On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping
Fei Hou, Huicheng Yin
TL;DR
The paper analyzes global existence versus finite-time blowup for the multi-dimensional compressible Euler equations with time-dependent damping $\alpha(t)=\frac{\mu}{(1+t)^\lambda}$ in $d=2,3$; it reformulates the system via $\theta$ and the vorticity $w$, and develops time-weighted energy estimates and damped-wave analysis to obtain global regularity in Case 1 and, in 2D, Case 2 under $\curl u_0\equiv0$. For Cases 3 and 4, the authors prove finite-time blowup by a moment method that uses a Riemann representation and a hypergeometric function, yielding explicit lifespan bounds that depend on $\varepsilon$. The results identify $\lambda=1$ and $\mu=3-d$ as critical thresholds for the global existence of small-amplitude smooth solutions and extend prior irrotational results to rotational, multi-dimensional flows. The work clarifies how time-dependent damping can prevent or fail to prevent singularity formation, with implications for damping mechanisms in compressible flows. Overall, the paper provides a detailed, rigorous framework for understanding the interplay between damping, vorticity, and nonlinear wave propagation in compressible Euler dynamics.
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.