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In this article, we study the higher-order regularity of the K\"ahler-Ricciflow on compact K\"ahler manifolds with semi-ample canonical line bundle. Weproved, using a parabolic analogue of Hein-Tosatti's work on collapsingCalabi-Yau metrics, that when the generic fibers of the Iitaka fibration arebiholomorphic to each other, the flow converges locally smoothly away fromsingular fibers to a negative K\"ahler-Einstein metric on the base manifold. Inparticular, we proved that the Ricci curvature of the flow is uniformly boundedon any compact subsets away from singular fibers when the generic fibers arebiholomorphic to each other.

Higher-Order Estimates of Long-Time Solutions to the K\"ahler-Ricci Flow