Sharp $L^1$-convergence rates to the Barenblatt solutions for the compressible Euler equations with time-varying damping

We study the asymptotic behavior of compressible isentropic flow when the initial mass is finite and the friction varies with time, which is modeled by the compressible Euler equation with time-dependent damping. In this paper, we obtain the best $L^1$-convergence rates to date, for any $\gamma \in (1,+\mathrm{\infty })$ and $\nu \in [0,1)$. Here, $\gamma$ is the adiabatic gas exponent, and $\nu$ is the physical parameter in the damping term. The key to the analysis lies in a new perspective on the relationship between the density function and the Barenblatt solution of the porous medium equation, and finding the relevant lower bound for the case of $\gamma <2$ is a tricky problem. Specialized to $\nu =0$, these convergence rates also show an essential improvement over the original rates. Moreover, for all $\gamma \in (1,+\mathrm{\infty })$, the results in this work are the first to present a unified form of $L^1$-convergence rates. Indeed, even for $\nu =0$, as noted in 2011, ``the current rate is difficult to improve with the current method". Our results are therefore an encouraging advancement.