# 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/221913>.

@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; doubly nonlinear},

language = {eng},

month = {2},

number = {1},

pages = {36-80},

publisher = {EDP Sciences},

title = {BV solutions and viscosity approximations of rate-independent systems∗},

url = {http://eudml.org/doc/221913},

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

DA - 2012/2//

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; doubly nonlinear

UR - http://eudml.org/doc/221913

ER -

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