Remarks on 1-D Euler Equations with Time-Decayed Damping

We study the 1-d isentropic Euler equations with time-decayed damping \begin{equation} \left\{ \begin{aligned} &\partial_t \rho+\partial_x(\rho u)=0, \\ &\partial_t(\rho u)+ \partial_x(\rho u^2)+\partial_xp(\rho)=-\frac{\mu}{1+t}\rho u,\\ &\rho|_{t=0}=1+\varepsilon\rho_0(x),u|_{t=0}=\varepsilon u_0(x). \end{aligned} \right. \nonumber \end{equation} This work is inspired by a recent work of F. Hou, I. Witt and H.C. Yin \cite{Hou01}. In \cite{Hou01}, they proved a global existence and blow-up result of 3-d irrotational Euler flow with time-dependent damping. In the 1-d case, we will prove a different result when the damping decays of order $-1$ with respect to the time $t$. More precisely, when $\mu>2$, we prove the global existence of the 1-d Euler system. While when $0\leq\mu\leq2$, we will prove the blow up of $C^1$ solutions.