GREEN'S FUNCTION OF BOLTZMANN EQUATION, 3-D WAVES

We study the Green's function for the linearized Boltzmann equation. For the short-time period, the Green's function is dominated by the particle-like waves; and for large-time, by the fluid-like waves exhibiting the weak Huygens principle. The fluidlike waves are constructed by the spectral analysis and complex analytic techniques, making uses of the rotational symmetry of the equation in the space variables. The particle-like waves are constructed by a Picard iteration, making uses of the exchange of regularity in the microscopic velocity with the regularity in the space variables through a Mixture Lemma. We obtain the pointwise estimates in the space and time variables of the Green's function through a long-short waves and particle-wave decompositions.