$L^1$-convergence to generalized Barenblatt solution for compressible Euler equations with time-dependent damping
The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(ρ)=κρ^γ, γ>1$) with time dependent damping like $-\frac{1}{(1+t)^λ}ρu$ ($0<λ<1$) 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 $λ$ increases in $(0, \fracγ{γ+2}]$, while is getting slower and slower in $[ \fracγ{γ+2}, 1)$.