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

TL;DR

This paper develops a detailed time-asymptotic expansion with pointwise error bounds for solutions to 1D viscous compressible flow, introducing higher-order diffusion waves to improve the understanding of solution decay and behavior.

Contribution

It introduces a novel family of higher-order diffusion waves for 1D compressible Navier--Stokes equations, providing precise pointwise and global asymptotic descriptions.

Findings

The expansion accurately describes solution decay near the origin.

It is valid both locally and globally in L^p norms.

The approach uses Green's function pointwise estimates.

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 newly introduced family of waves which we call \textit{higher-order diffusion 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 $1\leq p\leq \infty$. The proof is based on pointwise estimates of Green's function.