波尔兹曼方程是数学物理学中的一个基本方程，是非平衡态统计物理学的重要模型之一。它描述仅考虑二元碰撞这一相互作用的理想状态下，稀薄气体的运动行为。 波尔兹曼方程不仅具有深刻的物理背景，而且作为非线性偏微分方程具有重大的数学意义。 从数学上讲，波尔兹曼方程是一个积分微分方程。最近十几年，波尔兹曼方程解的正则性研究是许多工作的重点。 Lerner, Morimoto, Pravda-Starov, Xu在《非截断的径向对称的空间齐性波尔兹曼方程的Gelfand–Shilov光滑性》及Glangetas, Li, Xu在《非截断空间齐性波尔兹曼方程的最优正则性》中证明了非截断空间齐性波尔兹曼方程的柯西问题解属于Gelfand–Shilov类，受上述工作启发，本项目利用谱分解和傅里叶分析等技巧研究Debye-yukawa位势下空间齐性波尔兹曼方程解的定性理论，包括光滑性以及收敛于平衡态的速度等问题。

The Boltzmann equation is a basic equation of the mathematical physics and one of the most important models for the non-equilibrium statistical physics. It describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. The Boltzmann equation not only have rich physical backgrouds, but also as one of the nonliear partial differential equations, it is very important in mathematics . From mathematical viewpoint, the Botlzmann equation is an integro-differential equation. . Recently, the research about the regularity of the solutions for the Boltzmann equation has been the object of numerous articles. Inspired by the works of N. Lerner, Y. Morimoto, K. Pravda-Starov, C.-J. Xu, Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff and Leo Glangetas, Hao-Guang Li, Chao-Jiang Xu, Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation that the solution of the Cauchy problem for the non-cutoff spatially homogeneous Boltzmann equation belongs to the Gelfand-Shilov class, this foundation study qualitative theory which is concerned with the study of the speed of convergence to equilibrium and the smoothing properties for the spatially homogeneous Boltzmann equation with Debye-Yukawa potential by using the methods of Bobylov formula, Spectral decompostion and Fourier analysis.