R202, Astronomy-Mathematics Building, NTU
(台灣大學天文數學館 202室)
Boltzmann collision operator for the infinite range potential
Jin-Cheng Jiang (National Tsing Hua University)
The conventional Boltzmann collision operator for the infinite range inverse power law model is derived by adopting a kernel which is a limit of the calculation of collision kernel for the finite range model with assumption that glancing angles can be ignored. The purpose of this study is to clarify the physical meaning of this collision operator. We first prove that it is legitimate to ignore glancing angles under suitable conditions. Furthermore we prove that taking limit to collision operator with finite range potential directly will lead to the conventional one. These results endow a physical meaning to the conventional collision operator regarding the fact that Boltzmann equation has been rigorously derived when the inter-molecular potential is of finite range by Grad in 1958.
The analysis depends on a new method to estimate the upper bound of collision operator. For our purpose, this new method is required since the analysis must show how the quantity of upper bound varies with the range of potential. The core of analysis turns out relying on the study of a specify kind of Radon transforms which are of independent interest. The rich structure of these transforms are explored; each of it can expressed as the summation of a pseudo differential operator, Fourier integral operators, their transforms and degenerates. Each Radon transform as a whole satisfies the diffusion estimates in Sobolev space. The structural description of these transforms suggests the norm spaces to discuss the problem and enables us to see how the quantity of upper bound varies from finite range to infinite range in the suitable Sobolev spaces. This is a joint work with Tai-Ping Liu and Yoshio Sone.
