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.
Classification: 49Q20, 58E99
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}, publisher = {EDP-Sciences}, volume = {18}, number = {1}, year = {2012}, pages = {36-80}, doi = {10.1051/cocv/2010054}, zbl = {1250.49041}, mrnumber = {2887927}, language = {en}, url = {http://www.numdam.org/item/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, Volume 18 (2012) no. 1, pp. 36-80. doi : 10.1051/cocv/2010054. http://www.numdam.org/item/COCV_2012__18_1_36_0/
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