Nonlinear mobility continuity equations and generalized displacement convexity

J. A. Carrillo, S. Lisini, G. Savaré, D. Slepčev

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29 Citations (Scopus)

Abstract

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex. © 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)1273-1309
JournalJournal of Functional Analysis
Volume258
DOIs
Publication statusPublished - 15 Feb 2010

Keywords

  • Displacement convexity
  • Gradient flows
  • Nonlinear diffusion equations
  • Nonlinear mobility
  • Parabolic equations
  • Wasserstein distance

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