On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_d\right)=-α(t)ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), \end{equation*} where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is $α(t)=\fracμ{(1+t)^λ}$ with $λ\ge0$ and $μ>0$, $\barρ>0$ is a constant, $ρ_0,u_0 \in C_0^\infty(\Bbb R^d)$, $(ρ_0,u_0)\not\equiv 0$, $ρ(0,x)>0$, and $\varepsilon>0$ is sufficiently small. One can totally divide the range of $λ\ge0$ and $μ>0$ into the following four cases: Case 1: $0\leλ<1$, $μ>0$ for $d=2,3$; Case 2: $λ=1$, $μ>3-d$ for $d=2,3$; Case 3: $λ=1$, $μ\le 3-d$ for $d=2$; Case 4: $λ>1$, $μ>0$ for $d=2,3$. \noindent We show that there exists a global $C^{\infty}-$smooth solution $(ρ, u)$ in Case 1, and Case 2 with $\operatorname{curl} u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(ρ, u)$ blows up in finite time. Therefore, $λ=1$ and $μ=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(ρ, u)$ in $d-$dimensional compressible Euler equations with time-depending damping.