Green’s function and pointwise convergence for compressible Navier-Stokes equations

In this paper we introduce a program to construct the Green’s function for the linearized compressible Navier-Stokes equations in several space dimensions. This program contains three components, a procedure to isolate global singularities in the Green’s function for a multi-spatial-dimensional problem, a long wave-short wave decomposition for the Green’s function and an energy method together with Sobolev inequalities. These three components together split the Green’s function into singular and regular parts with the singular part given explicitly and the regular part bounded by exponentially sharp pointwise estimates. The exponentially sharp singular-regular description of the Green’s function together with Duhamel’s principle and results of Matsumura-Nishida on $L^\infty$ decay yield through a bootstrap procedure an exponentially sharp space-time pointwise description of solutions of the full compressible Navier-Stokes equations in ${\mathbb R}^n(n=2,3)$.