Time-Asymptotic Expansion with Pointwise Remainder Estimates for 1D Viscous Compressible Flow

Abstract

We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier–Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are a newly introduced family of waves which we call higher-order diffusion waves. In particular, these provide an accurate description of the power-law asymptotics of the solution around the origin x=0, where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the L^p({mathbb {R}})-norm for all 1le ple infty . The proof is based on pointwise estimates of Green’s function.