Abstract：

We study the propagation of singularities in the initial layer of the solutions of the Boltzmann equation in the space-time domain as well as the microscopic velocity domain. We draw upon the important works of Hilbert and Carleman on the collision operators. For the full Boltzmann equation, singularities transport in the space-time domain and interacts with nonlinear collision operator in the microscopic velocity domain.  We identify the essential features of the collision operators with the aid of the Carleman-Hilbert coordinates so that the Green’s function approach is sufficient to yield the explicit structure of the singularities in the initial layers