Convergence of weak K\"ahler-Ricci Flows on minimal models of positive Kodaira dimension

Studying the behavior of the K"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp`ere equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized K"ahler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models.