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.