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Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

For the compressible Euler equations, even when the initial data are uniformly away from vacuum, solution can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this paper, for the initial value problems of isentropic and full Euler equations in one space dimension, assuming initial density has positive lower bound, we prove that density functions in classical solutions have positive lower bounds in the order of $\textstyle O(1+t)^{-1}$ and $\textstyle O(1+t)^{-1-δ}$ for any $\textstyle 0<δ\ll 1$, respectively, where $t$ is time. The orders of these bounds are optimal or almost optimal, respectively. Furthermore, for classical solutions in Eulerian coordinates $(y,t)\in\mathbb{R}\times[0,T)$, we show velocity $u$ satisfies that $u_{y}(y,t)$ is uniformly bounded from above by a constant independent of $T$, although $u_{y}(y,t)$ tends to negative infinity when gradient blowup happens, i.e. when shock forms, in finite time.