WIAS Preprint No. 1845, (2013)

Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems



Authors

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

2010 Mathematics Subject Classification

  • 49Q20 58E99

Keywords

  • Doubly nonlinear equations, generalized gradient flows, rate-independent systems, vanishing-viscosity limit, variational Gamma convergence, energy-dissipation balance, arclength parameterized solutions

Abstract

Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for in nite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their di erentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on re ned lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

Appeared in

  • J. Eur. Math. Soc. (JEMS), 18 (2016), pp. 2107--2165.

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