  Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative
Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$
has nonnegative isotropic curvature. The lower bound of the Ricci flow's
existence time on $(M^n, g)$ is proved. This provides an alternative proof for
the uniform lower bound of a family of closed Ricci flows' maximal existence
times, which was firstly proved by E. Cabezas-Rivas and B. Wilking. We also get
an interior curvature estimates for $n= 3$ under $Rc\geq 0$ assumption among
others. Combining these results, we proved the short time existence of the
Ricci flow on a large class of $3$-dim open manifolds, which admit some
suitable exhaustion covering and have nonnegative Ricci curvature.
