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

Abstract

We study the 1-d isentropic Euler equations with time-decayed damping \begin{equation} \left\{ \begin{aligned} &\partial_t ρ+\partial_x(ρu)=0, \\ &\partial_t(ρu)+ \partial_x(ρu^2)+\partial_xp(ρ)=-\fracμ{1+t}ρu,\\ &ρ|_{t=0}=1+\varepsilonρ_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 $0\le \mu \le 2$, we will prove the blow up of $C^1$ solutions.