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arXiv · 0910.3360

BV solutions and viscosity approximations of rate-independent systems

Alexander Mielke·Riccarda Rossi·Giuseppe Savare'
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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 which 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 a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result 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.

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Alexander Mielke, Riccarda Rossi, Giuseppe Savare'. 2009-10-18. BV solutions and viscosity approximations of rate-independent systems. https://doi.org/10.1051/cocv%2F2010054

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