BV solutions and viscosity approximations of rate-independent systems

Alexander Mielke; Riccarda Rossi; Giuseppe Savaré

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 36-80
  • ISSN: 1292-8119

Abstract

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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 that 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 crucial roles in the description of the associated jump trajectories. We shall prove general convergence results 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.

How to cite

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Mielke, Alexander, Rossi, Riccarda, and Savaré, Giuseppe. "BV solutions and viscosity approximations of rate-independent systems." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 36-80. <http://eudml.org/doc/272906>.

@article{Mielke2012,
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 that 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 crucial roles in the description of the associated jump trajectories. We shall prove general convergence results 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.},
author = {Mielke, Alexander, Rossi, Riccarda, Savaré, Giuseppe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {doubly nonlinear; differential inclusions; generalized gradient flows; viscous regularization; vanishing-viscosity limit; vanishing-viscosity contact potential; parameterized solutions},
language = {eng},
number = {1},
pages = {36-80},
publisher = {EDP-Sciences},
title = {BV solutions and viscosity approximations of rate-independent systems},
url = {http://eudml.org/doc/272906},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Mielke, Alexander
AU - Rossi, Riccarda
AU - Savaré, Giuseppe
TI - BV solutions and viscosity approximations of rate-independent systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 1
SP - 36
EP - 80
AB - 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 that 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 crucial roles in the description of the associated jump trajectories. We shall prove general convergence results 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.
LA - eng
KW - doubly nonlinear; differential inclusions; generalized gradient flows; viscous regularization; vanishing-viscosity limit; vanishing-viscosity contact potential; parameterized solutions
UR - http://eudml.org/doc/272906
ER -

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