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

TL;DR

This paper investigates the conditions under which smooth solutions to multi-dimensional compressible Euler equations with time-dependent damping exist globally or blow up, identifying critical damping parameters that determine the solution behavior.

Contribution

It establishes the precise damping conditions that guarantee global existence or finite-time blowup of solutions, highlighting critical damping parameters for multi-dimensional Euler equations.

Findings

Global solutions exist for certain damping parameters

Solutions blow up in finite time under other damping conditions

Identifies critical damping thresholds for solution behavior

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_t\rho+\operatorname{div}(\rho u)=0, \quad \partial_t(\rho u)+\operatorname{div}\left(\rho u\otimes u+p\,I_d\right)=-\alpha(t)\rho u, \quad \rho(0,x)=\bar \rho+\varepsilon\rho_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 $\alpha(t)=\frac{\mu}{(1+t)^\lambda}$ with $\lambda\ge0$ and $\mu>0$, $\bar\rho>0$ is a constant, $\rho_0,u_0 \in C_0^\infty(\Bbb R^d)$, $(\rho_0,u_0)\not\equiv 0$, $\rho(0,x)>0$, and $\varepsilon>0$ is sufficiently small. One can totally divide the range of $\lambda\ge0$ and $\mu>0$ into the following four cases: Case 1: $0\le\lambda<1$, $\mu>0$ for $d=2,3$; Case 2: $\lambda=1$, $\mu>3-d$ for $d=2,3$; Case 3: $\lambda=1$, $\mu\le 3-d$ for $d=2$; Case 4: $\lambda>1$, $\mu>0$ for $d=2,3$. \noindent We show that there exists a global $C^{\infty}-$smooth solution $(\rho, u)$ in Case 1, and Case 2 with $\operatorname{curl} u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(\rho, u)$ blows up in finite time. Therefore, $\lambda=1$ and $\mu=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(\rho, u)$ in $d-$dimensional compressible Euler equations with time-depending damping.