无截断非齐次Boltzmann方程Gevrey正则性理论研究

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.

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。

在本项目中，我们以非齐次Boltzmann方程及其他相关类型方程为主要研究对象，重点研究非Maxwellian情形下，解在不同位势时的Gevrey正则性等性质。已完成的工作主要有在赋予初值满足一般性质量守恒、能量守恒以及熵守恒的情况下，经过对一般性质的碰撞算子进行适当的修正，我们得到了对应的带有Debye-Yukawa位势或强奇性逆幂律位势的线性化Cauchy问题解的局部Gevrey正则性。在其他诸如各类抽象空间及算子的性质、高维实欧氏空间中协方差不等式在多维随机变量的推广及其应用、带有变阶导数微分方程和Camassa-Holm型方程解的适定性等相关研究领域均取得了一定的突破。上述成果有一部分已发表或接受，有一部分仍在整理当中。此外，我们还整理并出版了其中所应用的泛函分析相关知识。