Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables

TL;DR

This paper investigates the long-time asymptotic behavior of damped compressible Euler equations, showing convergence to self-similar solutions and deriving explicit convergence rates using parabolic scaling and entropy methods.

Contribution

It introduces a novel analysis of the Euler equations with damping in parabolic variables, establishing convergence to porous medium solutions and Darcy's law, with explicit rates depending on profile flatness.

Findings

Density converges to a self-similar porous medium solution.

Momentum converges according to Darcy's law.

Explicit convergence rates depend on the flatness of the limit profile.

Abstract

We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting.