速度平均引理及其在波茲曼方程的應用

Abstract

1988年時，Golse、Lions、Perthame和Sentis證明了速度平均會增加函數正則性，這個現象後來被稱作「速度平均引理」，速度平均引理在DiPerna和Lions的波茲曼方程之柯西問題整體解存在性理論有重要的應用，而其自身也有許多富有意義的延伸，在此論文中，我們首先回顧經典的速度平均引理，再透過速度平均效應導出我們關於穩態線性化波茲曼方程正則性的新結果。

In 1988, Golse, Lions, Perthame and Sentis jointly proved that velocity averaging has regularizing effects. This phenomenon was later called “Velocity Averaging Lemmas.” The Velocity Averaging Lemmas have significant applications in the global existence theory of the Cauchy problem for Boltzmann equations by DiPerna and Lions and also have many meaningful extensions themselves. In this thesis, we first review classical Velocity Averaging Lemmas, and then we present our new regularity results for the stationary linearized Boltzmann equation by velocity averaging effects.