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.
Keywords: doubly nonlinear, differential inclusions, generalized gradient flows, viscous regularization, vanishing-viscosity limit, vanishing-viscosity contact potential, parameterized solutions
@article{COCV_2012__18_1_36_0,
author = {Mielke, Alexander and Rossi, Riccarda and Savar\'e, Giuseppe},
title = {BV solutions and viscosity approximations of rate-independent systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {36--80},
year = {2012},
publisher = {EDP Sciences},
volume = {18},
number = {1},
doi = {10.1051/cocv/2010054},
mrnumber = {2887927},
zbl = {1250.49041},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2010054/}
}
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 SP - 36 EP - 80 VL - 18 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2010054/ DO - 10.1051/cocv/2010054 LA - en ID - COCV_2012__18_1_36_0 ER -
%0 Journal Article %A Mielke, Alexander %A Rossi, Riccarda %A Savaré, Giuseppe %T BV solutions and viscosity approximations of rate-independent systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 36-80 %V 18 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2010054/ %R 10.1051/cocv/2010054 %G en %F COCV_2012__18_1_36_0
Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe. BV solutions and viscosity approximations of rate-independent systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 36-80. doi: 10.1051/cocv/2010054
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