WIAS Preprint No. 3207, (2025)

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


Authors

  • Eiter, Thomas
    ORCID: 0000-0002-7807-1349
  • Schindler, Stefanie
    ORCID: 0000-0002-4005-7314

2020 Mathematics Subject Classification

  • 35B40 35C06 35Q31 76N10 76N15 76S05

Keywords

  • Euler equations, compressible fluid, frictional damping, porous medium equation, Darcy's law, self-similarity, parabolic scaling, relative entropy, energy variational solutions

DOI

10.20347/WIAS.PREPRINT.3207

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.

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