The time asymptotic expansion for the compressible Euler equations with time-dependent damping

TL;DR

This paper develops a time asymptotic expansion for the compressible Euler equations with time-dependent damping, providing a rigorous justification of the expansion as a precise asymptotic profile for solutions.

Contribution

It introduces a novel asymptotic expansion around the GPME self-similar solution for the Euler equations with damping, valid for bb rac{1}{7}, and rigorously justifies this expansion.

Findings

Established the validity of the asymptotic expansion for bb , bb , and bb .

Identified the best asymptotic profile of solutions to the damped Euler equations.

Connected the behavior of solutions to the generalized porous media equation.

Abstract

In this paper, we study the compressible Euler equations with time-dependent damping $-\frac{1}{(1+t)^{\lambda}}\rho u$. We propose a time asymptotic expansion around the self-similar solution of the generalized porous media equation (GPME) and rigorously justify this expansion as $\lambda \in (\frac17,1)$. In other word, instead of the self-similar solution of GPME, the expansion is the best asymptotic profile of the solution to the compressible Euler equations with time-dependent damping.