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BV solutions and viscosity approximations of rate-independent systems∗

Published online by Cambridge University Press:  23 December 2010

Alexander Mielke ,
Riccarda Rossi  and
Giuseppe Savaré
Alexander Mielke
Affiliation:
Weierstraß-Institut, Mohrenstraße 39, 10117 Berlin, Germany. mielke@wias-berlin.de Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany
Riccarda Rossi
Affiliation:
Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, Italy; riccarda.rossi@ing.unibs.it
Giuseppe Savaré
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100 Pavia, Italy; giuseppe.savare@unipv.it
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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 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.

Keywords

Doubly nonlineardifferential inclusionsgeneralized gradient flowsviscous regularizationvanishing-viscosity limitvanishing-viscosity contact potentialparameterized solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 36 - 80
Copyright
© EDP Sciences, SMAI, 2010

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