Solving the non-isentropic Navier-Stokes equations in odd space dimensions: The Green function method

In this paper, we investigate the wave structure of the solution for the compressible Navier-Stokes equations in both one space dimension and three space dimensions. First, we give the pointwise estimate on the Green function for the linearized system by dividing the physical domain into two parts, which are the finite Mach number region and outside finite Mach number region. We use long-wave short-wave decomposition and a weighted energy estimate method in each region separately. The Green function provides a complete picture of the wave pattern, with explicit leading terms. Then with the help of the Duhamel principle, we give the pointwise estimate of the solution for the nonlinear problems (1.1) and (1.2). Specially, we find that the decay rate in L-p( R-3) (2 < p <= infinity) for the energy ! <(omega)over tilde> is faster than those for the density (rho) over tilde and the momentum (m) over tilde. Published by AIP Publishing.