$L^1$-convergence to generalized Barenblatt solution for compressible Euler equations with time-dependent damping

TL;DR

This paper proves that solutions to the compressible Euler equations with time-dependent damping converge in L^1 to a generalized Barenblatt solution of the porous media equation, with decay rates depending on the damping parameter.

Contribution

It introduces an iterative and entropy-based method to establish L^1 convergence of entropy solutions to a generalized Barenblatt solution for damped Euler equations.

Findings

Solutions converge strongly in L^1 to the generalized Barenblatt solution.

The decay rate in L^1 varies with the damping parameter λ.

Faster decay as λ increases within a specific range.

Abstract

The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma}, \gamma>1$) with time dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$ ($0<\lambda<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is getting faster and faster as $\lambda$ increases in $(0, \frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[ \frac{\gamma}{\gamma+2}, 1)$.