BV solutions and viscosity approximations of rate-independent systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 36-80.

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.

DOI : https://doi.org/10.1051/cocv/2010054
Classification : 49Q20,  58E99
Mots clés : doubly nonlinear, differential inclusions, generalized gradient flows, viscous regularization, vanishing-viscosity limit, vanishing-viscosity contact potential, parameterized solutions
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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. http://www.numdam.org/articles/10.1051/cocv/2010054/

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