  In this article, we prove a local Sobolev inequality for complete Ricci
flows. Our main result is that the local $\nu$-functional of a disk on a Ricci
flow depends only on the Nash entropy based at the center of the disk, and
consequently depends only on the volume of the disk. Furthermore, we introduce
some applications of this local Sobolev inequality. These applications reveal
the way in which the local geometry evolves along Ricci flow. In particular, we
show that several classical theorems related to Perelman's monotonicity formula
can be derived from our results.
