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manuscripta mathematica

, Volume 121, Issue 1, pp 1–50 | Cite as

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

  • Luigi AmbrosioEmail author
  • Stefano Lisini
  • Giuseppe Savaré
Article

Abstract

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans and Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum.

Keywords

Tangent Velocity Gradient Vector Internal Energy Homogeneous Neumann Boundary Condition Admissible Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Luigi Ambrosio
    • 1
    Email author
  • Stefano Lisini
    • 2
  • Giuseppe Savaré
    • 2
  1. 1.Scuola Normale Superiore PisaPisaItaly
  2. 2.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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