Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow

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 $m\in [n,\mathrm{\infty }]$ are two constants. Finally, we prove the Li-Yau and the Li-Yau-Hamilton Harnack inequalities for positive solutions to the heat equation ${\mathrm{\partial }}_{t}u=Lu$ associated to the time dependent Witten Laplacian on compact or complete manifolds equipped with variants of the $(K,m)$-super Ricci flow.