Time-asymptotic expansion with pointwise remainder estimates for 1D viscous compressible flow

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 newly introduced family of waves which we call $\mathit{\text{higher-order}}\mathit{\text{diffusion}}\mathit{\text{waves}}$. In particular, these provide 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 $1\le p\le \mathrm{\infty }$. The proof is based on pointwise estimates of Green’s function.