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G. Savaré Gradient Flows and Diffusion Semigroups in Metric Spaces under Lower Curvature Bounds Pavia (2007) - To appear on "Comptes rendus Mathematique" DVI PS Abstract
We present some new results concerning well-posedness of gradient flows generated by lambda-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L2-Wasserstein spaces.Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered.
These results have been announced during the workshop on ``Optimal Transport: theory and applications'' held in Pisa, November 2006.
U. Gianazza, G. Savaré, G. Toscani The Wasserstein gradient flow of the Fisher information and the Quantum Drift-Diffusion equation Pubbl. IMATI-CNR Pavia (2006) DVI PS Abstract
We prove the global existence of nonnegative variational solutions to the ``drift diffusion'' evolution equation under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, nonnegative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance between probability measures. We also study long time behaviour of the solutions.
