The time asymptotic expansion for the compressible Euler equations with damping

TL;DR

This paper rigorously justifies a refined time-asymptotic expansion for solutions of the compressible Euler equations with damping, improving understanding of their large-time behavior and convergence to diffusion waves.

Contribution

It introduces a rigorous justification of a new asymptotic expansion around the diffusion wave using Green function and energy methods, enhancing prior results.

Findings

The solution converges to the diffusion wave as time goes to infinity.

The asymptotic expansion provides a more accurate description of the large-time behavior.

The method combines Green function analysis with energy estimates.

Abstract

In 1992, Hsiao and Liu \cite{Hsiao-Liu-1} firstly showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave $(\bar v, \bar u)$ of the porous media equation. In \cite{Geng-Huang-Jin-Wu}, we proposed a time-asymptotic expansion around the diffusion wave $(\bar v, \bar u)$, which is a better asymptotic profile than $(\bar v, \bar u)$. In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time asymptotic expansion.