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

2025/07/04 by Eiter, Thomas, Schindler, Stefanie · 1 citation
#35B40 #35C06 #35Q31 #76N10 #76N15 #76S05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.03688

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.

Citations

• Energy-variational solutions for viscoelastic fluid models
• Convergence to self-similar profiles in reaction-diffusion systems
• Existence of energy-variational solutions to hyperbolic conservation laws
• Scaling Variables and Stability of Hyperbolic Fronts
• The second law of thermodynamics and stability

Cited by

• Solution concepts for a model of visco-elasto-plasticity with slight compressibility

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