Modeling solutions with jumps for rate-independent systems on metric spaces

Rate-independent systems allow for solutions with jumps that need
additional modeling. Here we suggest a formulation that arises as limit of
viscous regularization of the solutions in the extended state space. Hence,
our parametrized metric solutions of a rate-independent system are absolutely
continuous mappings from a parameter interval into the extended state space.
Jumps appear as generalized gradient flows during which the time is constant.
The closely related notion of BV solutions is developed afterwards. Our ap-
proach is based on the abstract theory of gradient flows in metric spaces, and
comparison with other notions of solutions is given.