Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation

The purpose of this paper is to study asymptotic behaviors of the Green function of the linearized compressible Navier-Stokes equation. Liu, T.-P. and Zeng, Y. obtained a point-wise estimate for the Green function of the linearized compressible Navier-Stokes equation in [Comm. Pure Appl. Math. 47, 1053--1082 (1994)] and [Mem. Amer. Math. Soc. 125 (1997), no. 599]. In this paper, we propose a new methodology to investigate point-wise behavior of the Green function of the compressible Navier-Stokes equation. This methodology consists of complex analysis method and weighted energy estimate which was originally proposed by Liu, T.-P. and Yu, S.-H. in [Comm. Pure Appl. Math., 57, 1543--1608 (2004)] for the Boltzmann equation. We will apply this methodology to get a point-wise estimate of the Green function for large $t>0$.