  We establish effective existence and uniqueness for the heat flow on
time-dependent Riemannian manifolds, under minimal assumptions tailored towards
the study of Ricci flow through singularities. The main point is that our
estimates only depend on an upper bound for the logarithmic derivative of the
volume measure. In particular, our estimates hold for any Ricci flow with
scalar curvature bounded below, and such a lower bound of course depends only
on the initial data.
