On the heat flow on metric measure spaces: existence, uniqueness and stability

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some ${lambdainmathbb {R}}$ . Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ?converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.