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

Abstract

AbstractWe 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 $$x=0$$ x = 0 , where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the $$L^p({\mathbb {R}})$$ L p ( R ) -norm for all $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . The proof is based on pointwise estimates of Green’s function.