The Lifespan of Classical Solutions to the (Damped) Compressible Euler Equations

Abstract

In this paper, the initial-boundary value problem of the original three-dimensional compressible Euler equations with (or without) time-dependent damping is considered. By considering a functional $F(t,\alpha ,f)$ weighted by a general time-dependent parameter function $\alpha$ and a general radius-dependent parameter function f, we show that if the initial value $F|_{t=0}$ is sufficiently large, then the lifespan of the system is finite. Here, f can be any $C^1$ strictly increasing function such that the sum of initial values of f and $\alpha$ is non-negative. It follows that a class of conditions for non-existence of global classical solutions is established. Moreover, the conditions imply that a strong $\alpha$ will lead to a more unrestrained necessary condition for classical solutions of the system to exist globally in time.