WIAS Preprint No. 1451, (2009)

BV solutions and viscosity approximations of rate-independent systems



Authors

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

2010 Mathematics Subject Classification

  • 49Q20 58E99

Keywords

  • doubly nonlinear, differential inclusions, generalized gradient flows, viscous regularization, vanishing-viscosity limit, vanishing-viscosity contact potential, parametrized solutions

Abstract

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.

Appeared in

  • ESAIM Control Optim. Calc. Var., 18 (2012) pp. 36--80.

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