Calculus of Variations and Geometric Measure Theory

A. Mielke - R. Rossi - G. Savaré

BV solutions and viscosity approximations of rate-independent systems

created by rossi on 14 Oct 2009


Submitted Paper

Inserted: 14 oct 2009

Year: 2009


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 which 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 a crucial role in the description of the associated jump trajectories.

We shall prove a general convergence result 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: vanishing viscosity, rate-independent systems, BV solutions