Optimal mass transportation methods for gradient flows in the weak topology.

摘要

First we address, among other things, a question posed by Kinderlehrer and Walkington which concerns the convergence of the time-interpolants arising from the time-discretizations of flows. We establish a rigorous convergence result and a comparison maximum/minimum principle for inhomogeneous parabolic initial-value problems with natural boundary conditions in convex, bounded domains and in arbitrary dimensions. Uniqueness of solutions is studied by a novel and surprising adaptation of a technique due to Brezis and Crandall. As a model, we have chosen an inhomogeneous one-phase Stefan problem to which we give special attention although, as mentioned, our reach is much broader.; Next we restrict ourselves to a one-dimensional setting. Due to the special representation of optimal transfer functions in dimension one, we are able to offer an alternative to the approach discussed above by introducing the concept of Wasserstein kernel. We show how it can be used to study the free boundary and the regularity of the solution in the homogeneous case and we support our conclusions by experiment.; Transport plays a fundamental role in the dynamical formulation of the Monge-Kantorovich theory. The reverse implication is less explored and, at least at the outset, less obvious. We analyze systems of transport equations. We show how arbitrary Fokker-Planck equations in one dimension conform to the mass transport paradigm. Additional examples are provided, including a simple existence result for velocity-jump processes.; Finally, we develop a comprehensive and constructive theory of existence of solutions for the nonlinear friction equation. We construct approximations to the solution by time-discretizing the flow in the Wasserstein space. The case gamma = 1 is analyzed first and in more detail. The -1 gamma 0 is of special interest since it requires a novel "adaptive" time-step algorithm. We show how to eliminate the restrictions on the initial data from the original paper of Benedetto et al. Rates of convergence of the approximate solution and optimal rates of decay of the interaction energy for the gamma > 0 case are given. We also run numerical simulations confirming our theoretical results.