WIAS Preprint No. 1704, (2012)

Variational convergence of gradient flows and rate-independent evolutions in metric spaces



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Rossi, Riccarda
  • Savaré, Giuseppe

2010 Mathematics Subject Classification

  • 49Q20 58E99

Keywords

  • Doubly nonlinear equations, evolution in metric spaces, generalized gradient flows, viscous regularization, vanishing-viscosity limit, BV solutions, rate-independent systems

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.

Appeared in

  • Milan J. Math., 80 (2012) pp. 381--410.

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