2019
DOI: 10.1137/17m115116x
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Cauchy Problem for the Spatially Homogeneous Landau Equation with Shubin Class Initial Datum and Gelfand--Shilov Smoothing Effect

Abstract: 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… Show more

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Cited by 15 publications
(22 citation statements)
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“…For Landau equation, Villani in [2] constructed a linear equation for the homogeneous Landau equation and MPX in [12] proved that the solution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R 3 ). Recently, Li and Xu in [13] showed that global existence and Gelfand-Shilov regularizing properties of the solution to the Cauchy problem (1.2) for homogeneous non-cutoff Landau equation in Maxwellian molecules. From now on, the Gelfand-Shilov smoothing effect have never been studied in the non-Maxwellian case.…”
Section: Introductionmentioning
confidence: 99%
“…For Landau equation, Villani in [2] constructed a linear equation for the homogeneous Landau equation and MPX in [12] proved that the solution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R 3 ). Recently, Li and Xu in [13] showed that global existence and Gelfand-Shilov regularizing properties of the solution to the Cauchy problem (1.2) for homogeneous non-cutoff Landau equation in Maxwellian molecules. From now on, the Gelfand-Shilov smoothing effect have never been studied in the non-Maxwellian case.…”
Section: Introductionmentioning
confidence: 99%
“…Morimoto and Xu [28] proved the ultra-analytic effect for the Cauchy problem of linear Landau equation in the case of γ = 0. Li and Xu [26] studied the nonlinear Landau operator by introducing the spectral analysis.…”
mentioning
confidence: 99%
“…Preliminary: Analysis of Landau collision operator. For convenience of reader, we present the spectral properties for Landau operator briefly, see [8,23,25,26] for more details. First of all, one has an explicit expression for the linearized Landau operator with Maxwellian molecules.…”
mentioning
confidence: 99%
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