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 k-geodesically convex for some real number k . Also, we prove a general stability result for gradient flows of geodesically convex functionals which Gamma−converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.