我們將介紹在古典分數微分算子與雙曲微分算子下，對於氣體動力方程式解的平滑度估計。在第一章，我們用傅立葉變換來瞭解速度平均算子的結構與性質；在第二章，我們得到一些半徑為R的球體與球面估計；在第三章，我們會介紹對偶分析並利用其來簡化問題；最後，介紹一些在研究過程中懸而未解的問題。

We will introduce smoothing estimates for kinetic transport equations with classical and hyperbolic derivative operator. In section 1, we use the Fourier transformation to understand the structure and property of average velocity operator. We get some estimates on R-sphere and R-ball in section 2. In section 3, we will introduce dual analysis and use it to reduce the problem. Finally, we will introduce some unsolved problems that was founded in the process of research.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Represented  function with  function . . . . . . . . . . . . . . . . 1
2 Average velocity with dierent ranges . . . . . . . . . . . . . . . . . 2
2.1 Averages on r-spheres for transport I.V.P. . . . . . . . . . . . . . 2
2.2 Averages on r-circle for transport I.V.P . . . . . . . . . . . . . . 7
2.3 Average in R-balls for transport I.V.P . . . . . . . . . . . . . . . 9
3 Dual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Dual analysis about . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Dual analysis about . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

[1] Jonathan Bennett, Neal Bez, Susana Gutierrez, and Sanghyuk Lee. Estimates
for the kinetic transport equation in hyperbolic sobolev spaces. Journal de
Mathematiques Pures et Appliquees, 114:1{28, 2018.
[2] Nikolaos Bournaveas and Benoit Perthame. Averages over spheres for kinetic
transport equations; hyperbolic sobolev spaces and strichartz inequalities.
Journal de mathematiques pures et appliquees, 80(5):517{534, 2001.