BV solutions and viscosity approximations of rate-independent systems

TL;DR

This paper investigates the limit of viscous regularizations of rate-independent systems, introducing BV solutions that capture jumps and comparing them with existing solution concepts.

Contribution

It develops a new framework for BV solutions via vanishing viscosity, providing convergence results and detailed jump analysis in rate-independent systems.

Findings

Established convergence of viscous approximations to BV solutions.

Provided a detailed description of jump trajectories.

Compared BV solutions with energetic and local solutions.

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 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.