, modulation spaces, are recognized as appropriate background for time-frequency analysis. In particular, the Gelfand-Shilov spaces are revealed as projective and inductive limits of modulation spaces. For the continuity and compactness properties of localization operators we employ the norms in modulation spaces. We define localization operators in terms of the Grossmann-Royer transform, and show that such definition coincides with the usual definition based on the short-time Fourier transform.

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“…The linearized operator L is a positive unbounded symmetric operator on L 2 (R 3 v ) (see [11,29,30,31]) with the kernel…”

Section: Linearization Of the Boltzmann Equationmentioning

confidence: 99%

“…for q ≥ 1. Following [31], we therefore derive from (2.1) the following infinite system of ordinary differential equations :…”

Section: 2mentioning

confidence: 99%

Numerical computation for the non-cutoff radially symmetric homogeneous Boltzmann equation

Glangetas

Jrad

2018

Communications in Mathematical Sciences

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Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

Cited by 42 publications

References 26 publications

Cauchy Problem for the Spatially Homogeneous Landau Equation with Shubin Class Initial Datum and Gelfand--Shilov Smoothing Effect

Cauchy Problem for the Spatially Homogeneous Landau Equation with Shubin Class Initial Datum and Gelfand--Shilov Smoothing Effect

The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces

Numerical computation for the non-cutoff radially symmetric homogeneous Boltzmann equation

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