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Variational Convergence of Gradient Flows and Rate-Independent Evolutions in Metric Spaces

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Abstract

We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth.

We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials.

As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p > 1, when the exponents p converge to 1.

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Authors and Affiliations

  1. Weierstras-Institut, Mohrenstraße 39, 10117, D–Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, D–12489, Berlin (Adlershof), Germany

    Alexander Mielke

  3. Dipartimento di Matematica, Università di Brescia, via Valotti 9, I–25133, Brescia, Italy

    Riccarda Rossi

  4. Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, I–27100, Pavia, Italy

    Giuseppe Savaré

Authors
  1. Alexander Mielke
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  2. Riccarda Rossi
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  3. Giuseppe Savaré
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Corresponding author

Correspondence to Giuseppe Savaré.

Additional information

R.R. and G.S. have been partially supported by PRIN08 grant from MIUR for the project Optimal transport theory, geometric and functional inequalities, and applications.

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Mielke, A., Rossi, R. & Savaré, G. Variational Convergence of Gradient Flows and Rate-Independent Evolutions in Metric Spaces. Milan J. Math. 80, 381–410 (2012). https://doi.org/10.1007/s00032-012-0190-y

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  • DOI: https://doi.org/10.1007/s00032-012-0190-y

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