Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$

This paper concerns the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in $\R^2$. We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. For this end, the vector fields $\widehat{Z}$ (defined below) are used instead of the usual one $Z$, to get better decay for the linear error terms. This idea may also apply to the long time behavior study of nonlinear wave equations with time-dependent damping.