The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect
The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect
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DOI：
10.1016/j.jde.2017.06.010
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发表时间：
2017-02
影响因子：
2.4
通讯作者：
Hao-Guang Li;Chao-Jiang Xu
中科院分区：
文献类型：
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作者：
Hao-Guang Li;Chao-Jiang Xu
本文研究了具有Maxwell分子的径向对称齐次非截断Boltzmann方程的Cauchy问题，初始数据属于负指数的Shubin空间，可以用谐振子的谱分解来刻画，它是Maxwell分布的一个小扰动.负指数的Shubin空间包含概率测度。在此谱分解的基础上，构造了具有Shubin类初值的Cauchy问题的弱解，并证明了该问题具有Gelfand-Shilov光滑效应，即光滑性质与分数阶谐振子演化方程所定义的Cauchy问题相同.
In this paper, we study the Cauchy problem for the radially symmetric homogeneous non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum belongs to Shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator, and it is a small perturbation of Maxwellian distribution. The Shubin space of the negative index contains the probability measures. Based on this spectral decomposition, we construct the weak solution with Shubin class initial datum, we also prove that the Cauchy problem enjoys Gelfand–Shilov smoothing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.