# 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

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topMielke, 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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