  In this paper, we prove logarithmic Sobolev inequalities and derive the
Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on
complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We
establish the $W$-entropy formula for the heat equation of the Witten Laplacian
and prove a rigidity theorem on complete Riemannian manifolds satisfying the
$CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent
Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman
Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are two constants.
Finally, we prove the Li-Yau and the Li-Yau-Hamilton Harnack inequalities for
positive solutions to the heat equation $\partial_t u=Lu$ associated to the
time dependent Witten Laplacian on compact or complete manifolds equipped with
variants of the $(K, m)$-super Ricci flow.
