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July  2009, 25(2): 585-615. doi: 10.3934/dcds.2009.25.585

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

Alexander Mielke 1, , Riccarda Rossi 2, and  Giuseppe Savaré 3,

1. 

Weierstraß-Institut, Mohrenstraße 39, 10117 D–Berlin, Germany
2. 

Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I–25133 Brescia, Italy
3. 

Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, I–27100 Pavia, Italy

Received  July 2008 Revised  January 2009 Published  June 2009

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 approach is based on the abstract theory of gradient flows in metric spaces, and comparison with other notions of solutions is given.
Keywords: Rate-independent systems, jumps, vanishing viscosity, gradient flows in metric spaces..
Mathematics Subject Classification: Primary: 49Q20; Secondary: 58E9.
Citation: Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585
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