Sharper L^1-convergence rates of weak entropy solutions to damped compressible Euler equations

We consider the asymptotic behavior of compressible isentropic flow when the initial mass is finite, which is modeled by the compressible Euler equation with frictional damping. It is shown in \cite{HUA} (resp.\cite{GEN}) that any $L^{\mathrm{\infty }}$ weak entropy solution of damped compressible Euler equation converges to the Barenblatt solution with finite mass in $L^1$ norm, with convergence rates depending on the adiabatic gas exponent $\gamma$ in the case of $1<\gamma <3$ (resp.$\gamma \ge 2$). Whether or not these convergence rates can be improved remains an interesting and challenging open question. In this paper, we obtain a better $L^1$ convergence rate than that in \cite{GEN}, for any $\gamma \ge 2$, through a new perspective on the relationship between the density function and the Barenblatt solution of the porous medium equation. Furthermore, making intensive analysis of some relevant convex functions, we are able to obtain the same form of $L^1$ convergence rate for $1<\gamma <9\mathrm{/}7$, which is better than that in \cite{HUA} as well.