非Grad截断条件下Boltzmann方程解的正则性研究

Without the Grad’s cutoff assumption, the Boltzmann operator behaves like a fractional Laplace operator. This means that the solutions of the Boltzmann equation may have smooth properties. Inspired by this conclusion, this project mainly studies the regularities of the solution of the corresponding Cauchy problem of the Boltzmann equation without angular cutoff in suitable hypotheses of the initial value. Those include the Sobolev regularity and the Gevrey regularity. Firstly, based on the work of R.Alexandre and others, we plan to further study the Sobolev regularity of the solution of the Boltzmann equation with Debye-Yukawa potential in the usual non-Maxwellian case. Secondly, we will consider the Gevrey regularity of the solution of the corresponding linearized and nonlinearized Cauchy problem with the inverse power law potential in the modified non-Maxwellian case. The solution of these problems will promote the development of the theory of the Boltzmann equation. The new ideas and methods will help the study of the regularity of the solutions for the related equations (e.g., Landau equation, Kac equation).

在无Grad截断假设情况中，Boltzmann算子类似于一个分数形式的Laplace算子，这蕴含了Boltzmann方程的解可能存在着某些光滑性质。受此启发，本项目主要研究一定初值条件下，无截断情形Boltzmann方程对应Cauchy问题解的正则性质，包括Sobolev正则性和Gevrey正则性。首先探讨通常的无修正非Maxwellian情形，基于R. Alexandre等人的工作，我们进一步研究带有Debye-Yukawa位势的Boltzmann方程解的Sobolev正则性。然后考虑修正后的非Maxwellian情形，带有逆幂律位势的对应线性化及非线性化Cauchy问题解的Gevrey正则性。这些问题的解决将促进Boltzmann方程理论的发展，从中开辟的新思路、新方法则有助于研究与Boltamann方程有紧密联系的诸如Landau方程、Kac方程等方程解的正则性。

在本项目中我们主要完成的工作是研究了与齐次Boltzmann方程相关的线性化问题解的Gevrey类光滑性质。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，我们对一般性质的碰撞算子进行适当的修正，并在此基础上推出齐次Boltzmann方程相关的线性化问题解是具有一定的Gevrey类光滑性质的。在其他相关研究领域，例如抽象空间的性质、各类不等式性质的推广及应用、分数阶微分方程解的定性理论等方面也取得了一系列成果，得到了新型的变指数Herz型Banach空间、Bochner-Lebesgue空间以及T-L空间的一系列性质。推广了Gronwall不等式，并将之应用在研究不同形式分数阶微分方程解的定性理论课题中。