  We study the heat equation on time-dependent metric measure spaces (as well
as the dual and the adjoint heat equation) and prove existence, uniqueness and
regularity. Of particular interest are properties which characterize the
underlying space as a super Ricci flow as previously introduced by the second
author. Our main result yields the equivalence of (i) dynamic convexity of the
Boltzmann entropy on the (time-dependent) $L^2$-Wasserstein space; (ii)
monotonicity of $L^2$-Kantorovich-Wasserstein distances under the dual heat
flow acting on probability measures (backward in time); (iii) gradient
estimates for the heat flow acting on functions (forward in time); (iv) a
Bochner inequality involving the time-derivative of the metric.
  Moreover, we characterize the heat flow on functions as the unique forward
EVI-flow for the (time-dependent) energy in $L^2$-Hilbert space and the dual
heat flow on probability measures as the unique backward EVI-flow for the
(time-dependent) Boltzmann entropy in $L^2$-Wasserstein space.
