Volume 18, Number 1, January-March 2012
|Page(s)||36 - 80|
|Published online||23 December 2010|
BV solutions and viscosity approximations of rate-independent systems∗
Weierstraß-Institut, Mohrenstraße 39, 10117
2 Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany
3 Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, Italy
4 Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100 Pavia, Italy
Revised: 11 August 2010
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.
Mathematics Subject Classification: 49Q20 / 58E99
Key words: Doubly nonlinear / differential inclusions / generalized gradient flows / viscous regularization / vanishing-viscosity limit / vanishing-viscosity contact potential / parameterized solutions
© EDP Sciences, SMAI, 2010
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.