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

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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.

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Authors and Affiliations

  1. Scuola Normale Superiore Pisa, Pisa, Italy

    Luigi Ambrosio

  2. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Stefano Lisini & Giuseppe Savaré

Authors
  1. Luigi Ambrosio
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  2. Stefano Lisini
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  3. Giuseppe Savaré
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Correspondence to Luigi Ambrosio.

Additional information

This work was partially supported by grants of M.I.U.R. and of IMATI-CNR, Pavia, Italy.

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Ambrosio, L., Lisini, S. & Savaré, G. Stability of flows associated to gradient vector fields and convergence of iterated transport maps. manuscripta math. 121, 1–50 (2006). https://doi.org/10.1007/s00229-006-0003-0

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