前期研究表明，Boltzmann方程在无截断情形中，其算子类似一个分数形式的Laplace算子，这意味着其解可能有良好的正则性。赋予各种不同的前提条件，解的Sobolev正则性已在不少文献中证实。然而对于更强的Gevrey正则性，仍有许多待解决的问题。申请人近年来研究齐次型Boltzmann方程解在局部及全欧氏空间中的Gevrey正则性，取得了一系列成果。因此在本项目中拟选取适当的拟微分算子并利用调和分析等方法，进一步探讨带有Debye-Yukawa位势或强奇性的逆幂律位势等位势的非齐次Boltzmann方程的Gevrey正则性问题。期望能在无截断假设及非Maxwellian的情形下，得到方程对应线性及非线性Cauchy问题解的Gevrey正则性。此外还希望在解决问题过程中找到有效的新思路，能够很好地应用于Landau方程、Kac方程和Fokker-Planck方程等相关方程的课题研究中。

Previous studies have shown that the Boltzmann operator is similar to a fractional Laplacian operator in the non-cutoff case. This condition means that the solutions of the Boltzmann equation may have some superior regularities. Under a variety of different preconditions, the Sobolev regularity of the solutions has been verified in many literatures. However, lots of unsolved problems exist in the more strongly typed Gevrey regularity. Recently, the applicant has studied the Gevrey regularity on the velocity variable of the solutions of the homogeneous Boltzmann equation in local and global Euclidean spaces and made a series of achievements. Therefore, by selecting several suitable pseudo differential operators and applying methods such as harmonic analysis, this project aims to further discuss the problem of Gevrey regularity for inhomogeneous Boltzmann equation with Debye-Yukawa potential or the inverse power law potential under the hypothesis of strong singularity. We expect that the Gevrey regularity for the solutions of the corresponding linearized and nonlinear Cauchy problems can be concluded in non-cutoff and non-Maxwellian cases. Moreover, we also wish to identify some novel useful ideas in the problem solving process which can be applied in studies on the issues of the equations that are closely related to the Boltzmann equation, such as Landau, Kac and Fokker-Planck equations.