Modeling solutions with jumps for rate-independent systems  onmetric spaces

Alexander Mielke; Riccarda Rossi; Giuseppe Savaré

doi:10.3934/dcds.2009.25.585

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2009, Volume 25, Issue 2: 585-615. Doi: 10.3934/dcds.2009.25.585

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

1.
Weierstraß-Institut, Mohrenstraße 39, 10117 D–Berlin
2.
Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I–25133 Brescia
3.
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, I–27100 Pavia

Received: July 2008

Revised: January 2009

Published: June 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 49Q20; Secondary: 58E99.

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