无角截断Boltzmann方程解的正则性研究

本项目拟研究不带角截断的空间非齐次Boltzmann方程解的高阶正则性以及Gevrey正则性。该方程的本质是退化椭圆方程，研究的难点在于速度方向的非局部性与空间方向的退化耦合作用。本项目拟具体研究的内容为：(1)利用次椭圆估计和速度加权方法，研究不带角截断的空间非齐次Boltzmann方程在软势情况下经典解的Gevrey正则性以及在速度方向最佳的Gevrey指标。(2)不带角截断的空间非齐次Boltzmann方程Mild解及L^\infty弱解的正则性。这里拟采用微局部分析和象征运算的方法，期待得到Mild解及L^\infty弱解的高阶正则性，并进一步得到Gevrey光滑性效应。

In this project, we will investigate the high-order regularity and Gevrey regularity of solutions to the spatially inhomogeneous and and non-cutoff Boltzmann equation. It is a kind of degenerate elliptic equation, the main difficulty comes from the non local property of collision operator coupled with the degeneracy in spatial variable. This project will focus on the following topics:(1)We make use of the subelliptic estimates and velocity weights, to explore the Gevrey regularity and the best Gevrey index to the classical solution of spatially inhomogeneous and non-cutoff Boltzmann equation with soft potential. (2)We will explore the regularity of Mild solution and L^\infty weak solution to spatially inhomogeneous and Non-cutoff Boltzmann equation. Here, the main tool shall be the microlocal analysis and the symbolic calculus. We aim to get the higher order regularity of Mild solution and L^\infty weak solution and then obtain Gevrey smoothing effects.