The book is devoted to the theory of gradient flows in the
general framework of metric spaces, and in the more specific
setting of the space of probability measures, which provide a
surprising link between optimal transportation theory and many
evolutionary PDE's related to (non)linear diffusion. Particular
emphasis is given to the convergence of the implicit time
discretization method and to the error estimates for this
discretization, extending the well established theory in Hilbert
spaces. The book is split in two main parts that can be read
independently of each other.
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