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

Abstract: 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 $\gamma\in(1,+\infty)$ 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 $\gamma\in(1,+\infty)$, 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.