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On the heat flow on metric measure spaces: existence, uniqueness and stability. (English) Zbl 1200.35178

Summary: We prove existence and uniqueness of the gradient flow of the entropy functional under the only assumption that the functional is \(\lambda \)-geodesically convex for some \(\lambda\in\mathbb R\). 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.

MSC:

35K90 Abstract parabolic equations
60B05 Probability measures on topological spaces
28A33 Spaces of measures, convergence of measures
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
35B35 Stability in context of PDEs
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