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By means of a space-time Wasserstein control, we show the monotonicity of theW-entropy functional in time along heat flows on possibly singular metricmeasure spaces with non-negative Ricci curvature and a finite upper bound ofdimension in an appropriate sense. The associated rigidity result on the rateof dissipation of the W-entropy is also proved. These extend known results evenon weighted Riemannian manifolds in some respects. In addition, we reveal thatsome singular spaces will exhibit the rigidity models while only the Euclideanspace does in the class of smooth weighted Riemannian manifolds.

Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces